allianz lebensversicherungs-ag · 19. mai 2003 das risikokapitalmodell der allianz...

Lebensversicherung

Das Risikokapitalmodell derAllianz Lebensversicherungs-AG

Ulm19. Mai 2003

Dr. Max HappacherAllianz Lebensversicherungs-AG

19. Mai 2003 Das Risikokapitalmodell der Allianz Lebensversicherungs-AG

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LebensversicherungTable of contents

1. Introduction: Motivation, Group-wide Framework

2. Internal Risk Model: Basics, Life Approach

3. Standalone Risks and Liability Response

4. Simulation of Assets & Fair Value of Liabilities

5. Liability Response: Dynamic Reseves/Benefits

6. Liability Response: Valuation of Liabilities

7. Risk Aggregation, Diversification and Mitigation


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Lebensversicherung

Main goals for Allianz internal risk analysisSupporting value management

• Systematic risk assessment based on internal data

• Improving risk adjusted profitability measurement

• Portfolio management / asset mix

• Capital allocation

• Compare outcome against rating agencies’ models

Meeting legal requirements

• Risk reporting → More detailed quantitative requirements by DRSC or SEC

• Solvency → New solvency requirements on the way; new Basel II-like requirements for insurance companies (Solvency II) and financial conglomerates allow for internal risk analysis models


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LebensversicherungRisk analysis project covering AZ Group flagshipsPC: SGD, AZRE, Cornhill, AZ Elementar, AZ Suisse, RAS, Lloyd Adriatico, AZ

Seguros, AGF, FFIC, AIC, AZ Australia, Hermes/Euler

Life: AZ Leben, AZ Elementar, AZ Suisse, RAS, Lloyd Adriatico, AZ Seguros, AGF, AZ Life, AZ First Life

Banking: Dreba Group

Consistent framework of risk categories for all segments

Respect material differences between & within segments by specific risk models


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Lebensversicherung

Consistent framework for risk analysis, risk capital & capital allocation

Risks of AZ Leben

Insurance Risks

ProspectiveClaims

Pricing

Retrospective Claims

PC Risk

PremiumRisk

ReserveRisk

Non-Cat

Cat

Link to S&P standard categoriesC3 C4 C5 C2 C1 C6

Operating Risk

LifeRisk

Market Risk

Counter-party Risk

ReinsurerCredit

Worthiness(Security)

Market RisksCredit RisksOther Risks

Business Risk- Lapse- Cost

Operational Event Risk- IT failure- Litigation- Fraud

Biometric Risks- Mortality- Longevity- Disability- Calamity


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Lebensversicherung

Capital requirement depends on point of view

Internal risk approach measures the required risk capital on an economic point of view

ASSETS

LIABILITIES

Required Capital Net Asset Value =

Economic Capital

Excess Capital

Regulatory Capital

Rating Agency Capital

(Economic) Risk Capital• Market Value

• Best Estimates


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Lebensversicherung

Economic Capital is volatile due to unexpected developments affecting assets and liabilities

ASSETS• Market Value

LIABILITIES• Fair Value

ECONOMICCAPITAL

(NAV)

Market Crisis

Large Claims

All expected profits and losses are included in the best estimate NAV, unexpected developments induce volatility or risk into the business

INSOLVENCY

Probability 0

Economic Capital

BE NAV


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LebensversicherungRisk Capital: Potential loss of expected NAV for the next 12 months (VaR approach)

NAV0

Probability Density

Required Risk CapitalExpected NAV

SOLVENCY STANDARD

AAA0.01%

AA0.03%

A0.07%

Worst Case

Risk capital is the capital required to run the business tied toa given solvency / worst case level

Key parameters• Confidence interval

/ solvency level• Risk distribution• Risk correlation• Risk concentration

Diversification

Worst case


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LebensversicherungModelling risks - conceptional overview

Equity

Interest rate

Real estate

Standalone risks

Liability response via simulation

CV Credit

CV Life

CV Business

CV Op. Event

TCV

TCV -- Total C

hange in Total C

hange in ValueValue

Risks shareable with policyholder(correlated)

Non-shareable risks

BE

WC IR

WC EQ/RE

WC Shar.

