static hedging effectiveness for longevity risk

8/14/2019 Static Hedging Effectiveness for Longevity Risk

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Static Hedging Effectiveness forLongevity Risk with

Longevity Bonds and Derivatives

Andrew Ngai26 January 2010


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Disclaimer

The material in this report is copyright of Andrew Ngai.

The views and opinions expressed in this report are solely that of the authors and donot reflect the views and opinions of the Australian Prudential Regulation Authority.

Any errors in this report are the responsibility of the author. The material in thisreport is copyright.

Other than for any use permitted under the Copyright Act 1968, all other rights arereserved and permission should be sought through the author prior to anyreproduction.


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Static Hedging Effectiveness for Longevity Risk withLongevity Bonds and Derivatives

Andrew Ngai

January 26, 2010

Abstract

This paper contributes towards a solution to the ageing population problem

in Australia and many developed nations by providing a rigorous analysis of the

longevity risk in a range of annuities. There are several types of annuities which

involve lifetime payments that would help protect individuals against outliving their

savings. Traditional products include life and indexed annuities, whereas recent

innovations include the variable annuity with a guaranteed lifetime withdrawal

benefit. The development of a market for annuities requires a strong understanding

of the underlying risks, yet there are currently few contributions analysing the

risks of annuities from the perspective of an insurer. This paper investigates theeffectiveness of a variety of static hedging strategies based on q-Forwards and

longevity bonds. These strategies were applied to a typical annuity portfolio and

were generally found to be cost effective in reducing longevity risk as measured by

expected shortfall. Several sources of basis risk were also analysed and were found

to have minimal impact on the effectiveness of the hedging strategies. Improved

market and mortality models were also used to better account for the underlying

risks in annuities.

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1. Introduction 2

1 Introduction

Longevity risk arises from the fact that future lifetimes are uncertain and may be longer

than expected. For an individual, this may result in insufficient savings to support theirretirement. With an ageing population in most developed nations and continual mortality

improvements, longevity risk is becoming an increasingly important issue in the developed

world as individuals seek insurance against outliving their savings (Blake et al, 2008 [7]).

Life insurers have traditionally offered life annuities that transfer longevity risk away from

the insured, such as immediate, deferred, and inflation-indexed annuities. A relatively new

innovation is the guaranteed lifetime withdrawal benefit (GLWB), a variable annuity rider

which has recently become popular in the US, Japan, and Europe. The attractive feature

of the GLWB is its insurance against longevity and market risks simultaneously whileallowing individuals to maintain flexibility in their investments (Ledlie et al, 2008 [25]).

However, the popularity and complexity of the GLWB also demands a detailed analysis of

its pricing and its risks. The recent financial crisis has already resulted large losses from

variable annuity writers in the US, and subsequent rating downgrades have caused insurers

to increase fees, reduce benefits, or withdraw from selling variable annuities altogether

(Burns, 2009 [9]; Lankford, 2009 [24]).

To ensure a smoothly functioning financial system, it is vital that financial institutions

understand the nature of their risks and have appropriate risk management proceduresin place. Thus, to facilitate a market for life annuities, insurers must also have effective

techniques to hedge their longevity risk. A potential solution is the use of mortality-linked

securities (MLS) such as q-Forwards to transfer longevity risk to the capital markets

(Coughlan et al, 2007 [12]), while others have suggested that longevity risk could be

transferred to the government through the use of longevity bonds (Blake et al, 2009 [5]).

The effectiveness of such methods will be a significant step towards a successful solution

to the longevity risk problem.

A potential flow of longevity risk is illustrated in Figure 1. Individuals have a need to

insure against longevity risk which could be done by purchasing an annuity, transferring

the risk to an insurer. In turn, insurers have a need to transfer some longevity risk to

other parties due to (forthcoming) regulatory capital requirements in Solvency II. This is

currently a key issue which must be solved to allow the successful provision of longevity

insurance to individuals (Blake et al, 2009 [5]). Thus, an analysis of potential hedging

techniques will be a significant contribution to the insurance industry and to society, and

will make one further step towards an effective and sustainable market for longevity risk

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1. Introduction 3

Individuals Life Insurers

Capital Mkts

Government

Annuities MLS

Figure 1: Role of Institutions in the Transferral of Longevity Risk

insurance.

Most of the existing literature on life annuities focus on the individuals perspective and

address pricing and consumption issues in an attempt to explain the annuity puzzle

the phenomenon that relatively few individuals choose to annuitise their wealth despite

studies showing that annuities increase expected utility. In an early paper, Yaari (1965)

[36] proved that individuals should choose to annuitise their entire liquid wealth in the

absence of a bequest motive, a result that was recently extended by Davidoff et al (2005)

[13] to a more general setting by relaxing a number of assumptions.

On the contrary, there is relatively little literature analysing annuities and their longevityrisk from an insurers perspective. Dowd et al (2006) [14] analyse the risks of various

mortality-dependent positions, such as an annuity book with a longevity bond hedge.

The longevity bond involves coupons in line with survival probabilities, and is therefore

structured to specifically match a typical life annuity book for a given cohort. A range

of risk measures such as expected shortfall and spectral measures are used to quantify

the risk in the portfolios. Bauer and Weber (2008) [3] focus on the risk in an annuity

book itself, discovering that the mortality risk premium charged by UK annuity providers

is relatively large. The paper focuses on the simplest form of annuity, a single premium

immediate life annuity. The authors choose to use relatively simple stochastic models to

describe the market and mortality processes, and also assume independence between the

short rate and stock processes.

The literature on GLWBs and other equity-linked guarantees focus mainly on pricing

and market risk issues. A detailed overview of variable annuities and all its guarantees

is provided by Ledlie et al (2008) [25], who use an economic scenario generator to

stochastically analyse the GLWB from the consumers perspective, price the guarantees,

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2. Market Model 5

by quantifying the basis risk due to uncertainty in the relationship between annuitant

and population mortality and analysing the effect of this basis risk on the overall hedge

effectiveness. The basis risk due to the bucketing of q-Forwards by age groups will also

be analysed with regards to their impact on hedge effectiveness.

