the iasb insurance project for life insurance contracts ...zurich 2005 15th international afir...

Zurich 2005 15th International AFIR Colloquium 1

The IASB Insurance Project for The IASB Insurance Project for life insurance contracts: impact life insurance contracts: impact

on reserving methods and on reserving methods and solvency requirementssolvency requirements

Laura BallottaLaura BallottaCass Business School, City University LondonCass Business School, City University London

GiorgiaGiorgia EspositoEspositoUniversity of Rome La University of Rome La SapienzaSapienza

Steven HabermanSteven HabermanCass Business School, City University LondonCass Business School, City University London


Motivation

MotivationFinancial stability of insurance industry affected by increasingly difficult economic climate:

Equity market crashes in 2001 and 2002Steady fall in bond yieldsIncreased longevity

Implication: increased focus of regulators onAccounting rules

Comprehensive financial reporting frameworkStandardization of approaches (where sensible)Improved transparency and comparability of accounting information

IASB Insurance projectIASB Insurance project: IASB & FASB promote a Fair Value based accounting framework

Fair Value: amount for which an asset could be exchanged, or a liability settled, between knowledgeable, willing parties in an arm’s length transaction

SolvencyCapital requirementsInternal models for risk assessment

Solvency IISolvency II


Motivation

Fair value reporting systemEurope:

IFRS 4 (March 2004), in force since January 2005Increased disclosure accounting informationLiabilities to be recorded at FVFV (embedded derivatives)

Assets: regulated under IAS 39Available for sale; or Held for trading M 2 MM 2 MHeld to maturity amortised cost if able to demonstrate intent to forego future profit opportunities

Other countries: UK: Twin Peaks/Individual Capital Adequacy Standards (January 2005)Netherlands: Dutch Solvency Review (January 2006)Switzerland: Swiss Solvency Test (January 2006)

Common features: full markfull mark--toto--market of assets and liabilitiesmarket of assets and liabilities + + risk capital assessmentrisk capital assessment

Critics of the Fair Value approachInconsistent with managing a long-term business such as those of insurance companiesHigh level of subjectivity (measurement of insurance liabilities)

Difficult to provide earningsDifficult to provide earnings’’ forecastsforecastsHigher cost of capitalHigher cost of capital

Increased volatility of reported earnings


AgendaIn the light of these points, we want to analyze the evolution over time of A & L originated by a simple participating contract with minimum guaranteeFocus

Valuation of liabilitiesFair Value“Traditional approach” based on the mathematical reserve

Shortfall probability at maturityRedefinition of the premiumModel and parameter riskCapital requirements and solvency profile of the insurer


Example

Working exampleParticipating contract with minimum guarantee (rG )Single premium (P0 ); ignore lapses, mortality and death benefits or terminal bonusPayoff at maturity

Accumulation scheme:β = participation rateA = equity portfolio

backing the policy

( ) ( ) ( ) ( ) ( )( ), ( )C T P T D T D T P T A T+

= − = −

( ) ( ) ( )( ) ( )

( ) ( ) ( )( )

01 1 , 0

1max ,

1

P

P G

P t P t r t P P

A t A tr t r

A tβ

= − + =

⎧ ⎫⎞⎛ − −⎪ ⎪= ⎟⎜⎨ ⎬⎟−⎪ ⎪⎝ ⎠⎩ ⎭

Cumulated benefitCumulated benefit Default optionDefault option


Valuation

Fair Valuation: 1st order basis

Contract fair value (contingent claim pricing theory)

Standard Black-Scholes framework (log-returns normal distributed, constant interest rates, constant volatility)Benefit “price”

Default option price: via Monte Carlo simulation (due to path dependency)

( ) ( ) ( ) ( ) ( ) ( )1 2

2

1,2

1

ln2

T tr rP G G

G

V t P t e r N d e r N d

rr

d

β β

β σβσ

−− −⎡ ⎤= + + − +⎣ ⎦⎞⎛

+ ± ⎟⎜+ ⎝ ⎠=

( ) ( ) ( ) ( )( ) ( ) ( )* r T tC t P DV t e P T D T V t V t− −⎡ ⎤= − = −⎣ ⎦tE F