CV IR

CV EQ/RE

CV Shar.

Aggregation

Diversification

CV

CV

IRIRC

VC

VEQEQ

Busin

Busin ..

Op.

Op.

LifeLife

Mitigation

ShareholderShareholder

PolicyholderPolicyholder

Credit

Life

Business

Operationalevent

FX

CV FX


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LebensversicherungRisk sharing with policyholder

... completely via profit sharing (´´liability response´´)• Equities (EQ)• Real estate (RE)• Interest rate (IR)

... partially via profit sharing:• Credit (Default)• FX• Life (calamity, volatility, trend)• Business (lapse, new business, cost inflation)

... not sharable via profit sharing:• Operational event

} via asset returns


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4. Simulation of Assets and Fair Value of Liabilities





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Lebensversicherung

Change in Value (CV) as difference between asset value impact and liability response

Value Best Estimate Asset Value

Worst Case

Asset Value Impact

BestEstimateLiabilityValue

LiabilityValueImpact

AssetValueImpact Liability

ValueImpact

InvestmentChange in Value (CV)

WorstCase

Modelling life risks: worst casechange in value


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Lebensversicherung

Standalone risks for life insuranceWorst case shock scenarios

• According to 0.07%-quantile of corresponding distribution

• Impact of shock in the first year• Propagation for next n years necessary for

valuation of liabilitiesThree shocks are considered:

• Equity: 1st year loss of roundabout 40 %• Interest rates: down shift of yield curve by 1,5 %• Biometrical, operational and business risks lead to

an impairment of gross profits


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Lebensversicherung

Equity risk... before liability response

• Market value view (after hedging)• Risk driver: volatility of stock returns• Equity portfolio modeled by 4 indices

(DAX100, MSCI: EMU, EUROPE, WORLD)• Lognormal distribution• Best estimate (BE) return: 8,5%• Total volatility:

» historical volatilities of indices» covariance matrix for correlation effects

• Issues: time frame, ex post / ex ante, private equity, derivatives

Worst Case

BestEstimate

Risk Capital

Probability

PortfolioValue


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Lebensversicherung

IR down risk: Risky present value due to volatile discount rateDown-side Risk

• Down shift of interest rate curve in year 1• Shifted IR curve also used in years 2 to n• Lower discount rate leads to higher values of (fixed income)

assets and liabilities

• Increase of liabilities higher due to higher duration• Impact of lower asset returns on the liabilities lessened by

guaranteed rate for P/H

Yield (%)

Time

Yield Curve

Time

PresentValue

t=0 t=1 t=2 t=3 t=4


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Lebensversicherung

Other sharable risks... before liability response

Riskcategory

Subrisks Method Value[Mio. €]

RatioPH:SH

Issues

Investment(cont’d.)

Credit

FX

Distribution/ SimulationAnalytic

> 1000

small

50:50 Hugeportfolio

Hedges

Life CalamityVolatilityTrend

GdV-ModelGdV-ModelGdV-Model

< 10*> 10*> 100*

100:080:2080:20

Availabilityof internaldata

Business LapseNew businessCost inflation

GdV-ModelGdV-ModelProxy

> 10> 100> 10

80:2080:2080:20

Availabilityof internaldata

Total aggregated sharable risks: ~ 1 Mrd. €

* Values after reinsurance


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LebensversicherungFair value of liabilities...

