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TRANSCRIPT
RESERVING FOR MATURITY GUARANTEES UNDER
UNITISED WITH-PROFITS POLICIES
By
Wenyi Tong
Submitted for the Degree of
Doctor of Philosophy
at Heriot-Watt University
on Completion of Research in the
School of Mathematical and Computer Sciences
November 2004.
This copy of the thesis has been supplied on the condition that anyone who consults
it is understood to recognise that the copyright rests with its author and that no quo-
tation from the thesis and no information derived from it may be published without
the prior written consent of the author or the university (as may be appropriate).
I hereby declare that the work presented in this the-
sis was carried out by myself at Heriot-Watt University,
Edinburgh, except where due acknowledgement is made,
and has not been submitted for any other degree.
Wenyi Tong (Candidate)
Professor Angus S. Macdonald (Supervisor)
Professor Howard R. Waters (Supervisor)
Doctor Mark Willder (Supervisor)
Date
ii
Contents
Acknowledgements xiii
Abstract xiv
Introduction 10.1 Changes in the Regulatory Environment . . . . . . . . . . . . . . . . 10.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1 GUARANTEES UNDER UNITISED WITH-PROFITS POLI-CIES 91.1 Operation of UWP Polices . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Reserving Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.2 Three Reserving Approaches . . . . . . . . . . . . . . . . . . . 14
1.3 Models Required . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.3.1 Valuation Model . . . . . . . . . . . . . . . . . . . . . . . . . 161.3.2 Bonus and Asset Allocation Model . . . . . . . . . . . . . . . 171.3.3 Real World Asset Model . . . . . . . . . . . . . . . . . . . . . 17
2 RESERVING FOR A SINGLE UWP POLICY HISTORICALLY 192.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Reserving Approach of Buying Options . . . . . . . . . . . . . . . . . 20
2.2.1 Notation and Assumptions . . . . . . . . . . . . . . . . . . . . 202.2.2 Mechanism of the Option Approach . . . . . . . . . . . . . . . 232.2.3 Results Using the Option Method . . . . . . . . . . . . . . . . 28
2.3 Dynamic Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3.1 Mechanism of the Hedging Approach . . . . . . . . . . . . . . 312.3.2 Results Using Discrete Hedging . . . . . . . . . . . . . . . . . 35
2.4 CTE Reserving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.4.1 Mechanism of the CTE Approach . . . . . . . . . . . . . . . . 372.4.2 Results under the CTE Approach . . . . . . . . . . . . . . . . 40
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 DYNAMIC BONUSES 443.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.2 A Dynamic Bonus Strategy . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.1 Dynamic Bonuses without Smoothing . . . . . . . . . . . . . . 463.2.2 Dynamic Bonuses with Smoothing . . . . . . . . . . . . . . . . 48
iii
3.3 Results for the Single Policy with Dynamic Bonuses . . . . . . . . . . 503.3.1 Case A: Without Smoothing or Allowance for Future Bonuses 513.3.2 Case B: Smoothing without Allowance for Future Bonuses . . 543.3.3 Case C: Smoothing with Allowance for Future Bonuses . . . . 57
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4 A RISK-FREE RATE CONSISTENT WITH THE WILKIEMODEL 634.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2 A Yield Curve for the Wilkie Model . . . . . . . . . . . . . . . . . . . 644.3 Results with the Consistent Risk-Free Rate . . . . . . . . . . . . . . . 66
4.3.1 Buying Options . . . . . . . . . . . . . . . . . . . . . . . . . . 674.3.2 Discrete Hedging . . . . . . . . . . . . . . . . . . . . . . . . . 684.3.3 CTE Reserving . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5 A DYNAMIC INVESTMENT STRATEGY 735.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.2 A Dynamic Model Containing Dynamic Investment and Bonus Strate-
gies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.3 Results with the Dynamic Model . . . . . . . . . . . . . . . . . . . . 82
5.3.1 Buying Options . . . . . . . . . . . . . . . . . . . . . . . . . . 825.3.2 Discrete Hedging . . . . . . . . . . . . . . . . . . . . . . . . . 855.3.3 CTE Reserving . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6 SENSITIVITY TESTING FOR THE SINGLE UWP POLICY 906.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.2 Sensitivity to Different Parameters . . . . . . . . . . . . . . . . . . . 92
6.2.1 EBRs and Bonus Rates . . . . . . . . . . . . . . . . . . . . . . 926.2.2 Reserves and Profitability . . . . . . . . . . . . . . . . . . . . 96
6.3 Sensitivity to Different Upper and Lower Probability Boundaries . . . 1016.3.1 EBRs and Bonus Rates . . . . . . . . . . . . . . . . . . . . . . 1016.3.2 Reserves and Profitability . . . . . . . . . . . . . . . . . . . . 102
6.4 Sensitivity to Different 10-Year Periods . . . . . . . . . . . . . . . . . 1046.4.1 Investment Performance of the Two Asset Classes . . . . . . . 1046.4.2 Asset Shares and Guarantees . . . . . . . . . . . . . . . . . . 1056.4.3 Reserves and Profitability . . . . . . . . . . . . . . . . . . . . 108
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7 RESERVING FOR A PORTFOLIO OF UWP POLICIES HIS-TORICALLY 1127.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127.2 Equity Proportions and Regular Bonuses . . . . . . . . . . . . . . . . 114
7.2.1 Case A: Without Smoothing or Allowance for Future Bonuses 1147.2.2 Case B: Smoothing without Allowance for Future Bonuses . . 1177.2.3 Case C: Smoothing with Allowance for Future Bonuses . . . . 118
7.3 Asset Shares and Guarantees in Cases A, B and C . . . . . . . . . . . 120
iv
7.3.1 Asset Shares . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1207.3.2 Guarantees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.3.3 Terminal Bonus . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.4 Reserves Using the Three Approaches . . . . . . . . . . . . . . . . . . 1287.4.1 Buying Options . . . . . . . . . . . . . . . . . . . . . . . . . . 1297.4.2 Discrete Hedging . . . . . . . . . . . . . . . . . . . . . . . . . 1337.4.3 CTE Reserving . . . . . . . . . . . . . . . . . . . . . . . . . . 1337.4.4 Comparison of the Portfolio Reserves Set up by Different Ap-
proaches in Case C . . . . . . . . . . . . . . . . . . . . . . . . 1407.5 Profitability of the UWP Policies . . . . . . . . . . . . . . . . . . . . 140
7.5.1 Buying Options . . . . . . . . . . . . . . . . . . . . . . . . . . 1417.5.2 Discrete Hedging . . . . . . . . . . . . . . . . . . . . . . . . . 1437.5.3 CTE Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 1447.5.4 Comparison of the Free Estate under Different Reserving Ap-
proaches in Case C . . . . . . . . . . . . . . . . . . . . . . . . 1467.6 Sensitivity Testing for the Portfolio in Case C . . . . . . . . . . . . . 147
7.6.1 Sensitivity to Different Parameters . . . . . . . . . . . . . . . 1477.6.2 Sensitivity to Different Probability Boundaries in the Invest-
ment Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . 1577.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8 RESERVING FOR THE PORTFOLIO WITHIN THE SIMU-LATED REAL WORLD 1678.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1678.2 A Single 10-Year Policy . . . . . . . . . . . . . . . . . . . . . . . . . . 1698.3 Average EBR and Average Regular Bonus Rate . . . . . . . . . . . . 1718.4 Portfolio Asset Share and Guarantee . . . . . . . . . . . . . . . . . . 1758.5 Maturing Policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1778.6 Portfolio Reserves Set up Using the Three Reserving Approaches . . . 180
8.6.1 Option and Hedging Approaches . . . . . . . . . . . . . . . . 1818.6.2 CTE Reserving . . . . . . . . . . . . . . . . . . . . . . . . . . 182
8.7 Free Estate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1838.7.1 Option Approach . . . . . . . . . . . . . . . . . . . . . . . . . 1838.7.2 Hedging Approach . . . . . . . . . . . . . . . . . . . . . . . . 1848.7.3 CTE Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 185
8.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
9 CONCLUSIONS AND FURTHER RESEARCH 1889.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1899.2 Suggestions for Further Research . . . . . . . . . . . . . . . . . . . . 194
A Wilkie Model 1995 Version 196
B Investment Data and Derived Initial Conditions 199
C Details of Calculations for the 1991 Policy with a 5% Bonus Rate,a 5% Risk-Free Rate and a 100% EBR 202
v
D Six Sample Paths in the Simulated Real World 204
References 209
vi
List of Tables
4.1 The comparison of the accumulated values of the cashflows for the1991 policy in Case C with the zero-coupon yield or 5% constant asa risk-free rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2 The equity backing ratios for the policy issued at the end of 1991 inCase C (%) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3 The comparison of the accumulated values of the cashflows for the1991 policy in Case C with the dynamic and static EBRs . . . . . . . 87
6.4 The assumptions for the parameters g, c, TB, σ and τ (%) . . . . . . 916.5 The EBRs for the 1991 policy in Case C based on the different bases
(%) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.6 The regular bonus rates declared on the 1991 policy in Case C using
the different bases (%) . . . . . . . . . . . . . . . . . . . . . . . . . . 946.7 The asset share, guarantee and terminal bonus rate at maturity of
the 1991 policy in Case C based on the different bases . . . . . . . . . 956.8 The reserves using the option method for the 1991 policy in Case C
under the different bases . . . . . . . . . . . . . . . . . . . . . . . . . 966.9 The accumulated values of the cashflows incurred using the option
method for the 1991 policy in Case C under the different bases . . . . 976.10 The accumulated values of the cashflows incurred by discrete hedging
for the 1991 policy in Case C under the different bases . . . . . . . . 986.11 The 95th CTE reserves for the 1991 policy in Case C using the dif-
ferent bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.12 The 99th CTE reserves for the 1991 policy in Case C under the dif-
ferent bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.13 The accumulated values of the cashflows incurred to set up CTE
reserves for the 1991 policy in Case C using the different bases . . . . 1006.14 The EBRs for the 1991 policy in Case C under the standard basis
with the different upper and lower probability boundaries (%) . . . . 1016.15 The bonus rates for the 1991 policy in Case C under the standard
basis with the different probabilities to adjust the EBRs (%) . . . . . 1016.16 The reserves for the 1991 policy in Case C under the standard basis
with the probability boundaries of 95% and 99% to adjust the EBRs 1036.17 The accumulated values of the cashflows for the 1991 policy in Case
C under the standard basis with the different probability boundaries . 1036.18 The EBRs of the policies in Case C issued at different times under the
standard basis with the probability boundaries of 97.5% and 99.5%in the investment strategy (%) . . . . . . . . . . . . . . . . . . . . . . 105
vii
6.19 The bonus rates declared on the policies in Case C under the standardbasis with the probabilities of 97.5% and 99.5% to adjust the EBRs(%) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.20 The asset shares, guarantees and terminal bonus rates at maturityfor the policies in Case C issued at different times under the standardbasis with the probability boundaries of 97.5% and 99.5% . . . . . . . 108
6.21 The reserves for the three policies in Case C issued at different timesunder the standard basis with the probabilities of 97.5% and 99.5% . 108
6.22 The accumulated values of the cashflows incurred for the three policiesin Case C under the standard basis with the boundaries of 97.5% and99.5% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.23 The EBRs of each policy in the portfolio in Case A (%) . . . . . . . . 1157.24 The regular bonus rates declared on each policy in the portfolio in
Case A (%) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1167.25 The EBRs of each policy in the portfolio in Case B (%) . . . . . . . . 1177.26 The regular bonus rates declared on each policy in the portfolio in
Case B (%) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1187.27 The EBRs of each policy in the portfolio in Case C (%) . . . . . . . . 1197.28 The regular bonus rates declared on each policy in the portfolio in
Case C (%) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1207.29 The asset shares of each policy in the portfolio in Case A . . . . . . . 1227.30 The asset shares of each policy in the portfolio in Case B . . . . . . . 1237.31 The asset shares of each policy in the portfolio in Case C . . . . . . . 1247.32 The guarantees of each policy in the portfolio in Case A . . . . . . . 1267.33 The guarantees of each policy in the portfolio in Case B . . . . . . . . 1277.34 The guarantees of each policy in the portfolio in Case C . . . . . . . 1287.35 The asset shares, guarantees and terminal bonus rates at maturity in
Cases A, B and C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1287.36 The reserves for each policy in the portfolio using the option method
in Case A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1307.37 The reserves for each policy in the portfolio using the option method
in Case B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1317.38 The reserves for each policy in the portfolio using the option method
in Case C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327.39 The comparison of the portfolio cashflow and the sum of the individ-
ual cashflows incurred by the insurer using the hedging approach inCase C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.40 The portfolio asset shares in Case C under the different bases . . . . 1487.41 The portfolio guarantees in Case C under the different bases . . . . . 1497.42 The asset shares and guarantees at maturity in Case C under the
different bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1507.43 The terminal bonus rates declared on the maturing policies in Case
C under the different bases (%) . . . . . . . . . . . . . . . . . . . . . 1507.44 The portfolio reserves using the option method in Case C under the
different bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517.45 The free estate of the insurer using the option method in Case C
under the different bases . . . . . . . . . . . . . . . . . . . . . . . . . 152
viii
7.46 The free estate of the insurer using the hedging approach in Case Cunder the different bases . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.47 The 95% portfolio CTE reserves in Case C under the different bases . 1557.48 The 99% portfolio CTE reserves in Case C under the different bases . 1567.49 The free estate of the insurer who sets up 95% CTE reserves in Case
C under the different bases . . . . . . . . . . . . . . . . . . . . . . . . 1577.50 The free estate of the insurer who sets up 99% CTE reserves in Case
C under the different bases . . . . . . . . . . . . . . . . . . . . . . . . 1587.51 The portfolio asset shares and guarantees in Case C under the stan-
dard basis with the 95% and 99% probability boundaries . . . . . . . 1597.52 The asset share, guarantee and terminal bonus rate at maturity in
Case C under the standard basis with the 95% and 99% probabilityboundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.53 The portfolio reserves in Case C under the standard basis with the95% and 99% probability boundaries . . . . . . . . . . . . . . . . . . 161
7.54 The amount of the free estate in Case C under the standard basiswith the 95% and 99% probability boundaries . . . . . . . . . . . . . 162
8.55 The mean and standard deviation of the simulated EBRs, bonus rates,asset shares and guarantees for the 10-year policy . . . . . . . . . . . 169
8.56 The statistics at the maturity of the 10-year policy . . . . . . . . . . 1708.57 The quantiles of the simulated reserves for the 10-year policy . . . . . 1708.58 The statistics of the accumulated values of the cashflows for the 10-
year policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.59 The statistics for the maturing policies during the 30-year extended
period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178B.60 The market indices at 31 December of each year during the period of
1964 to 2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200B.61 The derived initial conditions for the 1995 version of the Wilkie model
at 31 December of each year during the period of 1964 to 2002 . . . . 201C.62 The details of calculations for the 1991 policy under the option approach202C.63 The details of calculations for the 1991 policy under the hedging ap-
proach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203C.64 The details of calculations for the 1991 policy under the CTE approach203
ix
List of Figures
2.1 Maturity payout before and after declaring a bonus at time t, and theincrease in the maturity payout after the bonus declaration . . . . . . 27
2.2 The asset shares and guarantees for the policy issued at the end of1991 with static bonus and investment strategies . . . . . . . . . . . . 29
2.3 The reserves set up using the option method for the 1991 policy withstatic bonus and investment strategies and a constant risk-free rate . 30
2.4 The equity index and exercise price of the options bought for the 1991policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5 The hedging error and transaction costs incurred by discrete hedgingfor the 1991 policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.6 The 95% and 99% CTE reserves for the 1991 policy with static bonusand investment strategies and a constant risk-free rate . . . . . . . . 41
3.7 The unsmoothed bonus rates for the policy issued at the end of 1991with a static investment strategy . . . . . . . . . . . . . . . . . . . . 47
3.8 The asset share, force of inflation (multiplied by 10,000) and 25%projected maturity asset share for the 1991 policy with a static in-vestment strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.9 The comparison of the smoothed and unsmoothed bonus rates of the1991 policy with a static investment strategy . . . . . . . . . . . . . . 50
3.10 The asset shares and guarantees of the 1991 policy with a static in-vestment strategy in Case A . . . . . . . . . . . . . . . . . . . . . . . 52
3.11 The comparison of the reserves using the option method for the 1991policy in Case A and the case of static bonuses . . . . . . . . . . . . . 52
3.12 The comparison of the CTE reserves for the 1991 policy in Case Aand the case of static bonuses . . . . . . . . . . . . . . . . . . . . . . 53
3.13 The asset shares and guarantees of the 1991 policy with a static in-vestment strategy in Cases A and B . . . . . . . . . . . . . . . . . . . 55
3.14 The comparison of the reserves using the option method for the 1991policy in Cases A and B . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.15 The comparison of the CTE reserves for the 1991 policy in Cases Aand B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.16 The asset shares and guarantees of the 1991 policy with a static in-vestment strategy in Cases B and C . . . . . . . . . . . . . . . . . . . 59
3.17 The comparison of the reserves using the option method for the 1991policy in Cases B and C . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.18 The comparison of the CTE reserves for the 1991 policy in Cases Band C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
x
4.19 The yield on the zero-coupon bond with the same maturity date asthe 1991 policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.20 The comparison of the reserves using the option method for the 1991policy in Case C assuming the zero-coupon yield or 5% constant as arisk-free rate over the policy term . . . . . . . . . . . . . . . . . . . . 67
4.21 The comparison of the CTE reserves for the 1991 policy in Case Cassuming the zero-coupon yield or 5% constant as a risk-free rate overthe policy term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.22 The comparison of the bonus rates declared on the 1991 policy inCase C with the static and dynamic EBRs . . . . . . . . . . . . . . . 80
5.23 The comparison of the asset shares and guarantees of the 1991 policyin Case C with the static and dynamic investment strategies . . . . . 81
5.24 The comparison of the reserves using the option method for the 1991policy in Case C with the dynamic and static EBRs . . . . . . . . . . 84
5.25 The equity index and exercise price of the options bought for the 1991policy in Case C with the dynamic and static EBRs . . . . . . . . . . 84
5.26 The comparison of the CTE reserves for the 1991 policy in Case Cwith the dynamic and static EBRs . . . . . . . . . . . . . . . . . . . 87
6.27 The asset shares of the 1991 policy in Case C based on the differentbases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.28 The guarantees of the 1991 policy in Case C based on the differentbases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.29 The asset shares and guarantees for the 1991 policy in Case C underthe standard basis with the different upper and lower probabilityboundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.30 The comparison of the equity indices at each policy duration over thedifferent 10-year periods . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.31 The comparison of the yields on the zero-coupon bonds, each matur-ing at the end of the policy term, at each policy duration over thedifferent 10-year periods . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.32 The comparison of the asset shares of the policies in Case C issued atdifferent times under the standard basis with the boundaries of 97.5%and 99.5% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.33 The comparison of the guarantees of the policies in Case C issued atdifferent times under the standard basis with the boundaries of 97.5%and 99.5% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.34 The comparison of the total asset shares of the portfolio in Cases A,B and C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.35 The comparison of the total guarantees of the portfolio in Cases A,B and C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.36 The portfolio reserves using the option method in Cases A, B and C . 1297.37 The 95% portfolio CTE reserves in Cases A, B and C . . . . . . . . . 1387.38 The 99% portfolio CTE reserves in Cases A, B and C . . . . . . . . . 1387.39 The comparison of the portfolio CTE reserves and the sum of the
individual CTE reserves in Case C . . . . . . . . . . . . . . . . . . . 1407.40 The comparison of the portfolio reserves set up using different reserv-
ing approaches in Case C . . . . . . . . . . . . . . . . . . . . . . . . . 141
xi
7.41 The free estate of the insurer using the option method in Cases A, Band C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.42 The free estate of the insurer using the hedging approach in Cases A,B and C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.43 The free estate of the insurer who sets up 95% CTE reserves in CasesA, B and C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.44 The free estate of the insurer who sets up 99% CTE reserves in CasesA, B and C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.45 The comparison of the free estate under different reserving approachesin Case C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.46 The quantiles of the average EBRs over the policy term for each policyin Case C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8.47 The quantiles of the average regular bonus rates declared over thepolicy term for each policy in Case C . . . . . . . . . . . . . . . . . . 174
8.48 The quantiles of the portfolio asset shares in Case C . . . . . . . . . . 1758.49 The quantiles of the portfolio guarantees in Case C . . . . . . . . . . 1768.50 The quantiles of the AS/G ratios in Case C . . . . . . . . . . . . . . 1778.51 The quantiles of the portfolio reserves set up using the option method
in Case C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1818.52 The quantiles of the 99% portfolio CTE reserves in Case C . . . . . . 1828.53 The quantiles of the insurer’s free estate using the option method in
Case C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1838.54 The quantiles of the insurer’s free estate by discrete hedging in Case C1848.55 The quantiles of the insurer’s free estate by setting up the 99% port-
folio CTE reserves in Case C . . . . . . . . . . . . . . . . . . . . . . . 185D.56 Six sample paths of the simulated portfolio asset share and guarantee
in Case C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205D.57 Six sample paths of the simulated reserve required for the portfolio
in Case C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206D.58 Six sample paths of the simulated free estate in Case C . . . . . . . . 207
xii
Acknowledgements
I would like to express my gratitude to all my supervisors, Prof. Angus Macdonald,
Prof. Howard Waters, and Dr. Mark Willder for their invaluable guidance, advice
and encouragement during the course of my research. I must thank my leading
supervisor Mark Willder for his unfailing patience, his understanding in a very
difficult period of time, his wide knowledge in the field of life insurance solvency,
and his eagerness to share this knowledge.
The project carried out in this thesis is sponsored by Standard Life Assurance
Company. I would like to thank the sponsor, in particular Dr. David Hare and
Douglas Morrison, for the financial and technical support at various stages.
This thesis would not be possible without the love and encouragement from my
parents. My indebtedness to them for their understanding and respect to all my
decisions.
I have enjoyed working with my colleagues. Their work in related areas has
stimulated my research. No less is my gratitude to my friends in Scotland and
China for their great support and wonderful friendship.
xiii
Abstract
As a result of the move by the International Accounting Standards Board (IASB)
towards fair value accounting, there is increasing interest in establishing how to
value life insurance liabilities, in particular liabilities with embedded options, on
a market consistent basis. In the UK, the Financial Services Authority (FSA) is
developing a new prudential regulatory regime which moves from the traditional
valuation approach to a mark-to-market regime. The recent CP195 proposals (FSA
(2003)) develop a ‘twin peaks’ approach by which the total reserves and capital
required are set as the greater of that required under the current statutory basis
and that required applying a stress test to a market consistent valuation of assets
and liabilities.
The aim of this thesis is to investigate the reserves required to meet the maturity
guarantees under unitised with-profits (UWP) policies, within the realistic reporting
framework. Under the UWP policies, a growth rate in the unit value is promised for
the premiums already paid. In addition, the policies allow the holders to participate
in the profits of the company through regular and terminal bonus declarations. Thus,
the UWP product includes explicit investment guarantees which build up over the
policy term. It is to be expected that any guarantee provided will have a cost and
hence it should be charged (either to premiums or to asset shares) and reserved for.
Three reserving approaches are considered in the thesis. The first two ap-
proaches, buying options over-the-counter (OTC) and discrete hedging, apply mod-
ern option pricing theory. These approaches are consistent with the realistic peak of
the ‘twin peaks’ approach. The third approach calculates the conditional tail expec-
tation (CTE) reserves by stochastic simulation. The third approach uses the same
idea of quantile reserving, recommended by the Maturity Guarantees Working Party
xiv
in 1980, but with a different risk measure. However, this approach is not favoured
by the FSA in CP195 proposals (FSA (2003)). The purpose of applying these three
approaches in the thesis is to compare the amount of the reserves required in the
current realistic reporting regime with that using traditional stochastic valuation
techniques.
In the thesis we assume that a fixed 1% (a different percentage is considered in
sensitivity testing) of the policyholder’s fund is deducted at the end of each policy
year as a charge for the guarantees. Reserves are funded by the insurer from its
inherited estate. The cashflows incurred by the insurer are calculated, from which
profitability of the UWP policies in a 1% stakeholder environment is investigated.
We obtain numerical results using both historical data and stochastic simulation.
In the historical part, we start from a single UWP policy with a constant bonus
rate, a constant risk-free interest rate and a static investment strategy assuming
a 100% equity proportion of policyholder’s assets. Then we make the model more
complicated by adding in a dynamic bonus strategy, a more realistic yield curve and
a dynamic investment strategy. We also build up a portfolio which contains different
policyholder generations. Under the CTE approach, we can see the benefit of pooling
risks when reserving for the fund as a whole instead of setting up reserves separately
for each generation. Finally we extend the investigation period and consider the
portfolio within the simulated real world.
xv
Introduction
0.1 Changes in the Regulatory Environment
With-profits business has flourished for over a century as a long-term savings ve-
hicle in the UK. The interaction of guaranteed benefits, policyholders’ reasonable
expectations, smoothing, participation in the upside returns and substantial equity
exposure makes with-profits a unique investment option. Historically, it has pro-
vided an enhanced investment return to many investors. However, recently the need
to meet the costs of pensions mis-selling, the closure to new business of Equitable
Life, the attribution of AXA’s inherited estate, the declining nominal investment re-
turns, lack of transparency and other problems have put with-profits business under
increasing criticism.
Over the last two years with-profits regulation in the UK has been going through
one of the most significant periods of change in its history, to meet the demands
for greater transparency and comparability of reporting. Hare et al. (2003) have
outlined the development of realistic reporting through the UK regulatory frame-
work. The series of Consultation Papers (CP) published by the FSA, including the
recent CP195 proposals (FSA (2003)), demonstrate a move from the traditional val-
uation approach to a market consistent approach which places values on assets and
liabilities consistent with the market values of assets with similar cashflow patterns.
CP195 proposes a ‘twin peaks’ approach for with-profits business to achieve the
objective of realistic reporting within the constraints of the EC Third Life Direc-
tive. The approach has also been described in Muir and Waller (2003), and Dullaway
and Needleman (2003). The regulatory peak is very similar to the existing statutory
1
valuation. It is governed by EU rules and based on a comparison of the admissible as-
sets with the sum of the mathematical reserves, the Resilience Capital Requirement
(RCR) and the Long Term Insurance Capital Requirement (LTICR). The realistic
peak is based on a comparison of assets, including some inadmissibles, with realistic
liabilities plus a risk capital margin (RCM). The RCM is required on top of realistic
liabilities to provide some resilience to adverse experience. The total reserves and
capital required for each with-profits fund are set as the greater of that required
under a statutory approach and that required under a realistic, market-consistent
approach. Therefore, additional capital, known as the with-profits insurance capital
component (WPICC), is required to bring the regulatory surplus down to the level
of the realistic surplus if the former is larger.
The realistic balance sheet set out by the FSA is the core of the realistic peak. It
is essentially split into three important items: the realistic value of assets available
to support with-profits business, the with-profits benefit reserve and future policy
related liabilities. The realistic balance sheet has also been described in detail in
Muir and Waller (2003) and Hare et al. (2003). To determine a value of liabilities,
CP195 proposals (FSA (2003)) adopt a ‘put option approach’ which defines a liability
as equal to the value of some underlying asset (asset share) plus an additional option
value which reflects the fact that payout is subject to a certain minimum value. Hare
et al. (2003) and Dullaway and Needleman (2003) suggest an alternative approach,
known as a ‘call option approach’, starting with the present value of guarantees to
which the cost of options is then added. The addition would be the excess (if any)
of asset share over contractual guarantees, which corresponds to a call option. The
authors have also described the relative advantages and disadvantages of these two
approaches.
The FSA sets out three approaches to determine the cost of any guarantees,
options and smoothing embedded within the with-profits policies, namely:
• a stochastic approach using a market consistent asset model
• the market costs of hedging the guarantees or options
• a series of deterministic projections with attributed probabilities.
2
If the underlying guarantees or options can be hedged in the market then the
cost of guarantees or options can be set equal to the market value of the hedge.
Under simplifying assumptions direct analytical approaches or closed form solutions
might be used. However, to the extent that the value of the guarantees or options
is materially affected by management actions taken by the company it is unlikely
that a closed form solution can be found which allows for dynamic asset allocation,
dynamic bonuses and any cross-subsidy of guarantee costs.
Therefore, the FSA states a preference for a stochastic approach using a mar-
ket consistent asset model. The assets and liabilities of the with-profits fund are
projected under a large number of economic scenarios generated by the asset model
that has been calibrated to the market prices of financial instruments most rele-
vant to the business being valued. The model should incorporate formulaic rules
for the actions that the company’s management may take to reduce risks and mit-
igate costs, for example adopting a more closely matching investment strategy or
reducing bonus rates. Describing the characteristics of an asset model for the re-
alistic balance sheet calculation is relatively straightforward, but the key questions
are what asset model should be used and how should the model be calibrated? At
present, no guidance exists in either of these two areas other than that the results
produced should be market consistent. In order to investigate the significance of
the choice and calibration of the asset model, Hare et al. (2003) have carried out a
survey of option prices produced using six economic models. The authors conclude
that market consistent models do not necessarily produce the same result; different
models using the same calibration method can produce similar results; and similar
models that are calibrated in different ways can produce dissimilar results.
The FSA also states that when incorporating management actions into the pro-
jection of claims, the company should ensure consistency with its Principles and
Practices of Financial Management (PPFM), which indicate to policyholders how
an insurance company exercises its discretion in managing with-profits funds, and
takes into account its regulatory duty to treat its customers fairly. However, it is not
straightforward in practice to reflect the complex interactions between the insurer’s
financial strength, investment policy and bonus strategy, in a wide range of future
3
economic scenarios. In Dullaway and Needleman (2003), three approaches (in as-
cending order of sophistication) to incorporating management actions are described:
• A closed form approach. The guarantees or options are valued using a com-
bination of deterministic and closed form solutions (e.g. the Black-Scholes
formula) on a market consistent basis, with no or very limited allowance for
management actions.
• A stochastic simulation approach. The guarantees or options are valued using
market consistent stochastic projection models, but again with no or very
limited allowance for management actions.
• A dynamic simulation approach. The guarantees or options are valued using
market consistent stochastic projection models, with dynamic management
actions incorporated.
Under the third, i.e. the most complicated approach, if the management actions
depend on the insurer’s prospective solvency position, there are some practical issues
with conducting a nested stochastic investigation.
In addition to calculating the realistic values of assets and liabilities, the CP 195
proposals (FSA (2003)) require the calculation of a risk capital margin (RCM). It
is defined as the fall in realistic surplus (which is the excess of the realistic value
of assets over the realistic value of liabilities plus the risk capital margin) following
a specific stress test. The rationale for the RCM calculation is that the insurer
should still be in surplus on a realistic basis following an adverse event. The stress
test set out in CP195 proposals (FSA (2003)) specifies a scenario including a fall
in equity and property values, a widening of credit spreads and a shift in the yield
curve in the direction that gives the greatest reduction in the realistic surplus. As
commented in Muir and Waller (2003), the prescribed stress scenario only covers the
major risks that with-profits funds are likely to be exposed to and ignores other risk
factors such as adverse currency movements, operational risk etc. Hare et al. (2003)
suggest that in the cases where the with-profits fund holds different assets to the
hedge portfolio, a market consistent valuation of liabilities coupled with a short-term
quantile based calculation of additional capital requirement is more appropriate from
4
both policyholder protection and regulatory action viewpoints. Similarly, Hibbert
and Turnbull (2003) also define the minimum regulatory capital requirement as the
capital sufficient to give a 99% probability of meeting the realistic value of guarantees
after one year, in an illustrative example to demonstrate possible implications for
risk-based capital requirements.
This thesis carries out an investigation of the reserves required to meet the ma-
turity guarantees under unitised with-profits (UWP) policies, within the new regu-
latory realistic reporting framework. The following questions are addressed in our
numerical results:
• How should the maturity guarantees under UWP policies have been reserved
for in the past for the realistic balance sheet calculation but using a closed
form approach to incorporating management actions?
• How do these reserves compare with conditional tail expectation (CTE) re-
serves calculated using traditional stochastic valuation techniques?
• What amount of reserves will be required in the future under different reserving
approaches in the simulated real world?
• What is the benefit of reserving for the fund as a whole instead of setting up
reserves separately for each generation of business?
• Would those policies issued in the past be sustainable in a 1% stakeholder
environment if the reserves had been set up as required? How about the
sustainability in the future?
• What reasonable bonus and investment strategies could have been used in the
past? Will these strategies still be reasonable in the future?
• What is the effect of smoothing regular bonus rates from year to year?
• What is the effect of reserving for future bonuses as required by the FSA?
Following the FSA, this thesis also adopts a ‘put option approach’ to value lia-
bilities. We assume that all policies survive to maturity. Mortality and lapses are
ignored for simplicity. At maturity 100% of unsmoothed asset share, which is the
5
accumulated value of the policyholder’s fund, is paid to each policyholder. Hence we
can take advantage of this approach in that no additional valuation is required for
the calculation of the asset share because the market asset value can be used directly.
In other words, the thesis concentrates on reserving for the excess of the guaranteed
payout over the asset share at maturity. As defined in the realistic balance sheet
published by the FSA, we are interested in future costs of financial options less
planned deductions for guarantees from with-profits benefits reserve when using the
market consistent approaches.
0.2 Thesis Outline
Chapter 1 describes the operation of UWP policies and the simplified version con-
sidered in this thesis. We review some of the literature on reserving for policies with
financial guarantees. Then we briefly discuss the three reserving approaches used in
the thesis: buying options over-the-counter (OTC), dynamic hedging internally, and
CTE reserving. The chapter also introduces the models used throughout the thesis.
Three models with different purposes are mainly considered: valuation model, bonus
and asset allocation model, and real world asset model.
In Chapters 2 to 5 we build up the overall methodology of reserving for a single
UWP policy historically. The results obtained for the 10-year policy issued at the
end of 1991 using the three reserving approaches are compared in each chapter.
Chapter 2 starts with the simplest case in which we assume a 100% equity backing
ratio (EBR) which is the proportion of the asset share invested in equities, a constant
bonus rate and a constant risk-free interest rate. In addition, the chapter also
introduces the notation and assumptions for the policy.
In Chapter 3 a dynamic bonus strategy is introduced. We first assume that
regular bonuses are declared according to the bonus strategy without smoothing and
that future regular bonuses are ignored when setting reserves. Then we smooth the
regular bonus rates from year to year, but again ignore future bonuses. Finally the
effect of reserving for the minimum future regular bonuses implied by our smoothing
mechanism is investigated.
6
In Chapter 4, the yield on the zero-coupon bonds with the same maturity date
as the policy is used as a risk-free interest rate. The zero-coupon yield is derived
from the consols yield and short-term interest rate using a simple yield curve. We
concentrate on the case of smoothing with allowance for future bonuses as it fits
better with the current regulatory framework.
In Chapter 5 we assume that the policyholder’s fund is invested in two asset
classes: equities and zero-coupon bonds with the same maturity date as the policy. A
dynamic investment strategy is introduced by which the EBRs are adjusted annually.
As in Chapter 4 we only consider the case of smoothing with allowance for future
bonuses.
Chapter 6 investigates the sensitivity of the results on a single policy to differ-
ent parameters, different probability boundaries used in our dynamic investment
strategy and different 10-year periods.
In Chapters 7 and 8 we build up a portfolio of single premium 10-year UWP
policies which are at different durations.
Chapter 7 looks at the portfolio historically with a 20-year investigation period
starting at the end of 1982. The dynamic bonus and investment strategies are
applied to each generation separately. The risk-free interest rate equals the yield
on the zero-coupon bonds with the same maturity date as the policy. The effects
of smoothing and allowing for future bonuses are both considered. The benefit to
the insurer of pooling risks under the CTE reserving approach is investigated, by
comparing the amount of the reserves required for the whole portfolio with that set
up for each generation of business separately. The portfolio cashflows incurred by
the insurer are rolled up at the risk-free interest rate to calculate the amount of the
insurer’s free estate, from which we can discuss the profitability of the UWP policies
over the last 20 years. The chapter also investigates the sensitivity of the portfolio
results to different parameters as we do in Chapter 6 for the single policy issued at
the end of 1991.
Chapter 8 extends the investigation period to the end of 2032. The real world
during the 30-year period starting at the end of 2002 is simulated stochastically. We
concentrate on the case of smoothing with allowance for future bonuses. Different
7
quantiles of the simulated results for the EBRs, regular bonuses, portfolio reserves,
and insurer’s free estate are given in the chapter. For those policies matured in
the following 30 years, we calculate the probability that the guarantees will bite
at maturity. Summary statistics of the simulated maturity payouts and terminal
bonuses are also calculated.
Chapter 9 gives conclusions and some suggestions for further research.
8
Chapter 1
GUARANTEES UNDER
UNITISED WITH-PROFITS
POLICIES
1.1 Operation of UWP Polices
With-profits business provides guaranteed and smoothed benefits which protect the
policyholders’ investment value against fluctuations in the financial market. In addi-
tion, unitised with-profits (UWP) policyholders can easily work out the value of their
investment at any time given the number of units and the current unit price. They
can also change their premiums or sum assured when their circumstances change.
These characteristics meet today’s increasing demand for greater transparency and
flexibility, and hence have made UWP contracts very popular in the market in recent
years.
The operation of UWP policies is different between different insurers, particularly
between mutual and proprietary companies. However, basically the product works
in the same way as a unit-linked policy. The policyholder pays a premium, from
which a charge is deducted via a reduced allocation rate, bid-offer spread and policy
fee to cover expenses. The rest is then converted into a certain number of units
according to the current unit price. Each year some units are deducted to pay for
mortality charges and fund management charges.
9
The unit price is set by a mixture of guarantees and bonuses, rather than directly
related to the performance of the underlying assets as under the unit-linked policy.
The unit price for the premiums already paid grows at a guaranteed rate. It is
possible for the guaranteed growth rate to be as low as 0%, and in this case the
guarantee still exists as the unit value is not allowed to fall in the future. Usually,
the insurer reserves the right to change the guaranteed rate on future premiums.
In the same way as conventional with-profits (CWP) policies declare reversionary
bonuses, UWP policies apply regular bonuses to increase the unit price beyond the
guaranteed rate. The bonus is actually added on a daily basis, but the bonus rate
declared is normally x% p.a. until further notice. Nevertheless, the insurer is not
forced to add the same bonus throughout the year. If the stock market falls, the
insurer may declare a new lower rate applying from that date.
Similar to the terminal bonus declared in the CWP product, at maturity the
UWP policyholder may receive a payment bigger than his guaranteed unit value
obtained by multiplying the number of his units by the current unit price. In order
to keep the guarantee at a low level and hence increase investment freedom, the
insurer usually promises a low growth rate and declares small regular bonuses to
leave room for a large terminal bonus.
UWP policies usually include a guaranteed sum assured which is payable on
death. Thus, the policyholder on death receives the greater of his unit value, and
the guaranteed sum assured. The policyholder can choose the guaranteed sum as-
sured independently of the premium. The insurer can deal with this flexibility by a
mortality charge to cover the expected cost of death claims.
To protect itself against the risk of financial selection, the insurer retains the right
to impose a market adjustment factor to surrenders which may bring the value of
their units down to a level close to the asset share (value of the underlying assets)
when the stock market falls.
The above description shows the general operation of UWP policies in practice.
However, in this thesis we look at a simplified version. In order to concentrate on
the maturity guarantees, we ignore mortality, lapses and expenses. We also assume
that after policy inception the valuation is conducted just after declaring a regular
10
bonus.
1.2 Reserving Approaches
In this section we first summarise some of the literature on reserving for policies
with financial guarantees. Then we describe the reserving approaches used in this
thesis for our UWP policies.
1.2.1 Literature Review
Willder (2004) has given a detailed literature review on pricing and reserving for
the financial guarantees under unit-linked and participating policies. Here we con-
centrate on reserving and only review the papers whose reserving approaches are
similar to ours.
Ford et al. (1980)
Ford et al. (1980) produced the report of the Maturity Guarantees Working Party.
The working party consider reserving for maturity guarantees under unit-linked
policies. The initial reserves are calculated from a large number of stochastically
simulated future outcomes, with the assumption that the reserves would be sufficient
in 99% of cases. The working party consider both a single policy and a portfolio of
policies with different terms. In the single policy case, the authors conclude that
smaller reserves are required for longer term policies. The reduction in the portfolio
reserves shows the benefit of risk diversification.
Collins (1982)
Collins (1982) explores an immunization approach to reserving for unit-linked
policies with maturity guarantees. A hedge portfolio is constructed and rebalanced
discretely to meet the guaranteed maturity benefit. The author finds that the im-
munization strategy requires the largest rebalancing of assets when the asset share is
in the ‘volatile region’ of the current value of the maturity guarantees discounted at
a risk-free interest rate. To reduce the intensity of the asset rebalancing, the author
sets up the hedge portfolio as if the policy had a longer term and larger guarantees.
11
Boyle and Hardy (1997)
Boyle and Hardy (1997) consider a stochastic simulation approach and an option
pricing approach to reserving for Canadian segregated funds which are unit-linked
policies with maturity guarantees. Under the stochastic simulation approach the
authors simulate the performance of the unit fund using the Wilkie model. Quantile
reserves are set up at a chosen probability level. Under the option pricing approach
they construct a hedge portfolio derived from the Black-Scholes equation. As con-
tinuous hedging is not possible in practice, the authors consider time-based and
move-based hedging strategies. The comparison of the two strategies is conducted
using a simulation approach. The authors conclude that the move-based strategy
provides superior hedging performance because the tracking errors are smaller for
the same expected hedging costs. The authors also consider buying correspond-
ing options externally from a bank or another financial institution. The reserves
required are the cost of buying these options. This method provides a 100% proba-
bility that the guarantee will be honoured assuming that there is no counter-party
risk, which is the risk that the option provider defaults.
Hardy (1999)
Hardy (1999) compares the quantile reserves calculated using different invest-
ment models. A lognormal model and a regime switching lognormal (RSLN) are
considered and both of them are calibrated to Toronto Stock Exchange data. The
author demonstrates a danger of being under-reserved if using an insufficiently fat
tailed distribution to model the investment performance.
Hardy (2000)
As in Boyle and Hardy (1997), Hardy (2000) also calculates reserves for segregated
funds using the stochastic simulation and option pricing approaches. Under the
simulation approach the author compares the initial quantile reserves obtained using
the Wilkie Model and the lognormal model, with and without allowance for fund
management charges. To calculate the additional capital required to increase the
reserves or the capital released from the reserve fund, the author adopts a corridor
approach whereby the reserves are strengthened if the probability of sufficiency falls
below 92.5% and are weakened if the probability rises above 99.8%. Under the option
12
pricing approach the author considers both discrete hedging and buying options.
Hardy (2001)
Hardy (2001) compares the conditional tail expectation (CTE) reserves calculated
using the lognormal model and the RSLN model. As in Hardy (1999), the author
demonstrates a danger of being under-reserved if the investment performance is
simulated by an insufficiently fat tailed distribution.
Hare et al. (2000)
Hare et al. (2000) consider maturity guarantees under conventional and unitised
with-profits policies at different durations. The authors first calculate reserves on the
current UK statutory reserving basis including the minimum solvency margin and
resilience reserve. Then they find that the statutory reserves are mostly inadequate
if reserves are required at a 99% probability level using the stochastic simulation
approach. The investment performance is simulated by the Wilkie model with ad-
justed low inflation parameters. The authors also calculate the ratio of the 99%
quantile reserve to the statutory minimum reserve at each duration.
Yang (2001)
Yang (2001) considers guaranteed annuity options (GAOs) attached to pensions
policies. The initial reserves are calculated using a stochastic simulation approach
and a hedging approach. Under the simulation approach, both quantile and CTE
reserves are calculated. Under the hedging approach, the author derives closed
form solutions for the hedging price. The tracking error and transaction costs are
compared under the hedging strategies with annual and monthly rebalancing.
Wilkie et al. (2003)
Wilkie et al. (2003) extend the work carried out in Yang (2001). The authors
consider reserving for a pensions-type contract with GAOs both historically and in
the stochastically simulated real world. The stochastic simulation and option pricing
approaches are used. Under the simulation approach, the quantile and CTE reserves
are both calculated. The authors also compare the initial reserves calculated for
the whole portfolio with different terms to maturity with those calculated for each
individual policy separately. Then the reserves required each year are calculated
using both marking-to-market and corridor approaches.
13
Hibbert and Turnbull (2003)
Hibbert and Turnbull (2003) mainly calculate the fair value of guarantees under
a single conventional with-profits policy using a market-consistent asset model. The
authors also investigate the sensitivity of this fair value to a set of decision rules for
bonuses, equity backing ratio and policyholder behaviour. The authors suggest that
reserves should be set up so that the fund would be sufficient to meet the realistic
value of guarantees after one year with a 99% probability.