This paper is structured as follows. Sections 2 & 3 develop the market and mortality

models which are applied in Section 4, where the longevity risk in a portfolio of annuities

is assessed. Longevity risk management techniques involving q-Forwards and longevity

bonds are applied and evaluated in terms of hedge effectiveness. Sensitivity analyses are

also performed, including an analysis of the effect of basis risk on hedge effectiveness. The

results are presented in Section 5, while Section 6 provides a conclusion.

2 Market Model

A number of multivariate time series models have been used in the literature for market

modelling purposes. A vector autoregressive (VAR) model was used by Wilkie (1986,

1995) [32, 33], while Harris (1997) [17] and Sherris and Zhang (2009) [30] incorporate

regime switching into the VAR model. Regime switching was found to be effective in

accounting for the long-tail nature (heteroskedacity) of market data. In addition, a

cointegration relationship can be incorporated by using a vector error correction model

(VECM). A regime switching VECM was used by Maysami and Koh (2000) [27] for

modelling several stock indices, and Krolzig et al (2002) [23] for modelling the UK labour

market.

2.1 Model Specification

This paper will use a regime switching VECM due to its ability to account for short-run

dynamics through its autoregressive structure, long-run equilibria using a cointegration

relationship (Johansen, 1995 [21]), and heteroskedacity through the use of regime switch-

ing. The RS-VECM was also found to be the best model according to AICc and SBC.

A general RS-VECM with an underlying lag of p is expressed as:

yt = +

p1i=1

Aiyti + BCytp + t(t) (1)

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2. Market Model 6

where yt is a d-dimensional vector of observations, yt = yt yt1 is the first differenced

series, is the mean vector, A is a d d parameter matrix of coefficients, B and C are

d r matrices of rank r describing the cointegration (equilibrium) relationship between

the variables, and t(t) is a vector of regime-dependent multivariate normal randomerrors with covariances t:

t(t) Nd (0, t) (2)

A lag ofp = 2 was found to be appropriate according to AICc and SBC and for parsimony.

Two regimes are used in the model, representing a normal state and high-volatility state

respectively:

t = 0 normal regime

1 high-volatility regime

(3)

The probabilities of switching between regimes are described in a Markov chain with

transition matrix P:

P =

p0 1 p0

1 p1 p1

(4)

where the probabilities p0 and p1 are assumed to be constant over time.

The model was fitted using a two-stage procedure detailed by Krolzig (1997) [22]. The firststep in fitting a RS-VECM is to determine the cointegration rank r and matrix C using

the Johansen (1988, 1995) [20, 21] methodology. The second stage of the estimation

procedure involves the use of an Expectation-Maximisation algorithm to estimate the

remaining parameters.

2.2 Data

The financial and economic series included in the model are:

ln Gt = Log Gross Domestic Product (GDP)

ln Bt = Log Bond Index (Accumulated 90-day Bank-Accepted-Bill Yields)

ln St = Log Stock Price Index (ASX All Ordinaries)

ln Ft = Log Inflation Index (CPI)

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2. Market Model 7

Statistic ln Gt ln Bt ln St ln FtMean 0.0078 0.0227 0.0137 0.0146Standard Deviation 0.0101 0.0101 0.1010 0.0115Skewness -0.2010 0.7783 -1.8734 0.9470

Excess Kurtosis 1.4224 -0.5803 7.6267 1.15431st Quartile 0.0021 0.0140 -0.0262 0.0060Median 0.0082 0.0191 0.0258 0.01253rd Quartile 0.0129 0.0296 0.0692 0.0216Minimum -0.0299 0.0107 -0.5719 -0.0046Maximum 0.0367 0.0483 0.2298 0.0566

Table 1: Summary Statistics for Market Log Quarterly Returns

Each market variable is one component of the time series vector yt. These are the variablesconsidered important for modelling economic scenarios for the purposes of this research.

The bank bill yield is representative of the risk-free interest rate, and is used in place

of a short-term T-bill yield due to data availability. The stock index and interest rates

are required for the investments (and possibly claims) of the insurer and policyholders,

and inflation is required for inflation-linked products. GDP is considered an important

macroeconomic variable which interacts with the other market variables, and is often

included in economic scenario generators in the literature (eg. Sherris and Zhang, 2009

[30]).

Market data was obtained from the Reserve Bank of Australia (RBA). The data consists

of observations over the period 1970 to 2009, the maximum period over which data was

available for all 4 variables. Quarterly data was used as GDP and CPI observations are

not available at a higher frequency. Note that the bond index is not directly observed,

but was obtained by accumulating at the 90-day bank-accepted-bill yield.

A statistical summary of the data (log quarterly returns) is provided in Table 1.

2.3 Simulation

The RS-VECM produces simulations of the economic series into the future that are

comparable with historical data. The model occasionally simulates negative bond returns

(around 1% of simulations) which is clearly unreasonable. In these cases, the bond return

was set to 0%. The following analysis is based on 100,000 simulations.

Figures 2-3 show the average simulation paths and returns along with 95% confidence

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3. Mortality Model 8

2010 2030 2050

400000

1000000

GDP

Year

Level

2010 2030 2050

0

40000

100000

BND

Year

Level

2010 2030 2050

0

2

00000

400000

SPI

Year

Level

2010 2030 2050

500

1000

2000

CPI

Year

Level

Figure 2: Average Simulation Path (and 95% CI) of the Market Variables

2010 2030 2050

0.0

2

0.

02

0.

06

GDP

Year

Level

2010 2030 2050

0.0

0

0.0

4

0.0

8

0.