... is the present value of future cash flows

... is influenced by asset movements:• Liability cash flows (usually) depend on future

portfolio (asset) returns• Profit sharing creates a tie between asset return

and liability value• Management rules to describe the

interdependence... depends on interest rates:

• Present value of future liability cash flows changes with moving interest rates


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LebensversicherungFair value of liabilities: Models

Deterministic world:• Future asset returns are assumed to be given• Future cash flows on basis of a known profit

sharing scheme• Options of insurer and PH are negligible

Real world:• Returns are volatile• Profit sharing according to actual return• Options of insurer and PH have to be considered


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Lebensversicherung

Guaranteed rates have negative financial value for the shareholder

Investment return paths may be below guaranteed rate

Investment Return

Guaranteed rate

Time

Best Estimate Return

The risk of investment return below guaranteed rate is pure shareholder risk


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Lebensversicherung

The cash flows of profit sharing products are only dependent on the investment return above the guaranteed rateInvestment

Return

Guaranteed rate

Time

LIABILITY CASH FLOWS(Schematic)

Thus financially the liability cash flow structure of this product is similar to a floor on the investment return


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Lebensversicherung

Pricing this “floor” can easiest be done by simulation

1. Simulate different scenarios of future interest rates and inv. returns

2.Calculate in each scenario the corresponding future liability cash flows

Year 1 Year 2 Year 3 Year 4 Year 5

Investment Return 7.0% 6,5% 5,5% 2,5% 3,0%Interest Rates 5.5% 4,5% 3,0% 2,0% 2,8%

3.Discount these cash flows at the simulated interest rates

Year 1 Year 2 Year 3 Year 4 Year 5

Cash Flow 70 65 55 30 30

Scenario i

Scenario i

4.Fair Value: Average value over all scenarios

∑ = +⋅= 5

1 ))(_1(1)(___

k kkRateInterestkFlowCashiScenarioValue

( ) niScenarioValueValueFair n

i/___

1∑ ==


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Lebensversicherung

Portfolio return: Combined return of asset classes

Return

Time

Return

Time

Return

Time

Return

Time

FIXED INCOME RET. EQUITY RETURN REAL ESTATE RETURN

PORTFOLIO RETURNS


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LebensversicherungSimulation of equity / real estate

Input:• Expected asset performance• Volatility of the portfolio (cont. compounded)

Output: 1-year performance P

Model: ;

therefore:

with i the cont. compounded return expectation

2σ

µ

2/1)(2+=+= σρµ ePEσρ,

+= LogNValue

ValuePt

t ~1

2−=

2−)+=

22 σσµρ i1log(


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LebensversicherungSimulation of FI-returns

Input:• Portfolio statement comprising the cash flows of

the FI portfolio (CFs for each investment)• Model for simulation of future interest rates

Assumptions:• Reinvestment in risk free bonds with same

duration as initial portfolio• Bonds are held to maturity

Simulation model for interest rates:• Crucial: IR model has to be arbitrage free


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LebensversicherungDynamic reserves and benefits

Problem: For each simulation run a new liability projection necessary

Retrospective Approach:• Two projections are used: declared profit sharing

(BE) and minimum guaranteed (MG)• Basis is the minimum guaranteed projection run• For each simulation run dynamic reserves are

generated retrospectively via interpolation• Dynamic benefits are generated via scaling


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Lebensversicherung

From simulated returns todynamic reserves and benefits

• PH credited rate crt as moving n-year average ofsimulated returns (after shareholder profit)

• Initial reserves R0 and initial cash flows Rim1identical for BE and MG

• Dynamic reserves calculated retrospectively:

• Dynamic benefits according reserves at b.o.y.:

• Valuation reserves are used as buffer

) Rim – Rim (R – R

R – R Rim Rim tMGtBEtMGtBE

tMGt tMGt 1111 ++++ +=

tttt Rim )) * R(mg, cr ( R ++= −1max1


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LebensversicherungValuation of cash flows

Central problem:

Appropriate discount factors for cash flows

Approach:• Simulate asset returns using actual return

expectations ( i. e. 5 % on RE, 8,5 % on EQ,...)• Discount cash flows at risk free rate

Problem:• Return expectations are risk adjusted• Discounting at risk free rates overstates PV


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LebensversicherungRisk neutral valuation

2 alternatives:a) Adjust discount factors to intrinsic risk of cash flowsb) Simulate asset returns using risk free rates and

discount liability cash flows at risk free rates

No applicable way to obtain option adjusted discount factors for liability cash flows in a).