Market Value Margin (2nd order basis): model risk

Valuation

The insurance company uses the Black-Scholes paradigm for pricing (simple to implement)The market dynamic of assets though is not log-normalWhat if, for example, the “true” asset in the market is

Parameter adjusted to maintain expected rate of growth and totalinstantaneous volatility constant

01

, t

t

NL

t t t kk

S S e L a t W Xγ=

= = + + ∑

Lévy process Brownian motion Compound Poisson process

Jumps sizeNormal distributed

( ) ( ) ( ) ( ) ( ) ( )1 21T tr r

P G GV t P t e r N d e r N dβ β−− −⎡ ⎤= + + − +⎣ ⎦


ValuationScenario generation

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700

800

A (t)

V P(t)

V D (t)

V C (t)

F u ll geom etric B row n ian m otion m odel: µ = 10% ; σ = 15% ; r = 4 .5% , rG = 4% ; β = 80% ; P 0 = 100

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700

S (t)

V P(t)

V D (t)

V C (t)

M ixed m odel: µ = 10% ; σ = 15% ; r = 4 .5% ; λ = 0 .68 ; µ X = -0 .0537 ; σX= 0 .07 ; rG = 4% ; β = 80% ; P 0 = 100

VVPP(0)(0) == 222.73222.73

VVDD(0)(0) == 122.73122.73

VVCC(0)(0) = V= VPP(0)(0)-- VVDD(0)(0)

== 100100 = P= P00

no arbitrageno arbitrage

ΠΠ(P(T) > A(T))(P(T) > A(T))

GBM GBM –– modelmodel= 74.42%= 74.42%

Mixed modelMixed model = 81.71%= 81.71%based on 100,000 scenariosbased on 100,000 scenarios

Hedging likely to failHedging likely to fail

??


Solvency loading: default option premium

Shortfall probability

Probability of default at maturity unacceptably highOverall contract sold at fair valueBenefit sold too cheaplyDefault option premium!

Solvency loading required to clear arbitrage

““ProtectionProtection”” against defaultagainst default

( ) ( ) ( ) ( )0 0 00 0 0 0 'P D P DP V V V P V P= − ⇒ = + =

Fair value condition for the contract Fair value condition for the benefit

Benefit related premiumBenefit related premium


Adjusted scenario

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700

800S tot(t)

V P(t)

V D (t)

V C (t)

M ixed m odel: µ = 10% ; σ = 15% ; r = 4 .5% ; λ = 0 .68 ; µ X = -0 .0537 ; σX=0 .07 ; rG = 4% ; β = 80% ; P 0 = 100

0 2 4 6 8 10 12 14 16 18 200

200

400

600

800

1000

1200A to t(t)

V P(t)

V D (t)

V C (t)

F u ll geom etric B row n ian m otion m odel: µ = 10% ; σ = 15% ; r = 4 .5% ; rG = 4% ; β = 80% ; P 0 = 100

ΠΠ(P(T) > (P(T) > AAtottot(T(T))))

GBM GBM –– modelmodel= = 6.976.97%%

Mixed modelMixed model = = 12.7412.74%%based on 100,000 scenariosbased on 100,000 scenarios

Any hedging strategy we Any hedging strategy we

can think of, has a higher can think of, has a higher

chance of successchance of success




Shortfall distribution

Significant reduction Significant reduction in the right tail in the right tail

of the distributionof the distribution

No solvencyloading

With solvency loading

GBM model Mixed model


Parameter risk

0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

µ

Sho

rtfal

l pro

babi

lity

0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

σ

Sho

rtfal

l pro

babi

lity

Mixed model

GBM model

Mixed model

GBM model

The higher the expected return The higher the expected return on asset, the lower the on asset, the lower the

probability of defaultprobability of default(assets grow more than liabilities(assets grow more than liabilities))

The higher the volatility,The higher the volatility,the higher the shortfallthe higher the shortfall

probabilityprobability(assets effect only)(assets effect only)


Alternative reporting system: mathematical reserve

Starting point: the contract is issued at a “fair premium”, i.e. there is a solvency loading equal to VD(0), which is invested in the equity fundReserve value (equivalence principle)

The “fictitious account” PR(T) ?