12

BND

Year

Level

2010 2030 2050

0.

2

0.

2

0.6

SPI

Year

Level

2010 2030 2050

0.0

0

0.

04

0.

08

CPI

Year

Level

Figure 3: Average Simulation Returns (and 95% CI) of the Market Variables

intervals, whereas Figure 4 shows the overall distribution of returns for each variable

obtained from these simulations and provides a comparison with the observed (historical)

distribution. Table 2 provides summary statistics of the simulated quarterly returns,

which are generally similar to the historical statistics in Table 1.

3 Mortality Model

The model for fitting and simulating mortality rates into the future is largely based

on the Wills and Sherris (2008) [34] model due to its applicability to pricing and riskmanagement. A further improvement is made by including a model for the relationship

between annuitant and population mortality. This is important when analysing a portfolio

of annuitants, and also facilitates an analysis of basis risk in hedging strategies.

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GDP

Return

Frequency

0.05 0.05 0.10

0

20000

50000

BND

Return

Frequency

0.00 0.10 0.20

0

10000

25000

SPI

Return

Frequency

0. 5 0 .0 0 .5 1 .0

0

20000

40000

CPI

Return

Frequency

0. 05 0. 05 0 .1 5

0

10000

30000

Figure 4: Overall Distribution of Simulated Annual Returns(Note: Red = Historical Distribution)

Statistic Gt Bt St FtMean 0.0077 0.0175 0.0250 0.0106Standard Deviation 0.0104 0.0078 0.1062 0.0091Skewness 0.0319 0.2159 0.4904 0.0311Excess Kurtosis 1.0849 0.2565 1.610 0.88101st Quartile 0.0017 0.0124 -0.0372 0.0052Median 0.0077 0.0175 0.0196 0.01063rd Quartile 0.0136 0.0223 0.0796 0.0160Minimum -0.0468 0.0000 -0.4250 -0.0370Maximum 0.0658 0.0571 0.8259 0.0580

Table 2: Summary Statistics for Simulated Quarterly Returns

Age70

80

90

100

Year

1970

1980

1990

2000

R

ate

4

3

2

1

0

1

Age70

80

90

100

Year

1970

1980

1990

2000

R

ate

5

4

3

2

1

0

1

Figure 5: Observed Logit Mortality Rates for Males (Left) and Females (Right)

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3.1 Model Specification

The model for the force of mortality for a life currently aged x at time t, denoted x,t, is

a linear model with an age factor x:

logit x,t lnx,t

1 x,t= a + bx + x,t (5)

where a and b are constant parameters and x,t is an error term. As in Wills and Sherris

(2008) [34], the difference operator is applied in the cohort direction:

logit x,t = logit x,t logit x1,t1 (6)

The vector of errors at each time t are multivariate normal with zero mean, and can bewritten as:

t = (x1,t, . . . , xN,t) = Wt (7)

where is a deterministic volatility parameter, and Wt NN(0, D). The matrix D

captures the dependence structure between ages.

The logit transform was chosen based on Cairns et al (2006) [10] and observation of the

data. Figure 5 shows the historical logit mortality rates which can be observed to be

approximately linear.

The parameters were estimated in a two-stage procedure as in Wills and Sherris (2008)

[34]. Firstly, maximum likelihood is used to estimate the parameters a, b and assuming

independent observations of logit x,t. The covariance matrix D is then determined

using principal components analysis on the standardised residuals.

3.2 Data

The mortality model is based on population data as Australian annuitant data is not

available. Mortality data was obtained from the Human Mortality Database (HMD) and

the Australian Bureau of Statistics (ABS). The data contains annual central rates of

mortality, mx,t, for ages 65 to 109 and years 1970 to 2007. The minimum age of 65 is

chosen to reflect the retirement age in Australia. Note that the data at higher ages does

not consist of raw mortality rates, as the rates have been smoothed as described in the

HMD Methods Protocol (Wilmoth et al, 2007 [35]) due to a lack of observed deaths at

higher ages.

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Age70

80

90

100Ye

ar

1980

2000

2020

2040

Rate

0.0

0.2

0.4

0.6

Age70

80

90

100Ye

ar

1980

2000

2020

2040

Rate

0.0

0.2

0.4

0.6

Figure 6: Observed and Projected Mortality for Males (Left) and Females (Right)

3.3 Simulation

One of the issues noted by Wills and Sherris (2008) [34] was the ridges effect in the

projection which arose from the use of unsmoothed mortality rates, x,T. To prevent this,

the observed logit mortality rates for the last year in the data (2007) are smoothed using

cubic splines before simulation.

Realisations of the random vector Wt were simulated from the multivariate normal

distribution, NN(0, D). The average mortality rates for all ages 65109 are shown in

Figure 6. The average (projected) mortality rates and 95% confidence intervals for the

cohort aged 65 in 2007 are shown in Figure 7.

3.4 Annuitant Mortality

To apply the model to a portfolio of annuitants, the mortality rates will be adjusted based

on the observed historical relationship between population and annuitant mortality. As

there is no Australian annuitant data available, an investigation of Australian annuitant

mortality was not possible. UK annuitant data was available from the Continuous Mor-

tality Investigation (CMI) for years 1947, 1968, 1980, 1992, and 2000. This data wascompared to UK population mortality available from the Human Mortality Database

(HMD) and the Office for National Statistics (ONS).

The relationship between annuitant (or pensioner) mortality and population mortality

can be described using the ratio between initial rates:

x,t =qAx,t

qx,t(8)

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0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

Force of Mortality

Years

Rate

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

Initial Death Rate

Years

Rate

Figure 7: Projected Male Mortality Rates for Cohort 65 (with 95% CIs)

where qAx,t and qx,t are the annuitant and population mortality rates for age x at time t.