Haberman et al. (2003)
Haberman et al. (2003) consider reserving for unitised with-profits contracts using
stochastic simulation techniques. Assets are modelled with a Brownian motion. The
authors consider three smoothing schemes for regular bonuses. The initial reserves
are calculated for the guaranteed benefit, future regular bonuses and terminal bonus.
Then the authors investigate the sensitivity of the required amount of reserves to
changes in the model parameters.
Summing up, Collins (1982), Boyle and Hardy (1997), Hardy (2000), Yang (2001),
Wilkie et al. (2003) and Hibbert and Turnbull (2003) set up reserves based on option
pricing theory. Ford et al. (1980), Boyle and Hardy (1997), Hardy (1999), Hardy
(2000) and Hare et al. (2000) calculate quantile reserves using stochastic simulation
techniques. CTE reserves are considered in Hardy (2001). Yang (2001) and Wilkie
et al. (2003) calculate both quantile and CTE reserves.
1.2.2 Three Reserving Approaches
Having reviewed some of the literature on approaches to reserving for policies with
financial guarantees, here in this section we briefly describe our reserving approaches
used throughout the thesis. The overall methodology is similar to that of Boyle and
Hardy (1997), though they considered unit-linked policies.
The maturity payout under a UWP policy corresponds to that of a combination
of shares and European put options. Thus, the first approach adopted in this thesis
is to buy the corresponding options from a third party so that the guarantees can
always be met at maturity. We only look at the part of the reserve fund held in
addition to the asset share, so the amount of the reserves calculated in the thesis
14
does not include the asset share. Under this approach, the asset and liability can
be exactly matched assuming that there is no counter-party risk. Therefore, the
required amount of reserves equals the cost of buying these options. The guarantee
builds up over the policy term through regular bonuses, hence the exercise price
of the options increases after each bonus declaration. We also consider how the
reserve changes through time on a marking-to-market basis. After declaring a regular
bonus, we should sell all the options bought one year ago with the lower exercise
price determined before the bonus declaration and buy some new options (whether
the number of options changes depends on the investment strategy) with the higher
exercise price determined just after the bonus declaration. We will explain this point
in later chapters. In practice buying or selling options incurs transaction costs, but
they are ignored in the thesis for simplicity.
Instead of buying options externally, the insurer can also construct a hedge port-
folio which replicates the payoff under the maturity guarantees. This is the second
approach considered in the thesis, known as dynamic hedging. Under this approach,
the required amount of reserves equals the value of the replicating portfolio. In the-
ory we should rebalance the hedge portfolio continuously as the value of underlying
assets changes through time. However, in practice rebalancing can only occur at
discrete intervals. Throughout the thesis the hedge portfolio is rebalanced annually.
The departure from continuous rebalancing introduces hedging error. Constructing
and rebalancing the hedge portfolio incurs transaction costs. The more frequently
the hedge portfolio is rebalanced, the smaller the hedging error but the more trans-
action costs are incurred.
The above two approaches are based on modern option pricing theory and they
calculate the reserves as the realistic peak of the ‘twin peaks’ approach. The third
approach uses stochastic simulation techniques which were introduced to the actu-
arial profession by the Maturity Guarantees Working Party in Ford et al. (1980).
At each valuation date, 10,000 possible future scenarios are simulated stochastically.
The cashflows incurred by the insurer during the remaining policy term in each simu-
lated scenario are projected. Then we discount these projected cashflows back to the
current valuation date to work out the CTE reserve. The CTE approach calculates
15
the amount of reserves required so that the probability of having sufficient assets to
cover liabilities at maturity is above a certain level, instead of the current value of
future liabilities as calculated by the other two reserving approaches. Therefore, the
third approach is thought to be not market consistent.
As mentioned before, the purpose of applying all these approaches is to compare
the amount of reserves required in the current realistic reporting regime with that
required using traditional stochastic valuation techniques.
1.3 Models Required
The models required throughout the thesis can be divided into three types:
• valuation model whereby the required amount of reserves is calculated
• bonus and asset allocation model whereby the investment performance of the
unit fund is projected for the bonus declaration and asset allocation
• real world asset model whereby the performance of the assets is simulated.
They are discussed in turn as follows.
1.3.1 Valuation Model
The three approaches to reserving for maturity guarantees under UWP policies
have been described in Section 1.2.2. The first two approaches are based on option
pricing theory. We assume that the options are priced using the Black-Scholes
equation. In the Black-Scholes option pricing model, the assets follow a lognormal
distribution and the option is valued as the discounted expectation of the payout
under a martingale measure.
The CTE reserving approach requires the availability of a stochastic investment
model. The Wilkie model is widely used in the actuarial profession. Wilkie (1995)
covered retail price index, equity dividend amount, equity dividend yield, consols
yield, short-term interest rate, wages index, property yield, property income, index-
linked yield and currency exchange rate. Many researchers have applied the Wilkie
model and also given a clear description of the model in their work, for example,
16
Hardy (1994), Macdonald (1995), Boyle and Hardy (1997), Hardy (2000), Hare
et al. (2000), Yang (2001), Wilkie et al. (2003) and Willder (2004). We follow
these authors, and simulate future possible scenarios for valuation using the Wilkie
model. In Appendix A we give details of the part of the Wilkie model (1995 version)
which is relevant to the variables considered in the thesis, i.e. retail price index,
equity dividend amount, equity dividend yield, equity price index, consols yield and
short-term interest rate.
1.3.2 Bonus and Asset Allocation Model
Our dynamic bonus strategy uses a bonus earning power mechanism based on the
projected value of the maturity asset share, and our dynamic investment strategy
allocates the asset share in different asset classes according to the insurer’s prospec-
tive solvency position. Detailed description of the mechanism of both strategies
will be given in later chapters. The two strategies both require the availability of a
stochastic investment model. We assume that the bonus and asset allocation model
is the same as the valuation model under the CTE reserving approach. In other
words, future possible scenarios are simulated using the Wilkie model for the bonus
declaration, asset allocation and CTE reserving.
However, there is no reason why in practice different models could not be used
for internal management and for reserving.
1.3.3 Real World Asset Model
The thesis starts with calculating reserves using historical data. The investment
performance of the assets under consideration is already known, hence no model is
needed. After investigating a portfolio of policies historically for 20 years, we will
extend the investigation period to the future to calculate what amount of reserves
will be required for different quantiles. In this case, we need to know how the
investment market will perform in the future. To be consistent with our valuation
model under the CTE reserving approach and also our bonus and asset allocation
model, we assume that the real world will follow the Wilkie model.
17
In future research we could look at the effect of model error in our internal models
by using a different real world model.
18
Chapter 2
RESERVING FOR A SINGLE
UWP POLICY HISTORICALLY
2.1 Introduction
In this chapter we go back to the end of 1991 and consider how the maturity guaran-
tees under a single UWP policy with a 10-year term, issued on 31 December 1991,
would have been reserved for within today’s realistic reporting regime and using
traditional stochastic simulation techniques. The policy we look at is artificially
simple. The policyholder’s fund is entirely invested in equities. Constant regular
bonus rates are declared over the policy term. The risk-free interest rate is also
constant over the 10-year period. The artificial simplifications help us concentrate
on the maturity guarantees under the policy. Later on we will make the overall
methodology more complicated and realistic.
In Sections 2.2, 2.3 and 2.4 we first introduce the required notation and assump-
tions. Then we describe the detailed mechanism of the three reserving approaches in
equations and also present the numerical results. Finally in Section 2.5 a summary
is given.
19
2.2 Reserving Approach of Buying Options
This section looks at the option method. Before describing its mechanism in detail,
we first introduce the notation and assumptions.
2.2.1 Notation and Assumptions
We define the following notation for the option approach,
• T : the policy term
• SP : the single premium of the policy
• P (t): the equity price index at time t
• D(t): the dividend amount at time t
• A(t): the asset share at time t
• e(t): the equity backing ratio of the asset share at time t
• G(t): the maturity guarantees at time t
• c: the fixed percentage of the units deducted at the end of each policy year as
a charge for the guarantees
• g: the guaranteed growth rate in the unit price
• b(t): the regular bonus rate declared at time t
• S(t): the value of a single unit of equity index at time t with all dividends
immediately reinvested and without allowance for tax
• GC(t): the amount of the guarantee charge deducted from the policyholder’s
fund at time t
• N(t): the number of put options held at time t
• N ′(t): the number of units of equity index held at time t
• E(t): the exercise price of the put options bought at time t
20
• r: the constant risk-free rate of interest
• δ(t): the risk-free force of interest at time t
• σ: the volatility of the equity index
• O(t): the price of each put option at time t with exercise price E(t)
• O′(t): the price of each put option at time t with exercise price E(t− 1)
• V (t): the amount of the reserve set up at time t
• CF (t): the value of the cashflows incurred by the insurer at time t
• AV CF : the accumulated value at policy termination of the cashflows incurred
during the policy term.
The policy we are looking at is a single premium UWP policy with a term of 10
years issued on 31 December 1991. The single premium is assumed to be £100.
At this stage, we assume that the policyholder’s fund is entirely invested in
equities and that constant regular bonus rates are declared during the policy term.
Ford et al. (1980), Collins (1982), Boyle and Hardy (1997), Hardy (1999), Hardy
(2000), Hardy (2001), Yang (2001), Jørgensen (2001), Grosen and Jørgensen (2002)
and Wilkie et al. (2003) all assume a 100% equity proportion of the asset share. We
will consider a dynamic investment strategy in Chapter 5.
Wilkie (1987) and Willder (2004) both consider a static bonus mechanism before
introducing dynamic bonuses. Hare et al. (2000) adopt a static bonus strategy.
Deterministic bonus rates are declared up to the valuation date, and the bonuses
declared afterwards are ignored when setting up reserves.
Throughout the thesis we assume that a fixed percentage of the asset share is
deducted at the end of each policy year as a guarantee charge. This is similar
to the ‘asset share charging approach’ used in Hare et al. (2000). The paper also
considers a ‘capital support charging approach’ whereby the charge is related to
the excess of the reserve over the asset share, and a ‘put spread strategy’ which is
similar to Wilkie (1987) and Willder (2004) that put options are bought to ensure
the guarantees can be met at maturity. Jørgensen (2001) and Grosen and Jørgensen
21
(2002) set the charge equal to the cost of guarantees under the equivalent martingale
measure. The fixed percentage deduction assumed in the thesis is unlikely to reflect
the actual cost of guarantees. However, the approach is simple to apply and it avoids
dealing with the problem of ‘iterative solution’ discussed in Wilkie et al. (2003) that
the more the charge is deducted, the more onerous is the guarantee as less can be
invested in the units. Also, this thesis is mainly concerned with reserving for the
guarantees rather than charging.
The guaranteed growth rate in the unit price, percentage of the asset share de-
ducted as a charge and regular bonus rate are important contract design parameters
which define the value of the contract. Ideally, these parameters should be set aim-
ing for a fair design so that the value of the contract equals the premium paid by
the policyholder according to the no-arbitrage principle. In Jørgensen (2001) and
Grosen and Jørgensen (2002), the charges and guarantees have the same expected
present value under the equivalent martingale measure. Haberman et al. (2003)
consider the trade-off among the parameters of the guaranteed growth rate, partici-
pation coefficient, smoothing parameter and terminal bonus rate for a fair contract.
Here, however, we simply assume a 2% guaranteed rate, 1% charge and 5% regular
bonus rate over the policy term. These assumptions are artificial in some sense,
though they are not unreasonable given current market practice.
The risk-free rate of interest and volatility of the equity index are important
parameters in the Black-Scholes equation. At this stage, we assume a 5% constant
risk-free interest rate, and in Chapter 4 we will introduce a more realistic rate. Boyle
and Hardy (1997) assume a risk-free rate of 6% p.a.
A main problem with this option method is that the policy term is too long
and beyond the duration of most OTC markets and the market for long-dated
equity options is too small at present. Therefore, the put options cannot be actually
purchased at their theoretical price, i.e. the price derived using the Black-Scholes
equation. Hardy (2000) deals with this problem by adding a margin of 5% in the
volatility of the underlying assets when calculating the option price. We assume a
20% volatility for the equity index in this thesis, and will investigate the sensitivity
of the results to a different volatility.
22
Summing up, we have made the following assumptions for the contract design
and market parameters:
• T = 10
• SP=£100
• e(t) = 100%, t=0, 1, ..., T − 1
• b(t) = 5%, t=0, 1, ..., T − 1
• g = 2%
• c = 1%
• r = 5%
• σ = 20%.
2.2.2 Mechanism of the Option Approach
The basic idea of the option method has been described in Section 1.2.2. The asset
share is entirely invested in equities which provide volatile returns. The guarantees,
however, build up over the policy term through regular bonuses. If the market value
of the equities falls below the guaranteed payout at maturity, the insurer bears the
cost of the shortfall. In the new realistic reporting framework, the optionality of the
contract should be allowed for explicitly using a market consistent valuation. The
maturity payout of the contract described in Section 2.2.1 corresponds to that of
a combination of shares and European put options. A closed form solution can be
used under some strong assumptions: future regular bonuses are ignored and the
equity backing ratio is maintained at the 100% level. The valuation model assumes a
Geometric Brownian Motion for shares and a constant risk-free interest rate, which
leads to the Black-Scholes equation. We describe the detailed mechanism of the
approach in the following equations.
At policy inception, the policyholder pays a single premium which is converted
into units. The initial asset share has a value of
A(0) = SP. (2.1)
23
The value of an equity index at the outset of the policy equals the current equity
price index, i.e.
S(0) = P (0). (2.2)
Hence the initial number of equity index held in the policyholder’s fund is given
by the equation
N ′(0) =A(0)
S(0). (2.3)
The maturity payout to the policyholder is the asset share subject to the guarantee.
Hence, for each unit of equity index held at maturity, we should have one put option
so that the guarantees can be met by exercising the options if necessary. We know
in advance how many units of equity index will be held at maturity when the policy
is issued, as the number of units changes in a deterministic way assuming that the
EBR is always 100% and that a fixed percentage of the units are cashed to pay for
the guarantees. Therefore, the number of options is constant over the policy term
and it equals the number of equity index units held at maturity. Expressed as an
equation,
N(t) = N ′(T ) (2.4)
= N ′(0) · (1− c)T
for t=0, 1, ..., T . However, equation 2.4 does not apply once we introduce a dynamic
investment strategy.
The initial value of the maturity guarantees, ignoring all future regular bonuses,
follows the equation
G(0) = SP · (1 + g)T . (2.5)
If the put options are exercised at maturity, the payout of the options should be
exactly enough to cover the guarantees under the policy. Thus, we have the required
exercise price of the options
E(0) =G(0)
N(0). (2.6)
24
Using the Black-Scholes formula (Hull (1999)), each put option at the outset has
a value of
O(0) = E(0) · e−δ(0)·T · Φ(−d2(0))− S(0) · Φ(−d1(0)) (2.7)
d1(0) =log(S(0)/E(0)) + (δ(0) + 1
2· σ2) · T
σ · √T
d2(0) = d1(0)− σ ·√
T
where δ(0) is the risk-free force of interest at policy inception. It is assumed in this
chapter that the risk-free rate δ(t) is constant over the policy term. Hence,
δ(t) = log(1 + r) (2.8)
for t=0, 1, ..., T .
Therefore, the initial reserves have a value of
V (0) = N(0) ·O(0). (2.9)
In the thesis we ignore the transaction costs of buying and selling options, so
the only cashflow incurred by the insurer at policy inception is to set up the initial
reserves, i.e.
CF (0) = V (0). (2.10)
Throughout the thesis, a positive cashflow means the money paid by the insurer and
a negative cashflow means the money paid to the insurer.
At the end of each policy year, the value of the equity index with the dividends
reinvested immediately and without allowance for tax is given by the equation
S(t) = S(t− 1) · P (t) + D(t)
P (t− 1)(2.11)
for t = 1, 2, ..., T .
The asset share is rolled up with the total nominal return in the equity index.
A fixed percentage of the units is deducted to pay for the cost of guarantees. The
amount of the guarantee charge is given by the equation
GC(t) = A(t− 1) · P (t) + D(t)
P (t− 1)· c. (2.12)
25
The asset share has a value of
A(t) = A(t− 1) · P (t) + D(t)
P (t− 1)−GC(t). (2.13)
After declaring a regular bonus at time t, t = 1, 2, ..., T − 1, the guarantee is
increased, i.e.
G(t) = G(t− 1) · (1 + b(t)). (2.14)
Given the number of options by equation 2.4, the exercise price of the option
after the bonus declaration follows the equation
E(t) =G(t)
N(t). (2.15)
Thus, the same number of put options are held before and after the bonus declara-
tion, but the exercise price is increased by the declared bonus rate. Figure 2.1 shows
the maturity payout of the policy before and after declaring the bonus at time t (in
the top graph), and the increase in the maturity payout after the bonus declaration
(in the bottom graph).
The top graph shows that before declaring the bonus at time t, the payout equals
the maturity asset share subject to the guarantee built up to time t− 1. Just after
the bonus declaration at time t, the guarantee is increased by the bonus rate and has
a value of G(t). Thus, on top of the reserve set up one year ago we need some options
which can provide a payoff at maturity as shown in the bottom graph. According to
Hull (1999), the payoff corresponds to that of a bear spread created by buying a put
option with a high exercise price (implied by G(t) in our case) and simultaneously
selling a put option with a low exercise price (implied by G(t− 1)). In other words,
after declaring a bonus we should sell all those put options bought one year ago with
the exercise price determined before the bonus declaration, and at the same time
buy the same number (in the case of our static investment strategy with a 100%
EBR) of new put options with the exercise price calculated by equation 2.15.
A put option with the exercise price E(t), using the Black-Scholes formula, has
a value of
O(t) = E(t) · e−δ(t)·(T−t) · Φ(−d2(t))− S(t) · Φ(−d1(t)) (2.16)
26
-
6
-
6
payout atmaturity
asset shareat maturityG(t− 1) G(t)
G(t− 1)
G(t)
increase inmaturitypayout
asset shareat maturityG(t− 1) G(t)
G(t) −G(t− 1)
Figure 2.1: Maturity payout before and after declaring a bonus at time t, and theincrease in the maturity payout after the bonus declaration
d1(t) =log(S(t)/E(t)) + (δ(t) + 1
2· σ2) · (T − t)
σ · √T − t
d2(t) = d1(t)− σ · √T − t.
Hence, the required amount of reserves at time t is given by the equation
V (t) = N(t) ·O(t). (2.17)
Each put option bought one year ago with the exercise price E(t−1) has a current
value of
O′(t) = E(t− 1) · e−δ(t)·(T−t) · Φ(−d′2(t))− S(t) · Φ(−d′1(t)) (2.18)
d′1(t) =log(S(t)/E(t− 1)) + (δ(t) + 1
2· σ2) · (T − t)
σ · √T − t
d′2(t) = d′1(t)− σ · √T − t.
27
With no allowance for the transaction costs of buying and selling options, the
value of the cashflows incurred by the insurer at time t is the cost of buying the new
options less the gains from selling the old options less the guarantee charge deducted
from the policyholder’s fund, i.e.
CF (t) = V (t)−N(t− 1) ·O′(t)−GC(t). (2.19)
At maturity, i.e. t = T , no regular bonus is declared and hence the amount of
guarantees remains the same as one year before maturity, i.e.
G(T ) = G(T − 1). (2.20)
The put options bought at time T − 1 expire worthless if the asset share is
bigger than the guarantee at maturity; otherwise the options are exercised and the
guaranteed amount is paid to the policyholder. Therefore, the only cashflow incurred
by the insurer at maturity is the guarantee charge deducted from the units, i.e.
CF (T ) = −GC(T ). (2.21)
Given the cashflows incurred during the policy term, we can calculate the accu-
mulated value at policy termination of these cashflows to investigate the profitability
of the policy. We accumulate the cashflows at a risk-free rate r, assuming that cap-
ital is provided by the insurer’s free estate which is invested in a risk-free asset. If
we assume that capital is provided by shareholders or other companies, the insurer
needs to pay a cost of capital rate on what it borrows. Thus, here we discuss prof-
itability by means of whether the insurer’s estate is increased or decreased by this
particular policy, and we ignore the cost of capital. We have the following equation
AV CF =T∑
t=0
CF (t) · (1 + r)T−t (2.22)
2.2.3 Results Using the Option Method
Figure 2.2 shows how the asset share and guarantee build up over the policy term,
under the assumptions we have made in Section 2.2.1.
The insurer has promised a 2% growth rate in the unit price, so we see in Figure
2.2 that the guarantee is bigger than the asset share at policy inception. How-
ever, the asset share builds up more rapidly than the guarantee most of the time.
28
year
1992 1994 1996 1998 2000
100
150
200
250
300
1991 1993 1995 1997 1999 2001
asset shareguarantee
Figure 2.2: The asset shares and guarantees for the policy issued at the end of 1991with static bonus and investment strategies
Although the asset share falls dramatically at later durations due to the poor eq-
uity performance, the insurer can still declare a terminal bonus rate of 40.94% at
maturity with the asset share and guarantee of £266.54 and £189.11 respectively.
In Figure 2.3 we show the amount of reserves at each valuation date under the
option approach. The details of the calculations for the 1991 policy are given in
Appendix C.
Equation 2.17 shows that the required amount of reserves equals the cost of
buying the put options. Hence the pattern of the reserves shown in Figure 2.3 can
be explained by the relative movement in the equity index and exercise price which
are shown in Figure 2.4.
Figure 2.4 shows that the equity index starts from a lower level than the exercise
price but increases more rapidly most of the time. The guarantees are deeply in-
the-money at policy inception, which explains why a large initial reserve is required
as shown in Figure 2.3. At the end of 1993 the equity index rises slightly above the
exercise price, but the fall in the equity index in the next year raises the value of the
put options. Afterwards, the amount of reserves decreases because the equity index
mostly goes up rapidly. The index falls at the end of 2000, but it has little effect
on the reserves because the policy is not far from maturity and the guarantees are
29
year
rese
rves
1992 1994 1996 1998 2000
02
46
810
1214
1991 1993 1995 1997 1999
Figure 2.3: The reserves set up using the option method for the 1991 policy withstatic bonus and investment strategies and a constant risk-free rate
year
1992 1994 1996 1998 2000
1500
2000
2500
3000
3500
4000
1991 1993 1995 1997 1999
equity indexexercise price
Figure 2.4: The equity index and exercise price of the options bought for the 1991policy
30
deeply out-of-the-money before the index falls.
The accumulated value at policy termination of all the cashflows incurred by the
insurer during the policy term equals £7.93. Given the convention that positive
cashflow represents the money paid by the insurer, we can conclude that the insurer
has made a loss of £7.93 by selling the 1991 policy. We have seen in Figure 2.3 that
a large reserve is required at policy inception. Hence if we allow for cost of capital
by assuming a higher rate of return required by the capital provider, the insurer
will make a larger loss. For example, if we accumulate the cashflows at 8%, the
loss will be £14.46. The risk-free interest rate and volatility of the equity index are
important parameters in the Black-Scholes equation. The value of the options might
have been overstated because of the wrong assumptions made for these parameters.
Later in Chapter 4 we will derive the risk-free rate from the historical consols yield
and short-term interest rate using a yield curve.
2.3 Dynamic Hedging
In this section we consider another reserving approach of dynamic hedging. As in
the previous section, we first describe its mechanism in equations and then give the
numerical results.
2.3.1 Mechanism of the Hedging Approach
We will continue to use the notation of Section 2.2.1. The extra notation for the
hedging approach is given as follows,
• H(t): the value at time t of the hedge portfolio constructed at time t to hedge
N(t) options with exercise price E(t)
• H ′(t): the value at time t of the hedge portfolio which should be held at time
t to hedge N(t− 1) options with exercise price E(t− 1)
• H ′′(t): the value at time t of the hedge portfolio constructed at time t− 1 to
hedge N(t− 1) options with exercise price E(t− 1)
• M(t): the value of the total mismatch in the hedge portfolio at time t
31
• M ′(t): the value of the mismatch at time t due to the hedging error
• M ′′(t): the value of the mismatch at time t due to the bonus declaration (and
asset allocation when adopting a dynamic investment strategy)
• τ : the transaction costs as a percentage of the change in the equity component
of the hedge portfolio
• TC(t): the amount of the transaction costs incurred at time t to construct or
readjust the hedge portfolio.
The mechanism of dynamic hedging is also based on option pricing theory. In-
stead of buying options from a third party, the insurer can also hedge the risk
internally by constructing a hedge portfolio which replicates the payoff under the
maturity guarantees. The required amount of reserves is equal to the value of the
replicating portfolio. To hedge the risk completely, the hedge portfolio should be
rebalanced continuously. The same results for the reserves and cashflows will be
obtained under the approaches of continuous hedging and buying options if transac-
tion costs are ignored. In practice, however, rebalancing can only occur at discrete
intervals. Throughout the thesis the hedge portfolio is rebalanced annually. The
departure from continuous rebalancing incurs hedging error. The transaction costs
incurred to construct or readjust the portfolio are allowed for under this approach.
We derive the following equations to describe the mechanism in detail.
The hedge portfolio is also valued using the Black-Scholes formula. Thus, the
required amount of reserves at each valuation date is the same under the option and
hedging approaches. Hence, for t=0, 1, ..., T − 1,
V (t) = H(t) (2.23)
= N(t) ·O(t)
in which the number of options N(t) follows equation 2.4, and the value of each
option O(t) is given by equations 2.7 and 2.16. Hence, at time t the hedge portfolio
contains
N(t) · E(t) · e−δ(t)·(T−t) · Φ(−d2(t))
32
amount of cash where the exercise price E(t) is given by equations 2.6 and 2.15, and
−N(t) · Φ(−d1(t))
number of equity index.
The cashflows incurred by the insurer using the option and hedging approaches
are different because discrete hedging incurs hedging error and transaction costs are
allowed for under the hedging approach. We only consider the transaction costs
incurred to change the equity component of the hedge portfolio. The transaction
costs on bonds are negligible compared to those on equities.
The amount of transaction costs at policy inception follows the equation
TC(0) = τ ·N(0) · S(0) · Φ(−d1(0)). (2.24)
The initial cashflow incurred by the insurer is to set up the initial reserves and
pay for the transaction costs. Hence,
CF (0) = V (0) + TC(0). (2.25)
Just before declaring a bonus at time t, t=1, 2, ..., T − 1, the value of the hedge
portfolio which should be held to hedge N(t−1) options with exercise price E(t−1)
follows the equation
H ′(t) = N(t− 1) ·O′(t) (2.26)
where O′(t), the value of each option with exercise price E(t−1), is given by equation
2.18. However, the portfolio constructed at time t − 1 to hedge N(t − 1) options
with exercise price E(t− 1) has a current value of
H ′′(t) = N(t− 1) · E(t− 1) · e−δ(t)·(T−t) · Φ(−d2(t− 1)) (2.27)
−N(t− 1) · S(t) · Φ(−d1(t))
because the cash part has been accumulated at δ(t) and the equity part has been
rolled up with S(t)S(t−1)
.
The difference between H ′(t) and H ′′(t) is the hedging error at time t which is
introduced by the departure from continuous rebalancing, i.e.
M ′(t) = H ′(t)−H ′′(t). (2.28)
33
The difference between H(t) and H ′(t) is the mismatch in the hedge portfolio due
to the current bonus declaration, i.e.
M ′′(t) = H(t)−H ′(t). (2.29)
M ′′(t) has the same value as the increase in the reserves using the option method
whereby the hedging error is transferred to the third party. The total mismatch in
the portfolio has a value of
M(t) = M ′(t) + M ′′(t) (2.30)
= H(t)−H ′′(t).
The amount of the transaction costs incurred at time t to rebalance the hedge
portfolio follows the equation
TC(t) = τ · S(t)· | N(t) · Φ(−d1(t))−N(t− 1) · Φ(−d1(t− 1)) | . (2.31)
The cashflows incurred at time t are the total mismatch, transaction costs, and
guarantee charges. Hence,
CF (t) = M(t) + TC(t)−GC(t) (2.32)
= V (t)−N(t− 1) ·O′(t)−GC(t) + M ′(t) + TC(t)
where GC(t) follows equation 2.12. Comparing equations 2.19 and 2.32, we see that
the difference between the cashflows at time t, t=1, 2, ..., T−1, using the option and
hedging approaches is the hedging error and transaction costs incurred by discrete
hedging.
At maturity, i.e. t = T , the hedge portfolio constructed at time T −1 has a value
of
H ′′(T ) = N(T − 1) · E(T − 1) · Φ(−d2(T − 1)) (2.33)
−N(T − 1) · S(T ) · Φ(−d1(T − 1)).
No regular bonus, except for the guaranteed 2%, is declared at maturity, so the
only mismatch in the hedge portfolio is caused by the hedging error which equals
34
the difference between the payoff under the maturity guarantees and the value of
the hedge portfolio constructed at time T − 1. Expressed as an equation,
M(T ) = M ′(T ) (2.34)
= max(G(T )− A(T ), 0)−H ′′(T ).
We assume that all the shares held short in the hedge portfolio are purchased at
maturity, so the transaction costs follow the equation
TC(T ) = τ ·N(T − 1) · S(T ) · Φ(−d1(T − 1)). (2.35)
The cashflows incurred at maturity have a value of
CF (T ) = M(T ) + TC(T )−GC(T ). (2.36)
While the only cashflow incurred at maturity using the option method is the guar-
antee charge deducted from the units.
The accumulated value at policy termination of all these cashflows incurred dur-
ing the policy term follows equation 2.22.
From the above equations, we can see that the hedging strategy requires the
insurer to short equities which is not allowed under the current regulations. There
are also some practical issues with hedging, for example trading might cause the
market to move. The feasibility of hedging is not discussed further in this thesis.
2.3.2 Results Using Discrete Hedging
We use the same assumptions as in Section 2.2.1 for the policy and market param-
eters. In addition, we assume that transaction costs are 0.2% of the change in the
equity component of the hedge portfolio, i.e. τ = 0.2%.
The details of the calculations are shown in Appendix C. The required amount
of reserves is the same under the option and hedging approaches, so the results are
not repeated here. However, the cashflows incurred by the insurer are different.
By hedging, the cashflows have a final value of £2.94 at policy termination. Thus,
the insurer has made a loss of £2.94 from the policy sold at the end of 1991 which
is smaller than £7.93 using the option approach. However, we have allowed for
35
year
1992 1994 1996 1998 2000
-1.0
-0.6
-0.2
0.0
0.2
0.4
1991 1993 1995 1997 1999 2001
hedging errortransaction costs
Figure 2.5: The hedging error and transaction costs incurred by discrete hedging forthe 1991 policy
the transaction costs incurred to construct and readjust the hedge portfolio, but
they have been ignored when buying and selling the options. Even so, the insurer
has made a smaller loss by hedging the risk internally. This can be explained by
Figure 2.5 which shows the hedging error and transaction costs incurred by discrete
hedging.
The rate of transaction costs is assumed to be 0.2%, so the transaction costs
shown in Figure 2.5 are negligible. The hedging error is very volatile over the
policy term, and it is mostly negative. The negative hedging error means that
the replicating portfolio brought forward is worth more than that required to be
set up and it is probably because the assumed 20% volatility for the equity index
might be higher than in reality. From Figure 2.5 we can infer that the sum of
the transaction costs and hedging error mostly has a negative value. Thus, some
cashflows are released back to the insurer and hence the loss is smaller under the
hedging approach.
2.4 CTE Reserving
In this section we calculate reserves using stochastic simulation techniques. The
methodology is similar to that proposed by the Maturity Guarantees Working Party
36
in its report, Ford et al. (1980), but we use a conditional tail expectation risk measure
instead of a quantile measure. The two risk measures are related in a way that the
CTE reserve is equal to the quantile reserve plus the expected excess loss. Expressed
in formulae,
P(X < Qα) = α (2.37)
and
Tα = E[X | X ≥ Qα] (2.38)
= Qα + E[X −Qα | X ≥ Qα]
in which X is a variable representing the incurred loss, α is a security level, for
example 99%, Qα is the quantile reserve, and Tα is the CTE reserve.
The quantile risk measure has the advantages of easy application and simple
interpretation. However, it has been criticised for being incoherent, in Artzner et al.
(1999), Wirch (1999) and Wirch and Hardy (1999). For a highly skewed distribution
it is possible that the quantile is less than the mean. Another problem is that when
combining losses, the quantile measure might be super-additive which means that
there is an incentive to divide a portfolio into subportfolios. From equation 2.38,
we can easily see that the CTE reserve can never be smaller than the mean or the
quantile reserve.
CTE reserving has also been considered in Hardy (2001), Yang (2001), Panjer
(2001) and Wilkie et al. (2003).
2.4.1 Mechanism of the CTE Approach
Here is the extra notation for the CTE approach,
• P ′(t, p, i): the projected equity price index at time p in the ith simulation
using the Wilkie model, given the index at time t < p
• D′(t, p, i): the projected dividend amount at time p in the ith simulation using
the Wilkie model, given the index at time t < p
37
• A′(t, p, i): the projected value of the asset share at time p in the ith simulation,
projected at time t < p
• GC ′(t, p, i): the projected amount of the guarantee charge deducted at time p
in the ith simulation, projected at time t < p
• CF ′(t, p, i): the projected value of the cashflows incurred at time p in the ith
simulation, projected at time t < p
• PV CF ′(t, i): the present value at time t of the projected cashflows incurred
during the remaining policy term in the ith simulation before sorting
• PV CF ′′(t, i): the present value at time t of the projected cashflows incurred
during the remaining policy term in the ith simulation after sorted into as-
cending order
Basically, the methodology includes three steps. First, possible future scenarios
are generated. As described in Section 1.3.1, we use the Wilkie model as our val-
uation model. Then, in each generated scenario the projected cashflows incurred
by the insurer are valued. Finally, reserves are set up according to our reserving
principles. We describe the detailed mechanism in the following equations.
At each valuation date t, t=0, 1, ..., T − 1, starting with the current asset share
and market indices, we project forward the performance of the unit fund using the
Wilkie model. 10,000 future scenarios are simulated. In the ith simulation and at
each future time point p, p = t + 1, t + 2, ..., T , given the projected equity price
index and dividend amount, we can project the amount of the guarantee charge as
follows,
GC ′(t, p, i) = A′(t, p− 1, i) · P ′(t, p, i) + D′(t, p, i)P ′(t, p− 1, i)
· c. (2.39)
Hence the projected asset share has a value of
A′(t, p, i) = A′(t, p− 1, i) · P ′(t, p, i) + D′(t, p, i)P ′(t, p− 1, i)
−GC ′(t, p, i) (2.40)
with the current asset share
A′(t, t, i) = A(t)
38
and the current equity price index
P ′(t, t, i) = P (t).
The CTE reserves are calculated with full allowance for the future guarantee
charges deducted from the unit fund. The projected value of the cashflows incurred
by the insurer before maturity, i.e. at time p, p = t + 1, t + 2, ..., T − 1, follows the
equation,
CF ′(t, p, i) = −GC ′(t, p, i). (2.41)
The projected value of the cashflows incurred at maturity equals the projected payoff
under the guarantees less the projected guarantee charge, i.e.
CF ′(t, T, i) = max(G(t)− A′(t, T, i), 0)−GC ′(t, T, i). (2.42)
The asset share is entirely invested in equities, so the reserves are more likely
to be used when the equity index falls. Therefore, it is not suitable for the reserve
fund to be also invested in equities. We assume that the reserve is invested in a
risk-free asset, so the projected cashflows should be discounted at the risk-free rate
to calculate the present value. Hence,
PV CF ′(t, i) =T∑
p=t+1
CF ′(t, p, i)(1 + r)p−t
. (2.43)
Equation 2.43 has calculated the amount of money that the insurer should hold
at current time t in the ith simulation. The 10,000 simulated present values are
sorted from the smallest to the largest so that
PV CF ′′(t, 1) ≤ PV CF ′′(t, 2) ≤ ... ≤ PV CF ′′(t, i) ≤ ... ≤ PV CF ′′(t, 10000).
The CTE reserves with a security level of α can be worked out from the average of
the largest
10, 000 · (1− α)
simulated present values. Although the guarantee costs are always non-negative,
PV CF ′(t) includes the future guarantee charges, so the net result may be either
39
positive or negative. The CTE reserve might therefore also be negative. In this case
it is set to zero, i.e.
V (t) =
∑10,000i=10,000·α+1 PV CF ′′(t,i)
10,000·(1−α)if
∑10,000i=10,000·α+1 PV CF ′′(t, i) > 0
0 otherwise. (2.44)
Once we have calculated the reserves using the simulated cashflows, we can work
out the actual amount of the cashflows incurred at time t.
The only cashflow incurred by the insurer at policy inception is to set up the
initial reserves, i.e.
CF (0) = V (0). (2.45)
The value of the cashflows incurred at time t, t=1, 2, ..., T−1, equals the increase
in the reserves less the guarantee charge, i.e.
CF (t) = V (t)− V (t− 1) · (1 + r)−GC(t) (2.46)
where GC(t) is given by equation 2.12.
At maturity, 100% of the unsmoothed asset share is paid to the policyholder
subject to the guarantee. Hence the cashflows incurred at maturity have a value of
CF (T ) = max(G(T )− A(T ), 0)− V (T − 1) · (1 + r)−GC(T ). (2.47)
Equation 2.22 can be used to calculate the accumulated value at policy termina-
tion of all these cashflows incurred during the policy term.
2.4.2 Results under the CTE Approach
The 95% and 99% CTE reserves are shown in Figure 2.6, under the assumptions
given in Section 2.2.1. The details of calculations are given in Appendix C.
With a higher security level, larger reserves are required in order to be more
cautious. Hence the 99% CTE reserves are greater than the 95% reserves, as shown
in Figure 2.6.
Comparing Figures 2.6 and 2.3, we notice a big difference in the pattern of the
reserves set up using the different approaches. The CTE reserves start from a very
low level at policy inception when the current guaranteed payout is small compared
40
year
rese
rves
1992 1994 1996 1998 2000
05
1015
2025
30
1991 1993 1995 1997 1999 2000
95% CTE reserves99% CTE reserves
Figure 2.6: The 95% and 99% CTE reserves for the 1991 policy with static bonusand investment strategies and a constant risk-free rate
with the projected asset share at maturity. Afterwards, the guarantee builds up with
the declared bonuses. Under the CTE approach, we are interested in the projected
values of the maturity asset share in the worst cases. These projected values do not
increase as rapidly as the guarantee probably due to the decreasing dividend yield
during this particular 10-year period, which is a dominant variable in the projection
of investment performance using the Wilkie model. When the dividend yield on
shares is very high, the share price is expected to increase according to the Wilkie
model; and projected share price will decrease when the dividend yield is low. In
the Black-Scholes model, however, the projected rate of return in equities is always
the same as the risk-free rate. Thus, the amount of the money required currently
to pay for the future guarantee costs shows an increasing trend, i.e. larger CTE
reserves are set up at later durations.
The reserves set up using the option pricing approach show a decreasing trend
over the policy term. The required amount of reserves equals the cost of buying
the corresponding options or constructing a hedge portfolio which depends on the
relative change in the equity index and exercise price of each put option. We have
noticed in Figure 2.4 that the equity index increases more rapidly than the exercise
price most of the time. Thus, the put options are cheaper and hence the reserves
41
are smaller at later durations.
However, the CTE reserves are not directly comparable with the reserves set up
using the option or hedging approach. The option method provides a 100% security
level assuming that there is no counter-party risk. Dynamic hedging is also 100%
secure if the hedge portfolio can be rebalanced continuously. Reserves are always
required using the option pricing approach. The CTE reserves are calculated from
the average of the projected maturity asset shares in the tail of the distribution.
There might be no reserves required at all under the CTE approach, particularly
with a low security level. There is an inconsistency in the way the future guarantee
charges are allowed for when setting up reserves under the different approaches.
Also, the assumption for the volatility in the equity market is different in the different
valuation models.
The cashflows incurred to set up 95% and 99% CTE reserves both have an ac-
cumulated value of £-26.61 at policy termination. The guarantees are not called
up for this particular policy, and we roll up the cashflows at the same risk-free rate
as we value the future loss. Therefore, the insurer earns the same amount of profit
which equals the accumulated value of the guarantee charges, although it sets up
CTE reserves at different security levels. Comparing these accumulated values with
£7.93 using the option approach and £2.94 using discrete hedging, we can conclude
that selling the policy at the end of 1991 is profitable only to the insurer who sets
up CTE reserves.
2.5 Summary
In this chapter we have considered a very simple case where a 100% EBR, a constant
regular bonus rate and a constant risk-free interest rate are assumed. The required
amount of reserves have been calculated for the 10-year policy issued at the end of
1991, using the three reserving approaches of buying options, discrete hedging and
CTE reserving. The main conclusions we have drawn are summarised as follows:
• The three reserving approaches are not directly comparable, because they
provide a different security level; there is inconsistency in the way the future
42
guarantee charges are allowed for; the assumption for the volatility in the
equity market is different in the different valuation models.
• Reserves are always required using the option pricing approach. It is possible
that no CTE reserve is required, in particular with a low security level.
• Under the option pricing approach, the reserves are calculated by comparing
the guarantees with the current asset share which increases rapidly most of
the time during these particular 10 years. While under the CTE approach,
the reserves are calculated by comparing the guarantees with the projected
maturity asset share which depends on the projected equity returns during the
remaining policy years as well as the actual returns in the past. The projection
is performed using the Wilkie model starting with the current market indices.
The dividend yield on shares is a dominant variable which declines over this
10-year period. Therefore, the reserves show a decreasing trend under the
option pricing approach but an increasing trend under the CTE approach.
• Selling the 1991 policy is profitable only to the insurer who sets up CTE
reserves. However, this conclusion depends on the fact that the guarantee
does not bite at maturity and also on those assumptions we have made in this
chapter, particularly the assumed 5% risk-free rate, which might be too low
compared with the realistic value so that the reserves are overstated.
• The insurer makes a smaller loss by hedging the risk internally instead of
buying options from a third party, because the replicating portfolio brought
forward is mostly worth more than that required to be set up.
43
Chapter 3
DYNAMIC BONUSES
3.1 Introduction
In Chapter 2 we considered a very simple bonus strategy that the regular bonus
rates declared during the policy term are fixed at 5%, taking no account of the in-
vestment performance. In reality, however, the insurer declares a bigger bonus after
experiencing a high return on assets. If the investment market performs badly, the
insurer cuts bonuses to reduce the cost of guarantees and maintain solvency. Usually
there is some smoothing mechanism to stop changing the bonuses too significantly.
This chapter considers a more complicated and realistic bonus mechanism than the
previous chapter. The three reserving approaches are applied to the same policy as
in Chapter 2 except that dynamic bonuses are introduced.
The dynamic bonus strategy uses a bonus earning power mechanism. Here the
bonus earning power is defined as the bonus rate that can be declared now and at
each future bonus declaration date given the current guarantee with a 75% prob-
ability of achieving at least a terminal bonus target. The detailed mechanism is
described in Section 3.2.
The bonus earning power mechanism has also been considered in Limb et al.
(1986), Forfar et al. (1989), Ross (1991), Ross and McWhirter (1991) and Macdonald
(1995). Limb et al. (1986) and Forfar et al. (1989) calculate the bonus rate by
comparing a smoothed discounted value of future income with a bonus reserve.
Ross (1991) and Ross and McWhirter (1991) set the bonus rate with a terminal
44
bonus target based on the smoothed asset values. In Macdonald (1995), the bonus
is declared according to the projected smoothed asset share allowing for a terminal
bonus where the projection is based on a geometric average of gilt yields.
Two different bonus strategies are considered in Wilkie (1987) and Willder (2004),
a fixed bonus mechanism and a dynamic mechanism linked to investment returns
directly. The latter has also been considered by Chadburn (1997), Chadburn and
Wright (1999), Hairs et al. (2002) and Haberman et al. (2003). In Hibbert and
Turnbull (2003) the bonus strategy is a combination of the mechanism linked directly
to investment performance and the bonus earning power mechanism.
We first calculate the bonus rates ignoring smoothing. In this case, no bonus
is declared if the bonus earning power is negative, otherwise the bonus rate equals
the bonus earning power. Then we add in a smoothing mechanism which sets a
constraint that the bonus rates are not allowed to increase by more than 20% or
decrease by more than 16.67% from year to year. Hence a maximum increase and
a maximum decrease will cancel out. In Limb et al. (1986), Forfar et al. (1989),
Ross (1991), Ross and McWhirter (1991), Macdonald (1995), Chadburn (1997),
Chadburn and Wright (1999), Hibbert and Turnbull (2003), Haberman et al. (2003)
and Willder (2004), the bonuses are smoothed to some extent.