( ) ( ) ( )r T tR RV t P T e − −=

Discount rate = risk free rate of interest

Best estimate of the benefit at maturity


Mathematical reserves

6 possible alternatives for the projection method of expected liabilities at maturity

Static methodDynamic method (4 alternative schemes)Retrospective method

Comparison with FV technique to assessAdequacy in terms of covering of the liabilityCost of implementationVolatility of the balance sheetCost of capitalSolvency profile


Mathematical reserves

PR(T)Static: ( ) ( )0 1 ; 8.5%T

R R RP T P r r= + =

Dynamic: ( ) ( ) ( )( )1 ; ,2 ,..., ; 1,3,4,5kT tR k R k

TP T P t r t k n n nn

− ⎡ ⎤= + = =⎢ ⎥⎣ ⎦

• 1 ( ) ( ){ }knRkR trtr µβ,max= • 2 ( ) ( ){ }knGkR trtr µβ,max=

( ) ( ) ( )( )

1

1 1

1 ; 80%; 8.5%; 4%n

tot k tot kn k R G

k tot k

A t A tt r r

n A tµ β−

= −

−= = = =∑

• 3 • 4( ) ( ) ( ){ }kP

nGkR trtr µ,max=

( ) ( ) ( ) ( )( )

1

1 1

1 nP k k

n kk k

P t P tt

n P tµ −

= −

−= ∑

( ) ( )( )R k R n kr t r tα µ µ= + −1 0 %

i f ( ) 0 i f ( ) 0

n k

n k

s tb s t

µβ

αβ

=

>⎧= ⎨ < <⎩


Scenario generation: static case

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 00

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

1 2 0 0a) G eom etr ic B row n ian m otion

tim e (years )

A to t( t)P (t) V P (t) V R ( t)

• The The static reservesstatic reserves are are always below the fair valuealways below the fair value

•• At maturity the reserves At maturity the reserves are clearly insufficient to are clearly insufficient to cover the liabilitycover the liability

••The The retrospective reserveretrospective reserveis, as expected, always is, as expected, always below the fair value of the below the fair value of the liability; however, at liability; however, at maturity it converges to maturity it converges to the benefit duethe benefit due

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 01 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

7 0 0

8 0 0b ) L evy p roc es s :

tim e (ye a rs )

S to t( t) P (t) V P (t) V R (t)


Scenario generation: dynamic case

0 5 10 15 200

1000

2000

3000

4000a) n=1

time (years)

Atot(t) P(t) VP(t) V1

R(t)V2

R(t)V3

R(t)V4

R(t)

0 5 10 15 200

200

400

600

800

1000

1200b) n=3

time (years)

0 5 10 15 200

200

400

600

800

1000

1200c) n=4

time (years)0 5 10 15 20

0

200

400

600

800

1000

1200d) n=5

time (years)

Geometric Brownian motion model

• The evolution of The evolution of reserves is very reserves is very unstable and unstable and volatilevolatile

•• As As nn increases, increases, the smoothing the smoothing effect becomes effect becomes stronger and the stronger and the peaks reduce in peaks reduce in terms of frequency terms of frequency and magnitudeand magnitude


Scenario generation: dynamic case

0 5 10 15 200

500

1000

1500

2000

2500a) n=1

time (years)

Stot(t) P(t) VP(t) V1

R(t)V2

R(t)V3

R(t)V4

R(t)

0 5 10 15 200

200

400

600

800

1000

1200b) n=3

time (years)