Alternative b):

Concept of risk neutral valuation


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LebensversicherungSummary:Liability CF simulation and valuation

(200)(150)(100)

(50)-50

Year

CF's

3Determine liability

cash flows

1 Generate risk free rates

0%4%8%

12%16%

0 20 40 60Year

Rate

2 Generate risk free asset returns

-30%

20%

0 20 40 60Year

Return

4 Discount @ risk freerate

liability valuei = X Mio. €

5

liability value =

Y Mio. €

Average liability value

Liability value: Done 1000 times in each simulation


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Lebensversicherung

Simulation used to calculate the liability value in a given shock scenario

Portfolio ofAssets

Portfolio ofLife Policies

Simulation of investment returns

after next year

DynamicLiabilities

PV of Assets

PV of

Liabilities

PV of the Net

Position•BE Case•WC Shocks

Shock Scenarios


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Lebensversicherung

Boundary conditions for WC scenarios:

Scaling of assets and liabilities:• A: 100% = capital investments + cash• L: 100% = insurance reserves - free RfB

Asset mix: constant

Liability response & ALM model

WC scenario EQ/RE FI IR curveBE BE BE BE

EQ/RE WC BE BE

IR down BE WC WC (down)

Sharable BE –WC Shar.

BE BE


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Lebensversicherung

Standalone risks ...before and after liability reponse

ScenarioAsset Value

Impact

Liability Value Impact

Change in

ValueEQ -100 +33 -66RE -100 +33 -66

IR down +100 -150 -50Shareable -100 +10 -90

Liability response is a 1st order effect


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LebensversicherungAggregation to total risk: TCVStandalone change in value after liability response: CViare aggregated to total change in value (TCV) viaparametric aggregation:

jji

iji CVCCV TCV ∑∑∑∑ ⋅⋅⋅⋅⋅⋅⋅⋅====,

with covariance matrix Cij expressing interdependence between standalone risks.

Disaggregation for risk analysis: ii

divi CVCVTCV CV ⋅⋅⋅⋅∂∂∂∂∂∂∂∂====,

Diversification benefits: ∑∑∑∑∑∑∑∑ ≤≤≤≤====i

ii

divi CVCVTCV ,


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Lebensversicherung

Risk mitigation:Hierarchy of buffers

S/H Risk Capital

Withoutrisk mitigation

TCV

With risk mitigation

TVR fr. RFB

1/2 LTD

1/2 LTD

SE

Σ1 Σ2 Σ3Shareholder part qSH = 0,1 * TVR/ ΣΣΣΣ1

Shareholder risk capital: RC = qSH • TCV if TCV < ΣΣΣΣ1

TVR: Total valuation reservesfr.RfB: free RfBLTD: Liability for terminal dividendsSE: Shareholder Equity


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Lebensversicherung

Review: standalone risks... before and after ...E.g. credit risk: 100 €

• After 50:50 risk sharing with P/H: 50,0 + 50,0

• After liability response: 50,0 + 45,0 = 95,0

• After diversification: 65

• After risk mitigation: 5


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LebensversicherungBackup: Interest rate model

Y0(t): current, continuously comp. zero coupon yield => Discount factor:=> Continuously compounded forward rate:

For tenor structure T define forward rate processes as

with

Choose volatility to meet 5-year swap volatility, (5 = average duration of FI portfolio)

2

1 0 1( , ) ( , ) , where ,2

tX it i i i i t if T T f T T e X N σ σ+ +

= × −

�

[ ] [ ] ( )221 0 1 1 0 1( , ) ( , ), ( , ) ( , ) e -1 t i i i i t i i i iE f T T f T T Var f T T f T T σ

+ + + += =

~

)/()()((),( 12110220210 ttttYttYttf −−=

))(exp( 0 ttY ⋅−

allianz lebensversicherungs-ag · 19. mai 2003 das risikokapitalmodell der allianz...

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