Stevenson and Wilson (2008) [31] investigate the relationship between Australian pen-

sioner and population mortality for the period 2005-07, concluding an approximate linear

increasing relationship between age and the ratio. The relationship between UK annuitant

and population mortality is also approximately linear and increasing, as shown in Figure 8.

However, Stevenson and Wilson found that the ratio for the pensioner data (significantly)

exceeds 1 at higher ages, which is inconsistent with the UK data. There is currently no

explanation for pensioner mortality exceeding population mortality at higher ages.

Due to the similarity between the pensioner and UK data at earlier ages, and the lack

of explanation for the higher observed ratio in the pensioner data at higher ages, the

following model will be based solely on the UK annuitant data for conservatism. Given

the observed linearity of the ratio and the relative scarcity of data (only 5 observations

per age), a simple linear model was used to model the ratio:

x,t = + x + x,t (9)

x,t N(0, 2) (10)

The variation in the ratio from year to year due to the error terms will result in basis

risk. Note that the error terms are unlikely to be independent by age, as the year to

year variation in actual mortality would depend on factors that affect the population as

a whole. However, since the analysis in the following chapter will based on a portfolio

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4. Risk Analysis and Management 13

0.4

0.6

0.8

1.0

1.2

1.4

65 70 75 80 85 90 95 100

Ratio

Age

Male

Pensioner UK '00 UK '92 UK '80

UK '68 UK '47 Fitted

0.4

0.6

0.8

1.0

1.2

1.4

65 70 75 80 85 90 95 100

Ratio

Age

Female

Pensioner UK '00 UK '92 UK '80

UK '68 UK '47 Fitted

Figure 8:Observed Ratios of Annuitant/Pensioner to Population Mortality

of lives of the same initial age, and the ratio x,t does not depend on past error terms

x,s (s < t), correlation of the errors between ages will not affect the results of this analysis

and is therefore ignored in this model.

The parameters are estimated using maximum likelihood assuming independent observa-

tions ofx,t. The fitted ratios are superimposed in the graphs of observed ratios (Figure 8)

for comparison. Furthermore, the linear relationship is extrapolated to age 110. For both

males and females, the extrapolated ratio remains below 1, and therefore no adjustment

is deemed necessary.

4 Risk Analysis and Management

The following products were stochastically analysed using the market and mortality

models developed in the previous sections:

Fixed Life Annuity (Immediate)

Fixed Life Annuity (Deferred to Age 85)

Inflation-Indexed Life Annuity (Immediate)

Inflation-Indexed Life Annuity (Deferred to Age 85)

Variable Annuity with GLWB

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These products have potential in insuring individuals against longevity risk as they involve

an income stream for life.

Longevity hedging strategies involving mortality-linked securities were considered due

to their advantages of liquidity and transparency. The effect of basis risk (the key

disadvantage of mortality-linked securities) on hedge effectiveness was also analysed. The

strategies use longevity bonds and q-Forwards, which are considered to have potential in

transferring longevity risk away from insurers based on previous literature (eg. Coughlan

et al, 2007 [12]; Dowd et al, 2006 [14]). q-Forwards have already been used by insurers to

hedge longevity risk, whereas longevity bonds could be issued by the government in the

future (Blake et al, 2009 [5]).

An effective hedge will reduce the longevity risk in the insurers final surplus. This is

measured by analysing the simulated distribution of the final surplus and calculating risk

measures such as expected shortfall. The decision to focus on expected shortfall is based

on its qualities as a risk measure (eg. coherency, focus on tail-risk, robustness) which

have led to its increasing popularity (Hardy, 2003 [16]). The risk measure should be

significantly reduced when an effective hedge strategy is applied.

The analysis focuses on static hedges due to their low cost and a lack of liquidity in

the MLS market. At present, dynamic hedges would be difficult to implement due to

illiquidity and high transaction costs.

4.1 Annuity Portfolio

The life insurers portfolio consists of 1,000 male policies of initial age 65, similar to Bauer

and Weber (2008) [3] and Dowd et al (2006) [14]. This would resemble a typical insurers

portfolio resulting from the sale of annuities to newly-retired individuals. In practice, an

insurer may write and hedge such a portfolio each year as each cohort of lives enter their

retirement.

The purchase of each annuity involves an initial lump-sum payment. For immediate and

variable annuities, the initial payment is assumed to be $100,000. For deferred annuities,

individuals will generally require a significant amount of their savings for consumption

during the deferral period, and therefore a lower initial investment of $10,000 is assumed.

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4.1.1 Insurer Surplus

The risk analysis is largely based on the framework of Bauer and Weber (2008) [3].

The aim is to obtain a distribution for the insurers final surplus at time T after allpolicyholders have died.

Define the following notation:

Ut = insurers surplus at time t

Pt = total premiums (fees) at time t

Ct = total claims at time t

Ht = net hedging cash flows at time t

The surplus Ut refers to the accumulated net cash flows at time t. The claims (and

premiums for a GLWB) are random variables that depend on actual mortality (and market

outcomes for indexed and variable annuities). The insurer is also assumed to purchase

assets in an attempt to hedge (match) the liabilities. These may include bonds, longevity

bonds, and q-Forwards, and will depend on the specific hedging strategy. The hedging

cash flows Ht are a result of the cash flows from these assets.

Each year, premiums are received, claims are paid, and the net cash flows from assets

purchased in the hedging strategy are received or paid. The surplus is assumed toaccumulate at a rate of return RUt , which is detailed in Section 4.1.2. Therefore:

Ut = Ut1

1 + RUt

+ Pt Ct + Ht (11)

Due to product and model complexity, the analysis was done numerically using simulation

based on the fitted market and mortality models under a real-world probability measure

P. Each simulated path of market and mortality variables and policyholder experience

produces one realisation of the final surplus UT. The risk of the portfolio was assessed

by analysing the simulated distribution of UT, since the final surplus represents the

accumulated profit/loss of the portfolio. The terminal age used in this analysis is = 111,

hence T = 46.