0 5 10 15 20100

200

300

400

500

600

700

800c) n=4

time (years)0 5 10 15 20

100

200

300

400

500

600

700

800d) n=5

time (years)

Lévy process model


Adequate reserves? Π(VR(t)<VP(t))

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1P(V1

R(t)<VP(t))

time (years)

VR(t)(static method)

n=1n=3n=4n=5

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1P(V2

R(t)<VP(t))

time (years)


0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1P(V3

R(t)<VP(t))

time (years)


0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1P(V4

R(t)<VP(t))

time (years)


Levy process model

At maturityAt maturity

The dynamic The dynamic reserves are reserves are consistently consistently above the Fair above the Fair Value Value more more expensive to expensive to implementimplement

During the During the lifetime of the lifetime of the contractcontract

Static method:Static method:

ΠΠ(V(VRR(T)<P(T))=92%(T)<P(T))=92%

Dynamic method:Dynamic method:VVRR(T)=P(T)(T)=P(T)


An important aspect of Solvency II project: Target capital

The target capital is the amount needed to ensure that the probability of ruin of the insurance company, within a given period, is lowLikely recommendations:

0.5% ruin probability1 year time horizon

We consider the Risk Bearing Capital (SST)

The best estimate of liabilities is given by both the FV approach and the mathematical reserves approach

Market value of assets - Best estimate of liabilityBest estimate of liability


0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1a) V1

R(t)

time (years)


n=1 n=3 n=4 n=5 VP(t)

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1b) V2

R(t)

time (years)


0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1c) V3

R(t)

time (years)


0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1d) V4

R(t)

time (years)


The Risk Bearing Capital: The Risk Bearing Capital: ΠΠ(RBC > 0)

The fair value approach generates the lowest shortfall probabiliThe fair value approach generates the lowest shortfall probability and the more stable ty and the more stable RBCRBC


Mean of Risk Bearing CapitalMean of Risk Bearing Capital

0 5 10 15 20-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8Risk Bearing Capital - V1

R(t)

time (years)

mea

n

n=1 n=3 n=4 n=5 VP(t)

0 5 10 15 20-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6


R(t)

time (years)

mea

n

0 5 10 15 20-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6


R(t)

time (years)

mea

n

0 5 10 15 20-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6


R(t)

time (years)

mea

n

The mean of RBC is always positive by using the fair valueThe mean of RBC is always positive by using the fair value


0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35


R(t)

time (years)

varia

nce

n=1 n=3 n=4 n=5 VP(t)

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8


R(t)

time (years)

mea

n

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8


R(t)

time (years)

varia

nce

0 5 10 15 200

5

10

15

20

25Risk Bearing Capital - V4

R(t)

time (years)

varia

nce

Variance of Risk Bearing CapitalVariance of Risk Bearing Capital

Irregular pattern of the annual variance of the dynamic reservesIrregular pattern of the annual variance of the dynamic reserves


0 5 1 0 1 5 2 00

0 .5

1

1 .5

2

2 .5

R i s k B e a r i n g C a p i ta l - V R ( t) (s ta ti c m e th o d )

ti m e (ye a rs )

mea

n

0 5 1 0 1 5 2 00

1

2

3

4

5

6

t i m e (ye a rs )

varia

nce

Moments of Risk Bearing Capital for static Moments of Risk Bearing Capital for static method of calculating reservesmethod of calculating reserves

This result is affected by the low level of static reserves (which are

unable to cover the actual amount)

This result is due to the fact that the reserves are completely

uncorrelated with the trend of assets


ConclusionsThe example shows

The market-based valuation approach provides awareness of default risk and suggests possible rule for the redefinition of the premia chargedReserving systems need to be consistent with, and appropriate for, the nature of the liabilities

More specific guidelines needed for fair valuation of insurance liabilities

Target accounting model (impact on MVM, shortfall probability, solvency capital requirements) … work in progress

The fair value approach can guaranteeEffective and cost efficient ALM policyStable solvency profile

the iasb insurance project for life insurance contracts ...zurich 2005 15th international afir...

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