In Section 3.3 we calculate reserves for the 10-year policy issued at the end of
1991 with dynamic bonuses. We first ignore future bonuses when setting up reserves
for the two cases of with and without smoothing. Then we reserve for future bonuses
as required by the FSA in the realistic regulatory regime, but in a simple way such
that the closed form solution can still be used under the option pricing approach.
Finally, a summary is given in Section 3.4.
3.2 A Dynamic Bonus Strategy
As in the previous chapter, we assume that the whole policyholder’s fund is invested
in equities. Equities are risky due to the uncertainty of the future equity index.
We calculate the bonus earning power based on a cautious projected value of the
maturity asset share.
45
The extra notation is as follows,
• A′′(t, T, i): the projected value of the maturity asset share in the ith simulation
after being sorted into ascending order, projected at time t < T
• TB: the terminal bonus target
• b′(t): the bonus earning power at time t
In the majority of the thesis, we assume a 30% terminal bonus target. A different
target will be considered in the sensitivity test in Chapter 6.
We first consider the case of without smoothing, then we will add in a smoothing
mechanism.
3.2.1 Dynamic Bonuses without Smoothing
A regular bonus is declared at the end of each policy year except at maturity. Thus,
no projection is required at policy inception for the bonus earning power calculation.
At each bonus declaration date t, t = 1, 2, ..., T − 1, we project forward the
performance of the unit fund given the current asset share and market indices. As
described in Section 1.3.2, 10,000 future possible scenarios are simulated using the
Wilkie model to calculate the bonus earning power. The methodology is close to
that of the CTE reserving approach. In the ith simulation, the projected maturity
asset share, A′(t, T, i), is calculated by equation 2.40. The 10,000 simulated values
are then sorted into ascending order, i.e.
A′′(t, T, 1) ≤ A′′(t, T, 2) ≤ ... ≤ A′′(t, T, i) ≤ ... ≤ A′′(t, T, 10000).
The bonus earning power is calculated from the 25th percentile of the projected
maturity asset share, i.e. the 2,501st smallest value. In other words, the calculated
bonus rate can be declared at the current and each future bonus declaration date in
75% of the cases allowing for a 30% terminal bonus target. We have the following
equation for the bonus earning power
b′(t) =
(A′′(t, T, 2501)
G(t− 1) · (1 + TB)
) 1T−t
− 1. (3.48)
46
year
bonu
s ra
te (
%)
1992 1994 1996 1998 2000
23
45
6
1993 1995 1997 1999
unsmoothed dynamic bonus rate5% constant
Figure 3.7: The unsmoothed bonus rates for the policy issued at the end of 1991with a static investment strategy
Smoothing is not allowed for at this stage, hence the declared bonus rate equals the
bonus earning power if the latter is positive and no bonus is declared otherwise, i.e.
b(t) =
b′(t) if b′(t) ≥ 0
0 otherwise. (3.49)
Figure 3.7 shows the unsmoothed bonus rates declared on the 1991 policy in
addition to the 2% guaranteed rate.
We see in Figure 3.7 that the unsmoothed dynamic bonus rates are mostly lower
than the 5% constant rate assumed in our static bonus strategy. The unsmoothed
bonus rates are quite volatile from year to year. The bonus earning power depends
on the projected equity returns which are simulated using the Wilkie model given
the current market indices. The Wilkie model has a ‘cascade’ structure in which all
variables are based on the retail price index.
Figure 3.8 shows the asset share, force of inflation (multiplied by 10,000 to adjust
the scale), and 25th percentile of the projected maturity asset share at each bonus
declaration date.
Figure 3.8 confirms that the force of inflation is an important variable in the
projection of future investment performance using the Wilkie model. In a low in-
flationary environment, the investment return is expected to be poor and hence a
47
year
1992 1994 1996 1998 2000
100
200
300
400
1993 1995 1997 1999
asset shareforce of inflation*10,00025% projected maturity asset share
Figure 3.8: The asset share, force of inflation (multiplied by 10,000) and 25% pro-jected maturity asset share for the 1991 policy with a static investment strategy
small maturity asset share is projected in the worst 25% of the cases. For example,
at the end of 1993 the asset share goes up because the equity market has achieved a
high return over the last policy year. However, the force of inflation goes down and
it is still far away from maturity, hence the maturity asset share has a low projected
value. At later durations, the current asset share has more effect on the projected
maturity values than the force of inflation.
So far, smoothing has not been allowed for. Thus, the fluctuations in the 25th
percentile of the projected maturity asset share are shown in the unsmoothed bonus
rates. As the policy duration increases, the full volatility is spread over fewer years
and hence the fluctuations in the bonus rates are more significant.
3.2.2 Dynamic Bonuses with Smoothing
In Section 3.2.1, the regular bonus rate equals the bonus earning power subject to a
minimum of zero. Hence the bonuses fluctuate significantly along with the volatile
investment return. In reality, the insurer usually sets a smoothing mechanism in
the bonus strategy in order to limit the range of movements in the bonus rates from
year to year so that policyholders are protected from the fluctuations in the stock
market. In this section the bonus rates are declared using the same bonus policy
48
but with a smoothing mechanism that the bonus rates are not allowed to increase
by more than 20% or decrease by more than 16.67% from year to year.
To apply this mechanism, a positive initial value for the bonus rate needs to be
determined as a starting point. We calculate this initial value in a consistent way
with our bonus strategy, i.e. using a bonus earning power mechanism.
Thus, the projection of the investment performance is required at policy incep-
tion. The asset share at maturity in each future possible scenario is projected in the
same way as described in Section 3.2.1. Given the initial guarantee from equation
2.5, the initial bonus earning power has a value of
b′(0) =
(A′′(0, T, 2501)
G(0) · (1 + TB)
) 1T−1
− 1 (3.50)
where A′′(0, T, 2501) is the 25th percentile of the projected maturity asset share.
Notice that the total projected returns are spread over T − 1 years because there
are T − 1 regular bonus declaration dates in total during the policy term.
The initial bonus rate must be positive to apply our smoothing mechanism, so
we set a lower boundary of 0.5% as the minimum initial bonus rate, i.e.
b(0) =
b′(0) if b′(0) > 0.005
0.005 otherwise. (3.51)
The bonus earning power at each bonus declaration date is calculated in the same
way as in Section 3.2.1. The declared bonus rate, constrained by the smoothing
mechanism, follows the equation
b(t) =
b(t− 1)× 1.2 if b′(t) > b(t− 1)× 1.2
b(t− 1)/1.2 if b′(t) < b(t− 1)/1.2
b′(t) otherwise
(3.52)
where b′(t) is given by equation 3.48.
Figure 3.9 compares the smoothed and unsmoothed bonus rates for the 1991
policy.
Clearly, the smoothed bonus rates are less volatile from year to year than the
unsmoothed rates. Figure 3.9 shows that the bonus rates frequently hit the upper
and lower boundary because the projected equity returns are very volatile from one
49
year
bonu
s ra
te (
%)
1992 1994 1996 1998 2000
23
45
6
1993 1995 1997 1999
with smoothingwithout smoothing
Figure 3.9: The comparison of the smoothed and unsmoothed bonus rates of the1991 policy with a static investment strategy
year to the next. The smoothing mechanism has stopped the insurer from cutting
the bonuses too significantly at the end of 1993 and 1998. The initial bonus rate for
the 1991 policy is 5.45%, which is quite high compared with the bonuses actually
declared. The large initial value has set up a large lower boundary for the bonus
rates declared at the end of 1992 and hence the smoothed bonus rate starts with a
higher level.
3.3 Results for the Single Policy with Dynamic
Bonuses
In this section we calculate reserves for the same policy as in Chapter 2, but with
dynamic bonuses declared according to the bonus strategy described in Section 3.2.
We consider three cases in turn,
A: without smoothing, and future bonuses are not reserved for,
B: with smoothing, and future bonuses are not reserved for,
C: with smoothing, and the minimum of future bonuses implied by our smoothing
mechanism are reserved for.
50
Here future bonuses refer to future regular bonuses. The insurer aims to pay
100% of the asset share to the policyholder at maturity subject to the guaranteed
payout. The excess of the asset share over the guarantee is a terminal bonus. In
this thesis we concentrate on the required amount of reserves in addition to the
asset share. Therefore, the terminal bonus has already been reserved for in the asset
share.
In each case, we look at how the guarantees build up, the amount of the reserves
required to be set up, and the profitability of the policy under the three reserving
approaches.
3.3.1 Case A: Without Smoothing or Allowance for Future
Bonuses
In this case the amount of guarantees follows equations 2.5 at policy inception, 2.14
after each bonus declaration, and 2.20 at maturity. The guarantees are rolled up
with the unsmoothed bonus rates as shown in Figure 3.7.
Figure 3.10 shows how the asset share and guarantee build up over the policy
term in Case A. As a comparison, the guarantee built up with the 5% constant
bonus rate is also shown in the figure. Notice that the asset share is not affected by
the bonus strategy.
We see in Figure 3.10 that the guarantee starts with the same initial value under
the static and dynamic bonus strategies but builds up at a lower speed with the
dynamic bonus rates which are mostly lower than 5%.
Given the maturity asset share and guarantee of £266.54 and £178.11 respec-
tively, a 49.65% terminal bonus rate can be declared which is higher than the 30%
target and the rate declared using the static bonus strategy. Although a larger ter-
minal bonus can be declared in Case A, the policyholder receives the same amount
of maturity payout because the guarantees are not called upon in either case.
Figure 3.11 compares the required amount of reserves using the option method
for the 1991 policy in Case A with that in the case of static bonuses.
The reserves show a similar pattern for the 1991 policy in the two cases. The
guarantees build up less rapidly with the dynamic bonuses, so smaller reserves are
51
year
1992 1994 1996 1998 2000
100
150
200
250
300
1991 1993 1995 1997 1999 2001
asset shareguarantee in Case Aguarantee with static bonuses
Figure 3.10: The asset shares and guarantees of the 1991 policy with a static invest-ment strategy in Case A
year
rese
rves
1992 1994 1996 1998 2000
05
1015
1991 1993 1995 1997 1999
Case Athe case of static bonuses
Figure 3.11: The comparison of the reserves using the option method for the 1991policy in Case A and the case of static bonuses
52
year
CT
E r
eser
ves
1992 1994 1996 1998 2000
010
2030
1991 1993 1995 1997 1999
95% in Case A95% in the case of static bonuses99% in Case A99% in the case of static bonuses
Figure 3.12: The comparison of the CTE reserves for the 1991 policy in Case A andthe case of static bonuses
required in Case A after policy inception.
The accumulated value at policy termination of the cashflows is £3.41 in Case A
and £7.93 in the case of static bonuses. Thus, the insurer makes a smaller loss by
declaring unsmoothed dynamic bonuses instead of the 5% constant rate.
The reserves are of the same amount under the option and hedging approaches,
but the cashflows are different because we allow for both hedging error and trans-
action costs incurred by discrete hedging. The insurer makes a loss of £0.36 by
hedging in Case A. A larger loss of £2.94 is made by the insurer declaring the 5%
static bonus rates. For the same reason that the replicating portfolio brought for-
ward is mostly worth more than that required to be set up, hedging internally is
cheaper to the insurer than buying options.
The 95% and 99% CTE reserves for the 1991 policy with the dynamic and static
bonuses are compared in Figure 3.12.
The CTE reserves show a similar pattern in the two cases but the reserves are
much smaller with dynamic bonuses. Very few reserves are required with a 95%
CTE measure in Case A.
The CTE reserving approach and our dynamic bonus strategy are both based on
the projected investment performance. The projection uses the same model with
53
the same parameters starting with the same initial conditions. If the equity market
booms in the projection, a large maturity asset share is projected and the guarantee
is increased by a high bonus rate. Conversely, if the equity market collapses in the
projection, a small maturity asset share is projected and the guarantee is slightly
increased by a low bonus rate (or remains the same if the declared bonus is zero).
Therefore, the reserves might not change too much from year to year. However,
with static bonuses the guarantee increases at a constant rate taking no account
of the projected investment performance. Hence, we see in Figure 3.12 that the
increasing trend in the CTE reserves in Case A is not so obvious as in the case of
static bonuses.
The cashflows incurred to set up the 95% and 99% CTE reserves have an ac-
cumulated value at policy termination of £-26.61 in Case A, which is the same as
in the case of static bonuses because the guarantee charges have the same amount
in the two cases. Again, selling the 1991 policy with the unsmoothed bonuses is
profitable only to the insurer who sets up CTE reserves.
Again, the 1991 policy is more profitable to the insurer who declares the un-
smoothed dynamic bonuses instead of the 5% constant rate. Smaller reserves are
required in Case A as shown in Figure 3.12, so less capital is locked up and the
insurer earns a bigger profit or makes a smaller loss.
Although our dynamic bonus strategy is better than the static one to the insurer,
it makes no difference to the policyholder. The maturity guarantees do not bite for
this policy, so the policyholder gets the same maturity payout in either case.
Recall that in this chapter we still assume a 5% risk-free interest rate, which might
be too low compared with the realistic value so that the reserves are overstated. We
will consider a more realistic risk-free rate in Chapter 4.
3.3.2 Case B: Smoothing without Allowance for Future
Bonuses
The smoothed bonus rates have been shown in Figure 3.9. The amount of guarantees
follows equations 2.5, 2.14 and 2.20.
Figure 3.13 shows the asset shares and guarantees during the policy term. The
54
year
1992 1994 1996 1998 2000
100
150
200
250
300
1991 1993 1995 1997 1999 2001
asset shareguarantee in Case Bguarantee in Case A
Figure 3.13: The asset shares and guarantees of the 1991 policy with a static invest-ment strategy in Cases A and B
guarantees in Cases A and B are compared. Notice that the asset shares are the
same with and without smoothing.
In Figure 3.13 we see that the smoothing mechanism has mostly increased the
guarantees, but the difference is not very obvious. As shown in Figure 3.9, the
smoothed bonus rate starts from a higher level and then fluctuates less significantly
for this 1991 policy. The guarantees build up more rapidly at early durations with
smoothing but at a lower speed afterwards. At maturity, the guarantees have a
smaller value of £177.86 in Case B versus £178.11 in Case A. So a slightly higher
terminal bonus rate of 49.86% can be declared in the case of smoothing.
The reserves set up using the option method in Cases A and B are compared in
Figure 3.14.
Larger reserves are mostly required with smoothing, but the difference is small
because the smoothing mechanism does not change the guarantees very much for
this particular policy. The most remarkable increase in the reserve is from £9.76
without smoothing to £10.59 with smoothing (by around 8.5%) at the end of 1994.
As explained in Section 2.2.3, the guarantees are not deeply out-of-the-money at the
end of 1993 when the policy is still far from maturity. The fall in the equity index
at the end of 1994 raises the value of the put options.
55
year
rese
rves
1992 1994 1996 1998 2000
05
1015
1991 1993 1995 1997 1999
Case BCase A
Figure 3.14: The comparison of the reserves using the option method for the 1991policy in Cases A and B
The cashflows incurred by the insurer have an accumulated value of £4.36 in
Case B (versus £3.41 in Case A). Smoothing has mostly increased the guarantees
for this policy, hence the insurer makes a greater loss.
Using discrete hedging, the insurer makes a loss of £1.21 in Case B (versus £0.36
in Case A). Again, we conclude that selling the 1991 policy was more profitable
to the insurer who hedges the risk internally rather than buying options, and that
smoothing has reduced the insurer’s profit.
Figure 3.15 compares the 95% and 99% CTE reserves in Cases A and B.
As under the option pricing approach, smoothing mostly increases the CTE re-
serves because the guarantees are slightly increased.
The cashflows incurred to set up 95% and 99% CTE reserves have an accumulated
value of £-26.61. Again, the 1991 policy is profitable only to the insurer who sets up
CTE reserves and the profit has the same amount in the cases of with and without
smoothing.
56
year
CT
E r
eser
ves
1992 1994 1996 1998 2000
05
1015
2025
1991 1993 1995 1997 1999
95% in Case B95% in Case A99% in Case B99% in Case A
Figure 3.15: The comparison of the CTE reserves for the 1991 policy in Cases Aand B
3.3.3 Case C: Smoothing with Allowance for Future
Bonuses
In Cases A and B, we ignore future regular bonuses when setting up reserves because
future bonuses are not promised to the policyholder. If the investment market
performs very badly, the insurer can cut off future bonuses to improve its solvency
position. However, in practice cutting bonuses can give rise to bad publicity.
The recent CP195 proposals (FSA (2003)) state that the insurer’s decision rules
for future bonuses should be incorporated into the projection of claims, and that
the decision rules should be consistent with its Principles and Practices of Finan-
cial Management and take into account the policyholder’s reasonable expectations
(PRE). Although the meaning of PRE is being debated intensively by actuaries and
regulators, no ultimate definition has been interpreted. Shelley et al. (2002) have
reviewed some key issues in relation to PRE.
However, it is not straightforward to model the decision rules because they can
be influenced by many factors, for example the prospective financial strength of the
insurer and competitive pressures. Here we consider a simple methodology. Our
smoothing mechanism has set a constraint that the regular bonus rates are not
allowed to increase by more than 20% or decrease by more than 16.67% from year
57
to year. Once a bonus is declared, the minimum of future bonuses is determined.
Reserving for the guarantees including the minimum future bonuses, we can still use
a closed form solution (i.e. Black-Scholes equation) to value the cost of guarantees
assuming that shares follow a Geometric Brownian Motion and that the risk-free
force of interest is a constant.
At policy inception, the guarantees have an initial value of
G(0) = SP · (1 + g)T ·T−1∏n=1
(1 + b(0)/1.2n) (3.53)
in which the initial bonus rate b(0) is given by equations 3.50 and 3.51.
After declaring a bonus rate of b(t), t = 1, 2, ..., T − 1, according to equations
3.48 and 3.52, the guarantees have a value of
G(t) = SP · (1 + g)T ·(
t∏n=1
(1 + b(n))
)·(
T−t−1∏n=1
(1 + b(t)/1.2n)
). (3.54)
At maturity, no regular bonus except for the guaranteed 2% is declared. Hence,
G(T ) = SP · (1 + g)T ·T−1∏n=1
(1 + b(t)) (3.55)
i.e.
G(T ) = G(T − 1).
Figure 3.16 shows how the asset share and guarantee build up over the policy
term in Case C. The guarantees in Case B are also shown as a comparison. Notice
that the asset shares are not affected by whether or not future bonuses are reserved
for.
As expected, the guarantees are increased when allowing for future bonuses. The
difference between the guarantees in Cases B and C is most significant at policy
inception because the minimum of all future regular bonuses are included in the
initial guarantees in Case C. At maturity, the guarantees are of the same amount in
the two cases because they are rolled up with the same smoothed bonus rates.
Figure 3.16 shows an obvious difference in the pattern of the guarantees in the
two cases. Constrained by the smoothing mechanism, the regular bonus rates are
always positive. Thus, in Case B the guarantee increases at each bonus declaration
58
year
1992 1994 1996 1998 2000
100
150
200
250
300
1991 1993 1995 1997 1999 2001
asset shareguarantee in Case Cguarantee in Case B
Figure 3.16: The asset shares and guarantees of the 1991 policy with a static invest-ment strategy in Cases B and C
date. Equation 3.54 shows that if a minimum bonus rate implied by the smoothing
mechanism is actually declared, the guarantees in Case C remain at the same level
before and after the bonus declaration.
Since the guarantees are the same at maturity in the two cases, the terminal
bonus rate is not affected by whether or not future bonuses are reserved for. Hence
49.86% of the guarantees can still be declared as a terminal bonus in Case C.
Figure 3.17 compares the amount of reserves using the option method in Cases
B and C.
Figure 3.17 shows that the initial reserves are greatly increased with allowance for
future bonuses. The options are very expensive at the outset when the guarantees
are deeply in-the-money. The difference between the two cases becomes smaller as
the policy duration increases because the guarantees include less and less future
bonuses in Case C.
The patterns in the reserves are similar in the two cases. For this 1991 policy,
the reserves decline over the policy term except for a temporary increase at the end
of 1994. The increase is 21.19% in Case C and 35.42% in Case B. The fall in the
equity index at the end of 1994 has less impact on the value of the options in Case
C where the guarantee does not increase after the bonus declaration at that time.
59
year
rese
rves
1992 1994 1996 1998 2000
05
1015
2025
1991 1993 1995 1997 1999
Case CCase B
Figure 3.17: The comparison of the reserves using the option method for the 1991policy in Cases B and C
The cashflows incurred to set up reserves using the option approach have an
accumulated value of £17.51 in Case C (versus £4.36 in Case B). Thus, the loss
made by the insurer is greatly increased when allowing for future bonuses.
The insurer makes a loss of £11.50 by discrete hedging in Case C (versus £1.21
in Case B). Again, we conclude that selling the 1991 policy incurs a smaller loss if
the insurer hedges the risk internally instead of buying options from a third party,
and that the loss is increased if future bonuses are reserved for.
Figure 3.18 compares the 95% and 99% CTE reserves in Cases B and C.
Initially there is no need to set up 95% CTE reserves even if future bonuses are
allowed for. The initial force of inflation is at the highest level during the policy
term, hence large equity returns are projected at policy inception and so in 95%
of the simulations the cost of guarantees can be covered by the guarantee charges.
However, the initial 99% CTE reserves are greatly increased with allowance for future
bonuses. The difference between the two cases is less obvious at later durations
because smaller future bonuses are included in the guarantees.
The cashflows incurred to set up CTE reserves have the same accumulated value
in Cases B and C, i.e. £-26.61.
60
year
CT
E r
eser
ves
1992 1994 1996 1998 2000
010
2030
40
1991 1993 1995 1997 1999
95% in Case C95% in Case B99% in Case C99% in Case B
Figure 3.18: The comparison of the CTE reserves for the 1991 policy in Cases Band C
3.4 Summary
This chapter has considered a slightly more complicated and realistic case than
Chapter 2. The asset share is still entirely invested in equities and a constant risk-
free rate of 5% p.a. is assumed, as in Chapter 2. But in this chapter, the insurer
declares dynamic bonuses according to a bonus earning power mechanism. The
smoothing mechanism sets a constraint on the maximum change in the bonus rates
from year to year.
Three cases have been considered:
A: the bonus rates are not smoothed, and future regular bonuses are not reserved
for,
B: the bonus rates are smoothed, and future regular bonuses are not reserved for,
C: the bonus rates are smoothed, and the minimum of future regular bonuses
implied by the smoothing mechanism are reserved for.
The main conclusions are summarised as follows.
• The unsmoothed dynamic bonus rates are mostly smaller than 5% assumed
in the static bonus strategy. Therefore, the guarantees and hence the reserves
are smaller in Case A than those in the case of static bonuses.
61
• Smoothing regular bonues in Case B slightly increases the required amount of
reserves because the guarantees are slightly increased.
• Reserving for future regular bonuses in Case C greatly increases the guaran-
tees and hence the reserves. The difference between Cases B and C is more
significant at early durations when the guarantees include more future bonuses
in Case C.
• The amount of reserves decreases over the policy term under the option pricing
approach except a temporary increase at the end of 1994. The CTE reserves
do not show an obvious trend because the guarantees, which are rolled up with
the dynamic bonuses, move in the same direction as the projected value of the
maturity asset share.
• The insurer makes a loss by selling the 1991 policy if reserves are set up using
the option pricing approach. Smoothing regular bonuses and reserving for
future bonuses both increase the amount of the loss, and the latter has a much
stronger effect.
62
Chapter 4
A RISK-FREE RATE
CONSISTENT WITH THE
WILKIE MODEL
4.1 Introduction
In the previous two chapters we assumed a constant risk-free interest rate of 5%
over the policy term regardless of the actual investment performance in the bond
market. We concluded in those chapters that 5% might be too low compared with
the realistic rate so that the reserves were overstated.
In this chapter we assume that the risk-free rate equals the current yield on a
zero-coupon bond with the same maturity date as our UWP policy. We derive
the historical zero-coupon yield from the consols yield and short-term rate which
are given in Appendix B. We could derive the zero-coupon yield more accurately
from the historical data on the gross redemption yields on the bonds with different
terms. However, our methodology is used for two reasons. First, the British Gov-
ernment Securities Database (www.ma.hw.ac.uk/∼andrewc/gilts/) had only existed
for a short period by the time our calculations were performed. Second, in Chapters
5 to 7 the projected rate of return on zero-coupon bonds will be needed. In a bonus
and asset allocation model, future possible scenarios are simulated using the Wilkie
model. Also, in Chapter 8 we will look at reserving in the simulated real world.
63
The real world is simulated by the Wilkie model. Wilkie (1995) does not model a
yield curve explicitly. However, the consols yield C(t) and short-term interest rate
B(t) in Wilkie (1995) give two ends of the yield curve. Yang (2001), Wilkie et al.
(2003) and Willder (2004) describe a way of constructing a full yield curve based on
C(t) and B(t) from the Wilkie model. We will follow their approach to derive the
projected return on a risk-free asset in Chapter 8. For consistency, we derive the
historical risk-free rate in the same way as the projected rate, i.e. from the consols
yield and short-term rate.
Section 4.2 describes a yield curve consistent with the Wilkie model and then
derives the zero-coupon yields during the policy term.
In Section 4.3 reserves are calculated for the 1991 policy under the three ap-
proaches. At this stage we still assume a 100% equity proportion of the asset share.
Dynamic bonuses are declared following the bonus earning power mechanism as con-
sidered in Chapter 3. The risk-free rate equals the yield on the zero-coupon bond
with the same maturity date as the policy. We concentrate on the case of smoothing
with allowance for future bonuses, i.e. Case C defined in Chapter 3.
Finally, Section 4.4 gives a summary.
4.2 A Yield Curve for the Wilkie Model
Before describing the yield curve in detail, we first define some notation as follows,
• PY (t, n): the par yield at time t for term n
• C(t): the consols yield at time t
• B(t): the short-term interest rate at time t
• v(t, n): the value at time t of a zero-coupon bond with duration n
• Z(t, n): the yield at time t on a zero-coupon bond with duration n.
We assume that coupons are payable annually in arrears. We model the par yield
(i.e. the redemption yield which is equal to the coupon) at time t for term n by the
64
following equation
PY (t, n) = C(t) + [B(t)− C(t)] · e−β·n (4.56)
in which β is a constant.
Hence, the value of a coupon bond at time t for term n, with coupon PY (t, n)
and redemption proceeds of unity, can be expressed as
1 = PY (t, n) ·n∑
d=1
v(t, d) + 1 · v(t, n). (4.57)
The value at time t of a zero-coupon bond with duration n can be derived recursively,
starting at n = 1, as follows,
1 = PY (t, 1) · v(t, 1) + 1 · v(t, 1).
So,
v(t, 1) =1
PY (t, 1) + 1. (4.58)
Then continuing year by year, we have
1 = PY (t, n)n−1∑d=1
v(t, d) + [1 + PY (t, n)] · v(t, n).
Thus,
v(t, n) =1− PY (t, n) ·∑n−1
d=1 v(t, d)
1 + PY (t, n). (4.59)
The yield at time t on a zero-coupon bond with term n can be derived from
v(t, n) as follows,
Z(t, n) =1
v(t, n)1/n− 1. (4.60)
As assumed, the risk-free rate at time t equals the current yield on a zero-coupon
bond with term T − t, i.e. Z(t, T − t).
A problem of this approach is that v(t, n) might be negative. This happens when
the value of β is too low, for the particular values of B(t) and C(t). However, a
high value of β produces a flat yield curve which is not very different from C(t).
Wilkie et al. (2003) use a value of β of 0.39, and Yang (2001) and Willder (2004)
65
year
risk-
free
inte
rest
rat
e (%
)
1992 1994 1996 1998 2000
56
78
910
1991 1993 1995 1997 1999
yield on zero-coupon bonds5% constant
Figure 4.19: The yield on the zero-coupon bond with the same maturity date as the1991 policy
use a value of 0.5. We follow Yang (2001) and Willder (2004) so that β = 0.5 is
used throughout the thesis.
Figure 4.19 compares the zero-coupon yield with the 5% constant level.
Figure 4.19 shows that the risk-free rate could be very volatile during the policy
term. The zero-coupon yield is mostly higher than 5%. Only at the end of 1998 is
the zero-coupon yield slightly lower. At policy inception the zero-coupon yield is
almost twice the constant rate.
4.3 Results with the Consistent Risk-Free Rate
We have seen in the previous section that sometimes the zero-coupon yield is very
different from 5%. In this section we calculate the reserves and investigate the
profitability of the 1991 policy with the consistent risk-free rate.
We only consider the case of smoothing with allowance for future bonuses, i.e.
Case C as defined in Chapter 3. Thus, the guarantee follows equations 3.53 at policy
inception, 3.54 after each bonus declaration and 3.55 at maturity.
At this stage we still assume that the whole asset share is invested in equities, so
a different assumption for the risk-free rate has no effect on the asset share. Figure
3.16 has shown how the asset share and guarantee build up over the policy term.
66
year
rese
rves
1992 1994 1996 1998 2000
05
1015
2025
1991 1993 1995 1997 1999
zero-coupon yieldconstant risk-free rate
Figure 4.20: The comparison of the reserves using the option method for the 1991policy in Case C assuming the zero-coupon yield or 5% constant as a risk-free rateover the policy term
At maturity, a terminal bonus rate of 49.86% can still be declared.
4.3.1 Buying Options
The reserves and cashflows are calculated in the same way as described in Section
2.2, but the risk-free force of interest δ(t), t = 0, 1, ..., T−1, is given by the following
equation
δ(t) = log(1 + Z(t, T − t)) (4.61)
Figure 4.20 shows the required amount of reserves for the 1991 policy under the
option approach with the consistent risk-free rate. As a comparison, the figure also
shows the reserves calculated assuming a constant rate of 5% over the policy term.
Figure 4.20 shows that the reserves are greatly reduced at early durations when
the zero-coupon yield replaces the constant rate of 5%. The difference between
the initial reserves is £16.94 for a £100 single premium policy. We have seen in
Figure 4.19 that the zero-coupon yield is mostly higher than 5%. A higher interest
rate reduces the present value of any future cashflows received by the holder of the
option. So the put options are cheaper and hence smaller reserves are required with
a higher risk-free rate. At later durations the reserves are of a similar amount in the
67
two cases because the zero-coupon yield is then not very different from 5%. Also,
at later durations the guarantees are deeply out-of-the-money and the policy is not
far from maturity, so the risk-free rate has less impact.
The patterns of the reserves are different in the two cases. The fall in the zero-
coupon yield at the end of 1993, shown in Figure 4.19, increases the amount of
reserves. The effect of the fall in the equity index in the following year is offset by
the restoration of the zero-coupon yield. Afterwards the interest rate declines but
the equity market performs very well most of the time, so the reserve decreases.
However, with the constant risk-free rate, the relative change in the equity index
to the exercise price is the dominant factor which affects the value of the options.
Therefore, in that case a hump appears at the end of 1994 after a temporary decrease
in the equity index.
Accumulating all the cashflows incurred by the insurer to policy termination,
using equation 2.22 with r replaced by Z(t, T − t), we have a final value of £-7.66.
Thus, the insurer earns a profit of £7.66 from the 1991 policy if the risk-free rate
is derived from the actual consols yield and short-term interest rate; whereas the
insurer makes a loss of £17.51 with the constant risk-free rate of 5%. This confirms
the conclusion we have drawn in the previous two chapters that the assumed rate of
5% is too low so that the reserves are overstated and the profitability of the policy
is distorted.
4.3.2 Discrete Hedging
The insurer earns a profit of £18.72 by selling the 1991 policy with the more realistic
risk-free rate; whereas it makes a loss of £11.50 with the constant rate of 5%.
Again, we conclude that changing the assumption for the risk-free interest rate has
a significant impact on the profitability of the policy, and that the insurer earns a
larger profit by hedging rather than buying options.
4.3.3 CTE Reserving
The detailed mechanism of calculating the reserves and cashflows is similar to that
described in Section 2.3. The difference here involves discounting. We have assumed
68
that the reserve fund is invested in a risk-free asset. In the previous two chapters,
we assumed a constant rate of 5% earned on the reserve fund. Here the risk-free
rate equals the yield at the valuation date t, t = 0, 1, ..., T − 1, on the zero-coupon
bond which will be redeemed at the same time as the projected cashflow is incurred.
Hence in equation 2.43, the constant risk-free rate r should be replaced by Z(t, p−t).
Calculating the cashflows is more complicated with the variable risk-free rate.
The initial cashflow follows equation 2.45. Afterwards, the value of the cashflows
equals the increase in the reserve fund less the guarantee charge deducted from the
unit fund. Thus, we need the current value of the previous year’s reserves. With
the constant risk-free rate as assumed before, the reserves are simply rolled up with
the constant rate. This is not the case when we use the variable rate. We need to
know the exact time of the occurrence and the amount of each projected cashflow
to calculate the previous year’s CTE reserves. Therefore, when we sort the 10,000
present values of the projected cashflows into ascending order, we have to record
the original position (i.e. the simulation number before sorting) of these sorted
present values. We use the notation ps(i) to represent the original position of the
ith simulation after sorting, so we have the following relation at the valuation date
t, t = 0, 1, ..., T − 1,
PV CF ′′(t, i) = PV CF ′(t, ps(i)). (4.62)
Equation 2.44, which calculates the CTE reserves at time t, can also be expressed
as
V (t) =
∑10,000i=10,000·α+1 PV CF ′(t, ps(i))
10, 000 · (1− α)
=
∑10,000i=10,000·α+1
∑Tp=t+1
CF ′(t,p,ps(i))[1+Z(t,p−t)]p−t
10, 000 · (1− α)
=T∑
p=t+1
∑10,000i=10,000·α+1 CF ′(t,p,ps(i))
10,000·(1−α)
[1 + Z(t, p− t)]p−t.
From the above derivation, we can see that at the valuation date t, the CTE
reserve is set up so that at each future point in time p, p = t + 1, t + 2, ..., T ,∑10,000i=10,000·α+1 CF ′(t, p, ps(i))
10, 000 · (1− α)
69
year
CT
E r
eser
ves
1992 1994 1996 1998 2000
010
2030
40
1991 1993 1995 1997 1999
95%, zero-coupon yield95%, constant risk-free rate99%, zero-coupon yield99%, constant risk-free rate
Figure 4.21: The comparison of the CTE reserves for the 1991 policy in Case Cassuming the zero-coupon yield or 5% constant as a risk-free rate over the policyterm
amount of money can be withdrawn from the reserve fund to pay for the projected
cashflows incurred at that time. This can be achieved by investing∑10,000
i=10,000·α+1 CF ′(t,p,ps(i))
10,000·(1−α)
[1 + Z(t, p− t)]p−t
in a zero-coupon bond at time t which will be redeemed at time p.
Therefore, at time t, t = 1, 2, ..., T , the current value of the previous year’s
reserves, denoted as V ′(t), follows the equation
V ′(t) =T∑
p=t
∑10,000i=10,000·α+1 CF ′(t−1,p,ps(i))
10,000·(1−α)
[1 + Z(t− 1, p− (t− 1)]p−(t−1)· v(t, p− t)
v(t− 1, p− (t− 1))
(4.63)
in which v(t, p− t) is the value at time t of a zero-coupon bond with duration p− t.
The cashflows incurred by the insurer at time t have a value of
CF (t) = V (t)− V ′(t)−GC(t). (4.64)
Figure 4.21 compares the CTE reserves with the variable and constant risk-free
interest rates.
As under the option pricing approach, the CTE reserves are smaller at early
durations if the risk-free interest rate is derived from the consols yield and short-
term rate. However, the reduction in the CTE reserves is not so significant as that
70
in the reserves required under the other two approaches. For example, the initial
reserves are reduced by 70.55% under the option pricing approach; whereas the
reduction in the 99% CTE reserves is 43.97%.
The CTE reserves are calculated from the discounted projected cashflows incurred
during the remaining policy years. The projected cashflows are negative before
maturity given our convention that a negative cashflow means the money paid to
the insurer. The projected cashflows at maturity have a positive value only if the
guarantee charge deducted at maturity is not sufficient to cover the payoff under the
guarantees. These projected cashflows are discounted back to the current valuation
date at the zero-coupon yields. The zero-coupon yields are mostly higher than 5%.
Both the positive and negative cashflows are discounted at a higher rate, but the
present value of the positive cashflow is reduced by a high discount rate to a greater
extent than the negative cashflows because the positive cashflow is only likely to be
incurred at maturity and hence it is discounted for a longer period. Therefore, the
CTE reserves are not greatly reduced by the higher zero-coupon yield.
Comparing Figures 4.21 and 4.20, we again see different patterns of the reserves
using the different approaches. The reserves show a decreasing trend under the
option pricing approach, whereas no obvious trend can be seen in the CTE reserves
because the amount of guarantees moves in the same direction as the projected
maturity value of the asset share.
Table 4.1 compares the accumulated values at policy termination of the incurred
cashflows, accumulated at the constant and variable risk-free rates.
Table 4.1: The comparison of the accumulated values of the cashflows for the 1991policy in Case C with the zero-coupon yield or 5% constant as a risk-free rate
Approach Option Hedging CTE (95%) CTE (99%)5% 17.51 11.50 -26.61 -26.61Z(t, T − t) -7.66 -18.72 -30.51 -31.00
With the variable risk-free rate, setting up CTE reserves at different security
levels results in different profits. Here setting 99% CTE reserves is slightly more
profitable than setting 95% CTE, but note we have ignored the cost of capital.
Comparing the profits earned under the different reserving approaches, we again
71
conclude that this 1991 policy is most profitable to the insurer who sets up CTE
reserves.
4.4 Summary
This chapter has made a more realistic assumption for the risk-free interest rate
that it equals the current yield on the zero-coupon bond with the same maturity
date as our 10-year policy issued at the end of 1991. The historical zero-coupon
yield has been derived from the consols yield and short-term interest rate. The
equity backing ratio of the asset share remains at 100%. Regular bonuses have been
declared according to the dynamic bonus strategy as considered in Chapter 3. We
have only considered Case C where the bonuses are smoothed and the minimum of
future bonuses implied by the smoothing mechanism are reserved for.
The main conclusions we have drawn in this chapter are summarised as follows.
• The yield on the zero-coupon bond with the same maturity date as the policy
is mostly higher than 5%. The difference is most significant at policy inception.
• The reserves set up using the option pricing approach are greatly reduced at
early durations by the higher zero-coupon yield. A temporary increase in the
reserves appears at the end of 1993 when the zero-coupon yield drops sharply.
• The 95% and 99% CTE reserves are also reduced by the higher zero-coupon
yield, but the reduction is not so significant as under the option and hedging
approaches.
• The reserves show a decreasing trend under the option pricing approach except
for a temporary increase at the end of 1993. However, no obvious trend is
shown in the CTE reserves.
• The 1991 policy is profitable to the insurer under all three reserving approaches
with the variable risk-free rate. The policy is most profitable to the insurer
who sets up reserves using the 99% CTE measure, followed by the 95% CTE
measure, discrete hedging, and finally the option approach.
72
Chapter 5
A DYNAMIC INVESTMENT
STRATEGY
5.1 Introduction
So far, we have assumed that the asset share is entirely invested in equities. However,
it is unrealistic for the investment strategy to be isolated from the bonus strategy,
the insurer’s financial strength, and the projected performance of different asset
classes. The market value of equities can be very volatile, but the guarantees build
up over the policy term. So in practice, the insurer usually invests the policyholder’s
fund in both a risky asset such as equities which can provide an enhanced return,
and a risk-free asset which can cover part of the guarantees with certainty.
In this chapter, we consider two asset classes: equities and zero-coupon bonds
with the same maturity date as our UWP policy. The percentage of the unit fund
in either asset is adjusted at the beginning of each policy year according to an
investment strategy.
The investment strategy can be either static or dynamic. In a static strategy,
different asset classes (e.g. equities and gilts) are invested at a fixed proportion (e.g.
100/0%, or 80/20%). This fixed proportion can be switched each year to restore
the mix or left drifting. The papers which assume a 100% equity backing ratio have
been listed in Section 2.2.1.
Forfar et al. (1989), Ross (1991), Ross and McWhirter (1991) and Hare et al.
73
(2000) also adopt a static investment strategy, but they consider two asset classes.
In Forfar et al. (1989), at each duration the net cashflow, defined as the premiums
received less expenses and death claims, is assumed to be invested in the equities and
gilts in fixed proportions. The ‘fixed investment strategy’ considered in Ross (1991)
assumes that 80% of the assets are held in equities and 20% in fixed interest stocks
with a 15-year term. The assets are switched each year to restore the 80/20% mix.
Ross and McWhirter (1991) construct a model office using typical experience since
1951 to build a portfolio of business in force at the beginning of 1990. Prior to 1990,
assets were invested in equities and fixed interest stocks in fixed proportions which
were different during different periods. From 1990 onwards the office operates a
simple strategy of investing 80% of assets into equities, regardless of the prospective
returns available on the two investment types. In Hare et al. (2000) the asset share
is invested in equities and gilts. Up to the current duration their asset share is
accumulated deterministically, weighted by the assumed equity backing ratio. In
the projection from the current time onwards their asset share is not rebased to the
starting EBR, to avoid the artificial inflation of returns due to the mean-reverting
feature of the Wilkie model.
In a dynamic investment strategy, the insurer has discretion over the choice of
investments to back the asset share. Wilkie (1987) assumes that the policyholder’s
fund is actually invested in shares and put options so that the guarantees can be
exactly matched. In Ross (1991), a ‘variable investment strategy’ is also considered
in which the 80/20% mix is permitted as long as the asset to liability ratio (A/L)
is at least 1.25. If A/L falls below 1.25, the asset share is progressively switched
out of equities into gilts. As the ratio falls towards 1.05, the whole asset share is
held in fixed interest stocks. In Hibbert and Turnbull (2003), a proportion of the
policyholder’s fund is switched from equities to gilts whenever the guarantee is too
large in relation to the asset share.
In this chapter we consider a dynamic investment strategy in which the EBR
changes according to the probability that the projected asset share is sufficient to
cover the guarantee at maturity. The detailed mechanism is described in Section
5.2. The dynamic bonus strategy considered in Chapter 3 uses a bonus earning
74
power mechanism based on the 25th percentile of the projected maturity asset share.
Hence, the bonuses declared to the policyholder are affected by the insurer’s invest-
ment strategy. We assume that the bonus earning power is calculated just after the
decision on the investment.
Section 5.3 continues to apply the three reserving approaches to the same 1991
policy, but in the most complicated case up to now. The policyholder’s fund is
invested in both the risky and risk-free assets, and the percentage in either asset is
adjusted according to the dynamic investment strategy. Dynamic regular bonuses
are declared using a bonus earning power mechanism. The risk-free rate of interest
equals the yield on a zero-coupon bond with the same maturity date as the policy
which is derived from the historical consols yield and short-term interest rate. Only
the case of smoothing with allowance for future bonuses, i.e. Case C as defined in
Chapter 3, is considered.
Finally in Section 5.4 a summary is given.
5.2 A Dynamic Model Containing Dynamic In-
vestment and Bonus Strategies
As mentioned in Section 5.1, two asset classes are considered: equities and zero-
coupon bonds with the same maturity date as our 10-year policy issued at the
end of 1991. We consider the zero-coupon bonds instead of consols because the
redeemable value of the former at policy maturity is known in advance.
Our dynamic investment strategy adjusts the EBRs according to the prospective
solvency position of the insurer. A corridor approach is used. We run simulations
and calculate the probability that the projected asset share is sufficient to pay the
guarantee at maturity. The asset share is switched out of equities to zero-coupon
bonds if the probability falls below 97.5%, and switched from zero-coupon bonds
back into equities if the probability rises above 99.5%. This rule, together with
the dynamic bonus strategy, reflects that the management actions taken by the
insurer are driven by its financial strength. This is due to the general logic that
the lower the margin between the assets and guarantees, the more low risk assets
75
should be held and the less bonuses should be declared. The corridor approach has
also been considered in Hardy (2000) and Wilkie et al. (2003), but for the purpose
of readjusting reserves so that adding in new capital to increase the reserves occurs
less frequently.