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4.1.2 Investment Strategy

The investment strategy of the insurer is similar to that used in Bauer and Weber (2008)

[3]. As in Bauer and Weber, the insurer is assumed to have access to risk-free bonds of allmaturities, a savings account (which accumulates at the short rate) and a stock portfolio.

The latter two types correspond to investing in the bond and stock indices (Bt and St)

modelled in Chapter 2.

At time 0, the insurer will receive an initial premium and will purchase a series of assets

in an attempt to hedge the liabilities. The remaining initial premium is then assumed to

be invested in a T-year risk-free bond. Although insurers in practice are unlikely to invest

their remaining capital in long-term assets due to the resulting illiquidity, purchasing

short-term assets will introduce considerable investment risk due to fluctuations in short-term returns as shown by the results of Bauer and Weber. The use of long-term bonds

will reduce the investment risk, allowing the analysis to focus on longevity risk.

In each future year, the surplus (accumulated net cash flows) is assumed to be invested

in a liquid portfolio of savings which accumulates at the risk-free rate. In contrast to

Bauer and Weber, no investment in stocks is made to reduce investment risk, allowing

the analysis to focus on longevity risk.

RUt = RBt =

Bt

Bt1 1 (12)

where RBt is the return on the bond index.

4.1.3 Annuities

The life and indexed annuities involve a single upfront premium paid at time 0 in exchange

for regular annual payments (claims). Australian average market prices were used for the

immediate annuities as in Ganegoda and Bateman (2008) [15]. As there are currently

no deferred annuities available in Australia, the moneys worth ratio from the immediate

annuities were assumed to apply for deferred annuities.

In a variable annuity, the lump-sum payment is invested in a portfolio, with the individual

assuming all the market risk. Withdrawals are made and deducted from the funds

invested. As in Holz et al (2007) [19], the withdrawals are assumed to be level and

made annually. If a GLWB rider is attached, the embedded option will be exercised by

the individual when the funds are depleted, causing withdrawals (claims) to be paid by

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the insurer. All policies will be invested in an index which contains a proportion of

shares and 1 of savings. The proportion is assumed to be 75%, which is similar to

the asset allocation observed in the US market and that assumed in Hill et al (2008) [18].

The guarantee fee is assumed to be 40bps. This is reflective of the market price in the

US, and is similar to the fees calculated/assumed by Holz et al (2007) [19] and Hill et al

(2008) [18] in their analyses of the GLWB. The withdrawal percentage for an individual

initially aged 65 is assumed to be 5%. This is representative of the market in the US, and

is also the percentage used by Holz et al (2007) [19] and Hall et al (2008) [18].

4.2 Hedging Instruments

The hedging instruments available for the insurer to purchase are zero coupon bonds,

q-Forwards, and longevity bonds, and will be purchased in an attempt to match assets

with liabilities.

As stated in Section 4.1.2, the insurer is assumed to have access to risk-free zero coupon

bonds of all maturities. A zero coupon bond of maturity will pay a cash flow of $1 at

time , and will involve no transfer of longevity risk.

The first type of mortality-linked securities considered will be q-Forwards. These are

forward contracts which pay an actual (floating) mortality rate qx,t in exchange for a

fixed (forward) mortality rate qFx,t. q-Forwards are used as building blocks to construct

a longevity swap. This will involve payments in line with the insurers expected claims

according to a fixed mortality index, in return for receipts according to a standardised

but variable index. The strategies aim to match the cash flows of the insurer with those

from the swap, however there will be some basis risk as the standardised index does not

align exactly with the insurers true experience.

As an alternative, a longevity swap using q-Forwards that are bucketed by 5 year age

groups (6569, . . . , 105109) will also be considered. In practice, q-Forwards are bucketed

to improve liquidity by reducing the number of contracts, although this may result in

additional basis risk (Cairns et al, 2008 [11]).

Longevity bonds have been proposed as a possible mechanism through which longevity

risk in an annuity portfolio can be transferred away, possibly to the government (Blake

et al, 2009 [5]). The longevity bond structures that have been proposed (including the

EIB/BNP longevity bond) generally involve payments in line with the survival experience

of a particular cohort (Dowd et al, 2006 [14]; Blake et al, 2006 [8]). Deferred longevity

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bonds have also been proposed by Blake et al (2006, 2009) [5, 6], who note that deferred

bonds have an advantage in requiring less upfront capital to purchase.

The first type of longevity bonds considered will be coupon bonds which pay (in proportion

to) the actual survival rate Sx(t) at each time t, similar to those analysed in Dowd et al

(2006) [14]. Longevity bonds with maturities of 20 and 40 years will be considered, as

these are approximately the average and maximum term of life annuities for policyholders

initially aged 65. The 20-year bond will be an immediate bond, whereas the 40-year bond

will be deferred for 20 years. Note that an immediate 40-year bond can be constructed

by purchasing both the 20-year immediate and 40-year deferred bonds.

Dowd et al (2006) [14] also consider zero coupon longevity bonds which pay a single

amount proportion to the actual survival rate Sx() at maturity . These were also be

analysed, and were used as building blocks for constructing a longevity swap, similar to

q-Forwards.

4.3 Pricing of Hedging Instruments

The pricing of the hedging instruments involves taking discounted expectations of cash

flows under a risk neutral measure Q, which incorporates a price of risk for both market

and mortality risks. In determining the Q measure, market and mortality risks will be

assumed independent. Due to the incomplete mortality market, the Q measure will notbe unique.

As in Bauer and Weber (2008) [3], market (interest rate) risk is accounted for by discount-

ing cash flows using the current market term structure. Market yield data is available for

terms up to 15 years. For higher maturities, the yield curve is assumed (conservatively)

to be flat. The market yield and spot curves were obtained from market bond yield data

available from the Reserve Bank of Australia (RBA).

