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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
Zurich 2005 15th International AFIR Colloquium 2
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
Zurich 2005 15th International AFIR Colloquium 3
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
Zurich 2005 15th International AFIR Colloquium 4
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
Zurich 2005 15th International AFIR Colloquium 5
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
Zurich 2005 15th International AFIR Colloquium 6
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
Zurich 2005 15th International AFIR Colloquium 7
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−− −⎡ ⎤= + + − +⎣ ⎦
Zurich 2005 15th International AFIR Colloquium 8
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
??
Zurich 2005 15th International AFIR Colloquium 9
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
Zurich 2005 15th International AFIR Colloquium 10
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 probability
Zurich 2005 15th International AFIR Colloquium 11
Shortfall probability
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
Zurich 2005 15th International AFIR Colloquium 12
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)
Zurich 2005 15th International AFIR Colloquium 13
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
Zurich 2005 15th International AFIR Colloquium 14
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
Zurich 2005 15th International AFIR Colloquium 15
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
=
>⎧= ⎨ < <⎩
Zurich 2005 15th International AFIR Colloquium 16
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)
Zurich 2005 15th International AFIR Colloquium 17
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
Zurich 2005 15th International AFIR Colloquium 18
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
Zurich 2005 15th International AFIR Colloquium 19
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)
VR(t)(static method)
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)
VR(t)(static method)
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)
VR(t)(static method)
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)
Zurich 2005 15th International AFIR Colloquium 20
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
Zurich 2005 15th International AFIR Colloquium 21
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)
VR(t)(static method)
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)
VR(t)(static method)
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)
VR(t)(static method)
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)
VR(t)(static method)
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
Zurich 2005 15th International AFIR Colloquium 22
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
0.8Risk Bearing Capital - V2
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
0.8Risk Bearing Capital - V3
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
0.8Risk Bearing Capital - V4
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
Zurich 2005 15th International AFIR Colloquium 23
0 5 10 15 200
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4Risk Bearing Capital - V1
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
0.9Risk Bearing Capital - V2
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
0.9Risk Bearing Capital - V3
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
Zurich 2005 15th International AFIR Colloquium 24
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
Zurich 2005 15th International AFIR Colloquium 25
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