The extra notation for the dynamic model, which contains the dynamic invest-
ment and bonus strategies, is given as follows,
• PY ′(t, p, n, i): the projected par yield at time p for term n in the ith simulation,
projected at time t < p
• C ′(t, p, i): the projected consols yield at time p in the ith simulation using the
Wilkie model, given the index at time t < p
• B′(t, p, i): the projected short-term interest rate at time p in the ith simulation
using the Wilkie model, given the index at time t < p
• v′(t, p, n, i): the projected value of a zero-coupon bond at time p with term n
in the ith simulation, projected at time t < p
• R′e(t, p, i): the projected rate of return on equities with the dividends rein-
vested during the year (p − 1, p) in the ith simulation, projected at time
t < p
• R′z(t, p, i): the projected rate of return on a zero-coupon bond during the year
(p− 1, p) in the ith simulation, projected at time t < p
• e′(t): the suggested equity backing ratio at time t
• A′(t, T ): the projected value of the maturity asset share, projected at time
t < T
At the beginning of each policy year, i.e. t = 0, 1, ..., T − 1, the possible future
scenarios are generated using the Wilkie model starting with the current market
indices. In the ith simulation, i = 1, 2, ..., 10000, given the projected equity price
index and dividend amount, the projected rate of return on equities during the
future policy year (p− 1, p), p = t + 1, t + 2, ..., T , follows the equation
R′e(t, p, i) =
P ′(t, p, i) + D′(t, p, i)P ′(t, p− 1, i)
− 1 (5.65)
76
with
P ′(t, t, i) = P (t).
Given the projected consols yield and short-term interest rate, the par yield at
future time p for term n is modelled as follows
PY ′(t, p, n, i) = C ′(t, p, i) + [B′(t, p, i)− C ′(t, p, i)] · e−β·n (5.66)
in which β is a constant. Then, the projected value of a zero-coupon bond at time
p with duration n can be derived in the same way as described in Section 4.2. We
have the following equations:
v′(t, p, 1, i) =1
PY ′(t, p, 1, i) + 1(5.67)
for a zero-coupon bond with 1-year duration, and
v′(t, p, n, i) =1− PY ′(t, p, n, i) ·∑n−1
d=1 v′(t, p, d, i)
1 + PY ′(t, p, n, i)(5.68)
for a zero-coupon bond with duration n > 1.
The asset share is partly invested in the zero-coupon bonds with the same matu-
rity date as the policy. Hence the projected rate of return on the zero-coupon bonds
during the future policy year (p− 1, p) follows the equation
R′z(t, p, i) =
v′(t, p, T − p, i)
v′(t, p− 1, T − (p− 1), i)− 1 (5.69)
with
v′(t, t, T − t, i) = v(t, T − t)
and
v′(t, T, 0, i) = 1.
The insurer needs to decide on an initial EBR at policy inception so that the
single premium can be invested in the two asset classes. Ideally, the initial bonus
rate and EBR should be consistent. However, to simplify the calculations we suggest
an initial EBR of 80%. Afterwards, the suggested EBR equals the previous year’s
equity proportion. Hence,
e′(0) = 0.80 (5.70)
77
and
e′(t) = e(t− 1) (5.71)
for t = 1, 2, ..., T − 1.
Assuming that the assets are switched each year to restore the suggested EBR,
we can project forward the performance of the unit fund as follows,
A′(t, p, i) = [A′(t, p− 1, i) · e′(t) · (R′e(t, p, i) + 1) (5.72)
+ A′(t, p− 1, i) · (1− e′(t)) · (R′z(t, p, i) + 1)] · (1− c)
with
A′(t, t, i) = A(t).
The initial asset share equals the single premium paid by the policyholder. Af-
terwards, the actual asset share depends on the previous year’s EBR and rate of
returns achieved in the two asset classes, i.e.
A(t) =
[A(t− 1) · e(t− 1) · P (t) + D(t)
P (t− 1)(5.73)
+ A(t− 1) · (1− e(t− 1)) · v(t, T − t)
v(t− 1, T − (t− 1))
]· (1− c).
From the 10,000 simulated values of the maturity asset share, the probability
that the projected asset share is sufficient to pay the guarantee at maturity can be
calculated by the equation
P(A′(t, T ) ≥ G(t)) =
∑10,000i=1 I(A′(t, T, i) ≥ G(t))
10, 000(5.74)
where I is an indicator function which equals 1 if the statement is true and 0 if
wrong. As in Chapter 4, here we only consider the case of smoothing with allowance
for future bonuses. Therefore, the guarantee, G(t), follows equations 3.53 at policy
inception, 3.54 after each bonus declaration, and 3.55 at maturity. However, the
regular bonuses depend on the investment strategy adopted by the insurer.
If the probability is above 99.5%, we can infer that the insurer has a very strong
prospective solvency position and so it can invest more in risky assets to enhance the
investment value of the policy. Hence the EBR takes the suggested value increased by
10%. If the probability is below 97.5%, the insurer’s financial strength is weakened
78
and so the EBR equals the suggested value decreased by 10%. If the probability is
within the range of [97.5%, 99.5%], the EBR takes the suggested value. Expressed
in equation,
e(t) =
e′(t)× 1.1 if P(A′(t, T ) ≥ G(t)) > 0.995
e′(t)/1.1 if P(A′(t, T ) ≥ G(t)) < 0.975
e′(t) otherwise
. (5.75)
In this thesis we round the value of the EBR to two decimal places.
Based on the EBR just calculated, the unit fund is projected forward again for
the purpose of a bonus calculation. The bonus rates are calculated in the same way
as described in Section 3.2.2, but the projected value of the maturity asset share in
the ith simulation before sorted into ascending order, i.e. A′(t, T, i), is calculated
by equation 5.72 with e′(t) replaced by the actual EBR e(t) calculated above.
The above derivation shows that our dynamic investment and bonus strategies
are related to each other in that they are both based on the projected performance
of the unit fund.
Table 5.2 gives the EBRs for the 1991 policy.
Table 5.2: The equity backing ratios for the policy issued at the end of 1991 in CaseC (%)
31/Dec/ 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000e(t) 88 80 72 72 72 72 72 72 79 86
In Table 5.2 we see that the EBR starts from 0.88 which is the highest possible
ratio at policy inception. Then the EBR falls to 0.72 at the end of 1993 and remains
at that level for a few years before rising at later durations. The dividend yield on
shares falls during the particular 10 years, so the Wilkie model will suggest lower
equity proportions. The EBRs for this 1991 policy are not very different from the
initial suggested proportion of 80%, which implies that the insurer is in a relatively
strong prospective solvency position because the probability of having sufficient asset
share at maturity is mostly higher than 97.5%. Recall that the guarantee includes
the minimum of future regular bonuses.
79
year
bonu
s ra
te (
%)
1992 1994 1996 1998 2000
3.0
3.5
4.0
4.5
5.0
5.5
6.0
1993 1995 1997 1999
dynamic EBRs100% EBRs
Figure 5.22: The comparison of the bonus rates declared on the 1991 policy in CaseC with the static and dynamic EBRs
Figure 5.22 compares the regular bonus rates for the 1991 policy in Case C with
the static and dynamic EBRs.
The bonus rates declared before the end of 1999 have a similar pattern in the two
cases. Most of the time the bonus earning power is increased by the investment in
both equities and zero-coupon bonds. However, the final two bonus rates are much
lower with the dynamic EBRs. We have seen in Table 5.2 that the EBR increases
at later durations probably due to the high projected rate of return on equities. If
the whole asset share were invested in equities as in our static investment strategy,
the 25th percentile of the projected maturity asset share would have been larger.
Figure 5.23 shows how the asset share and guarantee build up over the policy
term with the dynamic and static investment strategies.
The actual rate of return on zero-coupon bonds is mostly lower than that on
equities, so we see a larger asset share with the 100% EBR. The difference is most
significant at the end of 1999, £330.36 versus £293.28. Afterwards, when the equity
market falls, the asset share drops less rapidly with some assets invested in zero-
coupon bonds.
Looking at Table 5.2 and Figure 5.23 together, we can see that our dynamic
investment strategy does not work well in reality. We never increase the equity
80
year
1992 1994 1996 1998 2000
100
150
200
250
300
1991 1993 1995 1997 1999 2001
asset share with dynamic EBRsguarantee with dynamic EBRsasset share with 100% EBRsguarantee with 100% EBRs
Figure 5.23: The comparison of the asset shares and guarantees of the 1991 policyin Case C with the static and dynamic investment strategies
proportion just before the equity index actually increases, except at policy inception.
Also, we make wrong decisions to increase the EBRs at the end of 1999 and 2000
just before the equity market actually performs very badly.
Figure 5.23 also shows that using a different investment strategy does not affect
the guarantees very much for this policy. At maturity the guaranteed payout is
slightly smaller with the dynamic EBR.
Given the maturity asset share and guarantee of £249.17 and £173.66 respec-
tively, a terminal bonus rate of 43.48% can be declared which is lower than the rate
declared with the 100% EBRs but still higher than the 30% target.
For this policy, the guarantees do not bite at maturity. The policyholder receives
his asset share of £249.17 with the dynamic EBRs, whereas if the whole fund is
invested in equities the policyholder can get a larger asset share of £266.54 at
maturity. Therefore, the dynamic investment strategy has reduced the investment
value of this particular policy. However, a part of the guaranteed payout is covered
by the redeemable value of the zero-coupon bonds with certainty. The insurer sets
up reserves to cover the risk for which it makes no variable charges, so the dynamic
investment strategy has increased the insurer’s safety.
81
5.3 Results with the Dynamic Model
In this section, we calculate the reserves and investigate the profitability of the 1991
policy with the dynamic EBR.
An investment strategy is one of the very important management actions that
the insurer can take to reduce risks and mitigate costs. According to the recent
CP195 proposals (FSA (2003)), the insurer’s decision rules for future asset alloca-
tions should be incorporated into the projection of claims. As in the case of allowing
for future bonuses, it is not straightforward to model the investment strategy. To
project for reserving purposes (contrast with our projections to make the bonus
and investment decisions in which the assets are rebalanced each year to a fixed
proportion), we simply assume that the assets will not be switched in the future
and the current asset mix is left drifting. With this assumption, the value of the
zero-coupon bonds at policy maturity is known in advance given the current EBR.
Hence the guaranteed payout can partly be met by the redeemable value of the
zero-coupon bonds, and only the remaining guarantees which are not covered by
the bonds need to be reserved for. Therefore, we can still use a closed form solu-
tion (i.e. Black-Scholes equation) to value the cost of guarantees under the option
pricing approach.
We only consider Case C as defined in Chapter 3, so the amount of guarantees
follows equations 3.53 at policy inception, 3.54 after each bonus declaration and 3.55
at maturity, but rolled up with the regular bonuses calculated in Section 5.2.
5.3.1 Buying Options
The detailed mechanism of calculating the reserves and cashflows has been described
in Section 2.2, but the number of put options and the exercise price of each option
should be calculated in a different way. Using either investment strategy, we should
have one put option for each unit of equity index held at maturity so that the
remaining guarantees, which are not covered by the zero-coupon bonds, can be met
by exercising the options if necessary. Thus, at each valuation date t, t = 1, 2, ...,
82
T − 1, the number of options is given by the equation
N(t) = N ′(T ) (5.76)
=A(t) · e(t)
S(t)· (1− c)T−t
in which N ′(T ) is the number of equity index held at maturity.
With the static investment strategy that the whole asset share is invested in
equities, the number of put options is a constant over the policy term. However,
once we introduce dynamic EBRs, the number of options changes each year after
the assets are rebalanced.
If the put options are exercised at maturity, the payout of the options should be
exactly enough to cover the remaining guarantees. Hence the exercise price follows
the equation
E(t) =G(t)− A(t)·(1−e(t))
v(t,T−t)· (1− c)T−t
N(t)(5.77)
in which v(t, T − t) is the value at time t of the zero-coupon bond with the same
maturity date as the policy.
If the remaining guarantees have a negative value, which means that the re-
deemable value of the zero-coupon bonds is bigger than the total guarantees that
should be reserved for, no reserve is required in addition to the asset share.
Figure 5.24 compares the reserves set up using the option method for the 1991
policy with the dynamic and static EBRs.
Figure 5.24 shows that the reserves are much smaller with the dynamic EBR,
which is intuitive because a part of the unit fund is in a risk-free asset with a
guaranteed redeemable value. A significant difference can be noticed in the pattern
of the reserves between the two cases. The reserves with the static EBRs show
a temporary increase at the end of 1993 because the risk-free interest rate drops
sharply. However, the amount of reserves with the dynamic EBRs decreases through
time.
Figure 5.25 shows the equity indices and exercise prices during the policy term
with the dynamic investment strategy. As a comparison, the exercise prices with
the 100% EBR are also shown.
83
year
rese
rves
1992 1994 1996 1998 2000
02
46
8
1991 1993 1995 1997 1999
dynamic EBRs100% EBRs
Figure 5.24: The comparison of the reserves using the option method for the 1991policy in Case C with the dynamic and static EBRs
year
1992 1994 1996 1998 2000
1500
2000
2500
3000
3500
4000
1991 1993 1995 1997 1999
equity indexexercise price with dynamic EBRsexercise price with 100% EBRs
Figure 5.25: The equity index and exercise price of the options bought for the 1991policy in Case C with the dynamic and static EBRs
84
We see in Figure 5.25 that the exercise price of the put option is lower with the
dynamic investment strategy. With the 100% EBR, the exercise price is a non-
decreasing function of the policy duration; whereas with the dynamic EBR, the
exercise price might decrease.
The change in the pattern of the exercise price has its impact on the reserves.
The downward adjusting of the equity proportion during the first two policy years
when the equity market actually performs very well has reduced the number of units
of the equity index and hence the number of options. The decrease in the exercise
prices during that period has reduced the value of each put option. Therefore, we
see an obvious decrease in the required amount of reserves at early durations.
The cashflows incurred by the insurer has an accumulated value at policy ter-
mination of £-18.88. The insurer can only earn a profit of £7.66 using the static
investment strategy. Hence the dynamic strategy has greatly increased the insurer’s
profit, but at the cost of a smaller maturity payout to the policyholder. Therefore, if
the insurer wants to increase its safety and profitability by investing more of the pol-
icyholder’s fund in a low risk asset, the investment value of the policy is reduced. In
other words, there exists a conflict of interest between the insurer and policyholder.
5.3.2 Discrete Hedging
The cashflows incurred by discrete hedging have an accumulated value of £-23.99
with the dynamic investment strategy; whereas the insurer earns a profit of £18.72
using the static strategy. Again, we can conclude that the insurer earns a larger profit
by using the dynamic investment strategy but at the cost of a smaller investment
value to the policyholder; and that the insurer earns a larger profit by hedging the
risk internally instead of buying options from a third party.
5.3.3 CTE Reserving
Assuming, for reserving purposes, that the assets will not be switched in the future
and the current asset mix is left drifting, the projected value of the maturity asset
85
share at the valuation date t in the ith simulation follows the equation
A′(t, T, i) = A(t) ·[e(t) ·
T∏p=t+1
(1 + R′e(t, p, i)) (5.78)
+ (1− e(t)) · 1
v(t, T − t)
]· (1− c)T−t
in which R′e(t, p, i) is the projected rate of return on equities over the future policy
year (p− 1, p), and v(t, T − t) is the value at time t of the zero-coupon bond with
the same maturity date as the policy.
The projected amount of the guarantee charges deducted at each future time
point p, p = t + 1, t + 2, ..., T , is given by the equation
GC ′(t, p, i) = A(t) ·[e(t) ·
p∏d=t+1
(1 + R′e(t, d, i)) (5.79)
+ (1− e(t)) ·p∏
d=t+1
(1 + R′z(t, d, i))
]· (1− c)p−t−1 · c
in which R′z(t, d, i) is the projected rate of return on zero-coupon bonds over the
future policy year (d− 1, d).
Then, a similar mechanism as described in Section 4.3.3 can be used to calculate
the CTE reserves and cashflows.
The 95% and 99% CTE reserves for the 1991 policy with the dynamic and static
EBRs are shown in Figure 5.26.
Figure 5.26 shows that the CTE reserves are greatly reduced if a part of the
unit fund is invested in a risk-free asset. Investing in both equities and zero-coupon
bonds increases the projected value of the maturity asset share in the worst cases.
Therefore, a smaller CTE reserve is required. In the 95% of the projections, the
payoff under the maturity guarantees can be covered by the future guarantee charges
so that no CTE reserve is required at all during the policy term.
We have already seen in Figure 5.23 that at later durations the asset share well
exceeds the guarantee. Hence, if 20% of the asset share is invested in zero-coupon
bonds, it covers a big part of the guarantee. Therefore, the reduction in the 99%
CTE reserves after introducing a risk-free asset is more obvious at later durations.
Table 5.3 compares the accumulated values of the cashflows incurred by the
insurer using the different investment strategies.
86
year
CT
E r
eser
ves
1992 1994 1996 1998 2000
010
2030
40
1991 1993 1995 1997 1999
95%, dynamic EBRs95%, 100% EBRs99%, dynamic EBRs99%, 100% EBRs
Figure 5.26: The comparison of the CTE reserves for the 1991 policy in Case C withthe dynamic and static EBRs
Table 5.3: The comparison of the accumulated values of the cashflows for the 1991policy in Case C with the dynamic and static EBRs
Approach Option Hedging CTE (95%) CTE (99%)100% static -7.66 -18.72 -30.51 -31.00dynamic -18.88 -23.99 -27.52 -29.52
Table 5.3 shows that using the dynamic investment strategy, the insurer who
sets up CTE reserves earns a smaller profit. The asset share is of a smaller amount
when a risk-free asset is introduced, so the guarantee charges deducted from the
units are also smaller. No reserve is required in the 95% of the simulations, so
the profit is equal to the accumulated value of all the guarantee charges with the
95% CTE measure. The 99% CTE reserves are set up and released frequently. At
some durations, the previous year’s reserves might have earned a high rate of return
and be released back to the insurer. Thus, the insurer earns a bigger profit by
setting up 99% CTE reserves due to the performance of the zero-coupon bonds in
the particular 10-year period. Again, the 1991 policy is still more profitable under
the CTE approach.
87
5.4 Summary
This chapter has looked at the single 10-year UWP policy in the most complicated
and realistic cases up to now. The unit fund is invested in two asset classes: equities
and zero-coupon bonds with the same maturity date as the policy. The percentage
in either asset is adjusted according to a dynamic investment strategy which uses
a corridor approach. The regular bonus rates are declared according to a dynamic
bonus strategy which is related to our investment strategy. The risk-free interest rate
equals the yield on a zero-coupon bond with the same maturity date as the policy
which is derived from the historical consols yield and short-term interest rate. We
have only considered Case C in which the bonuses are smoothed and the minimum
of future bonuses implied by the smoothing mechanism are reserved for.
We summarise the main conclusions for the 1991 policy as follows:
• The equity backing ratios are not very different from the initial suggested value
of 0.80, which implies that the insurer is in a relatively strong prospective
solvency position.
• Our dynamic investment strategy does not work well in reality. Sometimes
wrong decisions are made to increase (decrease) the equity proportion when
the equity market actually performs very badly (well).
• Only the remaining guarantees which are not covered by the zero-coupon bonds
should be reserved for. The reserves are greatly reduced after introducing a
risk-free asset.
• Reserves are always required under the option pricing approach. There is no
need to set up 95% CTE reserves during the policy term. The 99% CTE
reserves reduce to zero at later durations.
• The policy is profitable under all three reserving approaches with the dynamic
EBR. The largest profit can be earned by the insurer who sets up reserves using
the 99% CTE measure, followed by the 95% CTE measure, discrete hedging
and finally the option method.
88
• Under the option pricing approach the insurer earns a larger profit by investing
the unit fund in both risky and risk-free assets, but at the cost of a smaller
maturity payout to the policyholder.
89
Chapter 6
SENSITIVITY TESTING FOR
THE SINGLE UWP POLICY
6.1 Introduction
In Chapter 5 we presented our results for the 10-year policy issued at the end of
1991 based on a standard basis in which
• the guaranteed growth rate g=2%
• the fixed percentage of units deducted as a guarantee charge c=1%
• the terminal bonus target TB=30%
• the volatility of the equity index assumed in the Black-Scholes model σ=20%
• the transaction costs as a percentage of the change in the equity component
of the hedge portfolio τ=0.2%.
The upper and lower boundaries of probability used in our dynamic investment
strategy are 99.5% and 97.5% respectively.
Different assumptions for the contract design and market parameters lead to
different reserves and cashflows. Different upper and lower probability boundaries
to adjust the EBRs also affect the results. In addition, the results depend on the
investment performance during the particular 10-year period. This chapter is to test
90
the sensitivity of the results to different parameters, upper and lower boundaries of
probability in the investment strategy, and date of issue of the 10-year policy.
Section 6.2 investigates the sensitivity to different parameters. Ideally, we should
change the parameters in a consistent way aiming for a fair design in which the
expected present value of the maturity payout equals the single premium under the
risk neutral measure. However, we investigate the effect of changing one parameter
while keeping others fixed, in order to show a clear picture of how the contract design
and market parameters affect the results for the 1991 policy. Including the above
standard basis, named as Basis i, we consider six groups of assumptions which are
given in Table 6.4.
Table 6.4: The assumptions for the parameters g, c, TB, σ and τ (%)
Basis g c TB σ τi 2 1 30 20 0.2ii 0 1 30 20 0.2iii 2 0.5 30 20 0.2iv 2 1 50 20 0.2v 2 1 30 25 0.2vi 2 1 30 20 0.5
We investigate the effect of changing from our standard basis to each of the other
bases.
In Section 6.3 we adjust the EBRs according to different upper and lower proba-
bility boundaries. The results are given for the 1991 policy under the standard basis
with the probabilities of 99% and 95%.
Section 6.4 considers two 10-year policies issued at the end of 1964 and 1982
respectively. Comparing the results for the three policies issued at different times, we
can investigate the effect of different investment performance on the EBRs, bonuses,
reserves and profitability of the policy.
Finally in Section 6.5 the conclusions are summarised.
In this chapter, we only consider the case of smoothing with allowance for future
bonuses, i.e. Case C as defined in Chapter 3.
91
6.2 Sensitivity to Different Parameters
In this section we investigate how different assumptions for the contract design and
market parameters affect the results for the 1991 policy.
6.2.1 EBRs and Bonus Rates
The volatility of the equity index and transaction costs rate have no impact on the
EBRs, bonuses, asset shares or guarantees. So we only consider the first four groups
of assumptions given in Table 6.4.
Tables 6.5 and 6.6 give the EBRs and regular bonus rates for the 1991 policy
under Bases i, ii, iii and iv.
Table 6.5: The EBRs for the 1991 policy in Case C based on the different bases (%)
Basis31/Dec/ i ii iii iv1991 88 88 88 881992 80 88 80 801993 72 80 72 721994 72 80 72 721995 72 80 72 791996 72 80 72 791997 72 80 72 861998 72 80 72 861999 79 88 79 942000 86 96 86 100
Figures 6.27 and 6.28 show how the asset share and guarantee build up under
the different bases.
Basis ii assumes a lower guaranteed growth rate than our standard basis. The
insurer has promised a smaller initial guarantee with the lower guaranteed rate,
hence more of the asset share can be invested in equities. During this particular
10-year period the equity market mostly achieves a higher return than the bond
market, so we see in Figure 6.27 that the asset share is mostly larger under Basis
ii than Basis i. During the final two policy years, however, equities perform badly
and so the asset share drops more rapidly with the higher EBRs under Basis ii. The
92
year
asse
t sha
re
1992 1994 1996 1998 2000
100
150
200
250
300
1991 1993 1995 1997 1999 2001
iiiiiiiv
Figure 6.27: The asset shares of the 1991 policy in Case C based on the differentbases
year
guar
ante
e
1992 1994 1996 1998 2000
130
140
150
160
170
180
190
1991 1993 1995 1997 1999 2001
iiiiiiiv
Figure 6.28: The guarantees of the 1991 policy in Case C based on the differentbases
93
Table 6.6: The regular bonus rates declared on the 1991 policy in Case C using thedifferent bases (%)
Basis31/Dec/ i ii iii iv1992 4.46 6.42 4.96 3.081993 3.72 5.35 4.13 2.561994 3.49 5.72 4.10 2.141995 4.18 6.86 4.92 2.561996 4.72 7.12 5.32 2.691997 4.78 7.18 5.38 2.241998 3.98 5.99 4.48 1.871999 3.70 5.90 4.34 1.552000 3.08 4.92 3.62 1.30
bonus earning power is increased by the smaller initial guarantee and larger asset
share, hence higher bonus rates can be declared. Therefore, the guarantee shown in
Figure 6.28 increases more rapidly with the lower guaranteed rate and the difference
between the two bases becomes less obvious as the policy duration increases.
Basis iii assumes a smaller guarantee charge than the standard basis. The fewer
the charges deducted from the units, the larger the asset share will be and hence
the higher the bonus earning power. Table 6.6 shows that the bonus rates declared
under Basis iii are around 0.6% higher than those under the standard basis. We see
in Figure 6.28 that the initial guarantee, including the minimum of future regular
bonuses, is of a larger amount under Basis iii due to the higher initial bonus rate.
Afterwards, the guarantee builds up more rapidly with the higher bonuses declared
under Basis iii. The smaller charges lead to larger guarantees and also higher pro-
jected asset values, so the impact on the insurer’s prospective financial strength is
not so significant. The EBRs are the same under the two bases.
Basis iv sets a higher terminal bonus target than the standard basis. The purpose
of setting a terminal bonus target is to build up a margin between the asset share
and guarantee which will provide a cushion against falls in asset values. The higher
target reduces the bonus earning power and hence smaller bonuses are declared.
The guarantee, shown in Figure 6.28, starts from a lower level due to the lower
initial bonus rate and afterwards builds up less rapidly with the smaller bonuses.
94
The smaller guarantees improve the insurer’s solvency position, so we see in Table
6.5 that the EBR starts to increase at the end of 1995 and finally the whole asset
share is in equities under Basis iv. In other words, the lower risk bonus strategy
leads to a higher risk investment strategy. Figure 6.27 shows that the unit fund
achieves a higher rate of return over the period from the end of 1995 to the end
of 1999 with the higher EBRs under Basis iv. However, the equity index falls
dramatically during the final two policy years, so the asset share drops more rapidly
with the higher EBRs and at maturity the asset share is even smaller under Basis iv.
Although equities mostly perform better than bonds, we see in Table 6.6 that the
bonus earning power is reduced by the higher EBRs during this period. Appendix B
shows that the dividend yield on shares falls in this particular 10-year period. The
force of inflation is a dominant variable in the projection of investment performance
using the Wilkie model. Therefore, the 25th percentile of the projected maturity
asset share is reduced by the higher EBR under Basis iv and hence the bonus rate
remains at a lower level.
Table 6.7 shows the asset share, guarantee and terminal bonus rate at maturity
under the different bases.
Table 6.7: The asset share, guarantee and terminal bonus rate at maturity of the1991 policy in Case C based on the different bases
Basis31/Dec/2001 i ii iii ivasset share (£) 249.17 249.93 262.05 247.20guarantee (£) 173.66 171.23 182.45 148.52terminal bonus rate (%) 43.48 45.96 43.62 66.44
We see in Table 6.7 that the guarantee does not bite at maturity under any of the
bases. The policyholder receives the whole asset share which has the highest value
under Basis iii with the smaller guarantee charges. The higher terminal bonus target
in Basis iv leads to the smallest asset share at maturity. The maturity guarantee is
also the largest (smallest) under Basis iii (iv). The terminal bonus rates declared
under the different bases are all above the initial target.
95
6.2.2 Reserves and Profitability
Now we consider the effect of the different bases on the reserves and profitability of
the 1991 policy.
The option method needs an assumption for the volatility of the equity index,
but it ignores the transaction costs when buying and selling the options. So the first
five bases given in Table 6.4 are considered. The reserves set up using the option
method under the different bases are given in Table 6.8.
Table 6.8: The reserves using the option method for the 1991 policy in Case C underthe different bases
Basis31/Dec/ i ii iii iv v1991 4.88 3.12 4.60 3.73 7.801992 3.38 2.93 3.07 2.36 5.911993 2.51 2.10 2.17 1.54 4.811994 2.06 2.11 1.92 1.06 4.041995 1.74 2.15 1.72 1.32 3.681996 1.30 1.65 1.27 0.73 3.011997 0.56 0.84 0.56 0.48 1.671998 0.20 0.32 0.19 0.16 0.801999 0.02 0.04 0.02 0.01 0.142000 0.01 0.02 0.01 0.00 0.07
We see in Table 6.8 that smaller reserves are required at early durations under
Basis ii because the guarantees are smaller with the lower guaranteed rate. However,
more reserves are required after the end of 1994 because the higher EBRs, shown in
Table 6.8, increase the amount of the remaining guarantees which are not covered
by the zero-coupon bonds.
We concluded in Section 6.2.1 that the smaller guarantee charges in Basis iii lead
to larger asset shares and guarantees. The reserves are slightly reduced probably
because the increase in the guarantees is not so significant as that in the asset shares.
The bonus earning power is reduced by the higher terminal bonus target under
Basis iv, so the guarantees include smaller future bonuses initially and build up with
lower bonus rates afterwards. A large margin is built up between the asset share
and guarantee, so smaller reserves are required to meet the guaranteed payout. The
96
reserves increase temporarily at the end of 1995 because the EBR goes up by 10%
when the policy is still far from maturity. The upward adjustment in the EBRs
at later durations does not increase the reserves because the guarantees are then
deeply out-of-the-money.
Basis v assumes a higher volatility for the equity index than our standard basis,
which greatly increases the amount of reserves. The higher volatility increases the
uncertainty about the equity price movements in the future and hence the options
are more expensive. However, the difference is less obvious at later policy durations
because the options are then deeply out-of-the-money.
The cashflows incurred during the policy term are accumulated to the termination
of the policy using the risk-free rate which equals the yield on the zero-coupon bond
with the same maturity date as the policy. The accumulated values under the
different bases are given in Table 6.9.
Table 6.9: The accumulated values of the cashflows incurred using the option methodfor the 1991 policy in Case C under the different bases
Basis i ii iii iv vAV CF -18.88 -18.80 -5.68 -21.38 -12.86
Table 6.9 shows that under the option approach the lower guaranteed rate as-
sumed in Basis ii slightly reduces the profitability of the 1991 policy, but the ma-
turity payout to the policyholder is slightly larger. The smaller guarantee charges
under Basis iii lead to a larger maturity payout to the policyholder, but a greatly
reduced profit to the insurer because fewer units are deducted to pay for the cost
of guarantees which is borne by the insurer. The profitability is increased by the
higher terminal bonus target under Basis iv because smaller reserves are required
during the policy term. However, the greater profitability is at the cost of a smaller
maturity payout to the policyholder because the asset share is switched into equities
when the equity market performs badly. The policy is less profitable under basis v
which assumes a higher volatility for the equity index so that the options are more
expensive.
The reserves are of the same amount using the option and hedging approaches.
97
We allow for transaction costs when constructing or readjusting the hedge portfolio,
but the rate of transaction costs τ does not affect the amount of reserves. The
accumulated values of the cashflows under the six bases are given in Table 6.10.
Table 6.10: The accumulated values of the cashflows incurred by discrete hedgingfor the 1991 policy in Case C under the different bases
Basis i ii iii iv v viAV CF -23.99 -23.95 -10.64 -25.30 -22.27 -23.82
The conclusions drawn from Table 6.9 also apply under the hedging approach.
The lower guaranteed rate, smaller guarantee charges, higher volatility for the equity
index under Bases ii, iii and v respectively reduce the profitability of the policy. The
insurer earns a larger profit under Basis iv with the higher terminal bonus target.
Comparing Tables 6.9 and 6.10, we notice that the higher volatility for the equity
index in Basis v has a smaller effect on the profitability if the insurer hedges the
risk internally rather than buying options. We have inferred from the previous
chapters that the assumed 20% volatility might be too high compared with reality
so that the replicating portfolio brought forward is mostly worth more than that
required to be set up. However, Basis v assumes an even higher volatility of 25%.
By discrete hedging, the reserves are gradually released back to the insurer through
the negative hedging error. Clearly, the higher rate of transaction costs assumed in
Basis vi reduces the insurer’s profit.
Using the different assumptions from the standard basis does not change the
conclusion that the insurer earns a larger profit by hedging the risk internally rather
than buying options over-the-counter.
Now we consider the effect of the different bases on the 95% and 99% CTE
reserves. The volatility of the equity index and transaction costs rate do not affect
CTE reserves. So only the first four bases given in Table 6.4 are considered. Table
6.11 shows the 95% CTE reserves calculated under the different bases. Note that
negative CTE reserves are replaced by zero.
The impact on the 95% CTE reserves of the different assumptions is not obvious
because most of the time no reserve is required at all with the 95% CTE measure.
98
Table 6.11: The 95th CTE reserves for the 1991 policy in Case C using the differentbases
Basis31/Dec/ i ii iii iv1991 0.00 0.00 0.00 0.001992 0.00 0.00 0.00 0.001993 0.00 0.00 0.00 0.001994 0.00 0.00 0.00 0.001995 0.00 0.00 0.00 0.001996 0.00 0.00 0.00 0.001997 0.00 0.00 0.00 0.001998 0.00 0.00 0.00 0.001999 0.00 0.00 0.00 0.002000 0.00 0.79 0.01 0.00
Only at the final valuation date is a small reserve required with a lower guaranteed
rate under Basis ii or the smaller guarantee charges under Basis iii.
The 99% CTE reserves under the different bases are given in Table 6.12.
Table 6.12: The 99th CTE reserves for the 1991 policy in Case C under the differentbases
Basis31/Dec/ i ii iii iv1991 5.35 0.00 8.38 1.571992 5.90 4.23 8.63 1.461993 3.36 1.20 5.74 0.001994 1.75 2.05 4.55 0.001995 0.00 0.85 2.04 0.001996 1.81 4.70 4.83 0.001997 0.14 4.53 2.97 0.001998 0.00 4.76 2.09 0.001999 0.00 6.58 1.29 0.002000 2.88 10.16 3.77 0.06
The lower guaranteed rate in Basis ii reduces the amount of reserves at early
durations, but larger reserves are required at later durations when the guarantees
increase rapidly and a big part of the asset share is in equities. In fact, the lower
guaranteed rate has changed the trend of the 99% CTE reserves.
The smaller guarantee charges in Basis iii has different effect on the reserves under
99
the different approaches. We have seen in Table 6.8 that the smaller charges slightly
reduce the reserves under the option pricing approach. However, Tables 6.11 and
6.12 show that the CTE reserves are increased by the smaller charges. As mentioned
before, there is an inconsistency in the way the future guarantee charges are allowed
for when setting up reserves using the different approaches. The CTE reserves are
calculated from the projected cashflows. Smaller charges mean less money paid to
the insurer to cover the guarantee cost, hence more CTE reserves are required.
With the higher terminal bonus target in Basis iv, the guarantees are smaller
initially and increase less rapidly afterwards due to the reduced bonus earning power.
Hence smaller reserves are required during the policy term.
Table 6.13 shows the accumulated values of the cashflows incurred to set up 95%
and 99% CTE reserves under the different bases.
Table 6.13: The accumulated values of the cashflows incurred to set up CTE reservesfor the 1991 policy in Case C using the different bases
Basis i ii iii ivAV CF (95%) -27.52 -27.86 -14.09 -27.77AV CF (99%) -29.52 -29.29 -15.12 -30.06
We have seen in Table 6.11 that almost no reserves are required with the 95%
CTE measure, hence the incurred cashflows are mostly the guarantee charges. The
asset share is mostly larger under the assumptions other than our standard basis.
However, the smaller guarantee charges in Basis iii reduce the amount of the money
paid to the insurer. Therefore, the insurer earns a larger profit under Bases ii and
iv but a smaller profit under Basis iii.
The lower guaranteed rate and the smaller guarantee charges both reduce the
profitability to the insurer who sets up 99% CTE reserves, and the latter has a
greater impact. The higher terminal bonus target in Basis iv increases the prof-
itability.
100
6.3 Sensitivity to Different Upper and Lower
Probability Boundaries
In our dynamic investment strategy, the upper and lower boundaries of probability
to switch the asset share between the two asset classes have been set as 99.5% and
97.5% respectively. This section will consider different probabilities of 99% and 95%.
6.3.1 EBRs and Bonus Rates
We start by investigating the effect of the different boundaries on the equity backing
ratios and bonus rates for the 1991 policy under the standard basis. Table 6.14
compares the EBRs which are adjusted according to the different probabilities.
Table 6.14: The EBRs for the 1991 policy in Case C under the standard basis withthe different upper and lower probability boundaries (%)
31/Dec/ 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000(97.5%, 99.5%) 88 80 72 72 72 72 72 72 79 86(95%, 99%) 88 88 80 80 80 80 80 80 88 96
Table 6.15 shows the regular bonus rates declared on the 1991 policy with the
different probabilities.
Table 6.15: The bonus rates for the 1991 policy in Case C under the standard basiswith the different probabilities to adjust the EBRs (%)
31/Dec/ 1992 1993 1994 1995 1996 1997 1998 1999 2000(97.5%, 99.5%) 4.46 3.72 3.49 4.18 4.72 4.78 3.98 3.70 3.08(95%, 99%) 4.46 3.72 3.27 3.93 4.71 4.80 4.00 3.36 2.80
Figure 6.29 shows how the asset share and guarantee build up with the different
probabilities.
With the new upper and lower boundaries of 99% and 95%, the asset share is
more likely to be switched into equities and less likely to be switched into zero-
coupon bonds. Thus, we see in Table 6.14 that the equity proportions are mostly
higher with the new boundaries. Going back to Table 6.5, we see that the EBRs
happen to be the same under the standard basis with the probabilities of 95% and
101
year
1992 1994 1996 1998 2000
100
150
200
250
300
1991 1993 1995 1997 1999 2001
asset share with (97.5%, 99.5%)guarantee with (97.5%, 99.5%)asset share with (95%, 99%)guarantee with (95%, 99%)
Figure 6.29: The asset shares and guarantees for the 1991 policy in Case C underthe standard basis with the different upper and lower probability boundaries
99%, or under Basis ii with a lower guaranteed rate and the probabilities of 97.5%
and 99.5%.
Table 6.15 shows that changing the probabilities mostly reduces the regular
bonuses declared on the 1991 policy, which implies that the 25th percentile of the
projected maturity asset share is smaller with the higher EBRs. Therefore, we have
a more risky investment strategy compensated by a less risky bonus strategy.
The asset share shown in Figure 6.29 is mostly larger with the new probabilities
of 95% and 99% because more of the policyholder’s assets are invested in equities
which perform better than zero-coupon bonds most of the time. The policyholder
receives a slightly larger maturity payout with the new boundaries (£249.93 versus
£249.17). Also, the guarantee builds up less rapidly over the policy term with the
lower bonus rates. Hence a higher terminal bonus rate of 45.39% can be declared at
maturity (versus 43.48% with the probabilities of 97.5% and 99.5%).
6.3.2 Reserves and Profitability
Table 6.16 shows the amount of reserves under the different approaches with the
new probabilities of 95% and 99%.
No reserve is required for the 1991 policy under the standard basis with the 95%
102
Table 6.16: The reserves for the 1991 policy in Case C under the standard basiswith the probability boundaries of 95% and 99% to adjust the EBRs
Approach Option/Hedging CTE (95%) CTE (99%)31/Dec/ (97.5%, 99.5%) (95%, 99%) (97.5%, 99.5%) (95%, 99%) (97.5%, 99.5%) (95%, 99%)1991 4.88 4.88 0.00 0.00 5.35 5.351992 3.38 4.76 0.00 0.00 5.90 11.341993 2.51 3.96 0.00 0.00 3.36 10.411994 2.06 3.10 0.00 0.00 1.75 7.131995 1.74 2.60 0.00 0.00 0.00 3.631996 1.30 2.08 0.00 0.00 1.81 8.121997 0.56 1.01 0.00 0.00 0.14 7.091998 0.20 0.41 0.00 0.00 0.00 7.611999 0.02 0.04 0.00 0.00 0.00 7.192000 0.01 0.02 0.00 0.97 2.88 10.80
CTE measure if the EBRs are adjusted according to the probability boundaries of
97.5% and 99.5%. Changing the probabilities results in a small reserve required at
the final valuation date.
Clearly, Table 6.16 shows that changing the probability boundaries increases
the reserves. The guarantees are slightly reduced by the new probabilities, but
the equity proportions are higher after the outset of the policy. The remaining
guarantees which are not covered by the zero-coupon bonds are larger. Hence, more
reserves are required with the more risky investment strategy.
Table 6.17 compares the profitability of the 1991 policy under the standard basis
with the different probabilities.
Table 6.17: The accumulated values of the cashflows for the 1991 policy in Case Cunder the standard basis with the different probability boundaries
Approach Option Hedging CTE (95%) CTE (99%)(97.5%, 99.5%) -18.88 -23.99 -27.52 -29.52(95%, 99%) -16.20 -22.90 -27.86 -29.89
Table 6.17 shows that changing the boundaries increases the profitability of the
1991 policy to the insurer who sets up CTE reserves. The increase in the guarantee
charges due to the larger asset shares with the new boundaries has a larger accu-
mulated value at policy termination than the increase in the CTE reserves. Using
the option pricing approach, however, the increase in the charges is not sufficient to
cover that in the reserves with the more risky investment strategy, so the policy is
103
policy duration
equi
ty in
dex
0 2 4 6 8 10
010
0020
0030
0040
00
1 3 5 7 9
199119641982
Figure 6.30: The comparison of the equity indices at each policy duration over thedifferent 10-year periods
less profitable.
6.4 Sensitivity to Different 10-Year Periods
All the conclusions we have drawn so far are on the particular 10-year policy issued
at the end of 1991. This section will consider two policies, issued at the end of
1964 and 1982, respectively. We start by comparing the investment performance
during the different 10-year periods, and then investigate the impact of the different
performance on the payouts to the policyholder and profitability to the insurer.
6.4.1 Investment Performance of the Two Asset Classes
Two asset classes are considered in our dynamic investment strategy: equities and
the zero-coupon bonds with the same maturity date as the policy. Figure 6.30
compares the equity indices at each policy duration over the different 10-year periods.
A comparison of the zero-coupon yields is shown in Figure 6.31.
We see in Figures 6.30 and 6.31 that the equity index for the 1964 policy does not
show an obvious trend but it falls dramatically in the final two policy years. For the
1982 policy, the equity index increases over the policy term except for a temporary
104
policy duration
yiel
d on
zer
o-co
upon
bon
ds (
%)
0 2 4 6 8
46
810
12
1 3 5 7 9
199119641982
Figure 6.31: The comparison of the yields on the zero-coupon bonds, each maturingat the end of the policy term, at each policy duration over the different 10-yearperiods
decrease at the 8th duration. The zero-coupon yield is mostly higher during the
10-year period starting at the end of 1982. The yield for the 1964 policy starts from
a lower level than that for the 1991 policy but afterwards it increases very rapidly
particularly in the final two years.
6.4.2 Asset Shares and Guarantees
Now we consider how the different investment performance, shown in Figures 6.30
and 6.31, affects the asset shares and guarantees.
Table 6.18 compares the EBRs of the policies issued at different times under the
standard basis.
Table 6.18: The EBRs of the policies in Case C issued at different times under thestandard basis with the probability boundaries of 97.5% and 99.5% in the investmentstrategy (%)
issued at policy duration31/Dec/ 0 1 2 3 4 5 6 7 8 91991 88 80 72 72 72 72 72 72 79 861964 80 80 72 72 72 65 65 71 78 701982 88 88 96 96 96 96 100 100 100 100
105
policy duration
asse
t sha
re
0 2 4 6 8 10
100
200
300
400
500
1 3 5 7 9
199119641982
Figure 6.32: The comparison of the asset shares of the policies in Case C issued atdifferent times under the standard basis with the boundaries of 97.5% and 99.5%
Table 6.19 compares the bonus rates declared on the three policies.
Table 6.19: The bonus rates declared on the policies in Case C under the standardbasis with the probabilities of 97.5% and 99.5% to adjust the EBRs (%)
issued at policy duration31/Dec/ 1 2 3 4 5 6 7 8 91991 4.46 3.72 3.49 4.18 4.72 4.78 3.98 3.70 3.081964 4.20 3.50 2.94 3.53 2.94 3.53 4.23 5.08 4.231982 6.90 8.28 9.05 10.63 10.49 12.58 15.10 16.84 20.21
Figure 6.32 compares the asset shares of the three policies. The guarantees during
the different 10-year periods are shown in Figure 6.33.
The 1964 policy has a lower exposure to equities at some durations than the 1991
policy. The equity index does not increase rapidly in the period 1964-1974. The
lower rate of return in equities suggests the lower equity backing ratios and lower
regular bonuses. The asset share for the 1964 policy is mostly the smallest among
the three policies, and it falls from £239.92 after 8 years to £128.52 at maturity.
The guarantee for this policy is also of the smallest amount.
The EBR for the 1982 policy starts from the highest possible value of 0.88. After 6
years, the whole asset share is in equities. We see in Appendix B that the investment
106
policy duration
guar
ante
e
0 2 4 6 8 10
150
200
250
300
350
1 3 5 7 9
199119641982
Figure 6.33: The comparison of the guarantees of the policies in Case C issued atdifferent times under the standard basis with the boundaries of 97.5% and 99.5%
market is in a high inflationary environment during this particular 10-year period.