The Q measure is calibrated for mortality risk using q-Forward prices, as a q-Forward

involves the exchange of a fixed mortality rate qFx,t for a variable rate qx,t, and therefore

involves the risk neutral expectation of mortality rates:

qFx,t = EQ (qx,t|F0) (13)

The q-Forward prices are determined using a Sharpe ratio as in Loeys et al (2007) [26]:

qFx,t = (1 Sqxt)qEx,t (14)

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where Sq is the required Sharpe ratio on the q-Forward, qEx,t is the expected mortality

rate under the real-world P measure, and x is the historical (percentage) volatility of

the mortality rates:

qEx,t = EP (qx,t|F0) (15)

The required Sharpe ratio is chosen as 0.20, which is the implied Sharpe ratio from UK

annuity markets as given in Loeys et al. This is consistent with Bauer et al (2008) [1] who

argue that the Sharpe ratio should be less than the 0.25 observed in the equity markets.

4.4 Risk Assessment

The risk in the insurers portfolio was assessed by considering the distribution of the final

surplus UT when all contracts have been terminated. 100,000 simulations were performed

for each scenario, obtaining a simulated distribution for UT from which the risks are

assessed.

4.4.1 Scenario Analysis

The insurers surplus was simulated under a number of scenarios, including a range of

stress tests which were done to analyse the risk and hedge effectiveness under adverse

conditions. The scenarios analysed were:

Scenario Market Assumption Mortality Assumption

1 Stochastic Stochastic

2 Average Stochastic

3 Adverse Stochastic

4 Stochastic Average + Excess Imp. (2%/yr Accumulating)

5 Stochastic Average + Excess Imp. (25% Flat)

Scenario 1 involves simulating 100,000 paths using the stochastic market and mortality

models developed in Chapters 2 and 3. Scenario 2 involves using a deterministic market

(average case from simulated paths) but with stochastic mortality. Scenario 3 involves a

deterministic market based on the historical period of 19301975, beginning just prior to

the Great Depression. This scenario is primarily for the purpose of assessing the GLWB,

which was found to insure mainly against (extreme) market risk. Scenario 4 stress tests the

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5. Results 20

longevity hedging strategies by applying a deterministic mortality but with 2% annual

excess improvement above the projected rates. In reality, an underestimate of future

mortality trends is likely to be a gradual underestimation, where the effects of health

improvements (eg. cure for cancer) gradually impact on the mortality rates beyond theexpected level. Thus, Scenario 4 represents an adverse scenario that is quite realistic.

Finally, Scenario 5 stress tests the strategies with a 25% flat shock across mortality rates

of all ages and times. This is the scenario that must be applied in determining the

longevity risk capital charge according to Solvency II.

4.4.2 Hedge Strategies

The following hedge strategies were considered for each scenario:

No Longevity Hedging

No Longevity Risk

q-Forwards

q-Forwards (Bucketed)

20yr Longevity Bond

40yr Longevity Bond

Both Longevity Bonds

Longevity Swap

Only the 40yr longevity bond was considered (out of the three longevity bond strategies)

for deferred annuities and the GLWB due to the deferred nature of the claims.

The No Longevity Hedging and No Longevity Risk strategies involves using (only) zero

coupon bonds to match assets with liabilities and is largely based on the bond strategy

used in Bauer and Weber (2008) [3]. In the latter strategy, determininstic average-case

mortality rates are used. These provide two benchmarks against which the other longevity

hedging strategies can be evaluated.

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5. Results 21

qqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqq

q

0 10 20 30 40

0

2000

6000

Immediate Annuities

Year

Claims

('000)

qqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqq

q

q

qq

q

qqqqqq

q

qqqqqqqqqqqqqqqqqqqqqqqqq

q

20 25 30 35 40 45

0

1000

3000

Deferred Annuities

Year

Claims

('000) q

qqqqqqqqqqqqq

qqqqqqqqqqq

q qqqqqqqqqqqqqqqqqqq

q

q

q

q

q

q

q

q

q

qqqqqqq

q

q

q

q

q

q

q

q

qq

q

0 10 20 30 40

0

10

20

30

40

50

VA + GLWB

Year

Claims

('000)

Figure 9: Expected Total Claims of Portfolio(Note: Blue = Life Annuity, Green = Indexed Annuity)

600 200 0 200 600

0.0

00

0.0

04

0.0

08

Immediate Annuities

Surplus

Density

150 50 0 50 100

0.0

00

0.0

10

0.0

20

Deferred Annuities

Surplus

Density

0 100 200 300

0.0

00

0.0

04

0.0

08

VA + GLWB

Surplus

Density

Figure 10: Simulated Distribution of Final Surplus UT

(Note: Blue = Life Annuity, Green = Indexed Annuity)

5 Results

Figure 9 shows the expected total claims, whereas the simulated distribution of UT for

each annuity is shown in Figure 10.

The difference in claims between life and indexed annuities can be seen in Figure 9. The

difference is most significant for immediate annuities, as the survival probabilities are

large at earlier ages and hence the increase in payments due to indexation is the dominant

effect. Thus, in hedging the indexed annuity (in particular the immediate annuity), it is

important to take into account the increasing nature of the claims, and this is verified by

the results in Section 5.2.

From the distributions in Figure 10, the immediate life annuity and the GLWB appear to

be overpriced, as the surplus distributions are almost always positive. The high price of

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5. Results 22

Annuity Mean (No Hedging) ES0.05 (No Hedging) ES0.05 (No L Risk)Life 150.3 54.6 116.2Indexed Life 81.5 -642.3 -587.9Deferred 14.7 -30.6 7.7

Def. Indexed 7.5 -133.5 -106.2VA + GLWB 121.43 40.80 41.27

Table 3: Change in 5% Expected Shortfall (No Hedging vs No Longevity Risk)

life annuities is consistent with the results of Bauer and Weber (2008) [3]. One reason for

the apparent high GLWB price is the relatively higher level of interest rates in Australia

compared to many other developed nations. This means that higher (risk-free) returns

can be earned on the investment portfolio, and therefore higher guaranteed withdrawalscould potentially be supported.