The high force of inflation results in high projected values of the maturity asset
share. The insurer is in a strong prospective solvency position, hence the unit fund
is switched from zero-coupon bonds to equities. The large projected maturity asset
share increases the bonus earning power, so high bonuses are declared on the 1982
policy. After 9 years, a regular bonus rate of 20.21% is declared on top of the 2%
guaranteed rate. Our dynamic investment strategy works quite well for the 1982
policy as the equity market actually booms during this 10-year period. The asset
share increases rapidly most of the time except for a temporary decrease in the 8th
policy year. The guarantees also build up rapidly with the high bonus rates.
Table 6.20 compares the asset shares, guarantees, and terminal bonus rates at
maturity of the three policies.
We see in Table 6.20 that on the 1964 policy the guarantee bites at maturity
with a loss of £41.91 born by the insurer for the £100 premium. The equity market
crashed in 1973 and 1974, so the maturity asset share is not sufficient to pay the
guarantee. The policyholder receives the guaranteed payout of £170.43 with a zero
terminal bonus. The 1982 policy provides the highest investment value so that the
policyholder receives the asset share of £482.74.
107
Table 6.20: The asset shares, guarantees and terminal bonus rates at maturity forthe policies in Case C issued at different times under the standard basis with theprobability boundaries of 97.5% and 99.5%
1991 1964 1982maturity asset share (£) 249.17 128.52 482.74maturity guarantee (£) 173.66 170.43 342.30terminal bonus rate (%) 43.48 0.00 41.03
6.4.3 Reserves and Profitability
Table 6.21 compares the required amount of reserves for the different policies.
Table 6.21: The reserves for the three policies in Case C issued at different timesunder the standard basis with the probabilities of 97.5% and 99.5%
Approach Option/Hedging CTE(95%) CTE(99%)Duration 1991 1964 1982 1991 1964 1982 1991 1964 19820 4.88 10.29 5.12 0.00 0.00 0.00 5.35 9.88 1.831 3.38 10.29 4.66 0.00 0.00 0.00 5.90 9.52 3.592 2.51 10.63 5.41 0.00 0.00 0.00 3.36 9.71 2.623 2.06 4.40 5.60 0.00 0.00 0.00 1.75 5.55 9.214 1.74 0.66 4.59 0.00 0.00 0.00 0.00 0.00 10.035 1.30 0.66 6.41 0.00 0.00 0.00 1.81 0.62 19.856 0.56 1.19 9.14 0.00 0.00 0.00 0.14 0.00 14.517 0.20 0.24 3.23 0.00 0.00 0.00 0.00 0.00 4.318 0.02 0.07 8.66 0.00 0.00 3.34 0.00 0.00 28.889 0.01 0.52 3.21 0.00 2.12 0.94 2.88 10.36 17.70
We see in Table 6.21 that under the standard basis the 1964 policy requires much
larger reserves than the other two policies at early durations, but at later durations
more reserves are required for the 1982 policy. The zero-coupon yields at early
durations of the 1964 policy are at a low level compared to the other two policies, so
the zero-coupon bonds are expensive in those years. The guarantees which should be
covered by the options are of a larger amount and hence more reserves are required.
The guarantees for the 1982 policy build up rapidly with the high regular bonuses.
The EBR is adjusted upwards and the whole asset share is in equities after 7 years.
Thus, large reserves are required to meet the guarantees which are not covered by
the zero-coupon bonds. Conversely, the 1991 policy has smaller guarantees and its
asset share has a larger component of zero-coupon bonds. Hence, a larger part of
the guarantee has been covered by a risk-free asset with certainty and so a smaller
reserve is required.
108
Table 6.22 compares the profitability of the three policies under the different
reserving approaches.
Table 6.22: The accumulated values of the cashflows incurred for the three policiesin Case C under the standard basis with the boundaries of 97.5% and 99.5%
Approach Option Hedging CTE (95%) CTE (99%)1991 -18.88 -23.99 -27.52 -29.521964 -4.35 36.02 19.29 20.381982 5.04 -4.97 -41.42 -41.74
Table 6.22 shows that the profitability of the policies depends on the investment
performance and the reserving approaches. Under the option method, the insurer
makes a profit on the 1964 and 1991 policies, but the 1982 policy incurs a loss; under
the hedging and CTE approaches, the insurer makes a big loss on the 1964 policy, but
the other two policies are both profitable. The guarantee bites on the 1964 policy.
The payoff under its guarantees is met by the third party only in the option method,
while using the other two approaches the insurer has to pay the cost of guarantees.
The advantages and disadvantages to the insurer between different approaches are
similar to the 1982 and 1991 policies. The investment market performs very well
during the 10-year term of the 1982 policy, so the insurer collects a large amount of
guarantee charges from the policyholder. The CTE reserves are released back to the
insurer at maturity as the guarantees are not called upon. By discrete hedging, it is
very expensive to match the increase in the guarantees after each bonus declaration.
Therefore, the policy is more profitable to the insurer who sets up CTE reserves.
However, hedging is still cheaper than buying options because the reserves are partly
released back to the insurer through negative hedging errors.
6.5 Summary
This chapter has investigated the sensitivity of the results for a single policy to dif-
ferent parameters, different probability boundaries used in our investment strategy
and different 10-year periods. The main conclusions from the sensitivity test are
summarised as follows:
109
• For the 1991 policy, reducing the guaranteed growth rate from 2% to 0% leads
to higher EBRs and higher regular bonus rates. With the lower guaranteed
rate, the asset shares are mostly larger; the guarantees are smaller initially but
then build up more rapidly with the higher bonuses; the reserves are smaller
at early durations but larger later on; the insurer earns a slightly smaller profit
using the option or hedging approach or setting up 99% CTE reserves, but the
policy is slightly more profitable to the insurer who sets up 95% CTE reserves.
• For the 1991 policy, reducing the percentage of the units deducted as a guar-
antee charge from 1% to 0.5% has no impact on asset allocation, but higher
bonuses can be declared due to the increased bonus earning power. With the
smaller charges, the asset shares and guarantees are both increased; smaller
reserves are required under the option pricing approach but CTE reserves are
increased; the insurer earns a much smaller profit.
• For the 1991 policy, increasing the terminal bonus target from 30% to 50%
leads to higher equity proportions and lower regular bonus rates. With the
higher target, the asset shares are mostly larger and guarantees build up less
rapidly; smaller reserves are required under all three approaches; a larger profit
can be earned by the insurer who sets up reserves using the option or hedging
approach or sets up 95% CTE reserves, but a higher target has little impact
on the profitability to the insurer who sets up 99% CTE reserves.
• Increasing the volatility for the equity index, assumed in the Black-Scholes
formula, from 20% to 25% increases the required amount of reserves under
the option pricing approach. The CTE reserves are not affected. The higher
volatility reduces the profit earned by the insurer using the option and hedging
approaches. The impact is more significant under the option approach.
• Increasing the rate of transaction costs, assumed in dynamic hedging, from
0.2% to 0.5% has no impact on the reserves, but slightly reduces the profit
earned by the insurer using the hedging approach.
• Increasing the additional rate of return, required by the capital providers on
top of the risk-free rate, from 3% to 5% reduces the insurer’s profit.
110
• For the 1991 policy, changing the upper and lower probability boundaries in
our investment strategy, from 99.5% and 97.5% to 99% and 95% respectively
leads to higher equity proportions and lower regular bonuses. With the new
boundaries, most of the time the asset shares are larger but the guarantees are
smaller; larger reserves are required under all three reserving approaches; the
insurer earns a smaller profit using the option or hedging approach or setting
up 99% CTE reserves but a larger profit can be earned by the insurer who sets
up 95% CTE reserves.
• The investment market performs quite differently during different 10-year pe-
riods.
• Amongst the three policies considered, the 1982 policy provides the highest
investment value to the policyholder. The asset share is mostly invested in eq-
uities and high bonus rates can be declared. The guarantees build up rapidly
with the high bonuses. The 1982 policy requires more reserves than the other
two policies at later durations. The guarantees on the 1964 policy bites at ma-
turity and the insurer needs to cover the loss of £41.91 for the £100 premium.
Larger reserves are required on the 1964 policy at early durations because the
zero-coupon bonds are expensive then.
• The profitability of the policies depends on which reserving approach is used
and the investment performance. The insurer earns a profit from the 1964 pol-
icy using the option method, but a loss is incurred under the other approaches.
For the 1982 policy, however, the policy is not profitable if the insurer buys
options.
111
Chapter 7
RESERVING FOR A
PORTFOLIO OF UWP
POLICIES HISTORICALLY
7.1 Introduction
In the previous chapters we carried out an extensive investigation for a single 10-year
UWP policy based on historical data. The required amount of reserves has been
calculated using three approaches. The profitability of the policy under the different
reserving approaches has been considered.
This chapter considers a portfolio of UWP policies, again historically, with a 20-
year investigation period which starts on 31 December 1982 and ends at 31 December
2002. During the investigation period, one single premium policy with a term of 10
years is issued each year. Therefore, by the end of the 20-year period the insurer
has issued 21 policies in total, among which 11 policies (issued at the end of 1982,
1983, ..., 1992 respectively) have matured and 10 policies (issued at the end of 1993,
1994, ..., 2002 respectively) are still in force. The policies in the portfolio are similar
to the single policy considered in the previous chapters. We use the standard basis,
i.e. Basis i given in Table 6.4, for the contract design and market parameters. The
other bases will be considered later in a sensitivity test. The single premium for
the policy was assumed to be £100 in the previous chapters. Here in the portfolio
112
case we allow for the effect of inflation. The single premium for the earliest cohort,
issued at the end of 1982, is assumed to be £100, and the premium for the later
cohorts increases with the retail price index. The retail price indices during the 20
years, Q(t), t = 0, 1, ..., 20, are given in Appendix B.
Each policyholder’s assets are allocated according to the EBRs calculated for his
own policy. Also, regular bonuses are declared on each policy based on its own
bonus earning power. We use the same dynamic strategies as described in Chapter
5 for the asset allocation and bonus declaration. The aim of applying the investment
and bonus strategies to each generation separately is to treat policyholders fairly so
that there is no subsidisation between different generations. As in Chapter 3, we
consider the following three cases,
A: without smoothing, and future regular bonuses are ignored,
B: with smoothing, and future regular bonuses ignored,
C: with smoothing, and the minimum of future regular bonuses implied by our
smoothing mechanism are included in the guarantees.
The EBRs and bonus rates for each policy in the three cases are given in Section
7.2.
Section 7.3 shows the asset shares and guarantees for each policy in the three
cases.
In Section 7.4, the three reserving approaches of buying options, dynamic hedging
and CTE reserving are applied to calculate the portfolio reserves. The advantages
to the insurer of pooling risks are investigated.
Section 7.5 considers profitability of the UWP policies with a 1% guarantee
charge, by rolling up the cashflows incurred by the insurer at a risk-free interest
rate.
Section 7.6 investigates the sensitivity of the portfolio results to different pa-
rameters and to different probability boundaries in our investment strategy. We
concentrate on the case of smoothing with allowance for future bonuses, i.e. Case
C.
Finally, a summary is given in Section 7.7.
113
7.2 Equity Proportions and Regular Bonuses
The mechanism described in Chapter 5 to allocate the asset share and declare regular
bonuses for a single policy can be used here in the portfolio case because we consider
each generation separately. In Chapter 5, we only considered Case C in which the
regular bonuses are smoothed from year to year and the EBRs were calculated
assuming that the guarantee includes the minimum of future bonuses implied by
our smoothing mechanism. In this chapter, Cases A and B are also considered. As
assumed before, the calculation of the bonus earning power is performed just after
the asset allocation.
7.2.1 Case A: Without Smoothing or Allowance for Future
Bonuses
We first consider Case A in which future regular bonuses are ignored and the bonus
rate equals the bonus earning power subject to a minimum of zero. Table 7.23 gives
the EBRs for each policy in the portfolio. The changes in the equity proportions are
still restricted to 10% upward or downward. Each row of the table shows the equity
proportions at different durations for the same generation. Each column shows the
ratios at the same duration for different generations. Thus, each diagonal from SW
to NE gives the ratios at different durations for different generations at the same
time t, t = 0, 1, ..., 20, when the portfolio contains min(t+1, T ) policies (excluding
the one just matured at time t, t = 10, 11, ..., 20).
Table 7.23 shows that for most cohorts issued during the first 10 years, the EBR
starts from a high level of 0.88 and then it is adjusted upwards. The EBRs of the two
earliest cohorts stay at the highest possible level during the policy term according
to our investment strategy. We see in Appendix B that the force of inflation is
relatively high during the first decade of the 20-year investigation period. The high
inflation leads to high projected returns in the investment market using the Wilkie
model, so the insurer is in a strong prospective solvency position. For those policies
issued more recently, the asset share is switched out of equities to zero-coupon
bonds because the prospective financial strength of the insurer is weakened when
114
Table 7.23: The EBRs of each policy in the portfolio in Case A (%)
Issued at Duration31/Dec/ 0 1 2 3 4 5 6 7 8 9
1982 88 96 100 100 100 100 100 100 100 1001983 88 96 100 100 100 100 100 100 100 1001984 88 88 96 96 100 100 100 100 100 1001985 80 88 88 96 100 100 100 100 100 1001986 80 80 88 96 100 100 100 100 100 1001987 80 88 96 100 100 100 90 90 99 1001988 88 96 96 100 100 90 90 99 99 1001989 80 80 88 88 80 80 88 88 96 1001990 88 96 96 96 96 96 96 100 100 1001991 88 88 88 88 88 88 88 88 96 961992 80 80 80 80 80 80 80 80 80 881993 72 65 65 65 65 65 65 59 59 591994 80 80 80 80 80 72 65 65 651995 72 72 72 65 65 59 53 531996 72 72 65 59 53 53 531997 72 65 59 53 48 431998 72 65 59 53 481999 72 65 59 532000 72 65 592001 72 652002 72
the economy moves into a low inflation period. Notice that the equity proportions
for the six latest cohorts have stayed at the lowest possible level by the end of 2002.
Table 7.24 shows the unsmoothed bonus rates declared on each policy.
Each row in Table 7.24 shows the bonus rates declared at different durations for
the same generation. The unsmoothed bonuses are quite volatile from year to year,
particularly at later durations when the full volatility is spread over fewer years.
For most cohorts issued during the first decade, high bonus rates are declared at
later durations. The bonus earning power is calculated cautiously based on the 25th
percentile of the projected maturity asset share. During the first decade, the actual
return is mostly higher than the 25th percentile of the projected return. As the
policy duration increases, the actual return has more impact on the projected value
of the maturity asset share. Hence the bonus rates on the early cohorts increase
with duration.
For the policies issued at the end of 1992, 1993, ..., 1996, the bonus rate starts
from a low initial value and reduces to zero afterwards. For the later cohorts (issued
after the end of 1997), no bonuses can be declared. The economy has been experi-
encing a low inflation period recently, so the investment market gives low returns.
The low dividend yield on shares suggests low equity price according to the Wilkie
115
Table 7.24: The regular bonus rates declared on each policy in the portfolio in CaseA (%)
Issued at Duration31/Dec/ 1 2 3 4 5 6 7 8 9
1982 6.96 9.04 8.88 10.89 10.46 15.37 19.65 14.34 17.281983 6.97 6.75 8.35 8.11 11.97 14.94 13.04 14.35 6.471984 5.04 6.00 5.75 8.98 11.35 10.36 11.26 5.79 13.281985 5.01 4.80 7.28 9.07 8.77 9.36 5.06 4.68 5.271986 3.71 5.65 6.99 6.89 7.28 4.07 2.35 2.91 21.671987 5.42 6.52 6.55 6.88 4.10 2.87 3.68 9.70 14.851988 6.24 6.35 6.47 4.02 2.96 3.51 6.86 7.98 21.031989 4.49 4.42 2.68 1.65 1.98 4.14 4.13 5.57 9.611990 6.39 4.68 3.47 3.79 6.13 5.93 6.59 5.55 24.541991 3.99 2.81 3.18 4.78 4.54 5.09 3.36 3.64 3.771992 2.18 2.40 3.62 3.46 3.64 1.92 1.86 0.00 0.001993 0.79 1.59 1.48 1.41 0.29 0.00 0.00 0.00 0.001994 3.29 3.07 3.07 1.78 1.74 0.61 0.00 0.001995 2.00 1.99 1.12 0.50 0.00 0.00 0.001996 1.48 0.69 0.52 0.00 0.00 0.001997 0.00 0.00 0.00 0.00 0.001998 0.00 0.00 0.00 0.001999 0.00 0.00 0.002000 0.00 0.002001 0.002002
model, so the projected return is poor and hence the bonus earning power for the
later cohorts is reduced.
In Table 7.24, each column gives the bonus rates at the same duration for different
generations. We see that lower bonus rates are mostly declared on more recently
issued policies. The low force of inflation in recent years reduces the bonus earning
power. The zeros at the bottom of each column imply that the insurer cannot afford
to declare a bonus to those recently issued policies.
Each diagonal from SW to NE in the table shows the bonus rates at different
durations for different generations at the same time. The bonus earning power is
calculated from the projections starting from the same initial conditions but with
different projection periods. Moving along the diagonal from SW to NE, the policy
duration increases. As mentioned before, the actual return earned in the unit fund,
which is mostly higher than the 25th percentile of the projected return, has more
impact on the projected value of the maturity asset share as duration increases.
Thus, the bonus rates on most of the diagonals show an increasing trend. Zero
bonus rates are declared on most cohorts after the end of 1999, which implies that
the market conditions have been so poor recently that the 25th percentile of the
116
projected return is not sufficient to declare any bonus.
The zero bonuses on the recent cohorts suggest that a major feature of with-profits
policies, namely regular bonuses, are no longer sustainable in a low inflationary
environment, unless we reduce the guaranteed rate (from the current level of 2%
p.a.), or set the initial guarantee to be less than the single premium, or maybe
change investments or set a lower terminal bonus target.
7.2.2 Case B: Smoothing without Allowance for Future
Bonuses
Tables 7.25 and 7.26 give the EBRs and regular bonus rates for each policy in Case
B where the policyholder’s assets are allocated and the smoothed bonus rates are
declared without allowance for future bonuses.
Table 7.25: The EBRs of each policy in the portfolio in Case B (%)
Issued at Duration31/Dec/ 0 1 2 3 4 5 6 7 8 9
1982 88 96 100 100 100 100 100 100 100 1001983 88 96 100 100 100 100 100 100 100 1001984 88 88 96 96 100 100 100 100 100 1001985 80 88 88 96 100 100 100 100 100 1001986 80 80 88 96 100 100 100 100 100 1001987 80 88 96 100 100 100 90 90 99 1001988 88 96 96 100 100 90 90 90 99 1001989 80 80 88 88 80 80 80 80 88 961990 88 96 96 96 96 96 96 100 100 1001991 88 88 88 88 88 88 88 88 88 961992 80 80 80 80 80 80 80 80 80 801993 72 65 65 65 65 65 65 59 59 591994 80 80 80 80 80 72 65 65 591995 72 72 72 65 65 59 53 481996 72 72 65 59 53 48 481997 72 65 59 53 48 431998 72 65 59 53 481999 72 65 59 532000 72 65 592001 72 652002 72
The EBRs given in Tables 7.25 and 7.23 show a similar trend along each row,
column and diagonal. Smoothing the regular bonuses has no impact on the EBRs
during the first 5 policy years. For some policies, smoothing has reduced the equity
proportion by 10% at later durations. The smoothing mechanism has stopped the
insurer cutting bonuses too significantly, so the guarantees build up more rapidly
117
Table 7.26: The regular bonus rates declared on each policy in the portfolio in CaseB (%)
Issued at Duration31/Dec/ 1 2 3 4 5 6 7 8 9
1982 6.96 8.35 8.98 10.77 10.61 12.73 15.27 18.16 21.201983 6.30 6.83 8.20 8.24 9.88 11.86 14.23 16.73 13.941984 5.04 6.00 5.75 6.90 8.28 9.94 11.93 9.94 11.931985 5.01 4.80 5.75 6.91 8.29 9.94 8.29 6.91 5.751986 3.71 4.46 5.35 6.42 7.70 6.42 5.35 4.46 5.351987 4.38 5.26 6.31 7.31 6.09 5.07 4.23 5.07 6.091988 6.24 6.35 6.47 5.39 4.49 3.75 4.49 5.39 6.471989 4.49 4.42 3.69 3.07 2.56 3.07 3.69 4.42 5.311990 6.39 5.32 4.44 3.70 4.44 5.32 6.39 6.03 7.241991 4.49 3.74 3.12 3.74 4.47 5.02 4.18 5.02 4.181992 2.66 2.34 2.81 3.37 3.73 3.11 2.59 2.16 1.801993 0.66 0.79 0.95 1.14 0.95 0.79 0.66 0.55 0.461994 2.79 3.14 3.13 2.61 2.17 1.81 1.51 1.261995 2.00 1.99 1.66 1.39 1.15 0.96 0.801996 1.48 1.23 1.03 0.86 0.71 0.591997 0.42 0.35 0.29 0.24 0.201998 0.42 0.35 0.29 0.241999 0.42 0.35 0.292000 0.42 0.352001 0.422002
with the higher bonus rates in Case B. Thus, the prospective financial strength of
the insurer is weakened by smoothing and more of the asset share is invested in
zero-coupon bonds.
Comparing Tables 7.26 and 7.24 clearly shows that the smoothed bonus rates are
less volatile from year to year than the unsmoothed rates. Our smoothing mechanism
has set a lower boundary of the initial bonus rate to be 0.5%. Thus, positive bonuses
have to be declared on policies issued after the end of 1997 though they cannot be
afforded according to the bonus strategy. For these policies, smoothing has no effect
on asset allocation because their EBRs have already been at the lowest possible level
according to the investment strategy.
7.2.3 Case C: Smoothing with Allowance for Future
Bonuses
Tables 7.27 and 7.28 give the EBRs and bonus rates for each policy in Case C where
the regular bonuses are smoothed and the current guarantees include the minimum
of future regular bonuses implied by the smoothing mechanism.
The first column in Tables 7.27 and 7.25 gives the same equity proportions.
118
Table 7.27: The EBRs of each policy in the portfolio in Case C (%)
Issued at Duration31/Dec/ 0 1 2 3 4 5 6 7 8 9
1982 88 88 96 96 96 96 100 100 100 1001983 88 88 88 88 88 96 100 100 100 901984 88 88 88 88 88 96 96 96 87 951985 80 80 80 88 96 96 96 87 79 861986 80 80 80 88 88 88 80 72 72 791987 80 80 88 88 88 80 72 72 79 861988 88 88 88 88 80 72 72 79 86 941989 80 80 80 72 65 65 65 71 78 851990 88 88 80 72 72 79 79 79 79 861991 88 80 72 72 72 72 72 72 79 861992 80 72 65 65 65 65 65 65 65 711993 72 65 65 65 65 65 65 59 59 591994 80 80 80 72 65 65 59 59 531995 72 65 65 59 53 48 48 481996 72 65 59 53 48 48 431997 72 65 59 53 48 431998 72 65 59 53 481999 72 65 59 532000 72 65 592001 72 652002 72
Allowing for future bonuses has no impact on the initial EBRs. Afterwards, the
EBRs are mostly lower in Case C because the insurer is in a weakened solvency
position with the larger guarantees. Again, allowing for future bonuses has no
effect on asset allocation for those policies issued after the end of 1997. The EBRs
cannot be reduced any more since they have already been at the lowest possible
level according to our investment strategy.
The difference in the bonus rates between Cases B and C is mainly caused by
the different projected values of the maturity asset share due to the different equity
proportions.
Summing up, smoothing the regular bonuses has reduced the EBRs at later
durations for some cohorts in the portfolio and increased the bonus rates declared
on the recently issued policies. Including the minimum future bonuses in the current
guarantees has reduced the EBRs for most cohorts but has little impact on the
smoothed bonus rates.
119
Table 7.28: The regular bonus rates declared on each policy in the portfolio in CaseC (%)
Issued at Duration31/Dec/ 1 2 3 4 5 6 7 8 9
1982 7.01 8.41 8.78 10.54 10.31 12.37 14.84 17.57 20.601983 6.30 7.00 8.39 8.12 9.74 11.69 13.93 15.26 12.721984 5.04 6.04 6.00 7.20 8.64 10.36 11.95 9.96 9.291985 5.10 5.00 6.00 7.20 8.63 9.85 8.21 6.84 5.701986 3.71 4.46 5.35 6.42 7.61 6.34 5.29 4.41 5.291987 4.38 5.26 6.31 7.03 5.86 4.88 4.07 4.88 5.861988 6.27 6.45 6.55 5.46 4.55 3.79 4.55 5.46 6.551989 4.49 4.41 3.67 3.06 2.55 3.06 3.52 4.23 5.071990 6.43 5.36 4.46 4.30 5.16 5.74 6.57 5.72 6.871991 4.49 3.74 3.41 4.09 4.58 5.01 4.17 3.59 2.991992 2.66 2.66 3.19 3.54 3.65 3.04 2.53 2.11 1.761993 0.66 0.79 0.95 1.14 0.95 0.79 0.66 0.55 0.461994 2.79 3.14 3.42 2.85 2.38 1.98 1.65 1.381995 2.14 2.14 1.78 1.48 1.24 1.03 0.861996 1.63 1.36 1.13 0.95 0.79 0.661997 0.42 0.35 0.29 0.24 0.201998 0.42 0.35 0.29 0.241999 0.42 0.35 0.292000 0.42 0.352001 0.422002
7.3 Asset Shares and Guarantees in Cases A, B
and C
This section compares the asset shares and guarantees in the three cases.
7.3.1 Asset Shares
Figure 7.34 compares the asset shares of the whole portfolio in the three cases.
In Figure 7.34, we see that the portfolio asset shares show a similar pattern in
the three cases. The asset share builds up more rapidly during the first 10 years
when the number of policies in the portfolio increases each year. During the second
decade, one policy is issued and one policy has matured and gone off the books each
year. The portfolio asset share decreases more often during the second decade due
to negative returns earned in the unit fund. We will come back to this point later
when we look at the asset share for each individual policy. Another reason for the
increasing trend shown in the portfolio asset share is that the single premium of
each policy increases with the retail price index.
Smoothing regular bonuses has little impact on the portfolio asset share. A
120
year
asse
t sha
re
1985 1990 1995 2000
010
0020
0030
00
1982 1987 1992 1997 2002
Case ACase BCase C
Figure 7.34: The comparison of the total asset shares of the portfolio in Cases A, Band C
comparison of Tables 7.23 and 7.25 shows that smoothing has reduced the equity
proportion by 10% for some policies since the end of 1995 (except the end of 2000).
Thus, smoothing has no impact on the portfolio asset share until the end of 1996.
Whether the asset share is increased or reduced by smoothing depends on the relative
performance of equities and zero-coupon bonds.
We have seen in the previous section that as to the asset allocation, more policies
are affected by allowing for future bonuses than by smoothing, so Figure 7.34 shows
that the difference in the portfolio asset share between Cases B and C is more
obvious than that between Cases A and B. The actual rate of returns on equities
is mostly higher than that on zero-coupon bonds, so the lower equity proportions
in Case C result in smaller portfolio asset shares most of the time. Only in recent
years have the asset shares been slightly larger with the smaller equity component
in Case C.
The asset shares for each individual policy in the three cases are given in Tables
7.29, 7.30 and 7.31 respectively. As before, each row of the tables shows the individ-
ual asset shares at different policy durations for the same generation; each column
shows the asset shares at the same duration for different generations; and each di-
agonal from SW to NE shows the asset shares at different durations for different
121
generations at the same time.
Table 7.29: The asset shares of each policy in the portfolio in Case A
Issued at Duration31/Dec/ 0 1 2 3 4 5 6 7 8 9
1982 100.00 125.92 162.87 193.73 244.10 262.58 289.85 388.92 347.93 416.231983 105.31 134.25 159.21 200.60 215.79 238.20 319.61 285.93 342.06 405.741984 110.13 129.85 160.56 173.18 191.08 256.39 229.37 274.39 325.48 410.881985 116.41 142.05 154.17 170.09 226.14 202.31 242.03 287.09 362.42 337.541986 120.74 132.00 145.76 190.15 171.17 204.77 242.90 306.63 285.58 347.821987 125.20 138.39 180.48 162.40 194.28 230.45 290.92 273.11 328.99 377.061988 133.68 174.26 156.75 187.18 222.03 280.28 262.55 316.54 362.80 443.381989 143.98 132.12 156.78 185.23 233.07 218.61 261.29 297.37 358.65 404.371990 157.44 187.55 222.23 280.48 261.37 317.24 362.91 442.22 499.69 612.811991 164.46 194.59 246.13 229.05 276.34 314.46 380.63 428.85 516.06 484.431992 168.71 214.63 198.83 238.70 270.17 325.62 367.50 436.20 416.68 375.081993 171.98 158.13 188.24 210.89 252.57 286.69 331.30 321.94 301.72 266.231994 176.95 213.02 241.07 292.25 332.35 393.84 380.20 351.26 305.071995 182.64 205.57 249.25 285.69 329.27 320.86 298.73 270.131996 187.13 227.77 262.46 302.13 296.96 278.38 252.461997 193.92 224.64 258.30 254.28 237.54 219.491998 199.25 231.95 226.93 209.35 190.911999 202.76 196.79 179.45 160.832000 208.70 187.98 165.412001 210.16 180.702002 216.34
We see in Tables 7.29, 7.30 and 7.31 that for the early cohorts the asset share
builds up rapidly as the policy duration increases. The investment market provides
a high return during a relatively high inflation period. For the policies issued at the
end of 1999, 2000 and 2001, we notice that a negative return has been earned in
each policy year by the end of the investigation period.
Comparing the diagonals from SW to NE in the tables can explain the pattern of
the portfolio asset share shown in Figure 7.34. Each diagonal from SW to NE gives
the individual asset shares of all those policies in the current portfolio. The sum
of the individual asset shares given on one particular diagonal equals the current
portfolio asset share. We see in Tables 7.29, 7.30 and 7.31 that the individual
asset share mostly increases with the policy duration, i.e. the asset shares given
on a particular diagonal are mostly larger than those given on the previous (left)
diagonal. Hence the portfolio asset share shown in Figure 7.34 mostly increases
through time. However, the investment market gives a poor return in 1990, 1994,
2000, 2001 and 2002, and the portfolio asset share decreases in these years.
Comparing the corresponding rows in the three tables shows that the lower equity
proportions reduce the asset shares for the early cohorts because equities perform
122
Table 7.30: The asset shares of each policy in the portfolio in Case B
Issued at Duration31/Dec/ 0 1 2 3 4 5 6 7 8 9
1982 100.00 125.92 162.87 193.73 244.10 262.58 289.85 388.92 347.93 416.231983 105.31 134.25 159.21 200.60 215.79 238.20 319.61 285.93 342.06 405.741984 110.13 129.85 160.56 173.18 191.08 256.39 229.37 274.39 325.48 410.881985 116.41 142.05 154.17 170.09 226.14 202.31 242.03 287.09 362.42 337.541986 120.74 132.00 145.76 190.15 171.17 204.77 242.90 306.63 285.58 347.821987 125.20 138.39 180.48 162.40 194.28 230.45 290.92 273.11 328.99 377.061988 133.68 174.26 156.75 187.18 222.03 280.28 262.55 316.54 360.68 440.791989 143.98 132.12 156.78 185.23 233.07 218.61 261.29 295.83 353.31 396.591990 157.44 187.55 222.23 280.48 261.37 317.24 362.91 442.22 499.69 612.811991 164.46 194.59 246.13 229.05 276.34 314.46 380.63 428.85 516.06 488.671992 168.71 214.63 198.83 238.70 270.17 325.62 367.50 436.20 416.68 375.081993 171.98 158.13 188.24 210.89 252.57 286.69 331.30 321.94 301.72 266.231994 176.95 213.02 241.07 292.25 332.35 393.84 380.20 351.26 305.071995 182.64 205.57 249.25 285.69 329.27 320.86 298.73 270.131996 187.13 227.77 262.46 302.13 296.96 278.38 256.441997 193.92 224.64 258.30 254.28 237.54 219.491998 199.25 231.95 226.93 209.35 190.911999 202.76 196.79 179.45 160.832000 208.70 187.98 165.412001 210.16 180.702002 216.34
better than zero-coupon bonds in those years. Conversely, for those policies issued
recently, the reduction in the equity proportions increases the asset shares. Allowing
for future bonuses has greater impact on the EBRs and hence the asset shares than
smoothing bonuses.
7.3.2 Guarantees
Figure 7.35 compares the portfolio guarantees in the three cases.
The portfolio guarantee builds up very rapidly during the first 10 years mainly
for three reasons: the number of policies in the portfolio increases each year; the
initial guarantee for the newly issued policy increases with the relatively high rate
of inflation; the guarantee for each cohort builds up rapidly with the high bonuses.
During the second decade, however, one policy matures and goes off the books each
year; the economy moves into a low inflation period; low regular bonus rates are
declared on each policy. Thus, the portfolio guarantees in Cases A and B build
up less rapidly during the second decade. The guarantees in Case C do not show
an obvious increasing trend in this period because the minimum bonus rates are
actually declared on most of the policies. Notice that in the three cases, the portfolio
guarantees are of a larger amount than the portfolio asset shares at the end of the
123
Table 7.31: The asset shares of each policy in the portfolio in Case C
Issued at Duration31/Dec/ 0 1 2 3 4 5 6 7 8 9
1982 100.00 125.92 160.55 190.38 238.42 257.04 283.40 380.26 340.19 406.971983 105.31 134.25 158.25 195.73 212.09 233.55 310.57 277.83 332.38 394.261984 110.13 129.85 160.56 174.11 191.93 250.54 225.71 269.39 318.85 395.531985 116.41 142.05 155.08 171.04 223.20 201.00 239.91 284.02 353.32 337.211986 120.74 132.00 145.76 186.54 169.98 202.05 238.35 295.42 282.90 333.111987 125.20 138.39 177.00 161.12 191.60 226.12 281.46 267.99 316.39 357.681988 133.68 174.26 158.46 188.53 222.61 278.29 263.36 311.69 352.60 423.521989 143.98 132.12 156.78 184.74 231.43 218.21 257.11 288.26 340.45 380.031990 157.44 187.55 221.74 279.61 261.43 310.77 351.60 420.70 471.26 559.511991 164.46 194.59 246.45 229.10 272.91 307.29 366.49 411.31 482.34 461.191992 168.71 214.63 198.41 235.61 264.00 314.52 354.65 410.47 398.44 369.691993 171.98 158.13 188.24 210.89 252.57 286.69 331.30 321.94 301.72 266.231994 176.95 213.02 241.07 292.25 333.19 384.51 374.15 349.56 309.351995 182.64 205.57 248.68 286.04 326.05 322.18 305.82 280.821996 187.13 227.77 263.73 300.19 297.18 280.92 258.791997 193.92 224.64 258.30 254.28 237.54 219.491998 199.25 231.95 226.93 209.35 190.911999 202.76 196.79 179.45 160.832000 208.70 187.98 165.412001 210.16 180.702002 216.34
investigation period.
Figure 7.35 clearly shows that the portfolio guarantees in Case B are less volatile
from year to year than those in Case A. In recent years the minimum bonus rates
constrained by the smoothing mechanism have to be declared though they are not
affordable according to the bonus mechanism, so the portfolio guarantees have been
increased by smoothing over the last few years.
Allowing for future bonuses has greatly increased the portfolio guarantees. In-
terestingly, Figure 7.35 shows that the difference in the guarantees between Cases
B and C is most obvious at the end of 1991. We will explain this point later when
we look at the guarantees for each individual policy.
The individual guarantees in Cases A, B and C are given in Tables 7.32, 7.33
and 7.34 respectively. Each row of the tables shows the individual guarantees at
different durations for the same generation; each column gives the guarantees at the
same duration for different generations; and each diagonal from SW to NE gives the
guarantees at different durations for different generations at the same time. The
sum of the individual guarantees given on one particular diagonal equals the current
portfolio guarantee shown in Figure 7.35.
A comparison of the corresponding rows in Tables 7.32 and 7.33 shows that the
124
year
guar
ante
e
1985 1990 1995 2000
500
1000
1500
2000
2500
1982 1987 1992 1997 2002
Case ACase BCase C
Figure 7.35: The comparison of the total guarantees of the portfolio in Cases A, Band C
guarantee for each generation starts from the same initial amount in Cases A and B
but builds up more smoothly in Case B. The insurer cannot afford to declare a bonus
at some durations on the policies issued during the second decade, so the individual
guarantee does not increase after the bonus declaration in Case A. However, in the
smoothing case the minimum bonus rate implied by the smoothing mechanism has
to be declared so that the individual guarantee builds up gradually.
Comparing Tables 7.33 and 7.34 shows that the difference in the individual guar-
antees between Cases B and C is more obvious at early durations because the guar-
antees in Case C include more future bonuses. Notice that the individual guarantees
for recently issued policies have stayed at a constant level in Case C by the end of
the investigation period because the minimum bonuses included in the guarantees
are actually declared.
A comparison of the corresponding diagonals from SW to NE in Tables 7.33
and 7.34 explains the difference in the portfolio guarantees between Cases B and C
shown in Figure 7.35. The number of policies in the portfolio increases during the
first 10 years. The bonus rates for the early cohorts are mostly at a high level, so at
early durations the individual guarantees are much larger in Case C. Therefore, the
difference in the portfolio guarantees between the two cases becomes greater during
125
Table 7.32: The guarantees of each policy in the portfolio in Case A
Issued at Duration31/Dec/ 0 1 2 3 4 5 6 7 8 9
1982 121.90 130.39 142.17 154.79 171.65 189.61 218.76 261.76 299.29 351.011983 128.37 137.32 146.59 158.82 171.71 192.27 220.99 249.80 285.65 304.151984 134.25 141.01 149.47 158.07 172.27 191.83 211.71 235.54 249.17 282.261985 141.90 149.02 156.17 167.53 182.72 198.75 217.36 228.37 239.05 251.651986 147.18 152.64 161.27 172.54 184.44 197.87 205.92 210.75 216.88 263.881987 152.61 160.89 171.38 182.61 195.16 203.16 208.98 216.68 237.70 273.001988 162.96 173.13 184.11 196.03 203.91 209.95 217.33 232.23 250.77 303.511989 175.51 183.40 191.51 196.64 199.89 203.84 212.27 221.04 233.36 255.801990 191.91 204.17 213.72 221.14 229.53 243.60 258.04 275.05 290.30 361.541991 200.48 208.49 214.34 221.15 231.72 242.24 254.57 263.12 272.68 282.961992 205.65 210.14 215.19 222.98 230.70 239.09 243.68 248.22 248.22 248.221993 209.64 211.29 214.66 217.85 220.92 221.57 221.57 221.57 221.57 221.571994 215.70 222.79 229.64 236.69 240.90 245.08 246.57 246.57 246.571995 222.64 227.10 231.63 234.22 235.40 235.40 235.40 235.401996 228.11 231.48 233.09 234.30 234.30 234.30 234.301997 236.38 236.38 236.38 236.38 236.38 236.381998 242.88 242.88 242.88 242.88 242.881999 247.17 247.17 247.17 247.172000 254.41 254.41 254.412001 256.18 256.182002 263.71
the first decade. The portfolio contains 10 policies after it has been built up. In Case
B where no future bonuses are allowed for, the guarantees for each policy increase
monotonically over the policy term. However, in Case C it is possible that the
individual guarantees remain at the initial level over the policy term if the declared
bonus rates are always constrained by the lower limit. Table 7.34 shows that during
the second decade, the individual guarantees for most policies do not increase after
declaring a bonus. Thus, the difference in the portfolio guarantees between the two
cases becomes less obvious through time.
7.3.3 Terminal Bonus
Among the 21 policies issued during the 20 years, 11 policies (issued at the end
of 1982, 1983, ..., 1992 respectively) have matured. At maturity, the policyholder
receives his own asset share, subject to the guaranteed payout. In other words, there
is no smoothing of the maturity payout between different generations.
The asset shares, guarantees and terminal bonus rates (denoted as TBR) at
maturity in the three cases are given in Table 7.35.
We can infer from the positive terminal bonuses that the maturity guarantee
does not bite on any of the policies in the three cases. The terminal bonus rates on
126
Table 7.33: The guarantees of each policy in the portfolio in Case B
Issued at Duration31/Dec/ 0 1 2 3 4 5 6 7 8 9
1982 121.90 130.39 141.28 153.96 170.54 188.63 212.64 245.11 289.62 351.011983 128.37 136.46 145.78 157.74 170.73 187.61 209.86 239.73 279.84 318.861984 134.25 141.01 149.47 158.07 168.98 182.98 201.17 225.17 247.55 277.081985 141.90 149.02 156.17 165.15 176.56 191.19 210.20 227.62 243.34 257.341986 147.18 152.64 159.45 167.97 178.75 192.52 204.87 215.83 225.45 237.511987 152.61 159.31 167.69 178.27 191.30 202.94 213.24 222.26 233.53 247.751988 162.96 173.13 184.11 196.03 206.60 215.89 223.97 234.04 246.67 262.631989 175.51 183.40 191.51 198.57 204.66 209.90 216.35 224.32 234.24 246.671990 191.91 204.17 215.04 224.58 232.89 243.22 256.17 272.53 288.97 309.891991 200.48 209.48 217.31 224.09 232.47 242.87 255.06 265.73 279.07 290.741992 205.65 211.12 216.07 222.14 229.64 238.21 245.62 251.99 257.43 262.071993 209.64 211.02 212.69 214.71 217.15 219.21 220.94 222.40 223.62 224.641994 215.70 221.72 228.68 235.84 241.99 247.26 251.74 255.54 258.761995 222.64 227.10 231.63 235.48 238.74 241.49 243.82 245.771996 228.11 231.48 234.33 236.74 238.76 240.46 241.891997 236.38 237.37 238.19 238.88 239.46 239.941998 242.88 243.89 244.74 245.45 246.041999 247.17 248.20 249.06 249.782000 254.41 255.47 256.352001 256.18 257.252002 263.71
most maturing policies are higher than the initial target of 30%, which implies that
smoothing in earlier years may have stopped the insurer declaring high bonuses and
that the actual rate of return earned in the final policy year may be much higher than
the projected rate. Notice that the rate declared on the 1989 policy is about triple
the target. For the 1992 policy, however, the rate is only around 20%. Table 7.24
has shown that the insurer cannot afford to declare any bonus at later durations for
the 1992 policy, but the minimum bonuses set by the smoothing mechanism have to
be declared. Also the investment market performs badly just before the 1992 policy
matures. Hence the terminal bonus target cannot be achieved.
Table 7.35 also shows that smoothing does not make much difference to the
holders of these maturing policies as they receive similar payouts at maturity in
Cases A and B. The difference between the two cases can be explained by the
relative performance of equities and zero-coupon bonds. We have seen in Section
7.2 that allowing for future bonuses reduces the equity backing ratios at most policy
durations. The equity market mostly earns a higher return than the bond market, so
the increase in safety with greater risk-free assets has a cost for most policyholders.