Conversely, both indexed annuities and the deferred life annuity may be underpriced, as a

significant portion of the simulated distribution is negative. In particular, the riskiness of

indexed annuities can be seen from the variance of the surplus distribution, with a much

broader range than their fixed counterparts. The inflation risk in an immediate annuity is

also higher than in a deferred annuity, as the immediate annuity involves payments (and

their corresponding inflation risk) over a longer period.

5.1 Significance of Longevity Risk

Longevity risk was found to be much more significant for the life annuities than the indexed

and variable annuities. This is intuitive, as the life annuities have payments which depend

only on mortality, whereas the indexed and variable annuities have payments depending

on inflation and market forces. However, the extent to which the products are dominated

by market risk is an interesting observation.

The total risk in indexed annuities consists primarily of inflation risk, while the total riskin the GLWB is heavily dominated by market risk due to the claim amounts being highly

dependent on market performance. This is consistent with the results of Holz et al (2007)

[19], and is also intuitive due to the higher volatility in market variables as opposed to

changes in mortality rates.

Table 3 shows the expected shortfalls for the no hedging and no longevity risk scenarios,

from which the indexed and variable annuities can be seen to contain relatively little

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5. Results 23

0%

20%

40%

60%

80%

100%

Life Indexed Life Deferred Ind. Deferred VA + GLWB

Type of Annuity

Significance of Longevity Risk

Figure 11:

Percentage Difference in 5% ES (No Hedging vs No Longevity Risk)

longevity risk compared to other risks.

Figure 11 illustrates the percentage difference in expected shortfall (as a difference from

the mean) under no longevity risk and no hedging. If longevity risk dominates the risk

in a product, then the expected shortfall under no longevity risk should be significantly

closer to the mean than in the no hedging case, resulting in a high percentage difference.

The low significance of longevity risk is shown by the (very) low percentages (7%, 19%,

1%) for indexed and variable annuities when compared to the life annuities (64%, 84%).

The risk remaining in the life annuities consists of unsystematic risk (random mortality

variation within a portfolio) and investment risk (accumulation of surplus). These risks

are lower for deferred annuities due to the shorter period during which claims are paid

and cash flows are accumulated.

5.2 Hedge Effectiveness

The effectiveness of hedging strategies can be assessed by observing their effect on the

risk measures.

The percentage reduction in expected shortfall can be measured as:

ES ESNHESNR ESNH

(16)

where ESNH and ESNR are the corresponding risk measures under no longevity hedging

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5. Results 24

ESNRESNH ES ESNH

ESNRESNH

ES

No Hedging

No Longevity Risk

Hedged

Figure 12:

Illustration of Relative Reduction in Expected Shortfall

and no longevity risk respectively (See Figure 12). Figures 13-14 show the average

reduction in expected shortfall (1% and 5% levels) for each of the 5 scenarios due to each

hedging strategy.

The results show that all the hedging strategies are useful in transferring longevity risk

away from the insurer. The hedging strategies are most effective for the life annuities,

as their claims do not depend on market forces. Indexed annuities contain less longevity

risk and are more difficult to hedge, with the strategies being significantly less effective.

The GLWB contains relatively little longevity risk and are the most difficult to hedge

as the claim amounts and timing are both significantly influenced by the market. For

a GLWB, the strategies generally do not reduce the longevity risk significantly enough

to compensate for the cost of hedging, as shown by the zero net reduction in expected

shortfall for Scenario 1 (Figure 14).

For indexed annuities, the longevity swap is most effective as it allows for the expected

increase in claims due to inflation. The longevity bonds are still useful, but are less

effective due to the greater mismatch between the claims and bond payouts. q-Forwards

also allow for the expected increase in claims, and perform well in Scenarios 2 to 5, with

the poor performance in Scenario 1 mainly due to its higher cost when compared with

longevity bonds.

The 40yr bond is generally more effective than the 20yr, suggesting that the longevity risk

realised in years 2040 is more significant than in earlier years (020). This is reasonable,

as mortality rates are unlikely to differ significantly from expected in the short term, and

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5. Results 25

0

20

40

60

80

100

1 2 3 4 5 1 2 3 4 5

Life Indexed Life

Percentage

Annuity/Scenario

q Fwd q Fwd (B) L Bond (20y) L Bond (40y) L Bond (Both) L Swap

Figure 13: Avg Reduction in Expected Shortfall (Immediate Annuities)

0

20

40

60

80

100

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

Deferred Indexed Deferred VA + GLWB

Percentage

Annuity/Scenario

q Fwd q Fwd (B) L Bond (40y) L Swap

Figure 14: Avg Reduction in Expected Shortfall (Deferred Annuities and GLWB)

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5. Results 26

the number of deaths in earlier years is significantly lower. In years 2040, mortality rates

are likely to differ more significantly, and there is still a significant portfolio size which

is affected by the longevity risk. In addition, the deaths in years 20-40 have also been

subject to longevity improvements in years 20-40, and therefore the 40-year bond alsohedges against some of the longevity risk in earlier years. The only scenario when the

20yr bond performs better is Scenario 5, where a flat shock of 25% is applied to mortality

rates of all ages and times, causing shorter term mortality rates to differ significantly as

well.

q-Forwards are structured in a similar manner to the longevity swap, except that payouts

are based on initial mortality rates instead of survival probabilities. This causes the q-

Forwards to be less effective than the longevity swap, as it introduces additional basis

risk. This is due to a mismatch in timing of cash flows, as the survival probability Sx(t)depends on all mortality rates qx+s,s (s < t), and therefore the cash flows for hedging the

claim at time t will be received at all times s < t. These must then be invested until time

t, and are therefore subject to investment risk.