However, zero-coupon bonds perform much better than equities at later durations
for the 1992 policy so the policyholder receives a larger maturity payout with lower
127
Table 7.34: The guarantees of each policy in the portfolio in Case C
Issued at Duration31/Dec/ 0 1 2 3 4 5 6 7 8 9
1982 160.76 169.80 190.10 204.47 230.70 242.77 270.52 300.01 326.25 343.201983 158.16 173.05 186.98 207.72 216.79 239.71 265.66 292.47 310.41 310.411984 164.54 170.62 185.25 192.81 209.71 229.39 251.47 271.59 271.59 274.081985 168.09 180.88 187.02 201.89 219.59 240.19 259.45 259.45 259.45 259.451986 171.64 175.77 186.85 200.10 215.75 233.12 233.12 233.12 233.12 236.751987 176.58 188.11 202.13 219.07 234.38 234.38 234.38 234.38 240.64 244.781988 200.59 219.40 231.63 243.17 243.17 243.17 243.17 251.58 259.06 264.021989 208.49 217.44 224.02 224.02 224.02 224.02 230.49 236.06 241.53 245.151990 247.97 260.24 260.24 260.24 266.20 281.20 293.52 306.43 307.76 313.921991 248.32 248.32 248.32 251.37 263.90 274.95 284.93 284.93 285.50 285.501992 233.64 233.64 238.34 248.40 257.14 263.50 263.50 263.50 263.50 263.501993 214.32 216.41 218.82 221.57 224.64 224.64 224.64 224.64 224.64 224.641994 236.78 246.60 255.79 264.69 264.69 264.69 264.69 264.69 264.691995 242.85 246.70 250.74 250.74 250.74 250.74 250.74 250.741996 242.07 246.77 246.77 246.77 246.77 246.77 246.771997 241.19 241.19 241.19 241.19 241.19 241.191998 247.82 247.82 247.82 247.82 247.821999 252.19 252.19 252.19 252.192000 259.58 259.58 259.582001 261.39 261.392002 269.08
Table 7.35: The asset shares, guarantees and terminal bonus rates at maturity inCases A, B and C
issued at Case A Case B Case C31/Dec/ A(T ) G(T ) TBR (%) A(T ) G(T ) TBR (%) A(T ) G(T ) TBR (%)
1982 493.72 351.01 40.66 493.72 351.01 40.66 482.74 343.20 40.661983 512.21 304.15 68.41 512.21 318.86 60.64 489.98 310.41 57.851984 382.68 282.26 35.58 382.68 277.08 38.11 370.69 274.08 35.251985 411.10 251.65 63.36 411.10 257.34 59.75 403.28 259.45 55.441986 398.92 263.88 51.17 398.92 237.51 67.96 375.92 236.75 58.781987 461.42 273.00 69.02 461.42 247.75 86.25 429.31 244.78 75.391988 501.01 303.51 65.07 498.08 262.63 89.65 476.75 264.02 80.581989 495.90 255.80 93.87 483.48 246.67 96.00 455.73 245.15 85.901990 572.74 361.54 58.42 572.74 309.89 84.82 531.35 313.92 69.261991 421.70 282.96 49.03 425.38 290.74 46.31 409.80 285.50 43.541992 301.48 248.22 21.46 309.32 262.07 18.03 313.57 263.50 19.00
EBRs in Case C.
7.4 Reserves Using the Three Approaches
This section applies the three reserving approaches of buying options, discrete hedg-
ing and CTE reserving to calculate the portfolio reserves.
128
year
rese
rves
1985 1990 1995 2000
050
100
150
1982 1987 1992 1997 2002
Case ACase BCase C
Figure 7.36: The portfolio reserves using the option method in Cases A, B and C
7.4.1 Buying Options
Under this approach, the portfolio reserves are calculated by summing up the re-
serves for each individual policy in the current portfolio. Pooling risks gives no
advantage to the insurer using the option method. The individual reserves can be
calculated using the mechanism described in the previous chapters.
Figure 7.36 shows the portfolio reserves under the option approach in Cases A,
B and C.
We see in Figure 7.36 that the portfolio reserves show a similar pattern in the three
cases. The reserves increase over the first decade because the number of policies in
the portfolio increases each year and also because of the effect of inflation (recall the
assumption that the single premium for each cohort increases with the retail price
index). Over the second 10 years, the portfolio reserves first show a decreasing trend
because the guarantees for most policies are deeply out-of-the-money. In recent years
the portfolio reserves increase rapidly because the guarantees for the later cohorts
are in-the-money due to low investment returns.
The portfolio reserves are of a similar amount in Cases A and B, because smooth-
ing bonuses has little impact on the amount of the guarantees which are not covered
by the zero-coupon bonds. The effect of smoothing will be investigated later when
we look at each individual policy.
129
The portfolio reserves are greatly increased in the first decade with allowance
for future bonuses. The difference in the portfolio reserves between Cases B and C
becomes greater during the first 10 years. However, afterwards the reserves are of
a similar amount in the two cases. Again, the effect of allowing for future bonuses
will be investigated later when we look at each individual policy.
The individual reserves in the three cases are given in Tables 7.36, 7.37 and 7.38
respectively. For ease of comparison, we show the single premium for each cohort
in the first column of Table 7.36. The premiums are the same in the three cases.
As before, each row in the tables gives the individual reserves at different durations
for the same generation; each column (except the first column in Table 7.36) gives
the reserves at the same duration for different generations; and each diagonal from
SW to NE gives the reserves at different durations for different generations at the
same valuation date. The sum of the individual reserves on one particular diagonal
equals the current portfolio reserves shown in Figure 7.36.
Table 7.36: The reserves for each policy in the portfolio using the option method inCase A
Issued at Single Duration31/Dec/ Premium 0 1 2 3 4 5 6 7 8 9
1982 100.00 1.48 2.20 1.87 1.72 1.10 1.79 2.37 0.92 4.10 3.381983 105.31 1.98 2.08 2.42 1.57 2.41 3.02 1.22 4.14 4.09 1.051984 110.13 1.91 1.71 1.82 2.73 4.06 1.77 4.50 4.46 2.52 0.411985 116.41 1.24 1.36 2.17 4.13 2.36 4.99 5.03 3.66 1.04 0.871986 120.74 1.09 1.83 3.65 2.57 5.64 5.66 4.64 2.17 2.22 1.141987 125.20 1.78 3.65 2.67 5.48 5.69 5.10 1.88 2.18 1.67 0.701988 133.68 3.61 2.73 4.53 5.71 5.45 2.66 2.87 2.64 1.27 0.361989 143.98 1.76 3.08 4.82 4.68 2.83 2.71 2.63 1.39 0.58 0.121990 157.44 2.22 4.11 4.38 5.07 4.69 3.55 2.43 1.40 0.48 0.061991 164.46 3.15 3.48 4.51 3.87 3.05 2.15 1.13 0.59 0.10 0.021992 168.71 2.90 4.22 3.29 2.68 1.91 1.18 0.83 0.06 0.03 0.071993 171.98 6.49 3.05 2.43 1.61 1.03 0.85 0.06 0.01 0.03 0.221994 176.95 3.53 3.29 2.57 2.26 2.57 0.13 0.04 0.12 0.881995 182.64 3.40 2.62 2.49 1.74 0.26 0.13 0.12 1.151996 187.13 3.55 3.70 3.23 0.24 0.13 0.42 2.751997 193.92 7.94 8.71 1.90 1.75 2.22 6.521998 199.25 18.53 7.22 8.08 9.81 20.641999 202.76 16.36 18.92 22.48 38.972000 208.70 18.85 21.33 37.722001 210.16 15.79 29.362002 216.34 19.86
The pattern of the individual reserves as the policy duration increases depends
on the comparison of the guarantees and asset shares. We see in Table 7.36 that
larger reserves are required at later durations for the early cohorts (issued before
the end of 1986) because their guarantees build up rapidly with the high bonuses
130
Table 7.37: The reserves for each policy in the portfolio using the option method inCase B
Issued at Duration31/Dec/ 0 1 2 3 4 5 6 7 8 9
1982 1.48 2.20 1.82 1.67 1.06 1.74 1.98 0.54 3.15 3.381983 1.98 2.02 2.36 1.51 2.34 2.64 0.86 3.16 3.47 1.901984 1.91 1.71 1.82 2.73 3.71 1.35 3.38 3.33 2.38 0.311985 1.24 1.36 2.17 3.87 1.98 4.12 4.18 3.58 1.23 1.161986 1.09 1.83 3.45 2.24 4.92 4.96 4.51 2.56 3.05 0.271987 1.78 3.48 2.41 4.97 5.21 5.08 2.19 2.67 1.42 0.171988 3.61 2.73 4.53 5.71 5.79 3.19 3.53 1.58 1.14 0.041989 1.76 3.08 4.82 4.92 3.34 3.36 1.80 0.86 0.32 0.051990 2.22 4.11 4.51 5.46 5.04 3.51 2.32 1.30 0.46 0.001991 3.15 3.56 4.84 4.16 3.11 2.19 1.15 0.66 0.03 0.031992 2.90 4.33 3.37 2.61 1.84 1.14 0.90 0.08 0.06 0.071993 6.49 3.02 2.24 1.39 0.83 0.74 0.05 0.01 0.04 0.331994 3.53 3.20 2.50 2.20 2.66 0.15 0.06 0.24 1.141995 3.40 2.62 2.49 1.84 0.33 0.22 0.28 1.471996 3.55 3.70 3.38 0.29 0.20 0.27 2.331997 7.94 8.94 2.07 2.01 2.64 7.641998 18.53 7.42 8.53 10.58 22.241999 16.36 19.28 23.28 40.532000 18.85 21.72 38.772001 15.79 29.832002 19.86
and their asset shares are mostly invested in equities. The reserves set up for the
later cohorts (issued after the end of 1999) also show an increasing trend as the
duration increases due to the poor investment performance in recent years. For the
cohorts issued from the end of 1989 to the end of 1993, smaller reserves are required
at later durations because their guarantees are deeply out-of-the-money with a high
investment return.
Table 7.36 also shows that at early durations, larger reserves are required for more
recently issued policies due to the effect of inflation. However, the cohorts issued
after the end of 1998 require much larger reserves than the other cohorts. Comparing
the 1982 and 2002 cohorts, we see that the single premium has increased from £100
to £216.34 but the initial reserves for the 2002 cohort are over 12 times larger. The
zero-coupon yield at the inception of the 1982 cohort is over twice that of the 2002
cohort, so the zero-coupon bonds are much cheaper at the end of 1982. Hence, a
larger part of the guarantees for the 1982 cohort can be covered by the risk-free asset,
although it has a larger equity exposure. At later durations, the policies issued in
the first decade require larger reserves than those in the second decade because their
guarantees build up more rapidly with the higher bonus rates.
Comparing the corresponding diagonals from SW to NE in Tables 7.36 and 7.37
131
Table 7.38: The reserves for each policy in the portfolio using the option method inCase C
Issued at Duration31/Dec/ 0 1 2 3 4 5 6 7 8 9
1982 5.12 4.76 5.56 5.39 4.51 6.18 8.74 3.01 8.95 3.301983 4.97 3.88 4.04 3.27 4.70 8.49 4.51 12.11 9.03 1.031984 4.76 4.34 3.29 4.82 6.35 4.28 10.42 10.16 3.31 0.281985 3.26 2.37 3.66 6.98 5.03 10.90 11.53 5.36 0.56 0.491986 2.72 4.07 5.40 3.84 7.94 9.02 4.69 0.92 0.80 0.041987 3.96 5.44 4.13 8.17 9.22 5.31 1.70 1.60 0.61 0.061988 8.51 5.24 8.64 9.30 5.70 2.47 2.23 1.57 0.87 0.031989 4.58 7.20 7.52 4.72 2.62 2.24 1.12 0.89 0.24 0.021990 7.14 8.14 5.22 3.13 2.90 3.69 2.49 0.90 0.16 0.001991 8.06 5.60 4.17 3.31 2.77 2.02 0.97 0.35 0.02 0.011992 5.71 5.03 2.45 2.15 1.60 0.89 0.50 0.01 0.01 0.021993 7.30 3.64 2.85 1.91 1.25 1.02 0.08 0.02 0.05 0.331994 5.77 5.86 5.07 2.79 1.46 0.15 0.05 0.18 0.721995 5.78 2.90 2.75 1.94 0.05 0.02 0.10 1.071996 5.13 3.62 3.20 0.18 0.11 0.40 1.731997 8.82 9.84 2.36 2.26 2.90 8.061998 20.13 8.22 9.30 11.32 23.161999 17.83 20.72 24.62 41.992000 20.49 23.25 40.522001 17.25 31.672002 21.59
explains the similarity in the portfolio reserves in Cases A and B. Smoothing has
little impact on the allocation of the asset share, but the individual guarantees
are less volatile from year to year in the smoothing case. Generally, smoothing
has stopped the insurer declaring high bonuses on the early cohorts, and has also
stopped the insurer cutting off bonuses on the later cohorts. Therefore, we see in
Figure 7.36 that the portfolio reserves are reduced in early years and increased in
later years by smoothing.
A comparison of the corresponding diagonals from SW to NE in Tables 7.37 and
7.38 clearly shows that during the first decade, much larger reserves are required
for each policy in Case C where future bonuses are allowed for. In this period,
the bonus rates declared on each policy are at a high level in a relatively high
inflationary environment. Although more of the asset share is in zero-coupon bonds,
the guarantees are never deeply out-of-the-money with large future bonuses included.
Therefore, the reserves are greatly increased in Case C. During the second decade,
allowing for future bonuses only increases the individual reserves for those policies at
early durations. The reserves for the policies at later durations are even smaller with
allowance for future bonuses. The bonus earning power is reduced by the relatively
low force of inflation in this period, so the guarantees for each cohort build up less
132
rapidly with the lower bonuses. Allowing for future bonuses has mostly reduced the
EBRs. At later durations, the part of the guarantees which are not covered by the
zero-coupon bonds and hence should be met by the options are smaller. Therefore,
we see in Figure 7.36 that much larger reserves are required for the portfolio in
Case C during the first decade but afterwards the portfolio reserves are of a similar
amount in the two cases.
7.4.2 Discrete Hedging
As under the option approach, the portfolio reserves are of the same amount as the
sum of the individual reserves. Thus, in respect of reserving, there is no advantage
of pooling risks to the insurer using the hedging approach. However, we will see later
in Section 7.5.2 that less transaction costs will be incurred if the insurer readjusts
hedge portfolios between different generations rather than between each generation
and a third party separately.
The reserves for each individual policy are the same under the option and hedging
approaches, so the portfolio reserves are also the same. The results are not repeated
here.
7.4.3 CTE Reserving
The portfolio CTE reserves are calculated with a probability of loss measured until
the latest cohort in the current portfolio matures. Thus, under the CTE approach
the reserves are set up by looking at the whole portfolio instead of looking at each
individual policy separately and then summing up the individual reserves as we did
under the other two approaches. We will compare the portfolio CTE reserves with
the sum of the individual CTE reserves to investigate the advantage to the insurer
of pooling risks.
The notation is similar to that used in the single policy case, but we need to
differentiate each policy in the portfolio by the date of issue. The extra notation is
defined as follows,
• A′(d, t, d + T, i): the projected value of the maturity asset share for the policy
133
issued at time d in the ith simulation, projected at time t < d + T (T is the
policy term)
• R′z(d, t, p, i): the projected rate of return on zero-coupon bonds, invested for
the policy issued at time d, during the projection year (p − 1, p) in the ith
simulation, projected at time t < p
• GC ′(d, t, p, i): the projected amount of the guarantee charges deducted at time
p from the policy issued at time d in the ith simulation, projected at time t < p
• PGC ′(t, p, i): the projected amount of the guarantee charges deducted at time
p from the whole portfolio in the ith simulation, projected at time t < p
• PV PGC ′(t, i): the present value at time t of the projected portfolio guarantee
charges deducted during the projection period in the ith simulation
• L′(t, p, i): the projected value of the loss incurred at time p in the ith simula-
tion, projected at time t < p
• PV L′(t, i): the present value at time t of the projected losses incurred during
the projection period, in the ith simulation
• PV PCF ′(t, i): the present value at time t of the projected portfolio cashflows
incurred by the insurer during the projection period in the ith simulation
before sorting
• PV PCF ′′(t, i): the present value at time t of the projected portfolio cashflows
incurred during the projection period in the ith simulation after sorting.
As in the single policy case, the methodology includes three steps. First, possible
future scenarios are generated by the Wilkie model. Then, in each generated scenario
the projected cashflows incurred by the insurer for the whole portfolio are valued.
Finally, the portfolio reserves are set up according to the CTE reserving principles.
We describe the detailed mechanism in the following equations.
At each valuation date t, t=0, 1, ..., 20, the performance of the equity and bond
markets is projected forward to the maturity of the latest cohort in the current
portfolio (i.e. t+T ) using the Wilkie model. The projection starts with the current
134
market indices. 10,000 simulations are performed. In the ith simulation, given the
projected equity price index and dividend amount, the projected rate of return on
equities during the projection year (p− 1, p), R′e(t, p, i), p = t + 1, t + 2, ..., t + T ,
can be calculated by equation 5.65.
Given the projected consols yield and short-term interest rate, the projected value
at time p, p = t + 1, t + 2, ..., t + T , of a zero-coupon bond with duration n, i.e.
v′(t, p, n, i), can be calculated by equations 5.67 and 5.68, with the projected par
yield modelled by equation 5.66. The asset share is partly invested in the zero-
coupon bond with the same maturity date as the policy. Hence the projected rate
of return on the zero-coupon bonds, invested for the policy issued at time d, during
the projection year (p− 1, p) follows the equation
R′z(d, t, p, i) =
v′(t, p, d + T − p, i)
v′(t, p− 1, d + T − (p− 1), i)− 1. (7.80)
The current portfolio at time t contains min(t + 1, T ) policies. For the policy
issued at time d, d = max(t− T + 1, 0), max(t− T + 1, 0) + 1, ..., t, the projected
value of the maturity asset share follows the equation
A′(d, t, d + T, i) =
[A(d, t) · e(d, t) ·
d+T∏p=t+1
(1 + R′e(t, p, i)) (7.81)
+ A(d, t) · (1− e(d, t)) · 1
v(t, d + T − t)
]· (1− c)d+T−t
in which A(d, t) is the asset share at time t for the policy issued at time d, and
e(d, t) is its equity backing ratio at time t. The individual EBRs and asset shares
have been calculated in Sections 7.2 and 7.3 respectively. v(t, d + T − t) is the value
of a zero-coupon bond at time t with duration d + T − t.
The projected amount of the guarantee charges deducted at future time p, p =
t + 1, t + 2, ..., d + T , from the policy issued at time d is given by the equation
GC ′(d, t, p, i) = A(d, t) ·[e(d, t) ·
p∏s=t+1
(1 + R′e(t, s, i)) (7.82)
+ (1− e(d, t)) ·p∏
s=t+1
(1 + R′z(d, t, s, i))
]· (1− c)p−t−1 · c.
To calculate the CTE reserves, we allow for all future guarantee charges deducted
during the projection period. The projected charges from the whole portfolio are
135
calculated by summing up the projected charges from all those policies in the current
portfolio. Hence,
PGC ′(t, p, i) =t∑
d=max(t−T+1, 0)
GC ′(d, t, p, i). (7.83)
The present value at the current valuation date t of the projected portfolio charges
deducted in the projection period follows the equation
PV PGC ′(t, i) =t+T∑
p=t+1
PGC ′(t, p, i)[1 + Z(t, p− t)]p−t
(7.84)
in which Z(t, p− t) is the yield on the zero-coupon bond for term p− t.
In the single policy case, a loss can only be incurred at the end of the projection
period because the loss comes from the maturity payout which is bigger than the
asset share. In the portfolio case, however, losses are measured until the latest cohort
matures. Earlier cohorts (if there are any) mature before the end of the projection
period and a loss might be incurred if the guarantee of any of the maturing policies
is called upon. We should reserve for all these losses incurred during the projection
period.
The earliest cohort in the current portfolio is issued at time max(t−T +1, 0). No
loss is possible to be incurred until the earliest cohort matures at time max(t+1, T ).
Thus, the loss incurred at time p has a projected value of
L′(t, p, i) = 0 (7.85)
when p = t + 1, t + 2, ..., max(t, T − 1), and
L′(t, p, i) = max(G(p− T, t)− A′(p− T, t, p, i), 0) (7.86)
when p = max(t + 1, T ), max(t + 1, T ) + 1, ..., t + T . G(p− T, t) is the guarantee
at time t for the policy issued at time p − T , which has been calculated in Section
7.3.
The present value at the valuation date t of the projected losses incurred from
the whole portfolio is given by the equation
PV L′(t, i) =t+T∑
p=t+1
L′(t, p, i)[1 + Z(t, p− t)]p−t
. (7.87)
136
The present value at time t of the projected portfolio cashflows incurred by the
insurer equals the present value of the projected losses less the present value of the
projected guarantee charges, i.e.
PV PCF ′(t, i) = PV L′(t, i)− PV PGC ′(t, i). (7.88)
The 10,000 simulated present values are then sorted from the smallest to the
largest so that
PV PCF ′′(t, 1) ≤ PV PCF ′′(t, 2) ≤ ... ≤ PV PCF ′′(t, i) ≤ ... ≤ PV PCF ′′(t, 10000).
For later cashflow calculations, we have to record the original position, ps(i), (i.e.
the simulation number before sorting) of these sorted present values. Hence we have
the following relation at time t,
PV PCF ′′(t, i) = PV PCF ′(t, ps(i)). (7.89)
The CTE reserve Tα can be worked out from the average of the largest
10, 000 · (1− α)
present values of the projected cashflows. The required amount of reserves for the
portfolio at time t follows the equation
V (t) =
∑10,000i=10,000α+1 PV PCF ′′(t,i)
10,000·(1−α)if
∑10,000i=10,000α+1 PV PCF ′′(t, i) > 0
0 otherwise.(7.90)
Figures 7.37 and 7.38 show the 95% and 99% portfolio CTE reserves in Cases A,
B and C.
Intuitively, larger reserves are required with a higher security level. The insurer
only needs a very small reserve for the whole portfolio under the CTE approach
except in the recent two years. In most simulations, the projected guarantee charges
are sufficient to cover the projected losses. In later years, however, the low actual
and projected investment returns lead to large reserves even though a large part of
the asset share has been switched into zero-coupon bonds.
Here we concentrate on Figure 7.38 to discuss the effects of smoothing and al-
lowing for future bonuses.
137
year
rese
rves
1985 1990 1995 2000
020
4060
8010
012
0
1982 1987 1992 1997 2002
Case ACase BCase C
Figure 7.37: The 95% portfolio CTE reserves in Cases A, B and C
year
rese
rves
1985 1990 1995 2000
050
100
150
200
1982 1987 1992 1997 2002
Case ACase BCase C
Figure 7.38: The 99% portfolio CTE reserves in Cases A, B and C
138
The effect of smoothing on the 99% portfolio CTE reserves is similar to that on
the portfolio guarantees shown in Figure 7.35. The reserves are slightly increased to
meet the larger guarantees. Although smoothing has reduced the equity proportions
for some policies so that a larger part of the guarantees can be covered by a risk-free
asset, the reduction in the risky asset has little impact on CTE reserves probably
because the insurer does not incur a large projected loss from these policies.
Figure 7.35 has shown that allowing for future bonuses increases the portfolio
guarantees during the whole investigation period. However, we see in Figure 7.38
that at the end of 1993, 1994 and 1998, the portfolio reserves are smaller with
allowance for future bonuses. The equity backing ratios for most policies are smaller
in Case C. The increase in the risk-free asset reduces the need to set up large reserves
in those years.
A comparison of the portfolio CTE and the sum of the individual CTE
in Case C:
Under the CTE approach, the portfolio reserves are calculated by looking at
the whole portfolio instead of looking at each individual policy separately and then
summing up the individual reserves. For the purpose of investigating the advantage
to the insurer of pooling risks, we compare the portfolio CTE reserves with the
sum of the individual CTE reserves in Figure 7.39. We concentrate on the case of
smoothing with allowance for future bonuses.
Figure 7.39 shows that the portfolio CTE reserve is obviously smaller than the
sum of the individual CTE reserves. The worst 10, 000 · (1 − α) simulations might
be different for different cohorts. To calculate the sum of the individual reserves, we
always choose the worst 10, 000 · (1−α) simulations for each policy in the portfolio.
For the portfolio CTE reserve, however, the worst simulations are chosen for the
whole portfolio. Some of these bad simulations might be less bad or even good
simulations for some policies. Hence there is an advantage of pooling risks to the
insurer using the CTE approach, but pooling risks gives no advantages under the
option or hedging approaches.
139
year
rese
rves
1985 1990 1995 2000
050
100
150
200
250
1982 1987 1992 1997 2002
95% portfoliosum of 95% individuals99% portfoliosum of 99% individuals
Figure 7.39: The comparison of the portfolio CTE reserves and the sum of theindividual CTE reserves in Case C
7.4.4 Comparison of the Portfolio Reserves Set up by Dif-
ferent Approaches in Case C
Figure 7.40 compares the portfolio reserves set up using the option pricing approach
with the portfolio CTE reserves in Case C.
Figure 7.40 shows that the smallest reserves are required with the 95% CTE mea-
sure. The 99% CTE reserves show a similar pattern to the reserves under the option
pricing approach, but the latter is greater before the end of 1999. For each policy in
the portfolio, reserves are always required under the option and hedging approaches;
whereas it is likely that no CTE reserve is required for the whole portfolio. However,
the three reserving approaches are not directly comparable for the reasons given in
Chapter 2.
7.5 Profitability of the UWP Policies
In this section we calculate the actual cashflows incurred by the insurer during the
20-year period. We assume that at the beginning of the 20 years the insurer has
no inherited estate. Given the convention that a positive cashflow represents the
money paid by the insurer, the insurer’s free estate is reduced by positive cashflows
140
year
rese
rves
1985 1990 1995 2000
050
100
150
200
1982 1987 1992 1997 2002
option/hedging95% CTE99% CTE
Figure 7.40: The comparison of the portfolio reserves set up using different reservingapproaches in Case C
and increased by negative cashflows. We consider how the free estate has built up in
the last 20 years so that the profitability of the UWP policies with a 1% guarantee
charge can be investigated.
7.5.1 Buying Options
Under this approach, the portfolio cashflows can be calculated by summing up the
cashflows incurred for each policy in the current portfolio. The methodology for
calculating the individual cashflows has been described in the previous chapters.
The portfolio cashflows are then rolled up to each time point in the investigation
period to calculate the amount of the free estate EST (t), t = 0, 1, ..., 20. Hence,
EST (0) = −CF (0) (7.91)
and
EST (t) = EST (t− 1) · [1 + Z(t− 1, 1)]− CF (t) (7.92)
in which CF (t) is the portfolio cashflow incurred at time t, and Z(t − 1, 1) is the
yield on a 1-year zero-coupon bond at time t− 1.
Figure 7.41 shows the insurer’s free estate using the option method in Cases A,
B and C.
141
year
free
est
ate
1985 1990 1995 2000
-100
010
020
030
0
1982 1987 1992 1997 2002
Case ACase BCase C
Figure 7.41: The free estate of the insurer using the option method in Cases A, Band C
We see in Figure 7.41 that smoothing has little impact on the accumulation of
the free estate. The estate shows a similar pattern in Cases A and B. In early years,
the guarantee charges deducted from the in force business are not sufficient to cover
the increase in the portfolio reserves. After a few years, more policies have been sold
and the business starts to produce a surplus to the insurer. The surplus increases
rapidly during the second decade when the portfolio has been built up and negative
cashflows (i.e. profits) come out steadily from the portfolio.
In Case C where future bonuses are allowed for, much larger reserves are required
particularly at early policy durations. Hence the deficit increases rapidly in the first
decade. The business does not produce a surplus until the end of 1995.
Summing up, UWP policies have been profitable to the insurer in the last 20
years using the option method. However, the insurer does need some capital to
start the business, and the deficit lasts for a much longer period if future bonuses
are allowed for. We assume that 1% of the units is deducted to pay the cost of
guarantees. In a 1% stakeholder environment, all charges should come from 1% of
the units. Therefore, if a part of the charge is to cover expenses which are ignored
in this thesis, the free estate might end up with a loss at the end of 2002. Later in
a sensitivity test we will investigate the profitability of the policies with a smaller
142
guarantee charge.
7.5.2 Discrete Hedging
We have assumed that buying or selling equities incurs transaction costs. If the
hedge portfolios are readjusted between different cohorts instead of between each
cohort and a third party, the positions in equities can be partly offset and hence the
insurer can save some transaction costs.
Table 7.39 concentrates on Case C to compare the portfolio cashflow with the
sum of the individual cashflows under the hedging approach.
Table 7.39: The comparison of the portfolio cashflow and the sum of the individualcashflows incurred by the insurer using the hedging approach in Case C
year portfolio sum of individuals1982 5.1385 5.13851983 4.1613 4.17201984 6.1843 6.20371985 -0.1660 -0.14811986 -0.1726 -0.14411987 -4.3170 -4.31701988 9.3409 9.34091989 12.4042 12.44881990 9.8246 9.82471991 -11.4535 -11.37221992 -59.9146 -59.85971993 -44.2344 -44.16911994 -29.4926 -29.43181995 -20.6802 -20.61721996 -28.3236 -28.27311997 -30.2520 -30.17431998 -26.5322 -26.43061999 -27.5245 -27.39252000 -20.1587 -20.15332001 -16.9270 -16.92642002 12.9913 12.9923
As expected, the portfolio cashflow has a slightly smaller value than the sum of the
individual cashflows. However, the difference can only be noticed if we maintain at
least four decimal places. According to the Black-Scholes equation, the positions for
all in force policies move in the same direction when the equity index changes. The
maturing policy can buy equities from the new business, but the assumed transaction
costs are only 0.2% of the change in the equity component of the hedge portfolio.
Therefore, rebalancing between different cohorts or between each individual cohort
and a third party does not make a lot of difference.
143
year
free
est
ate
1985 1990 1995 2000
010
020
030
040
0
1982 1987 1992 1997 2002
Case ACase BCase C
Figure 7.42: The free estate of the insurer using the hedging approach in Cases A,B and C
Figure 7.42 shows the amount of the free estate under the hedging approach in
Cases A, B and C.
The patterns of the free estate are similar under the option and hedging ap-
proaches. However, the effects of smoothing and allowing for future bonuses are
less significant by hedging. For most cohorts, the hedge portfolios brought forward
are mostly worth more than those that need to be set up, which is probably due
to the assumed high volatility for the equity index. Thus, the reserves are gradu-
ally released back to the insurer and the difference between the three cases is less
obvious.
As was the case using the option method, we can conclude from Figure 7.42
that UWP policies have been profitable over the past 20 years with a 1% guarantee
charge, although a deficit lasts for a few years after the business starts.
7.5.3 CTE Approach
Under the CTE approach, the portfolio cashflow equals the increase in the portfolio
reserves less the guarantee charges from the portfolio and plus the payoff under the
guarantees for the maturing policy.
In a similar way as described in Section 4.3.3 for a single policy, we can prove
144
year
free
est
ate
1985 1990 1995 2000
010
020
030
040
050
060
0
1982 1987 1992 1997 2002
Case ACase BCase C
Figure 7.43: The free estate of the insurer who sets up 95% CTE reserves in CasesA, B and C
that the amount invested in the zero-coupon bond which is issued at time t and
expires at time p is ∑10,000i=10,000α+1 L′(t,p,ps(i))−PGC′(t,p,ps(i))
10,000·(1−α)
[1 + Z(t, p− t)]p−t
if a positive CTE reserve is set up at time t.
Therefore, at valuation date t, the previous year’s reserve (if it was greater than
zero) has a current value of
V ′(t) =t+T−1∑
p=t
∑10000i=10000α+1 PCF ′(t−1,p,ps(i))
10000·(1−α)
[1 + Z(t− 1, p− t + 1)]p−t+1· v(t, p− t)
v(t− 1, p− t + 1)
(7.93)
in which v(t, p − t) is the value of the zero-coupon bond at time t with duration
p− t.
Figures 7.43 and 7.44 show how the free estate accumulates under the CTE
reserving approach in Cases A, B and C.
As there is no need to set up 95% CTE reserves for the portfolio until the end of
1999 and the guarantee does not bite on any of the maturing policies, the amount
of the free estate equals the accumulated guarantee charges. Hence we see in Figure
7.43 that before the end of 1999 the surplus accumulates gradually. Afterwards,
the surplus increases less rapidly because reserves are required in later years. The
145
year
free
est
ate
1985 1990 1995 2000
010
020
030
040
050
0
1982 1987 1992 1997 2002
Case ACase BCase C
Figure 7.44: The free estate of the insurer who sets up 99% CTE reserves in CasesA, B and C
difference in the free estate between the different cases is not obvious particularly
when no reserve is required so that the difference is just due to different equity
proportions in the asset share.
Larger reserves are required with a higher security level, so the insurer who sets
up 99% CTE reserves mostly has a smaller surplus. Notice in Figure 7.44 that the
surplus reduces temporarily at the end of 2000. Figure 7.44 shows greater effects of
smoothing and allowing for future bonuses than Figure 7.43.
Figures 7.43 and 7.44 show that UWP policies have been profitable with a 1%
guarantee charge in the last 20 years if 95% or 99% CTE reserves have been set up.
Only in Case C does the insurer who sets up 99% CTE reserves has a small deficit
in the early years.
7.5.4 Comparison of the Free Estate under Different Re-
serving Approaches in Case C
Figure 7.45 concentrates on Case C to compare the insurer’s free estate under the
different reserving approaches.
Figure 7.45 shows that the business produces quite different surpluses if the in-
surer uses different reserving approaches. The approaches of buying options and
146
year
free
est
ate
1985 1990 1995 2000
020
040
060
0
1982 1987 1992 1997 2002
optionhedging95% CTE99% CTE
Figure 7.45: The comparison of the free estate under different reserving approachesin Case C
discrete hedging give the same amount of reserves, but the insurer has a much
larger surplus by hedging during the second decade due to the negative hedging
errors. The surplus is even larger under the CTE approach because smaller reserves
are required most of the time. However, different reserving approaches provide the
insurer with a different level of security and hence they are not directly comparable.
7.6 Sensitivity Testing for the Portfolio in Case
C
In this chapter, the results presented so far are based on the standard basis with the
97.5% and 99.5% probability boundaries to adjust the EBRs. This section inves-
tigates the sensitivity of the portfolio results to different parameters and different
probability boundaries in Case C.
7.6.1 Sensitivity to Different Parameters
As in the single policy case, we investigate the effect of changing one parameter
while keeping others fixed. Six groups of assumptions, including our standard basis,
are considered. They have been given in Table 6.4. We compare the results under
147
the standard basis, i.e. Basis i, with those under each of the other bases.
Asset share and guarantee
The volatility for the equity index assumed in the Black-Scholes model and the
rate of transaction costs have no impact on asset share or guarantee, so we only
consider the first four bases given in Table 6.4.
Tables 7.40 and 7.41 show the portfolio asset shares and guarantees under the
different bases.
Table 7.40: The portfolio asset shares in Case C under the different bases
Basis31/Dec/ i ii iii iv1982 100.00 100.00 100.00 100.001983 231.22 231.22 231.86 231.221984 404.93 404.93 407.24 404.931985 594.89 595.85 601.02 595.851986 857.50 863.00 873.24 861.151987 1055.51 1057.18 1074.50 1056.421988 1297.75 1300.41 1325.10 1299.701989 1846.34 1871.04 1899.07 1867.741990 1823.84 1836.12 1876.86 1836.211991 2339.62 2358.32 2415.43 2357.651992 2453.98 2480.72 2534.49 2479.581993 2748.11 2776.16 2828.54 2792.481994 2393.69 2406.08 2455.59 2398.731995 2621.48 2643.90 2686.90 2656.401996 2766.08 2801.03 2841.85 2805.081997 3080.86 3128.88 3168.87 3126.281998 3210.79 3257.69 3302.50 3255.561999 3487.39 3579.28 3591.12 3578.732000 3061.78 3111.63 3151.99 3085.762001 2632.19 2638.38 2706.33 2609.362002 2248.86 2214.21 2300.79 2203.10
The lower guaranteed rate in Basis ii leads to smaller initial guarantees and
hence more of the assets can be invested in equities. Equities mostly earn a higher
return than zero-coupon bonds, so we see in Table 7.40 that the portfolio asset share
increases more rapidly under Basis ii most of the time. At the end of 2002, however,
the asset share drops to a lower level with higher equity proportions for some policies
under Basis ii due to the poor equity performance. Table 7.41 shows that the lower
guaranteed rate leads to smaller portfolio guarantees and the difference becomes
more obvious in later years. We have seen in the single policy case that the individual
guarantees under basis ii start from a smaller initial amount but build up more
rapidly with higher regular bonuses. However, bonuses still cannot be afforded on
the later cohorts with the zero guaranteed rate. Thus, promising a lower guaranteed
148
Table 7.41: The portfolio guarantees in Case C under the different bases
Basis31/Dec/ i ii iii iv1982 160.76 144.57 164.55 150.521983 327.96 297.53 336.43 305.131984 527.69 489.81 544.47 482.981985 730.16 673.52 751.05 668.931986 976.19 909.79 1008.62 879.241987 1191.73 1114.17 1233.13 1077.551988 1497.40 1429.21 1557.77 1333.461989 1844.77 1794.53 1928.97 1616.611990 2242.22 2180.55 2351.37 1945.771991 2627.89 2547.53 2748.16 2273.891992 2518.33 2413.29 2626.28 2210.091993 2424.73 2293.90 2519.30 2168.021994 2403.25 2273.51 2489.96 2161.091995 2461.23 2348.45 2554.45 2205.011996 2531.64 2415.17 2631.12 2266.451997 2588.44 2465.34 2689.47 2326.641998 2577.18 2414.37 2667.40 2342.031999 2590.96 2408.55 2674.18 2388.862000 2536.62 2307.64 2603.27 2383.682001 2512.51 2232.51 2564.70 2399.722002 2518.09 2214.69 2562.57 2437.35
rate has greatly reduced the portfolio guarantees in recent years.
The smaller guarantee charges in Basis iii clearly lead to larger portfolio asset
shares. The bonus earning power is increased by the smaller charges, hence the
portfolio guarantees build up more rapidly with the higher bonus rates.
The higher terminal bonus target in Basis iv reduces the bonus earning power.
The portfolio guarantee, given in Table 7.41, starts from a smaller amount and
afterwards builds up less rapidly with the lower bonuses. The smaller guarantees
improve the insurer’s financial strength, hence more of the asset share can be invested
in a risky asset which mostly earns a higher return. Only in the final two years are
the portfolio asset shares smaller in Basis iv.
In Table 7.42 we compare the maturity asset shares and guarantees under the
four bases.
Table 7.42 shows that the guarantee does not bite at maturity under any of the
bases. Hence at maturity the policyholder receives the whole asset share. The
smaller guarantee charges in Basis iii lead to larger asset shares and also larger
guarantees at maturity. The lower guaranteed rate in Basis ii and the higher ter-
minal bonus target in Basis iv increase the investment value of the policy to most
policyholders due to the higher equity proportions when the equity market performs
149
Table 7.42: The asset shares and guarantees at maturity in Case C under the dif-ferent bases
issued at Basis i Basis ii Basis iii Basis iv31/Dec/ A(T ) g(T ) A(T ) g(T ) A(T ) g(T ) A(T ) g(T )1982 482.74 343.20 482.24 342.85 507.68 360.94 482.24 287.861983 489.98 310.41 501.99 316.87 527.92 333.86 509.07 264.061984 370.69 274.08 378.05 278.98 400.21 296.60 380.90 232.011985 403.28 259.45 415.03 260.44 427.26 273.44 411.10 218.101986 375.92 236.75 384.23 232.74 392.80 246.37 398.92 196.151987 429.31 244.78 445.95 243.89 451.50 255.94 462.23 201.851988 476.75 264.02 480.17 263.74 501.39 276.19 491.63 233.531989 455.73 245.15 474.01 244.11 479.04 256.24 486.54 208.391990 531.35 313.92 546.58 313.85 558.80 330.49 556.01 264.751991 409.80 285.50 421.49 289.56 430.97 299.96 406.56 245.361992 313.57 263.50 312.92 258.72 334.97 276.16 286.87 231.44
very well. Some policies have larger maturity guarantees with the lower guaranteed
rate because the guarantees build up more rapidly with the higher bonuses. The
maturity guarantees are smaller with the higher terminal bonus target so that a
larger margin has been built up between the asset share and guarantee.
At maturity, the excess of the asset share over the guarantee is paid out as a
terminal bonus. The terminal bonus rates declared on the maturing policies under
the different bases are compared in Table 7.43.
Table 7.43: The terminal bonus rates declared on the maturing policies in Case Cunder the different bases (%)
issued at Basis31/Dec/ i ii iii iv
1982 40.66 40.66 40.66 67.531983 57.85 58.42 58.13 92.781984 35.25 35.51 34.93 64.181985 55.44 59.36 56.25 88.491986 58.78 65.09 59.44 103.371987 75.39 82.85 76.41 129.001988 80.58 82.06 81.54 110.521989 85.90 94.18 86.95 133.481990 69.26 74.15 69.08 110.011991 43.54 45.56 43.68 65.701992 19.00 20.95 21.30 23.95
We see in Table 7.43 that the lower guaranteed rate, the smaller guarantee charges
and the higher terminal bonus target all lead to a higher terminal bonus rate for
most of the maturing policies. Even so, the rate declared on the 1992 policy is still
much lower than the initial target of 30% (in Bases i, ii and iii) or 50% (in Basis iv).
150
Reserves and free estate using the option method
Only the first five bases given in Table 6.4 are considered. The portfolio reserves
set up under the different bases are given in Table 7.44.
Table 7.44: The portfolio reserves using the option method in Case C under thedifferent bases
Basis31/Dec/ i ii iii iv v1982 5.12 3.30 4.83 3.91 8.101983 9.73 6.35 9.21 7.13 16.001984 14.19 11.59 15.29 10.91 24.351985 17.04 14.00 19.07 12.33 30.331986 16.16 16.27 19.27 12.23 31.401987 27.39 27.12 29.69 19.75 49.081988 49.91 48.77 52.60 34.80 82.021989 34.61 35.81 36.90 19.87 66.441990 81.48 81.54 87.00 42.72 128.021991 85.29 82.12 89.32 43.46 137.011992 46.64 45.32 47.46 32.95 84.391993 28.18 25.04 26.68 23.95 55.251994 25.43 26.06 24.75 19.85 50.621995 26.43 30.31 29.32 19.65 52.521996 22.92 26.17 24.10 15.84 47.781997 22.27 26.77 22.18 18.40 44.931998 38.61 25.84 33.18 37.61 63.171999 28.90 10.35 23.16 29.20 45.142000 53.00 18.30 43.00 53.10 74.512001 80.10 27.43 65.11 80.26 107.402002 170.84 80.46 147.57 172.66 206.08
Table 7.44 shows that the reserves required for the portfolio are mostly smaller
in Basis ii because the lower guaranteed rate leads to smaller guarantees. However,
for some policies, the EBRs are increased by the lower guaranteed rate and hence
the remaining guarantees which are not covered by the zero-coupon bonds are of a
larger amount in Basis ii. Therefore, in some years more reserves are required for
the portfolio in Basis ii.
The effect on the portfolio reserves of the smaller guarantee charges in Basis iii
is not so clear. The smaller charges increase both the asset shares and guarantees.
Thus, whether the portfolio reserves are larger or smaller in Basis iii depends on the
relative changes in the individual asset shares and guarantees.
Smaller reserves are mostly required in Basis iv because a large margin has been
built up between the asset share and the guarantee with a higher terminal bonus
target. In later years, however, the higher EBRs for some policies lead to larger
reserves, hence we see in Table 7.45 that slightly larger reserves are required for the
portfolio in Basis iv.
151
The higher volatility for the equity index in Basis v increases the uncertainty
about the equity price movements in the future and hence the options are more
expensive. Thus, larger reserves are required.
Table 7.45 compares the amount of the free estate using the option method under
the different bases.
Table 7.45: The free estate of the insurer using the option method in Case C underthe different bases
Basis31/Dec/ i ii iii iv v1982 -5.12 -3.30 -4.83 -3.91 -8.101983 -10.17 -6.32 -10.24 -7.27 -16.751984 -17.07 -12.35 -20.10 -12.23 -29.211985 -20.33 -14.32 -27.29 -13.01 -37.481986 -22.59 -17.86 -35.97 -13.29 -46.581987 -24.77 -19.22 -42.82 -11.50 -56.481988 -42.32 -35.71 -68.63 -19.74 -88.701989 -48.93 -43.64 -88.47 -14.07 -112.931990 -72.19 -64.37 -127.49 -15.32 -156.631991 -88.84 -77.49 -162.05 -9.83 -195.261992 -57.77 -48.09 -150.71 15.14 -170.221993 -25.17 -13.83 -139.16 49.46 -142.121994 -5.38 3.54 -139.25 76.17 -133.411995 11.50 17.14 -149.32 102.04 -134.011996 35.37 40.14 -150.49 135.13 -126.201997 64.27 66.81 -149.49 170.45 -112.441998 89.57 108.43 -151.51 200.60 -102.891999 123.59 153.59 -147.17 241.30 -79.832000 147.73 191.22 -151.01 272.29 -71.632001 170.27 226.31 -155.84 301.31 -68.012002 175.08 236.82 -176.77 309.73 -83.56
The lower guaranteed rate in Basis ii has mostly reduced the portfolio reserves,
hence the deficit lasts for a slightly shorter period before a surplus appears and the
surplus accumulates more rapidly in later years.
The percentage of the units deducted as a guarantee charge is reduced from 1%
to 0.5% in Basis iii. Table 7.45 shows that the deficit increases rapidly over the first
decade and afterwards it remains at around £150. The guarantee charges are not
sufficient to cover the increase in the reserves, so the policies have not been profitable
with a 0.5% guarantee charge under the option approach. In other words, in a 1%
stakeholder environment, if half of the charges are deducted to cover expenses, the
free estate will end up with a loss at the end of 2002. However, future charges on
the existing business might pay back the loss.
Smaller reserves are required in Basis iv with the higher terminal bonus target.
Hence the deficit is reduced and the accumulation of the surplus is accelerated.
152
The higher volatility for the equity index in Basis v has greatly increased the
portfolio reserves and the large reserves set up using the option method cannot
be released back to the insurer, so the profitability of the UWP policies is greatly
reduced. Although we see in Table 7.45 that the deficit shows a decreasing trend
over the second decade, no surplus has ever appeared by the end of 2002. Therefore,
the policies are not profitable if the options are priced assuming a high volatility for
the equity index.
Free estate using the hedging approach
The rate of transaction costs τ only affects the cashflows incurred by hedging
rather than the reserves. We allow for transaction costs incurred for the whole
portfolio assuming that the replicating portfolios are rebalanced between different
cohorts. The amounts of the free estate under the six bases are compared in Table
7.46.
Table 7.46: The free estate of the insurer using the hedging approach in Case Cunder the different bases
Basis31/Dec/ i ii iii iv v vi1982 -5.14 -3.31 -4.85 -3.93 -8.12 -5.171983 -9.82 -6.06 -9.89 -6.97 -16.12 -9.881984 -16.91 -12.30 -19.98 -12.13 -28.03 -17.001985 -18.38 -12.70 -25.23 -11.41 -33.28 -18.491986 -20.20 -15.90 -33.42 -11.43 -39.94 -20.351987 -18.03 -12.79 -35.31 -5.93 -42.63 -18.281988 -28.95 -22.48 -53.85 -8.74 -64.33 -29.401989 -44.67 -39.81 -83.42 -10.48 -90.59 -45.411990 -60.25 -52.57 -114.19 -6.30 -117.71 -61.591991 -56.40 -44.82 -126.81 10.74 -122.18 -57.981992 -2.23 8.02 -90.79 48.60 -57.31 -4.561993 41.83 53.88 -67.35 92.20 -5.56 38.931994 73.79 83.72 -55.17 128.72 27.22 70.711995 99.75 106.75 -55.83 161.31 46.52 96.371996 135.06 142.50 -44.70 202.80 78.05 131.371997 174.34 180.50 -32.58 245.51 114.13 170.311998 212.92 237.76 -20.36 285.63 150.68 208.581999 252.33 288.15 -10.20 329.59 189.05 247.582000 285.76 334.66 -4.56 367.97 216.68 280.632001 318.28 379.47 0.97 404.80 241.38 312.682002 319.32 390.68 -22.64 407.68 235.44 312.99
Some of the conclusions drawn from Table 7.45 also apply under the hedging
approach. The lower guaranteed rate in Basis ii and the higher terminal bonus
target in Basis iv reduce the deficit and increase the surplus. Although Table 7.46
shows that the policies have still not been profitable with the 0.5% guarantee charge
under the hedging approach, the deficit shows a decreasing trend over the second
153
decade and a very small surplus appears at the end of 2001. The effect of the higher
volatility for the equity index in Basis v on the cashflows incurred by discrete hedging
is different from that under the option approach. We concluded from Table 7.45 that
UWP policies are not profitable if the options are priced assuming a high volatility.
However, Table 7.46 shows that the higher volatility reduces the profitability but the
surplus appears at the end of 1994 and accumulates rapidly afterwards. By discrete
hedging, larger reserves are required with the higher volatility, but they are released
back to the insurer gradually if the assumed volatility is higher than reality. Clearly,
the higher rate of transaction costs under Basis vi leads to more money incurred to
rebalance the hedge portfolios and hence increases the insurer’s deficit or reduces
its surplus. However, the different rate of transaction costs does not make a lot
of difference because the transaction costs are not a large component of the total
cashflows incurred by the insurer.
Whichever basis is chosen, we have the same conclusion that the insurer has a
smaller deficit or a larger surplus by hedging the risk internally instead of buying
options over-the-counter.
Reserves and free estate using the CTE approach
The 95% and 99% portfolio CTE reserves calculated under the first four bases,
in Table 6.4, are given in Tables 7.47 and 7.48 respectively.
The effect on the portfolio CTE reserves of the lower guaranteed rate in Basis
ii is not clear. In the single policy case, the lower guaranteed rate reduces the
individual CTE reserves at early durations, but larger reserves are required at later
durations when the guarantee increases rapidly and more of the asset share has been
switched into equities. Whether larger or smaller reserves are required for the whole
portfolio with the lower guaranteed rate depends on the relative change in the equity
component of the asset share and the remaining guarantees which are not covered
by the zero-coupon bonds. Tables 7.47 and 7.48 clearly show that in later years the
portfolio CTE reserves are greatly reduced by the lower guaranteed rate.
The smaller guarantee charges in Basis iii lead to larger portfolio CTE reserves.
As in the single policy case, there is an inconsistency in the way the future guarantee
charges are allowed for when setting up reserves using the different approaches. The
154
Table 7.47: The 95% portfolio CTE reserves in Case C under the different bases
Basis31/Dec/ i ii iii iv1982 0.00 0.00 0.00 0.001983 0.00 0.00 0.00 0.001984 0.00 0.00 0.00 0.001985 0.00 0.00 0.00 0.001986 0.00 0.00 0.00 0.001987 0.00 0.00 0.00 0.001988 0.00 0.00 0.00 0.001989 0.00 0.00 0.00 0.001990 0.00 0.00 16.27 0.001991 0.00 0.00 9.37 0.001992 0.00 0.00 15.28 0.001993 0.00 0.00 0.00 0.001994 0.00 0.00 0.00 0.001995 0.00 0.00 0.00 0.001996 0.00 0.00 0.00 0.001997 0.00 0.00 0.00 0.001998 0.00 0.00 0.34 0.001999 0.00 0.00 15.44 0.002000 48.02 0.00 76.72 50.892001 81.96 0.00 101.09 83.982002 115.37 0.00 129.37 117.94
effect of deducting smaller charges on the CTE reserves is different from that on the
reserves set up using the option pricing approach. The CTE reserves are calculated
from the projected cashflows which are increased in value by the smaller charges, so
more reserves are required to meet the future guarantee cost.
The higher terminal bonus target in Basis iv mostly reduces the portfolio CTE
reserves because the guarantees build up less rapidly with the lower regular bonuses.
Due to a larger margin built up between the asset share and guarantee, more of the
assets can be invested in equities. The higher EBRs mostly lead to a larger asset
share, but in later years the equity market performs badly and hence the guarantee
increases more rapidly than the asset share. Therefore, larger reserves are required
for the portfolio in later years with the higher terminal bonus target.
Tables 7.49 and 7.50 compare the amount of the free estate under the different
bases.
The 95% portfolio CTE reserves are only required in later years, hence the cash-
flows incurred by the insurer are mostly the guarantee charges deducted from the
unit fund. The charges are a fixed percentage of the asset share which is mostly
larger with a higher EBR. Therefore, we see in Table 7.49 that the surplus accu-
mulates more rapidly with the lower guaranteed rate in Basis ii or with the higher
155
Table 7.48: The 99% portfolio CTE reserves in Case C under the different bases
Basis31/Dec/ i ii iii iv1982 1.83 0.00 4.99 0.001983 5.10 0.00 11.60 0.001984 0.00 0.00 16.72 0.001985 9.72 0.00 33.89 0.001986 0.00 0.00 29.65 0.001987 13.51 13.87 47.16 0.001988 0.00 0.00 28.70 0.001989 0.00 0.00 31.51 0.001990 49.06 51.05 95.21 0.001991 37.47 35.75 87.16 0.001992 37.86 38.39 84.05 0.001993 4.39 0.00 47.24 0.001994 0.00 0.00 19.97 0.001995 0.00 0.00 21.02 0.001996 0.00 0.00 24.96 0.001997 0.00 3.66 27.93 0.001998 12.74 0.00 53.45 10.631999 17.57 0.00 53.98 24.272000 94.68 0.00 121.14 99.092001 140.44 14.13 157.64 143.052002 191.46 47.70 202.89 195.60
terminal bonus target in Basis iv. As expected, the insurer has a smaller surplus
with the smaller guarantee charges in Basis iii.
Using a different basis has a larger impact on the 99% CTE reserves. Table
7.50 shows that the lower guaranteed rate in Basis ii accelerates the accumulation
of the surplus mainly due to the larger portfolio asset shares in early years and
smaller CTE reserves required in later years. The insurer incurs a deficit during the
first decade under Basis iii with the smaller guarantee charges because additional
money is required to increase reserves. Afterwards, the portfolio has been built
up and the reserves are partly released back to the insurer at some stage. Thus,
a surplus accumulates gradually in the second decade. Under Basis iv with the
higher terminal bonus target, no reserves are required until the end of 1998 with the
99% CTE measure. Under our standard basis, however, the reserves are set up and
released frequently in this period. The higher terminal bonus target mostly increases
the surplus in early years, but afterwards the surplus accumulates less rapidly under
Basis iv.
156
Table 7.49: The free estate of the insurer who sets up 95% CTE reserves in Case Cunder the different bases
Basis31/Dec/ i ii iii iv1982 0.00 0.00 0.00 0.001983 1.27 1.27 0.64 1.271984 4.37 4.37 2.19 4.371985 9.62 9.63 4.83 9.631986 18.11 18.17 9.14 18.151987 29.43 29.52 14.88 29.491988 43.77 43.90 22.17 43.861989 65.97 66.35 33.53 66.281990 91.31 91.87 30.22 91.791991 124.80 125.62 54.80 125.521992 165.47 166.64 70.73 166.521993 209.22 210.88 112.18 210.991994 247.70 249.66 132.26 249.731995 294.11 296.56 156.45 296.721996 344.56 347.62 182.72 347.971997 401.09 405.00 212.15 405.521998 464.04 468.74 244.59 469.391999 527.72 533.79 261.11 534.602000 541.64 596.71 233.40 546.512001 562.85 658.06 235.51 568.722002 587.88 710.42 240.08 592.83
7.6.2 Sensitivity to Different Probability Boundaries in the
Investment Strategy
As in the single policy case, we now produce results using a less cautious investment
strategy assuming that the asset share is switched between equities and zero-coupon
bonds based on the upper and lower probability boundaries of 99% and 95% respec-
tively.
Asset share and guarantee
Table 7.51 gives the portfolio asset shares and guarantees under the standard
basis with 95% and 99% probability boundaries.
A comparison of Tables 7.51 and 7.40 shows that changing the probability bound-
aries mostly increases the portfolio asset share. The investment strategy is less cau-
tious with the new probabilities. The asset share is more likely to be switched into
equities and less likely to be switched into zero-coupon bonds. Equities mostly earn
a higher rate of return than bonds, so with the new probabilities the asset share is
increased by the larger equity component. In later years, however, the equity market
crashes and so the asset share drops more rapidly with higher EBRs. At the end of
2002, the asset share is smaller with the new probability boundaries.
157
Table 7.50: The free estate of the insurer who sets up 99% CTE reserves in Case Cunder the different bases
Basis31/Dec/ i ii iii iv1982 -1.83 0.00 -4.99 0.001983 -3.50 1.27 -10.60 1.271984 4.54 4.37 -14.32 4.371985 0.09 9.63 -28.35 9.631986 17.45 18.17 -21.10 18.151987 15.19 15.65 -30.76 29.491988 44.31 45.12 -3.67 43.861989 66.57 67.72 2.43 66.281990 42.93 42.36 -52.19 91.791991 89.32 92.78 -25.63 125.521992 132.77 134.02 2.46 166.521993 218.91 224.35 73.67 210.991994 256.39 263.93 114.19 249.731995 303.43 311.84 140.30 296.721996 354.52 363.97 163.39 347.971997 411.72 418.79 195.55 405.521998 462.67 494.45 211.07 458.761999 519.16 560.94 241.48 507.482000 506.76 625.29 209.51 497.592001 515.07 674.07 198.89 507.202002 525.36 697.33 189.75 514.97
Comparing Tables 7.51 and 7.41 shows that changing the probability boundaries
mostly reduces the portfolio guarantee. Hence we can infer that for most policies in
the portfolio, the bonus earning power is lower with the new boundaries probably
because the 25th percentile of the projected maturity asset share is smaller with the
higher EBR.
Summing up, the 95% and 99% probability boundaries mostly increase the port-
folio asset share and reduce the portfolio guarantee, hence the insurer’s financial
strength is improved. However, the new boundaries also lead to larger equity com-
ponents, so the guarantees can only be covered by fewer zero-coupon bonds. Later
we will see this effect on the portfolio reserves.
Table 7.52 gives the individual asset share and guarantee at maturity under the
standard basis with the new probabilities. The terminal bonus rate is also shown in
the table.
The positive terminal bonuses show that the guarantee does not bite on any of
the maturing policies. Again, the bonus rate declared on the 1992 policy has not
met the initial target of 30%.
The policyholder receives the whole asset share at maturity. Comparing Tables
7.52 and 7.42 shows that except for the earliest and latest maturing policies, the
158
Table 7.51: The portfolio asset shares and guarantees in Case C under the standardbasis with the 95% and 99% probability boundaries
31/Dec/ asset share guarantee1982 100.00 160.761983 231.22 327.961984 404.93 527.691985 595.85 728.671986 863.00 972.341987 1056.65 1185.801988 1300.10 1489.131989 1870.63 1832.561990 1835.75 2227.791991 2357.87 2628.321992 2480.19 2519.271993 2784.95 2425.691994 2402.48 2390.091995 2648.37 2439.511996 2799.61 2515.071997 3130.38 2567.061998 3256.62 2562.041999 3586.61 2578.262000 3107.08 2522.972001 2634.08 2500.542002 2221.53 2507.55
more risky investment strategy increases the investment value to the policyholders.
Reserves and free estate
Table 7.53 gives the reserves for the portfolio under the three approaches based
on the standard basis with the new boundaries.
Comparing Table 7.53 with Tables 7.44, 7.47 and 7.48 clearly shows that under
the standard basis if the EBRs are adjusted according to the new probability bound-
aries, larger reserves are required under all three reserving approaches . Changing
the probability boundaries leads to larger equity components for some policies, al-
though the new boundaries slightly reduces the portfolio guarantees most of the
time. Therefore, the part of the guarantees which are not covered by the risk-free
asset is larger with the new boundaries, and hence more reserves are required to be
set up against the higher risk.
Changing the probability boundaries in our investment strategy does not change
the conclusion that the smallest reserves are required for the portfolio with the 95%
CTE measure and that the 99% portfolio CTE reserves are mostly smaller than
the reserves set up using the option pricing approach. However, the three reserving
approaches are not directly comparable.
159
Table 7.52: The asset share, guarantee and terminal bonus rate at maturity in CaseC under the standard basis with the 95% and 99% probability boundaries
issued at maturity maturity terminal31/Dec/ asset share guarantee bonus rate (%)1982 482.24 342.68 40.731983 509.07 316.93 60.631984 382.68 277.08 38.111985 411.26 262.26 56.821986 390.79 231.53 68.791987 446.20 247.33 80.411988 487.18 262.50 85.591989 471.85 243.03 94.151990 549.78 314.86 74.611991 411.04 283.82 44.831992 309.32 262.07 18.03
Table 7.54 gives the amount of the free estate with the new probability bound-
aries.
A comparison of Tables 7.54 and 7.45 shows that under the option approach, the
more risky investment strategy has postponed the appearance of the surplus. The
deficit accumulates more rapidly in the first 10 years to set up the larger reserves
with the new boundaries.
Comparing Tables 7.54 and 7.46, again we see that by discrete hedging, changing
the boundaries reduces the profitability of the policies. However, the impact is less
significant under the hedging approach than that using the option method because
the larger reserves are partly released back to the insurer through time.
Comparing Tables 7.54 and 7.49 shows that the surplus of the insurer, who sets
up 95% CTE reserves, accumulates slightly more rapidly with the new probability
boundaries. We have seen before that the 95% CTE reserves are not required until
the end of 2000 with either boundaries. The asset share is mostly larger with the
new boundaries and hence the deducted guarantee charges are of a larger amount.
The surplus is slightly reduced by the new boundaries at the end of 2000 because
larger reserves are then set up.
For the insurer who sets up 99% CTE reserves, a comparison of Tables 7.54
and 7.50 shows that the more risky investment strategy has mostly reduced the
profitability of the policies because larger reserves have been set up. In recent years,
however, the increase in the reserve fund is smaller with the new boundaries and
hence the surplus accumulates more rapidly.
160
Table 7.53: The portfolio reserves in Case C under the standard basis with the 95%and 99% probability boundaries
31/Dec/ option/hedging CTE (95%) CTE (99%)1982 5.12 0.00 1.831983 9.73 0.00 5.101984 15.70 0.00 4.451985 19.78 0.00 18.971986 22.84 0.00 21.771987 35.31 0.00 38.681988 57.89 0.00 6.011989 40.37 0.00 1.321990 86.82 0.00 55.881991 93.42 0.00 51.161992 61.62 0.00 81.011993 43.18 0.00 67.421994 39.72 0.00 26.731995 37.31 0.00 0.001996 35.18 0.00 18.331997 35.45 0.00 32.881998 51.10 0.00 80.221999 30.71 0.00 45.072000 54.18 56.89 113.522001 81.56 87.18 150.752002 177.23 122.61 204.44
7.7 Summary
In this chapter we have considered a portfolio of UWP policies historically with
a 20-year investigation period which starts on 31 December 1982 and ends at 31
December 2002. Each year one single premium policy with a term of 10 years is
issued. The dynamic investment and bonus strategies are applied to each generation
separately so that each policyholder’s assets have been allocated according to the
EBRs calculated for his own policy, and regular bonuses have been declared on each
policy based on its own bonus earning power. We have considered the following
three cases:
A: without smoothing, and future bonuses are ignored,
B: with smoothing, and future bonuses are ignored,
C: with smoothing, and the minimum future bonuses implied by the smoothing
mechanism are included in the guarantees.
The three reserving approaches of buying options, discrete hedging and CTE
reserving have been applied to calculate the portfolio reserves. The advantage to
the insurer of pooling risks under the CTE approach (and the hedging approach by
161
Table 7.54: The amount of the free estate in Case C under the standard basis withthe 95% and 99% probability boundaries
31/Dec/ option hedging CTE (95%) CTE (99%)1982 -5.12 -5.14 0.00 -1.831983 -10.17 -9.82 1.27 -3.501984 -18.57 -18.42 4.37 0.091985 -23.48 -21.40 9.63 -8.641986 -31.05 -28.44 18.17 -4.011987 -34.12 -26.03 29.51 -7.331988 -53.35 -37.22 43.89 41.501989 -63.48 -57.13 66.34 66.921990 -86.85 -72.08 91.85 34.611991 -108.05 -71.97 125.59 74.541992 -86.52 -25.70 166.61 89.471993 -59.10 14.94 211.01 161.151994 -42.99 47.63 249.80 230.011995 -30.67 71.67 296.72 308.461996 -14.60 102.19 347.84 342.071997 7.16 137.12 405.26 392.791998 30.05 177.86 469.07 422.341999 64.63 218.36 534.20 520.022000 87.02 252.42 540.24 518.442001 106.39 284.03 565.50 536.922002 106.02 280.78 588.57 545.68
means of saving transaction costs) has been investigated. We have also considered
the profitability of the UWP policies with a 1% guarantee charge by calculating
the insurer’s free estate. The sensitivity of the results to different parameters and
different probability boundaries in our investment strategy has been investigated.
The main conclusions under the standard basis and with the probability bound-
aries of 97.5% and 99.5% to adjust the EBRs are summarised as follows:
• The policies issued during the first decade of the 20-year period have higher
equity backing ratios, but for the later cohorts more of the asset share is
invested in zero-coupon bonds.
• Higher regular bonus rates are declared on the early cohorts, but the insurer
cannot afford to declare any bonus on the policies issued after the end of 1997.
The zero bonuses on the recent cohorts in Case A suggest that a major feature
of with-profits policies, namely regular bonuses, is not sustainable in a low
inflationary environment.
• Smoothing regular bonuses in Case B has increased the bonus rates for the
recent cohorts and reduced the equity backing ratios at later durations for
some cohorts.
162
• Allowing for future bonuses in Case C has reduced the EBRs for most cohorts
but has little impact on the smoothed bonus rates.
• The equity market mostly earns a higher return than the bond market, but in
recent years equities have performed badly. Thus, the portfolio asset share is
mostly of a larger amount with a larger equity component.
• The portfolio asset share shows an increasing trend because the number of
policies in the portfolio increases in the first decade; the investment market
mostly performs well; and the single premium for each cohort increases with
the retail price index. However, the asset share has dropped very rapidly in
recent years mainly because the equity market crashes.
• The portfolio guarantee builds up very rapidly in the first 10 years because the
number of policies in the portfolio increases each year; the initial guarantee
for each cohort increases with the relatively high rate of inflation; and the
guarantee for each policy builds up rapidly with the high regular bonuses.
During the second decade, however, the force of inflation is relatively low and
the bonus earning power is reduced. Hence the guarantee increases less rapidly.
• The guarantee does not bite on any of the maturing policies. The terminal
bonus rate on the latest maturing policy is lower than the initial target of 30%.
• At maturity, the policyholder receives the whole asset share. Smoothing regu-
lar bonuses in Case B does not make much difference to the maturity payouts.
Most policyholders receive a smaller payout in Case C where the asset share
has a smaller equity component, so the increase in the safety has a cost for
most policyholders.
• The portfolio CTE reserve is smaller than the sum of the individual CTE
reserves because some of the worst simulations for the whole portfolio might
be good simulations for some cohorts. Hence there is an advantage to the
insurer of pooling risks using the CTE approach. However, pooling risks gives
no advantage under the option approach.
163
• Under the option pricing approach, smoothing regular bonuses has little im-
pact on the portfolio reserves; whereas the reserves are greatly increased during
the first decade when the minimum future bonuses implied by our smoothing
mechanism are reserved for.
• Smoothing or allowing for future bonuses does not make much difference to
the 95% portfolio CTE reserves. The 99% CTE reserves are slightly increased
by smoothing, but the increase is more significant with allowance for future
bonuses particularly during the first decade.
• The smallest reserves are required with the 95% CTE risk measure. The 99%
portfolio CTE reserves show a similar pattern to the portfolio reserves set up
using the option and hedging approaches, but the latter is greater before the
end of 1999. For each policy in the portfolio, reserves are always required
under the option pricing approach; whereas it is likely that no CTE reserve is
required for the whole portfolio. However, the three reserving approaches are
not directly comparable.
• The policies have been profitable in the last 20 years with a 1% guarantee
charge under all three reserving approaches. However, the insurer does need
some capital to start the business under the option and hedging approaches.
The deficit lasts for a much longer period with allowance for future bonuses.
• The reserving approaches of buying options and discrete hedging set up the
same amount of reserves, but in the second decade the insurer has a much
larger surplus by hedging the risk internally because the replicating portfolios
brought forward are mostly worth more than those required to be set up. The
surplus is even larger using the CTE approach because smaller reserves are
mostly required.
• Using the hedging approach, the insurer can save transaction costs by rebal-
ancing the hedge portfolios between different cohorts instead of between each
cohort and a third party.
We summarise the main conclusions from the sensitivity testing as follows:
164
• Reducing the guaranteed growth rate from 2% to 0% leads to higher EBRs and
hence the asset share increases more rapidly most of the time; the portfolio
guarantees are smaller particularly in recent years; most policyholders receive
a larger maturity payout; smaller reserves are mostly required under the option
pricing approach but it is not so clear how the CTE reserves are affected; the
surplus appears earlier and accumulates more rapidly under all three reserving
approaches.
• Reducing the percentage of the units deducted as a guarantee charge from 1%
to 0.5% leads to a larger portfolio asset share and guarantee; the maturing
policies have a larger investment value to the policyholder; it is not clear how
the portfolio reserves are affected under the option pricing approach, but more
CTE reserves are required to be set up for the portfolio; the policies have not
been profitable in the last 20 years using the option or hedging approach but
future guarantee charges on the existing business might pay back the loss;
the deficit does show a decreasing trend over the second decade with discrete
hedging; the policies have still been profitable to the insurer who sets up CTE
reserves, but the surplus is greatly reduced.
• Increasing the terminal bonus target from 30% to 50% leads to higher equity
proportions and lower regular bonuses for most cohorts; the portfolio asset
shares are mostly larger and the guarantees are smaller; smaller reserves are
mostly required under all three approaches; under the option pricing approach
or with the 95% CTE measure, the surplus appears earlier and accumulates
more rapidly; to the insurer who sets up 99% CTE reserves, the surplus accu-
mulates less rapidly in later years.
• Increasing the volatility for the equity index, assumed in the Black-Scholes
formula, from 20% to 25% increases the reserves under the option pricing
approach; the policies have not been profitable in the last 20 years to the
insurer using the option method; with discrete hedging, the profitability is
reduced but a surplus appears at the end of 1994 and accumulates rapidly
afterwards.
165
• Increasing the rate of transaction costs, assumed in discrete hedging, from
0.2% to 0.5% has no impact on the required amount of reserves, but reduces
the profit earned by the insurer with hedging.
• Changing the probability boundaries in the investment strategy from 99.5%
and 97.5% to 99% and 95% respectively leads to higher EBRs for most cohorts;
the portfolio asset shares are mostly larger and the guarantees are smaller;
most policyholders receive a larger maturity payout; larger reserves are re-
quired under all three approaches; the appearance of the surplus is postponed
under the option and hedging approaches; the insurer who sets up 95% CTE
reserves has a slightly larger surplus; the surplus is mostly reduced with the
99% CTE measure, but it accumulates more rapidly in later years.
166
Chapter 8
RESERVING FOR THE
PORTFOLIO WITHIN THE
SIMULATED REAL WORLD
8.1 Introduction
In the previous chapters, the numerical results were obtained based on the historical
data. We concluded in Chapter 7 that the insurer could not afford to declare any
bonus on the recently issued policies, and that UWP policies have been profitable
in the last 20 years under our standard basis (i.e. Basis i in Table 6.4), though the
deficit lasts for many years before a surplus appears under the option approach.
However, if the insurer continues writing new business, how sustainable are reg-
ular bonuses, what amount of reserves will be required, and will the insurer’s free
estate keep building up in the future? This chapter extends the investigation period
to the end of 2032. As before, one single premium UWP policy with a term of 10
years is issued each year, and the single premium keeps increasing in line with the
retail price index. The assumptions for the contract design and market parameters
are the same as in the standard basis. The real world during the 30-year period
of 2002 to 2032 is simulated stochastically using the Wilkie model with the initial
conditions observed directly or derived from the market indices at the end of 2002.
167
Different quantiles of the simulated results are given in this chapter, and in Ap-
pendix D we show some sample paths. Only the case of smoothing with allowance
for future bonuses is considered in this chapter.
Before investigating the portfolio, we first consider a single 10-year policy in
Section 8.2. The real world is simulated starting with the Wilkie neutral initial
conditions which are set at their long-run means with the standard deviations equal
to zero. They are given in Appendix A.
By the end of the whole 50-year investigation period which starts on 31 December
1982, the insurer will have sold a total of 51 policies. Assets are allocated and
regular bonuses are declared for each cohort separately according to our dynamic
investment and bonus strategies. In Section 8.3 we calculate the arithmetic average
equity backing ratio and geometric average regular bonus rate over the policy term
for each policy issued during the 50 years. Different quantiles of the simulated results
are then picked out so that we can investigate how the asset share is allocated and
what bonuses are declared over the policy term with different probabilities.
Section 8.4 shows quantiles of the simulated portfolio asset shares and guarantees.
We concluded in Chapter 7 that the guarantee does not bite on any of the policies
maturing in the last 20 years. In Section 8.5 we look at all the policies maturing in
the period 2002-2032. The asset share and guarantee are compared at maturity in
each simulation. The number of simulations in which the guarantee bites is counted.
We also calculate the mean and standard deviation of the simulated payouts, asset
shares, guarantees and terminal bonus rates at maturity.
Section 8.6 simulates the required amount of reserves for the portfolio under the
three reserving approaches. Different quantiles of the simulated reserves are then
calculated.
Section 8.7 looks at the insurer’s free estate. We have calculated the amount of
the free estate at the end of 2002 in Chapter 7. The simulated portfolio cashflows
incurred by the insurer are rolled up to investigate the free estate required in the
future for different quantiles.
We give a summary of conclusions in Section 8.8.
168
8.2 A Single 10-Year Policy
In this section we consider a single 10-year policy in the simulated real world. As be-
fore, the single premium is assumed to be £100. The real world is projected forward
using the Wilkie model starting with the neutral initial conditions. We simulate
the equity backing ratios and regular bonus rates based on the dynamic strategy
described in Chapter 5. Table 8.55 shows the mean and standard deviation of the
simulated equity proportions, bonus rates, asset shares and guarantees including the
minimum of future bonuses.
Table 8.55: The mean and standard deviation of the simulated EBRs, bonus rates,asset shares and guarantees for the 10-year policy
policy EBR (%) bonus rate (%) asset share guaranteeyear mean s.d. mean s.d. mean s.d. mean s.d.
0 79.95 0.62 2.34 0.05 100.00 0.00 133.87 0.251 75.13 4.22 2.44 0.39 110.64 15.83 137.03 2.512 72.34 7.20 2.56 0.66 121.55 22.39 140.44 4.643 71.03 10.06 2.72 0.95 133.70 28.06 144.05 7.064 70.41 12.57 2.91 1.26 147.03 34.65 147.84 9.795 70.50 14.61 3.13 1.60 162.06 41.83 151.73 12.796 71.07 16.31 3.39 1.98 179.10 50.81 155.57 15.987 72.25 17.67 3.69 2.42 197.82 61.51 159.16 19.168 74.10 18.75 4.04 2.93 218.61 74.60 162.16 21.969 77.01 19.42 4.46 3.52 241.23 90.79 164.03 23.7510 265.47 107.01 164.03 23.75
Note that the initial EBR can only be 72%, 80% or 88% according to our invest-
ment strategy. Table 8.55 shows that the mean equity proportions starts at about
80% but then decreases to about 70% at duration 5. The mean EBR is above 70%
during the policy term, which implies that on average the investment market per-
forms very well and the insurer is in a strong solvency position. The mean bonus
rate increases through the policy term because a cautious bonus strategy has been
used so that the bonus rate can be afforded in 75% of the cases allowing for a 30%
terminal bonus target. The mean asset share increases rapidly, but the guarantee is
much bigger than the asset share at early durations because of the 2% guaranteed
rate and the future bonuses.
The 10,000 simulations start from the same neutral initial conditions, but the
standard deviation of the investment performance increases through time. There-
fore, Table 8.55 shows increasing standard deviations through time. The simulated
169
asset shares are much more variable than the guarantees because the latter is con-
strained by the smoothing mechanism.
Table 8.56 shows the mean and standard deviation of the simulated terminal
bonus rates and the payoff under the guarantees (i.e. the excess of the guarantee
over the asset share subject to a minimum of zero). We also count the number of
simulations in which the guarantee bites at maturity.
Table 8.56: The statistics at the maturity of the 10-year policy
terminal bonus rate payoff numbermean (%) s.d. (%) mean s.d. (A < G)
58.79 46.86 0.2 1.72 246
We see in the table that the insurer can on average declare a terminal bonus
rate of 58.79% which is much higher than the 30% target. However, the standard
deviation of the terminal bonus is also at a very high level. There are only 246 (of
10,000) simulations in which the guarantee bites at maturity, and the mean payoff
(over 10,000 simulations instead of 246) is only 20 pence. Therefore, in the majority
of the simulations the investment return is big enough to declare a high terminal
bonus.
In Table 8.57 we show the quantiles of the simulated reserves under the option
pricing approach and the 99% CTE reserves.
Table 8.57: The quantiles of the simulated reserves for the 10-year policy
policy option/hedging CTE (99%)year 1% 10% 50% 90% 99% 1% 10% 50% 90% 99%
0 4.56 4.60 4.64 4.69 4.75 6.12 6.76 7.66 8.39 8.891 0.56 1.49 3.62 6.73 10.23 0.00 0.00 4.39 9.38 15.462 0.17 0.89 2.97 6.90 11.51 0.00 0.00 2.52 8.99 16.843 0.07 0.58 2.53 6.75 11.78 0.00 0.00 1.23 8.89 17.754 0.03 0.40 2.17 6.46 12.66 0.00 0.00 0.62 9.06 17.835 0.02 0.25 1.77 6.14 12.80 0.00 0.00 0.47 9.80 19.786 0.01 0.15 1.33 5.59 13.45 0.00 0.00 0.33 10.80 22.777 0.00 0.06 0.86 4.55 12.79 0.00 0.00 0.32 11.61 25.148 0.00 0.02 0.38 3.01 11.76 0.00 0.00 0.00 12.53 25.659 0.00 0.00 0.05 1.29 10.08 0.00 0.00 0.00 9.87 28.88
Table 8.57 shows that the quantiles of the reserves under the two approaches show
a similar pattern through the policy term, but more reserves are required under the
CTE approach. In the worst 1% of the cases, the CTE reserve is almost 3 times the
170
reserve set by the option pricing approach at duration 9. However, if the guarantee
does not bite at maturity (which has happened in most simulations here), the CTE
reserves can be released back to the insurer.
Table 8.58 gives the mean, standard deviation and quantiles of the simulated ac-
cumulated values of the cashflows at policy termination under the different reserving
approaches.
Table 8.58: The statistics of the accumulated values of the cashflows for the 10-yearpolicy
approach mean s.d. 1% 10% 25% 50% 75% 90% 99%option -12.43 5.21 -24.90 -17.43 -15.22 -13.17 -10.13 -6.12 4.04hedge -11.77 7.39 -22.73 -18.66 -16.68 -13.59 -8.60 -2.48 14.13
CTE (99%) -24.08 5.80 -41.12 -31.22 -26.97 -23.33 -20.55 -18.36 -10.83
We see in Table 8.58 that on average the insurer can earn a larger profit using
the CTE approach. Only in rare cases does the guarantee bite at maturity, so in
most cases the insurer can have the CTE reserve back at maturity. There are only
246 cases where the loss at maturity can be covered by the options, so the standard
deviation of the accumulated values is slightly smaller under the option approach.
The table also shows that hedging internally brings extra risk to the insurer com-
pared to buying options in the cases where the investment market performs very
well or very badly. Under the CTE approach, however, even in the worst 1% of the
cases the insurer can still make a profit of £10.83.
8.3 Average EBR and Average Regular Bonus
Rate
Now we go back to our portfolio case. We assume that insurer continues writing new
business. In the period of 2002 to 2032, the investment performance is simulated
stochastically using the Wilkie model starting with the market indices at the end of
2002. In each simulation, each policyholder’s assets are allocated between equities
and zero-coupon bonds according to the EBRs calculated for his own policy; regular
bonuses are declared on each policy based on its own bonus earning power.
171
We look at the historical and simulated results together for all the policies issued
in the whole 50 years. A summary statistic for the EBRs and regular bonus rates over
each policy term is calculated so that we can compare the equity proportions and
regular bonuses between different cohorts. In the ith simulation and for the policy
issued at time d, we calculate the arithmetic average EBR e(d, i), and geometric
average regular bonus rate b(d, i) over the policy term T ,
e(d, i) =
∑T−1t=0 e(d, t, i)
T(8.94)
b(d, i) =
(T−1∏t=1
[1 + b(d, t, i)]
) 1T−1
− 1 (8.95)
in which e(d, t, i) is the EBR at duration t and b(d, t, i) is the regular bonus rate at
duration t.
Notice that for those policies whose final EBR has been decided and final regular
bonus has been declared by the end of 2002 (i.e. those policies issued by the end of
1993), the EBRs and bonus rates are calculated based on the historical data. The
policies issued after the end of 2023 are still in force at the end of the investigation
period. For these cohorts, we calculate the average EBR and average bonus rate
over their policy duration.
Figure 8.46 shows the quantiles of the arithmetic average EBRs for each policy
issued during the 50 years. The horizontal axis gives the time of policy issue. Notice
that different quantiles coincide for policies issued before the end of 1993.
The historical results show that the average equity proportion has decreased
through the time of policy issue. The EBRs for the early cohorts (issued at the end
of 1982, 1983 and 1984) are above 90% on average over the policy term, whereas the
average EBR is only around 64% for the 1993 policy. According to our investment
strategy, the minimum average EBR over the policy term is 47.3% and the maximum
average is 98.4%.
For policies issued between the end of 1994 and the end of 2002, the average EBR
is calculated from both the historical and simulated equity proportions. We see in
Figure 8.46 that the quantiles for these policies spread out because the simulated
EBRs have a greater impact on the average for the later cohorts.
172
time of issue
equi
ty b
acki
ng r
atio
(%
)
1980 1990 2000 2010 2020 2030
4060
8010
0
2023
99% quantile90% quantile75% quantile50% quantile
25% quantile10% quantile1% quantile
Figure 8.46: The quantiles of the average EBRs over the policy term for each policyin Case C
For policies issued between the end of 2003 and the end of 2023, the average is
calculated from the simulated EBRs over the whole policy term. Figure 8.46 shows
that if the insurer continues writing new business, in the worst 1% of the simulations
the average equity proportion will be at the minimum level; in the 50% quantile,
around 70% of the asset share will be in equities on average; and in the best 1% of
the simulations, the average EBR for most cohorts will be around 97%.
For policies issued after the end of 2023, the average is taken over the policy
duration. Hence, different quantiles converge to the three possible initial ratios of
72%, 80% and 88%.
We can infer from Figure 8.46 that the equity proportion in 1982 is too high. In
at least 1% of the simulations the EBR should have been cut more quickly, because
it reaches its minimum possible value.
Figure 8.47 shows the quantiles of the geometric average regular bonus rates
declared on each cohort. Again, different quantiles coincide for policies issued before
the end of 1993.
The average bonus rate, calculated from the historical results, has decreased very
rapidly from around 12% p.a. on the earliest cohort to less than 1% p.a. on the
1993 policy.
173
time of issue
bonu
s ra
te (
%)
1990 2000 2010 2020 2030
05
1015
1982 2023
99% quantile90% quantile75% quantile50% quantile25% quantile10% quantile1% quantile
Figure 8.47: The quantiles of the average regular bonus rates declared over thepolicy term for each policy in Case C
For policies issued between the end of 1994 and the end of 2001, the average
bonuses are similar for different quantiles. Figure 8.47 shows that in most cases the
insurer cannot afford to declare any bonus on recently issued policies (issued at the
end of 1997, 1998, ..., 2001) in a low inflationary environment.
If the insurer continues writing new business, Figure 8.47 shows that in the worst
25% of the simulations, the average bonus rate for policies issued between the end
of 2002 and 2023 will be less than 1% p.a. However, notice that the regular bonus
is declared on top of a 2% p.a. guaranteed growth rate in the unit price aiming for
a 30% terminal bonus target under our standard basis. If the insurer reduces the
guaranteed rate or sets the initial guarantee to be less than the single premium or sets
a lower terminal bonus target, regular bonuses will be sustainable with probability
higher than 75%. In the 50% quantile, a bonus rate of around 2% p.a. can be
declared in addition to the guaranteed rate; and, in the 75% quantile, the insurer
can on average declare a regular bonus of around 5% p.a.
The policies issued after the end of 2023 will not have matured by the end of
2032. We calculate their average bonus rate over their policy duration which de-
creases through the time of policy issue. In our bonus strategy, the bonus earning
power is calculated from the 25th percentile of the projected maturity asset share
174
year
asse
t sha
re
1980 1990 2000 2010 2020 2030
010
000
2000
030
000
4000
0 99% quantile90% quantile75% quantile50% quantile25% quantile10% quantile1% quantile
Figure 8.48: The quantiles of the portfolio asset shares in Case C
with allowance for a terminal bonus. In the simulated real world at high quantiles,
this bonus strategy is too cautious and hence we expect that higher bonuses can
be declared at later durations. Therefore, we see in Figure 8.47 that the average
bonus rate decreases at high quantiles and the 99% quantile shows the most obvious
decreasing trend.
8.4 Portfolio Asset Share and Guarantee
Figures 8.48 and 8.49 show the quantiles of the simulated portfolio asset shares
and guarantees respectively, during the extended 30-year period. Notice that the
corresponding quantiles are not directly comparable because they are not necessarily
picked out from the same simulation. The historical results in the last 20 years are
also given in the figures.
The quantiles of the simulated portfolio asset shares and guarantees show a sim-
ilar pattern that they spread out through time. The 10,000 simulations start from
the same initial conditions at the end of 2002, but the standard deviation of the
investment performance increases through time.
Figures 8.48 and 8.49 show that the 1% quantile of both the portfolio asset shares
and guarantees will increase slowly through time, and the 99% quantile will increase
175
year
guar
ante
e
1980 1990 2000 2010 2020 2030
010
000
2000
030
000
4000
0
99% quantile90% quantile75% quantile50% quantile25% quantile10% quantile1% quantile
Figure 8.49: The quantiles of the portfolio guarantees in Case C
very rapidly. However, notice that in one particular simulation, both the asset share
and guarantee could decrease at some time. We will look at a few sample paths
in Appendix D. The portfolio asset share increases partly due to the investment
return earned in the unit fund and partly due to the fact that the single premium
increases with the retail price index. Inflation also affects the portfolio guarantee
because regular bonuses are linked to investment return and the intial guarantee for
each cohort increases with inflation.
An easy way to compare the asset share with the guarantee is to look at the asset
share to guarantee (AS/G) ratio. If the ratio is below 1.0, the guaranteed payout
has a larger amount than the policyholders’ assets. Figure 8.50 shows the quantiles
of the simulated AS/G ratios during the extended 30-year period. The historical
ratios in the last 20 years are also shown in the figure.
We can see in Figure 8.50 that when the portfolio is building up during the first
10 years of the investigation period, the AS/G ratio is mostly well below 1.0. The
insurer has promised a 2% growth rate in the unit price, and the initial guarantee
for each cohort is 100% of the single premium rolled up at the guaranteed rate.
Therefore, at the beginning of the business with only one issued policy, the AS/G
ratio is 0.6220. Afterwards, the ratio shows an increasing trend and reaches 1.3460
at the end of 1999. In recent years the investment market has given a poor return,
176
year
AS
/G r
atio
1990 2000 2010 2020 2030
0.5
1.0
1.5
2.0
99% quantile90% quantile75% quantile50% quantile25% quantile10% quantile1% quantile
Figure 8.50: The quantiles of the AS/G ratios in Case C
so the ratio reduces to 0.8931 at the end of 2002. During the following 30 years, the
mean-reverting feature of the Wilkie model will pull up the AS/G ratios because the
starting force of inflation is below the long-run mean of 4.7%. After the end of 2010
the quantiles of the simulated ratios are stable. The 50% quantile will be around
1.0; in the worst 1% of the simulations the ratio will be below 0.70; and in the best
1% of the cases the ratio will be above 1.55. However, note the guarantee does not
need to be paid immediately so that a ratio below 1.0 does not mean insolvency.
8.5 Maturing Policies
One policy matures each year during the extended period. This section looks at all
the maturing policies in the period 2002-2032, which are issued at the end of 1993,
1994, ..., 2022 respectively.
At maturity, the policyholder receives the greater of the asset share and the
guarantee. The excess of the asset share over the guarantee is paid as a terminal
bonus which can never be negative. From the 10,000 simulations, we calculate
the mean and standard deviation of the simulated maturity asset shares, maturity
guarantees, maturity payouts and terminal bonus rates. We also count the number
of simulations in which the guarantee bites at maturity. These statistics for the 30
177
maturing policies are given in Table 8.59.