The two q-Forward strategies produce very similar results, with only a minor difference in

hedge effectiveness. This suggests there is little basis risk added by bucketing q-Forwards

by 5 year groups, which is an encouraging result as bucketing can help to improve liquidity.

Under the stress test scenarios (4 & 5), the most effective strategies are the longevity swap

(all products), q-Forwards (all products), both longevity bonds (immediate life annuity),and the 40-year longevity bond (deferred life annuity). These are the strategies which

have mortality-linked payments closely matching all the cash flows, and do not leave a

significant portion (eg. inflation-linked claims, earlier/later claims) of the longevity risk

unhedged. In such scenarios, leaving a portion of the longevity risk unhedged will greatly

reduce the hedge effectiveness, which is intuitive as these are severe adverse scenarios in

terms of longevity risk. These stress tests show that when managing extreme longevity

risk, it is important to implement a strategy that closely matches all cash flows without

leaving a significant portion unhedged.

5.3 Sensitivity to Basis Risk

Basis risk arising from the uncertainty in the relationship between annuitant and popu-

lation mortality is quantified using the model described in Section 3.4. The sensitivity of

hedge effectiveness to basis risk was analysed by considering annuitant-population ratio

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5. Results 27

No Basis Risk Basis Risk 150% Basis Risk0% 6.68% 10.02%

Table 4: Standard Deviation of Error Term in Annuitant-Population Ratio Model

0

20

40

60

80

100

q qB L2 L4 LB LS q qB L2 L4 LB LS q qB L4 LS q qB L4 LS q qB L4 LS

Life Indexed Life Deferred Ind. Deferred VA + GLWB

Percentage

Annuity/Strategy

No Basis Risk

Basis Risk

150% Basis Risk

Figure 15: Reduction in Expected Shortfall: Effect of Basis Risk

volatilities of 0 and 1.5 times the estimated value (Table 4). The first case corresponds

to the situation where basis risk is ignored by assuming the expected ratio of annuitant

to population mortality will hold with certainty. The second case corresponds to the

situation where the level of basis risk is 50% greater than expected.

The effect of basis risk on hedge effectiveness is illustrated in Figure 15. The results

show that basis risk arising from uncertainty in the annuitant-population ratio does not

significantly reduce hedge effectiveness.

5.4 Sensitivity to Price of Risk (Sharpe Ratio)

The strategies can be very effective in transferring longevity risk, but can also be rather

costly resulting in a significant shift in the surplus distribution towards the negative. As

there is currently an illiquid market for mortality-linked securities, the price at which

hedge instruments can be purchased is rather uncertain, and this could have a significant

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6. Conclusion 29

due to its strong influence on the amounts and timing of claims. Longevity risk is also

more dominant in deferred annuities than in immediate annuities, as the claims focus on

the higher ages where mortality improvements are more likely to differ from expected,

and will have a larger impact on the number of lives remaining.

Longevity bonds were found to be very effective in hedging the longevity risk in life

annuities. Zero coupon longevity bonds allow more flexibility in constructing a hedge

strategy, and are able to better hedge the longevity risk in indexed annuities. q-Forwards

are also effective in hedging longevity risk, but contain a significant amount of additional

basis risk over longevity bonds.

The cost of hedging was investigated, with all strategies shown to be sensitive to the price

of risk. Given the current illiquidity in the market for mortality-linked securities, the

cost of hedging may vary significantly, and thus the costs and benefits must be analysed

carefully by insurers seeking to manage their longevity risk.

Two sources of basis risk were analysed, and were found to have minimal impact on

the hedge effectiveness. The first source arises from the uncertainty in the relationship

between annuitant and population mortality, which was quantified based on UK data. The

second source arises from the bucketing of q-Forwards by 5-year age groups. The minimal

impact of these sources of basis risk is encouraging, and allows insurers to hedge their

longevity risk using mortality-linked securities with greater effectiveness and confidence.

This paper has further developed the work of Bauer and Weber (2008) [3] in providing

a more rigorous analysis of longevity risk for an annuity portfolio. Improved market

models have been used to better account for the underlying risks, incorporating the work

of Krolzig (1997) [22]. The work of Dowd et al (2006) [14] was also applied and extended

through an analysis of longevity risk hedging using the improved market and mortality

models. The effectiveness of q-Forwards were also investigated in addition to longevity

bonds.

A further contribution involves the investigation of a larger range of annuities, including

more complex products such as indexed annuities and the GLWB. These products have

potential in insuring against individual longevity risk, and are attractive due to their

flexibility (GLWB) and their insurance against a rising cost of living (indexed annuities).

Thus, these annuities could become useful tools in solving the ageing population problem.

Given the dominance of market risk in these annuities, insurers will need to have appro-

priate market risk management techniques in place, such as the use of inflation-indexed

bonds for hedging inflation risk.

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References 30

A rigorous analysis of the underlying risks is vital in the development of a market for

longevity risk insurance. As shown by the recent financial crisis, it is important that

financial institutions understand their risks to ensure a smoothly functioning financial

system. This analysis of longevity risk has resulted in a greater understanding of the risksinvolved in a range of annuities, and the extent to which these risks can be managed.

Finally, note that this paper focused on static hedging due to their simplicity, lower cost,

and practicability. As the market for mortality-linked securities develops further, dynamic

hedges may become more implementable. Dynamic hedges are likely to be more useful

in hedging indexed annuities and the GLWB, as the strategy can potentially match the

market-dependent claims more closely. Thus, the investigation of dynamic hedges and

their effectiveness would be a useful area for further research, especially in regards to

indexed annuities and the GLWB which were found to be significantly affected by marketrisk.

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static hedging effectiveness for longevity risk

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