Table 8.59: The statistics for the maturing policies during the 30-year extendedperiod
Issued at maturity asset share maturity guarantee maturity payout number terminal bonus rate31/Dec/ mean s.d. mean s.d. mean s.d. (A < G) mean (%) s.d. (%)
1993 276.29 28.37 224.64 0.00 276.41 28.13 161 23.04 12.521994 342.41 45.02 264.88 0.46 342.64 44.57 228 29.34 16.721995 331.62 47.56 251.08 0.66 331.82 47.19 198 32.13 18.621996 324.34 48.64 247.15 0.78 324.57 48.23 213 31.29 19.291997 291.66 45.91 241.23 0.16 293.01 44.02 998 21.46 18.201998 270.97 45.55 247.84 0.12 278.00 38.15 3224 12.17 15.361999 245.12 44.71 252.20 0.09 266.30 27.97 6047 5.59 11.062000 274.88 61.78 259.66 0.46 289.82 48.81 4460 11.59 18.612001 336.98 105.54 262.63 2.77 342.65 99.95 1998 30.23 36.602002 476.16 200.38 281.69 10.94 477.38 199.00 532 68.04 65.132003 538.53 262.23 339.18 91.01 539.72 261.09 504 56.26 50.542004 586.04 315.00 370.63 130.36 587.40 313.74 527 55.31 49.112005 629.10 358.74 397.80 157.96 630.51 357.50 541 55.09 48.932006 675.14 408.88 425.94 185.64 676.62 407.59 531 55.29 50.452007 718.28 457.51 454.70 210.52 719.92 456.17 573 54.35 48.912008 764.29 498.99 482.98 235.89 765.91 497.66 539 54.59 48.212009 814.93 539.58 514.12 262.43 816.54 538.33 511 54.98 48.562010 864.59 582.78 544.65 286.65 866.24 581.41 526 55.46 49.182011 919.46 631.90 577.94 310.60 921.13 630.71 518 55.65 48.962012 976.93 683.93 615.39 343.59 978.60 682.73 483 55.54 48.792013 1036.47 740.33 651.76 380.03 1038.14 739.11 463 55.50 47.412014 1093.81 789.48 687.82 408.52 1095.49 788.52 432 55.54 47.582015 1153.62 854.53 725.91 440.23 1155.23 853.68 420 55.24 47.432016 1220.98 919.52 769.51 483.34 1222.63 918.46 435 55.38 47.312017 1300.17 994.69 819.76 521.88 1301.86 993.67 396 55.17 46.382018 1376.75 1082.71 865.86 561.02 1378.43 1081.82 377 55.27 45.812019 1456.33 1173.69 914.93 608.17 1457.99 1172.85 375 55.37 46.022020 1532.60 1238.03 965.57 648.25 1534.46 1236.98 386 55.10 45.602021 1622.46 1334.14 1019.55 686.43 1624.59 1333.05 421 55.17 46.272022 1712.05 1440.79 1077.79 757.95 1714.33 1439.71 393 55.20 45.76
The mean of the maturity asset shares, guarantees and payouts increases through
the time of policy issue mainly due to inflation. Inflation also affects the standard
deviation. Notice that the standard deviation of the maturity asset shares, guaran-
tees, payouts and terminal bonus rates for the policies issued in the 1990s is much
lower than that for the other policies. The 1990s cohorts are already in force when
simulation starts, hence the simulated values for the maturity asset share and guar-
antee depend partly on the historical data. No regular bonus is declared at maturity,
hence the guarantees for the 1993 policy only depend on the historical investment
performance. Therefore, the standard deviation of the maturity guarantees for the
1993 policy is zero. The standard deviation of the maturity guarantees for policies
issued between the end of 1994 and the end of 2000 is very low because the insurer
almost always declares the smallest bonus allowed.
178
For most of the maturing policies the asset share is on average greater than
the guarantee at maturity. For the 1999 policy, however, the mean asset share is
smaller. There are 6,047 (among 10,000) simulations in which the guarantee will bite
for this particular cohort. Table 8.59 shows that at maturity the simulated asset
shares have much higher volatility than the guarantees. The guarantee for each
individual policy builds up with its regular bonuses which are constrained by the
smoothing mechanism, whereas the individual asset share bears the full volatility in
the investment market.
In Table 8.59, the mean of the maturity payouts is larger than that of both the
asset shares and guarantees at maturity, and the standard deviation of the payouts
is between that of the asset shares and the guarantees. This is intuitively reasonable
because the maturity payout equals the greater of the asset share and the guarantee.
At most policy maturity dates, the guarantee bites rarely and so the policyholder
receives only the asset share. Thus, the mean of the payouts is only slightly larger
than that of the asset shares; and the standard deviation of the payouts is slightly
smaller than that of the asset shares. However, for those cohorts whose guarantee
bites in a number of simulations, the mean of the payouts is much larger than that
of the asset shares; and the standard deviation of the payouts is much smaller than
that of the asset shares. In the long-run the maturity payout is only £2 more than
the maturity asset share on average, which is consistent with the results shown in
the single policy case. We have seen in Table 8.56 that only in 246 (among 10,000)
simulations does the guarantee bite at maturity if the real world follows the Wilkie
model starting with the neutral initial conditions.
We also see in Table 8.59 that for most of the maturing policies, the probability
of having insufficient asset share at maturity is less than 6%. It looks as if it was
unprofitable for the insurer to have written any business (under our standard basis)
in 1997, 1998, ..., 2001. We have seen in Chapter 7 that the insurer cannot afford
to declare any regular bonus in respect of these policies by the end of 2002. Here
Table 8.59 shows that when these cohorts mature, there is a high probability of the
guarantee being called upon. For the 1999 policy, in particular, there is a 60.47%
chance that the guarantee will bite. The economy has recently moved into a low
179
inflation period and hence the investment market has given a low return. During the
future 30 years, the real world is simulated by the Wilkie model which has a mean-
reverting feature. Thus, in the long term the force of inflation is expected to revert to
the mean level of 4.7% and the economy will recover. If the insurer continues writing
new business, the UWP policies are expected to provide an enhanced investment
return to the policyholders.
On most of the maturing policies, the mean of the terminal bonus rates is higher
than the initial target of 30%. The mean rate declared on the 1999 policy is only
5.59% because in most simulations no terminal bonus can be declared. Notice that
the probability of zero terminal bonus on the 2001 policy is 10% higher than the
same probability in respect of the 1997 policy, but the mean bonus rate on the 2001
policy is almost 10% higher. Hence we can infer that in some simulations a very
large terminal bonus is declared on the 2001 policy.
Recall that in Section 7.3.3 the insurer was able to pay more than the guaranteed
amount on the policies issued between 1982 and 1992. Hence the insurer has had
a long period in which the guarantee does not bite. It is only by performing the
projections that we can show the insurer’s potential losses.
Table 8.59 shows that even if we are back into a higher inflation period (the
mean force of inflation assumed in Wilkie (1995) is 4.7%), the probability of loss at
maturity is still around 4%. If we had re-parameterised the model for low inflation,
the situation would be even worse.
8.6 Portfolio Reserves Set up Using the Three
Reserving Approaches
The same methodology as in Chapter 7 can be used to simulate the required amount
of reserves for the whole portfolio under the three reserving approaches. This section
shows different quantiles of the simulated reserves.
180
year
rese
rve
1980 1990 2000 2010 2020 2030
020
040
060
080
010
0012
00 99% quantile90% quantile75% quantile50% quantile25% quantile10% quantile1% quantile
Figure 8.51: The quantiles of the portfolio reserves set up using the option methodin Case C
8.6.1 Option and Hedging Approaches
The first two approaches give the same amount of reserves, which is shown in Figure
8.51. The quantiles of the simulated portfolio reserves during the extended 30-year
period, as well as the historical results in the last 20 years, are given in the figure.
Some sample paths are shown in Appendix D.
In Figure 8.51, the quantiles spread out because the standard deviation of the
simulated reserves increases through time.
In the best 1% of the simulations, Figure 8.51 shows that the portfolio reserves
will be very small. However, reserves are always required under the option pric-
ing approach because the guarantee always has a cost. In the 50% quantile, the
portfolio reserves will be less than £200, but the 99% quantile shows a very rapid
increase. Most quantiles will increase through time, which is partly due to the effect
of inflation.
We have seen in the previous chapter that the guarantees for policies issued in the
late 1990s are mostly in-the-money by the end of 2002. Afterwards, these guarantees
continue being expensive in some simulations. Therefore, Figure 8.51 shows that the
portfolio reserve in the higher quantiles will keep increasing after the end of 2002
but will decrease in the late 2000s after these problematic policies mature.
181
year
rese
rve
1980 1990 2000 2010 2020 2030
020
040
060
080
0 99% quantile90% quantile75% quantile50% quantile25% quantile10% quantile1% quantile
Figure 8.52: The quantiles of the 99% portfolio CTE reserves in Case C
8.6.2 CTE Reserving
Here we consider only the 99% CTE reserves for the whole portfolio, because the
95% portfolio CTE reserves are often zero and 99% CTE reserves have been more
comparable to the reserves set up using the option pricing approach. In Appendix
D a few sample paths for both 95% and 99% CTE reserves are given. The quantiles
of the simulated 99% CTE reserves, as well as the historical results, are shown in
Figure 8.52.
The corresponding quantiles given in Figures 8.51 and 8.52 are not directly com-
parable. At each point in time during the following 30 years, they are probably
picked out from the different simulations.
Figure 8.52 shows that in the 50% quantile, no reserves will be required at all
after the end of 2018, and in the best 1% of the simulations, the portfolio reserve will
reduce to zero only one year after the simulation starts. However, the 99% quantile
will increase rapidly through time.
As under the option pricing approach, the portfolio reserves in the higher quan-
tiles will keep increasing after the end of 2002 but then decrease in the late 2000s for
a similar reason that the policies issued in the late 1990s have much larger guarantees
than their asset shares before the simulation starts.
The 99% CTE reserves are much smaller at all quantiles than the reserves set up
182
year
free
est
ate
1980 1990 2000 2010 2020 2030
020
0040
0060
00 99% quantile90% quantile75% quantile50% quantile25% quantile10% quantile1% quantile
Figure 8.53: The quantiles of the insurer’s free estate using the option method inCase C
under the option pricing approach. Hence CTE reserving is cheaper for the insurer,
but this does not mean that the CTE reserve is smaller in all simulations.
8.7 Free Estate
The insurer’s free estate in the last 20 years has been calculated in Chapter 7,
assuming that the insurer has no free estate before selling the first policy at the end
of 1982. The simulated cashflows incurred for the portfolio during the extended 30
years are rolled up at the simulated zero-coupon yields to investigate how the estate
will build up in each simulation. This section looks at the quantiles of the simulated
free estate under the three reserving approaches, and some sample paths are given
in Appendix D.
8.7.1 Option Approach
Figure 8.53 shows the quantiles of the simulated free estate under the option ap-
proach, as well as the historical results during the last 20 years.
The figure shows that in the worst 1% of the simulations, the free estate of
£175.08 at the end of 2002 will be used up to set reserves for the portfolio and a
183
year
free
est
ate
1980 1990 2000 2010 2020 2030
020
0040
0060
0080
00
99% quantile90% quantile75% quantile50% quantile25% quantile10% quantile1% quantile
Figure 8.54: The quantiles of the insurer’s free estate by discrete hedging in Case C
deficit will occur at the end of 2008. The deficit will increase rapidly to £436.34 at
the end of 2022. Afterwards cashflows will gradually come into the free estate and
the deficit will be reduced, but by the end of 2032, the insurer will still have a small
deficit of £1.71. The 10% quantile reflects a significantly different situation from the
1% quantile. The free estate decreases temporarily after the end of 2002, but from
the end of 2009 onwards the estate will build up steadily and it will end up with a
surplus of around £1296 at the end of 2032. In the best 1% of the simulations, the
free estate will keep increasing in the following 30 years and the surplus at the end
of 2032 will be over £6550.
Therefore, the UWP policies are profitable in all but the rarest cases, even though
the guarantees are very likely to bite for recently issued policies.
8.7.2 Hedging Approach
Figure 8.54 shows the quantiles of the simulated free estate and the historical results
under the hedging approach.
The corresponding quantiles in Figures 8.53 and 8.54 show a similar pattern.
We have seen in Chapter 7 that policies are more profitable to the insurer who
hedges the risk internally instead of buying options over-the-counter. The insurer
with discrete hedging has a much larger free estate at the end of 2002. However,
184
year
free
est
ate
1980 1990 2000 2010 2020 2030
020
0060
0010
000
1400
0
99% quantile90% quantile75% quantile50% quantile25% quantile10% quantile1% quantile
Figure 8.55: The quantiles of the insurer’s free estate by setting up the 99% portfolioCTE reserves in Case C
Figure 8.54 shows that in the worst 1% of the cases the simulated free estate de-
creases more rapidly under the hedging approach because of hedging losses. The
estate will be used up at the end of 2010 to set up reserves as required and a deficit
will occur. The amount of the deficit will increase to £666.22 at the end of 2028, and
the insurer will have a deficit of around £537 at the end of 2032. Hence hedging also
works well, but there is some extra risk to the insurer compared to buying options.
8.7.3 CTE Approach
The quantiles of the simulated free estate of the insurer who sets up 99% CTE
reserves, together with the historical results, are shown in Figure 8.55.
We see in Figure 8.55 that the quantiles of the simulated results decrease tem-
porarily at the end of 2003 (and 2004 for the 1% quantile) and afterwards the estate
will build up steadily. Even in the worst 1% of the simulations the free estate built
up in the last 20 years is big enough to set up the required amount of reserves. In
the best 1% of the cases, the insurer will have a surplus of £14270.51 at the end of
2032.
The corresponding quantiles in Figures 8.53, 8.54 and 8.55 are not directly com-
parable because they might be picked out from different simulations.
185
Thus, the UWP business is profitable as long as the real world does follow the
Wilkie model with the 4.7% mean force of inflation. The CTE approach works better
from the insurer’s point of view, but policyholders would prefer the 100% security
of options.
8.8 Summary
In this chapter we have extended the investigation period to the end of 2032 and
assumed that the insurer continues selling one policy each year. The real world in
the 30-year period of 2002 to 2032 has been simulated by the Wilkie model starting
with the initial conditions at the end of 2002.
We have combined the numerical results obtained in the whole 50-year investi-
gation period. For each cohort, we have calculated the average EBR and average
regular bonus rate. We have also simulated the required amount of reserves for the
portfolio under different reserving approaches and investigated how the insurer’s free
estate will build up in the future. Different quantiles of the simulated results have
been shown in this chapter.
The main conclusions for the business, under our standard basis and in the case
of smoothing with allowance for future bonuses, are summarised as follows:
• The arithmetic average EBR calculated only from the historical results has
decreased through the time of policy issue. The average EBR calculated only
from the simulated results will be at a minimum level in the worst 1% of the
simulations; at around 70% in the 50% quantile; and at around 97% in the
best 1% of the cases.
• The geometric average regular bonus rate calculated only from the historical
results has decreased through the time of policy issue. The average bonus rate
calculated only from the simulated results will be less than 1% p.a. (on top
of the 2% p.a. guaranteed growth rate and aiming for a 30% terminal bonus
target) in the worst 25% of the simulations; at around 2% p.a. in the 50%
quantile; and at around 5% p.a. in the 75% quantile.
186
• The quantiles of the simulated results spread out because the standard devia-
tion increases through time, and then become stable once the transition from
historical data to simulation is complete.
• The policies are not sustainable in a low inflationary environment. It is not
wise for the insurer to write any business in 1997, 1998, ..., 2001. The insurer
cannot afford to declare any regular bonus in respect of these policies by the
end of 2002. Also, there is a high probability that the guarantees on these
policies will bite at maturity.
• In the best 1% of the simulations, the portfolio reserves will be small under
the option pricing approach but they are always required; whereas the 99%
portfolio CTE reserves will reduce to zero only one year after the simulation
starts. The 50% quantile will be less than £200 under the option pricing
approach; whereas the 50% quantile of the simulated 99% CTE reserves will
reduce to zero after the end of 2018. The 99% quantile will increase rapidly
through time under all three reserving approaches.
• In the worst 1% of the simulations, the free estate held at the end of 2002 will
be used up in the future to set up reserves as required under the option and
hedging approaches and a deficit will occur; whereas the estate is big enough
to set up 99% CTE reserves. In the best 1% of the cases, the insurer will have
a large free estate at the end of 2032 under all three reserving approaches.
187
Chapter 9
CONCLUSIONS AND FURTHER
RESEARCH
A new prudential regulatory regime for with-profits funds, developed by the Finan-
cial Services Authority recently, demonstrates a move from the traditional valuation
approach to a market consistent approach. In this thesis, we have investigated
the reserves required to meet the maturity guarantees under unitised with-profits
policies, within the realistic reporting framework. The required amount of reserves
under the market consistent approach has been compared with that using traditional
stochastic valuation techniques.
The new research performed in this thesis can be summarised as follows:
• reserving for maturity guarantees under UWP policies for the realistic balance
sheet calculation but using a closed form approach
• comparing the required amount of reserves for UWP policies using modern
option pricing theory and traditional stochastic valuation techniques
• calculating reserves for UWP policies using both historical data and stochastic
simulation
• modelling the insurer’s decision rules for bonus and investment strategies
• investigating the sustainability of UWP policies
188
• considering dynamic investment and bonus strategies which are related to each
other in that they are both based on the projected performance of the unit
fund
• allocating assets and declaring bonuses for each cohort of business separately
so that there is no subsidisation between different generations.
We summarise the main conclusions of this thesis in Section 9.1. Then in Section
9.2 we give suggestions for further research.
9.1 Conclusions
Chapter 1 began by describing the operation of UWP policies. Then we reviewed
some of the literature on reserving for policies with financial guarantees. The three
reserving approaches and the models used throughout the thesis were discussed.
In Chapter 2 we looked at a single UWP policy with a term of 10 years issued
at the end of 1991. The unit price is guaranteed to grow at 2% p.a., and 1% of the
asset share is deducted at the end of each year to pay for the cost of guarantees. We
simply assumed a 100% equity backing ratio in the asset share, a 5% p.a. regular
bonus rate, and a 5% p.a. risk-free interest rate. The guarantee for the policy does
not bite at maturity. The reserves show a decreasing trend under the option pricing
approach but the CTE reserves show an increasing trend. The 1991 policy is only
profitable to the insurer who sets up 95% CTE reserves. The loss is smaller if the
insurer hedges the risk internally rather than buying options from a third party.
However, the three reserving approaches are not directly comparable.
Chapter 3 introduced a dynamic bonus strategy which uses a bonus earning power
mechanism. The bonus earning power is defined as the bonus rate which can be
declared now and at each future bonus declaration date given the current guarantee
with a 75% probability of achieving at least the 30% terminal bonus target. Three
cases were considered:
A: the bonus rates are not smoothed, and future regular bonuses are not reserved
for,
189
B: the bonus rates are smoothed so that they are not allowed to increase by more
than 20% or decrease by more than 16.67% from year to year, and future
regular bonuses are not reserved for,
C: the bonus rates are smoothed as in Case B, and the minimum of future regular
bonuses implied by the smoothing mechanism are reserved for.
The unsmoothed dynamic bonus rates on the 1991 policy are mostly lower than
the 5% p.a. assumed in our static bonus strategy. Hence the guarantees and reserves
are smaller in Case A than those in the case of static bonuses. Smoothing bonuses in
Case B slightly increases the guarantees and reserves, but they are greatly increased
with allowance for future bonuses in Case C. The difference between Cases B and C is
more significant at early durations when more future bonuses are included in Case
C. The reserves decrease over the policy term under the option pricing approach
except for a temporary increase at the end of 1994, but there is no obvious trend
shown in the CTE reserves. The 1991 policy is not profitable if reserves are set up
using the option pricing approach or if 99% CTE reserves are set up. Smoothing in
Case B and reserving for future bonuses in Case C both increase the amount of the
loss, and the latter has a much stronger impact. The policy is only profitable to the
insurer who sets up 95% CTE reserves ignoring future regular bonuses.
In Chapter 4, the yield on the zero-coupon bond with the same maturity date as
the policy was used as a risk-free interest rate. The zero-coupon yield was derived
from the consols yield and short-term interest rate using a simple yield curve. We
only considered the case of smoothing with allowance for future bonuses. The zero-
coupon yield is mostly higher than 5% in the particular 10-year period. The reserves
required under the option pricing approach are greatly reduced at early durations
by the dynamic risk-free rate. The reserves increase temporarily at the end of
1993 when the zero-coupon yield drops sharply. The CTE reserves are also smaller
with the zero-coupon yields, but the reduction is not so significant as under the
option pricing approach. The 1991 policy is profitable to the insurer under all three
reserving approaches with the variable risk-free rate.
Chapter 5 looked at the single policy in the most complicated case. We assumed
that the unit fund is invested in two asset classes: equities and the zero-coupon
190
bonds with the same maturity date as the policy. The percentage in either asset is
adjusted according to a dynamic investment strategy. The corridor approach was
used. We ran simulations and calculated the probability that the projected asset
share is sufficient to pay the guarantee at maturity. The asset share is switched
out of equities to zero-coupon bonds by 10% if the probability falls below 97.5%,
and switched from zero-coupon bonds back into equities by 10% if the probability
rises above 99.5%. We considered only the case of smoothing with allowance for
future bonuses. The EBRs for the 1991 policy are not very different from the
initial suggested ratio of 80%, which implies that the insurer is in a relatively strong
prospective solvency position. The required amount of reserves is greatly reduced
after introducing a risk-free asset. There is no need to set up 95% CTE reserve
during the policy term and the 99% CTE reserve reduces to zero at later durations.
The policy is profitable under all three reserving approaches and the insurer earns
a larger profit by investing the unit fund in both risky and risk-free assets, but at
the cost of a smaller maturity payout to the policyholder.
Chapter 6 investigated the sensitivity of the single policy results to different
contract design and market parameters, to different probability boundaries in the
dynamic investment strategy, and to different 10-year periods. The main conclusions
from the sensitivity test in the case of smoothing with allowance for future bonuses
are summarised as follows:
• For the 1991 policy, promising a 0% p.a. growth rate in the unit price instead
of 2% p.a. reduces the reserves at early durations but larger reserves are
required later on; deducting 0.5% of the units at the end of each year as a
guarantee charge instead of 1% greatly reduces the profitability of the policy;
aiming for a 50% terminal bonus target in a bonus declaration instead of 30%
reduces the required amount of reserves; increasing the volatility for the equity
index, assumed in the Black-Scholes equation, from 20% to 25% leads to larger
reserves and greatly reduced profitability of the policy under the option pricing
approach; increasing the rate of transaction costs, under the hedging approach,
from 0.2% to 0.5% slightly reduces the profitability of the policy; increasing
the additional rate of return, required by the capital providers on top of the
191
risk-free rate, from 3% p.a. to 5% p.a. reduces the profit earned by the insurer.
• Changing the upper and lower probability boundaries from 99.5% and 97.5% to
99% and 95% respectively leads to a less cautious investment strategy. For the
1991 policy, larger reserves are required under all three reserving approaches;
the insurer earns a smaller profit using the option pricing approach or setting
up 99% CTE reserves.
• The investment market performs quite differently during different 10-year pe-
riods. Among the three policies issued at the end of 1982, 1991 and 1992,
the 1982 policy provides the highest investment value to the policyholder; it
requires more reserves than the other two policies, particularly at later dura-
tions; it is least profitable to the insurer using the option pricing approach, but
the most profitable to the insurer who sets up CTE reserves; The profitability
of the 1991 and 1992 policies is not very different.
Chapter 7 considered a portfolio of UWP policies historically with a 20-year
investigation period starting at the end of 1982. Each year one single premium policy
with a term of 10 years is issued. The dynamic investment and bonus strategies are
applied to each cohort separately. As in Chapter 3, three cases were considered.
We investigated the benefits to the insurer of pooling risks. The sensitivity of the
portfolio results to different parameters and to different probability boundaries in
our investment strategy was also investigated. The main conclusions are summarised
as follows:
• Under the standard basis and with probability boundaries of 97.5% and 99.5%
to adjust EBRs, regular bonuses are no longer sustainable in a low inflationary
environment according to our bonus strategy, so the insurer needs to reduce
the guaranteed growth rate or set the initial guarantee to be lower than the
single premium or set a lower terminal bonus target in order to increase the
bonus earning power; the guarantee does not bite on any of the maturing
policies; smoothing regular bonuses in Case B does not make much difference
to the maturity payouts, but most policyholders receive a smaller payout in
Case C where future bonuses are allowed for; there is an advantage to the
192
insurer of pooling risks using the CTE approach (and discrete hedging by
means of saving transaction costs), but pooling risks gives no advantage under
the option approach; allowing for future bonuses greatly increases the reserves
in the first decade; the 99% CTE reserves show a similar pattern to the reserves
set up using the option pricing approach; the policies have been profitable in
the last 20 years, but the insurer does need some capital to start the business
under the option and hedging approaches and the deficit lasts for a much longer
period with allowance for future bonuses; the policies are more profitable to
the insurer who hedges the risk internally rather than buying options from a
third party.
• Promising a 0% p.a. growth rate instead of 2% p.a. mostly reduces the
portfolio reserves under the option pricing approach and the policies are more
profitable under all three reserving approaches; deducting 0.5% of the units
p.a. as a guarantee charge instead of 1% increases the portfolio CTE reserves
and the policies have not been profitable over the last 20 years under the
option pricing approach; increasing the terminal bonus target from 30% to 50%
mostly leads to smaller portfolio reserves under all three reserving approaches;
increasing the volatility for the equity index, assumed in the Black-Scholes
formula, from 20% to 25% increases the portfolio reserves under the option
pricing approach and the policies have not been profitable in the last 20 years
to the insurer using the option method; increasing the transaction costs rate,
assumed in discrete hedging, from 0.2% to 0.5% has no impact on the portfolio
reserves but reduces the profit earned by the insurer.
• Changing the probability boundaries in the investment strategy from 99.5%
and 97.5% to 99% and 95% respectively leads to higher EBRs for most cohorts;
larger reserves are required under all three reserving approaches; the appear-
ance of the surplus is postponed under the option and hedging approaches.
Chapter 8 extended the investigation period to the end of 2032 and assumed that
the insurer continues to sell one policy each year. The real world during the period
2002-2032 is simulated by the Wilkie model starting with the initial conditions at
193
the end of 2002. We combined the numerical results obtained in the whole 50-
year investigation period. For each cohort, we calculated the average EBR and
average regular bonus rate over the policy term. We also simulated the amount of
the portfolio reserves under different reserving approaches and investigated how the
insurer’s free estate would build up in the future. For policies issued before the end
of 1993, the average EBR and the average regular bonus rate decrease through the
time of policy issue. The average EBR calculated only from the simulated results
will be at a minimum level according to our investment strategy in the worst 1%
of the simulations, and the corresponding average regular bonus rate will be less
than 1% p.a. (on top of a 2% p.a. guaranteed rate and aiming for a 30% terminal
bonus target) in the worst 25% of the cases. The policies are not sustainable in a
low inflationary environment. There is a high probability that the guarantees on
the policies issued at the end of 1997, 1998, ..., 2001 will bite at maturity. In the
worst 1% of the simulations, the portfolio reserves will increase rapidly through time
under all three reserving approaches; the free estate held by the insurer at the end
of 2002 will be used up in the future to set up the reserves as required under the
option pricing approach and a deficit will occur. In the best 1% of the simulations,
the portfolio reserves will be of a small amount under the option pricing approach
but they are always required, and the 99% CTE reserve will reduce to zero only one
year after the simulation starts; the insurer will have a large free estate at the end
of 2032 under all three reserving approaches.
9.2 Suggestions for Further Research
Although this thesis has investigated the reserves required to meet the maturity
guarantees under UWP policies within the new realistic reporting framework, we
have used a simplistic closed form approach with very limited allowance for the in-
surer’s management actions. The options have been valued using the simple form
of the Black-Scholes equation where shares follow Geometric Brownian Motion with
constant volatility and the risk-free interest rate is also constant. To project for
reserving purposes, we have simply assumed that the assets will not be switched
194
in the future and that only the minimum of future regular bonuses implied by our
bonus strategy is allowed for. Further investigation can be undertaken on a dy-
namic simulation approach whereby the options are valued using market consistent
stochastic projection models with dynamic management actions incorporated.
Other possible lines for further improvements are indicated as follows:
• we could model the UWP policies in a more realistic way to allow for mortality,
lapses and expenses;
• we could smooth the maturity payout between different generations, then the
cost of future smoothing should be valued;
• we could set the charges to reflect the actual cost of guarantees;
• we could consider different investment and bonus strategies;
• we could use a different smoothing methodology for regular bonuses;
• we could calculate a single EBR for the whole unit fund;
• we could allow for subsidisation between different generations in a bonus dec-
laration;
• we could consider more frequent rebalancing in dynamic hedging;
• we could look at the effect of model error in our internal models by using
a different real world model, e.g. the regime switching lognormal model as
considered in Hardy (1999) and Hardy (2001);
• we could value the options using a model which assumes that the risk-free
interest rate and volatility of the equity index are stochastic.
195
Appendix A
Wilkie Model 1995 Version
The Wilkie investment model was first introduced in 1986 (Wilkie (1986)) and a
revised version was presented in 1995 (Wilkie (1995)). The 1986 version covered
retail price index, equity dividend index, equity dividend yield, and consols yield.
The parameters were estimated based on U.K. data over the period of 1919 to 1982.
The 1995 version extended the model to cover wages index, short-term interest rate,
property rentals, property yield and index-linked stock yield. The 1995 version
updated the parameter values based on U.K. data over the period of 1924 to 1994.
We only introduce the part of the Wilkie model 1995 version which is relevant
to the variables considered in the thesis: retail price index, equity dividend index,
equity dividend yield, consols yield and short-term interest rate. See Wilkie (1995)
for more details.
• Retail price index Q(t):
Q(t) = Q(t− 1) · eI(t)
with
I(t) = QMU + QA · (I(t− 1)−QMU) + QE(t)
where {QE(t)}∞t=1 is a sequence of i.i.d. (independently and identically dis-
tributed) random variables each distributed as Normal(0, QSD2). The esti-
mated parameters are:
QMU = 0.047, QA = 0.58, QSD = 0.0425.
196
• Equity dividend amount D(t):
D(t) = D(t− 1) · eDW ·DM(t)+DX·I(t)+DMU+DY ·Y E(t−1)+DB·DE(t−1)+DE(t)
with
DM(t) = DD · I(t) + (1−DD) ·DM(t− 1)
where {DE(t)}∞t=1 is a sequence of i.i.d. random variables each distributed as
Normal(0, DSD2). The estimated parameters are:
DW = 0.58, DD = 0.13, DX = 0.42, DMU = 0.016,
DY = −0.175, DB = 0.57, DSD = 0.07.
• Equity dividend yield Y (t):
Y (t) = Y MU · eY W ·I(t)+Y N(t)
with
Y N(t) = Y A · Y N(t− 1) + Y E(t)
where {Y E(t)}∞t=1 is a sequence of i.i.d. random variables each distributed as
Normal(0, Y SD2). The estimated parameters are:
Y W = 1.8, Y A = 0.55, Y MU = 0.0375, Y SD = 0.155.
The equity price index P (t) can be derived as follows:
P (t) =D(t)
Y (t).
• Consols yield C(t):
C(t) = CW · CM(t) + CMU · eCN(t)
with
CM(t) = CD · I(t) + (1− CD) · CM(t− 1)
CN(t) = CA · CN(t− 1) + CY · Y E(t) + CE(t)
197
where {CE(t)}∞t=1 is a sequence of i.i.d. random variables each distributed as
Normal(0, CSD2). The estimated parameters are:
CW = 1.0, CD = 0.045, CMU = 0.0305,
CA = 0.90, CY = 0.34, CSD = 0.185.
• Short-term interest rate B(t):
B(t) = C(t) · e−BD(t)
with
BD(t) = BMU + BA · (BD(t− 1)−BMU) + BE(t)
where {BE(t)}∞t=1 is a sequence of i.i.d. random variables each distributed as
Normal(0, BSD2). The estimated parameters are:
BMU = 0.23, BA = 0.74, BSD = 0.18.
The neutral initial conditions are defined as follows:
• I(0) = QMU = 4.7%
• Y (0) = exp(Y W ·QMU) · Y MU = 4.0811%
• C(0) = QMU + CMU = 7.75%
• B(0) = exp(−BMU) · C(0) = 6.1576%
• DM(0) = CM(0) = QMU = 4.7%
• Y E(0) = DE(0) = 0.
198
Appendix B
Investment Data and Derived
Initial Conditions
Table B.60 gives the market indices at 31 December of each year during the period
of 1964 to 2002.
Table B.61 gives the derived initial conditions for the 1995 version of the Wilkie
model at 31 December of each year during the period of 1964 to 2002.
The data shown in the two tables has been provided by Prof. Wilkie.
199
Table B.60: The market indices at 31 December of each year during the period of1964 to 2002
31 December Q(t) I(t)% Y (t)% D(t) C(t)% B(t)%1964 14.43 4.69 5.18 5.03 6.31 7.001965 15.08 4.39 5.22 5.42 6.48 6.001966 15.63 3.61 5.78 5.43 6.66 7.001967 16.02 2.42 4.38 5.31 7.06 8.001968 16.97 5.77 3.19 5.53 7.99 7.001969 17.76 4.57 3.85 5.67 8.76 8.001970 19.16 7.59 4.39 5.98 9.79 7.001971 20.89 8.65 3.25 6.29 8.46 5.001972 22.49 7.37 3.15 6.87 9.84 9.001973 24.87 10.05 4.77 7.14 12.25 13.001974 29.63 17.51 11.71 7.83 17.20 11.501975 37.01 22.23 5.47 8.65 14.67 11.251976 42.59 14.04 6.42 9.76 14.47 14.251977 47.76 11.46 5.28 11.33 10.46 7.001978 51.76 8.05 5.79 12.75 11.93 12.501979 60.68 15.90 6.87 15.79 12.23 17.001980 69.86 14.08 6.10 17.81 11.61 14.001981 78.28 11.37 5.89 18.44 13.43 14.501982 82.51 5.27 5.26 20.10 10.25 10.001983 86.89 5.18 4.62 21.74 9.71 9.001984 90.87 4.48 4.42 26.21 9.90 9.501985 96.05 5.53 4.33 29.57 9.80 11.501986 99.62 3.65 4.04 33.75 10.06 11.001987 103.30 3.63 4.32 37.59 9.21 8.501988 110.30 6.56 4.71 43.64 8.99 13.001989 118.80 7.42 4.24 51.08 9.66 15.001990 129.90 8.93 5.47 56.46 10.48 14.001991 135.70 4.37 5.02 59.62 9.71 10.501992 139.20 2.55 4.35 59.32 8.83 7.001993 141.90 1.92 3.37 56.69 6.52 5.501994 146.00 2.85 4.02 61.16 8.53 6.251995 150.70 3.17 3.80 68.52 7.78 6.501996 154.40 2.43 3.74 75.31 7.74 6.001997 160.00 3.56 3.23 77.88 6.39 7.251998 164.40 2.71 2.92 78.08 4.55 6.251999 167.30 1.75 2.36 76.37 4.89 5.502000 172.20 2.89 2.48 73.93 4.62 6.002001 173.40 0.69 2.92 73.75 5.04 4.002002 178.50 2.90 3.94 74.70 4.56 4.00
200
Table B.61: The derived initial conditions for the 1995 version of the Wilkie modelat 31 December of each year during the period of 1964 to 2002
31 December Y E(t) DM(t) DE(t) CM(t)1964 0.1815 0.0312 0.0321 0.02611965 0.0914 0.0328 0.0341 0.02691966 0.2001 0.0332 -0.0503 0.02741967 -0.1195 0.0321 -0.0052 0.02721968 -0.3561 0.0354 -0.0381 0.02861969 0.0612 0.0367 -0.0708 0.02941970 0.0226 0.0418 0.0320 0.03151971 -0.3393 0.0476 -0.0448 0.03391972 -0.1718 0.0510 -0.0211 0.03571973 0.1995 0.0575 -0.0709 0.03861974 0.7617 0.0728 0.0357 0.04481975 -0.5046 0.0922 0.0490 0.05281976 0.2684 0.0985 -0.1276 0.05671977 -0.0499 0.1006 0.1466 0.05931978 0.1856 0.0980 -0.0805 0.06031979 0.1309 0.1059 0.1478 0.06471980 0.0285 0.1104 -0.0798 0.06811981 0.0895 0.1109 -0.0428 0.07021982 0.0788 0.1033 0.0283 0.06941983 -0.0476 0.0966 -0.0180 0.06861984 -0.0087 0.0899 0.1021 0.06751985 -0.0309 0.0854 -0.0277 0.06701986 -0.0446 0.0790 0.0655 0.06561987 0.0424 0.0735 -0.0112 0.06431988 0.0390 0.0724 0.0775 0.06441989 -0.1003 0.0727 0.0307 0.06481990 0.1936 0.0748 -0.0317 0.06591991 0.0648 0.0708 0.0310 0.06491992 -0.0436 0.0649 -0.0757 0.06311993 -0.2269 0.0590 -0.0682 0.06121994 0.0670 0.0550 0.0153 0.05971995 -0.0829 0.0520 0.0571 0.05841996 -0.0513 0.0484 -0.0068 0.05691997 -0.2170 0.0467 -0.0297 0.05591998 -0.2107 0.0442 -0.0715 0.05461999 -0.3610 0.0407 -0.0652 0.05302000 -0.2223 0.0392 -0.1093 0.05192001 -0.0345 0.0350 -0.0183 0.04992002 0.1134 0.0342 -0.0309 0.0489
201
Appendix C
Details of Calculations for the
1991 Policy with a 5% Bonus
Rate, a 5% Risk-Free Rate and a
100% EBR
Tables C.62, C.63 and C.64 show the details of calculations in Chapter 2. In these
tables, V (t)− V ′(t) represents the increase in the reserve at time t.
Table C.62: The details of calculations for the 1991 policy under the option approach
31/Dec/ A(t) G(t) N(t) E(t) S(t) O(t) O′(t) V (t) V (t)− V ′(t) GC(t) CF (t)1991 100.00 121.90 0.0761 1600.80 1187.65 178.41 13.58 13.58 0.00 13.581992 118.62 127.99 0.0761 1680.84 1423.00 154.66 132.84 11.77 1.66 1.20 0.461993 149.74 134.39 0.0761 1764.89 1814.53 108.80 91.22 8.28 1.34 1.51 -0.181994 139.46 141.11 0.0761 1853.13 1707.05 156.53 131.88 11.91 1.88 1.41 0.471995 169.86 148.17 0.0761 1945.79 2100.08 109.65 89.70 8.34 1.52 1.72 -0.201996 194.81 155.58 0.0761 2043.08 2432.92 81.66 64.76 6.21 1.29 1.97 -0.681997 238.40 163.36 0.0761 2145.23 3007.30 38.61 28.81 2.94 0.75 2.41 -1.661998 269.38 171.52 0.0761 2252.49 3432.50 18.78 12.98 1.43 0.44 2.72 -2.281999 330.36 180.10 0.0761 2365.12 4252.01 2.18 1.24 0.17 0.07 3.34 -3.272000 308.76 189.11 0.0761 2483.37 4014.13 0.78 0.34 0.06 0.03 3.12 -3.092001 266.54 189.11 0.0761 3500.27 0.00 0.00 2.69 -2.69
202
Table C.63: The details of calculations for the 1991 policy under the hedging ap-proach
31/Dec H(t) H′(t) H′′(t) M ′(t) M ′′(t) TC(t) C(t)cash equity cash equity cash equity
1991 37.92 -24.33 0.00 0.00 0.05 13.631992 36.19 -24.42 31.97 -21.85 39.82 -29.16 -0.54 1.66 0.01 -0.071993 29.48 -21.20 25.45 -18.50 38.00 -31.14 0.08 1.34 0.02 -0.081994 41.19 -29.27 35.84 -25.80 30.96 -19.94 -0.97 1.88 0.02 -0.481995 33.79 -25.44 28.60 -21.77 43.25 -36.01 -0.41 1.52 0.02 -0.591996 29.03 -22.81 23.88 -18.95 35.48 -29.47 -1.08 1.29 0.01 -1.741997 17.30 -14.36 13.42 -11.23 30.48 -28.19 -0.09 0.75 0.03 -1.731998 10.58 -9.15 7.64 -6.65 18.17 -16.39 -0.79 0.44 0.01 -3.051999 1.86 -1.69 1.11 -1.02 11.11 -11.34 0.32 0.07 0.02 -2.932000 0.98 -0.92 0.45 -0.43 1.95 -1.60 -0.33 0.03 0.00 -3.412001 1.03 -0.81 -0.23 0.00 0.00 -2.92
Table C.64: The details of calculations for the 1991 policy under the CTE approach
31/Dec CTE (95%) CTE (99%)∑10000i=9501 PV CF ′′(t,i)
500V (t) V (t)− V ′(t) C(t)
∑10000i=9901 PV CF ′′(t,i)
100V (t) V (t)− V ′(t) C(t)
1991 -8.05 0.00 0.00 0.00 0.56 0.56 0.56 0.561992 -3.41 0.00 0.00 -1.20 12.94 12.94 12.35 11.151993 3.32 3.32 3.32 1.80 20.79 20.79 7.21 5.701994 4.32 4.32 0.84 -0.57 23.06 23.06 1.23 -0.181995 -1.11 0.00 -4.54 -6.25 17.62 17.62 -6.59 -8.301996 2.96 2.96 2.96 0.99 23.70 23.70 5.20 3.231997 2.32 2.32 -0.79 -3.20 20.97 20.97 -3.92 -6.331998 10.99 10.99 8.56 5.83 31.85 31.85 9.84 7.121999 6.98 6.98 -4.56 -7.90 26.22 26.22 -7.22 -10.562000 6.16 6.16 -1.16 -4.28 22.67 22.67 -4.86 -7.982001 0.00 -6.47 -9.17 0.00 -23.80 -26.50
203
Appendix D
Six Sample Paths in the Simulated
Real World
Figure D.56 gives six sample paths of the simulated portfolio asset share and guar-
antee during the 30-year period starting at the end of 2003. For ease of comparison,
we use the same scale in the six graphs.
We see in the figure that the patterns of the simulated asset share and guarantee
could be significantly different in different simulations. In a particular simulation,
the asset share will be more volatile from year to year than the guarantee. The
portfolio asset share might sometimes be much larger or smaller than the guarantee,
but it shows a similar trend in any simulation.
The reserves required for the portfolio in these six simulations are shown in Figure
D.57.
The required amount of reserves depends mainly on how the asset share changes
relative to the guarantee. The reserves set up using the option pricing approach and
the 99% CTE reserves show a similar trend, though in some simulations the former
will be of a much larger amount.
Figure D.58 shows how the insurer’s free estate will build up in these six simula-
tions.
In some simulations the free estate held at the end of 2002 will be used up in
the future to set up reserves as required under the option approach. The different
reserving approaches might make a lot of difference to the amount of the free estate.
204
Sample 1year
2005 2010 2015 2020 2025 2030
1000
020
000
3000
040
000
assetguarantee
Sample 2year
2005 2010 2015 2020 2025 2030
1000
020
000
3000
040
000
assetguarantee
Sample 3year
2005 2010 2015 2020 2025 2030
1000
020
000
3000
040
000
assetguarantee
Sample 4year
2005 2010 2015 2020 2025 2030
1000
020
000
3000
040
000
assetguarantee
Sample 5year
2005 2010 2015 2020 2025 2030
1000
020
000
3000
040
000
assetguarantee
Sample 6year
2005 2010 2015 2020 2025 2030
1000
020
000
3000
040
000
assetguarantee
Figure D.56: Six sample paths of the simulated portfolio asset share and guaranteein Case C
205
Sample 1year
rese
rve
2005 2010 2015 2020 2025 2030
050
010
0015
00
option/hedging95% CTE99% CTE
Sample 2year
rese
rve
2005 2010 2015 2020 2025 2030
050
010
0015
00
option/hedging95% CTE99% CTE
Sample 3year
rese
rve
2005 2010 2015 2020 2025 2030
050
010
0015
00
option/hedging95% CTE99% CTE
Sample 4year
rese
rve
2005 2010 2015 2020 2025 2030
050
010
0015
00
option/hedging95% CTE99% CTE
Sample 5year
rese
rve
2005 2010 2015 2020 2025 2030
050
010
0015
00
option/hedging95% CTE99% CTE
Sample 6year
rese
rve
2005 2010 2015 2020 2025 2030
050
010
0015
00
option/hedging95% CTE99% CTE
Figure D.57: Six sample paths of the simulated reserve required for the portfolio inCase C
206
Sample 1year
free
est
ate
2005 2010 2015 2020 2025 2030
020
0040
0060
0080
0010
000
optionhedging95% CTE99% CTE
Sample 2year
free
est
ate
2005 2010 2015 2020 2025 2030
020
0040
0060
0080
0010
000
optionhedging95% CTE99% CTE
Sample 3year
free
est
ate
2005 2010 2015 2020 2025 2030
020
0040
0060
0080
0010
000
optionhedging95% CTE99% CTE
Sample 4year
free
est
ate
2005 2010 2015 2020 2025 2030
020
0040
0060
0080
0010
000
optionhedging95% CTE99% CTE
Sample 5year
free
est
ate
2005 2010 2015 2020 2025 2030
020
0040
0060
0080
0010
000
optionhedging95% CTE99% CTE
Sample 6year
free
est
ate
2005 2010 2015 2020 2025 2030
020
0040
0060
0080
0010
000
optionhedging95% CTE99% CTE
Figure D.58: Six sample paths of the simulated free estate in Case C
207
In some simulations the free estate will build up less rapidly using discrete hedging
instead of buying options. More CTE reserves are required with a higher security
level, but the free estate of the insurer who sets up larger reserves might build up
more rapidly.
208
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