Pierre DEVOLDER Université Catholique de Louvain 03/ 09/2004

FAIR VALUATION OF INSURANCE

LIABILITIES

Pierre DEVOLDER

Fair value of insurance liabilities

1. INTRODUCTION TO FAIR VALUE

2. RISK NEUTRAL PRICING AND DEFLATORS

3. EXAMPLES : THE BINOMIAL and THE

BLACK/ SCHOLES CASES

4. FAIR VALUE OF LIFE INSURANCE

PARTICIPATING CONTRACTS

1. Introduction to FAIR VALUE

9International norms IAS / IFRS for all financial institutions in Europe soon ( …)

9…A lot of discussions linked to accounting principles and artificial volatility

1. Introduction to FAIR VALUE

‰ Basic principle : from an historical or statutory accounting point of view to fair value bases

‰ Fair value : price at which an instrument would be traded

if a liquid market existed for this instrument

‰ ASSETS : market values

‰ LIABILITIES : ???

1. Introduction to FAIR VALUE

‰ Need to develop good models of valuation especially for actuarial liabilities where there is no market price

‰ Consistency between modern financial pricing theory and classical actuarial models

‰Important example : guarantees in life insurance and pension:one of the most challenging risk nowaday

2. Risk neutral pricing and Deflators

Purpose :

- introduction to the modern financial paradigm of risk neutral pricing

- link with deflator methodology

- link with classical actuarial principle of discounting

2. Risk neutral pricing and Deflators

Paradigm of risk neutral pricing:

Purpose: to compute the present price of a future

stochastic cash flow correlated with financial market in an uncertain environment

What we could expect : price = discounted expected value of the future cash flow:

)) T ( L ( E )

i 1 (

1 )

0 (

L _T

+

2. Risk neutral pricing and Deflators

You have to change either the probability measure, either the discounting factor

Change of probability measure : risk neutral method:

You can stay with a classical discounting factor but the expectation of the cash flows must be done

using another probability measure than the real one

Change of discounting factor : deflators method:

2. Risk neutral pricing and Deflators

Risk neutral pricing :

)) T ( L ( E )

r 1 (

1 )

0 (

L _{T Q}

+ =

Q= risk neutral measure

Deflator method :

)) T ( L ) T ( D ( E )

0 (

L =

2. Risk neutral pricing and Deflators

Single period market model : 9 one riskless asset :

rate riskfree

r with r

S

S₀(1) = ₀(0)(1+ ) =

9 d risky assets defined on a probability space :

N j

P p

with _{j j}

N} ({ }) 1,...,

,..., ,

{ _{1 2} = =

=

Ω

ω

ω

ω

ω

d i

S S

S

S_i(1) = ( _i(1,

ω

ω

ω

2. Risk neutral pricing and Deflators

9STATE PRICE : Basic property :

if the market is without arbitrage there exists a random variable ψ such that for any asset :

∑

=

〉 Ψ

= Ψ Ψ

= N

j

j j

j i

j

i S with

S

1

0 ) (

) ,

1 ( )

0

(

ω

ω

If the market is complete, the state price is unique.

Consequence : for asset i =0 ( risk free asset)

∑

= +

= Ψ =

Ψ N

1 j

j

r 1

2. Risk neutral pricing and Deflators

RISK NEUTRAL MEASURE

Definition : artificial probability measure given by:

j j

j

j Q( ) (1 r)

q = + Ψ

Ψ Ψ = ω

= ω =

Properties :

∑

=

= ≤

≤ N

1 j

j

j 1 and q 1

q 0

2°) for each asset: mean return= risk free rate 1°)

∑

=

ω +

= N

1 j

j i

j

i q S (1, )

) r 1 (

1 )

2. Risk neutral pricing and Deflators

9DEFLATOR : random variable D defined by :

j j j

j

p D

D(

ω

Properties :

∑

= +

= =

N 1 j

j j

r 1

1 )

D ( E D

p .i

( expected value of the deflator = classical discount )

∑

= N p D S (1, )

2. Risk neutral pricing and Deflators

∑

j X(1, j) ) 0 ( X : vision price State .i

Generalization : if X is a financial instrument on this market (replicable by the underlying assets) and giving for scenario j

a cash flow X(1,j) then the initial value of this instrument can be written as :

)) 1 ( X ( E r 1 1 ) j , 1 ( X q r 1 1 ) 0 ( X : vision neutral Risk .

ii N _Q

2. Risk neutral pricing and Deflators

Multiple periods model : discrete time model ( t=0,1,…, T) 9Riskfree asset:

rate riskfree

r with r

S t

S₀( ) = ₀(0)(1+ )t =

9Risky assets :

d i

t S t

S t

S t

S_i( ) = ( _i( ,

ω

ω

ω

9STATE PRICE :

∑

=

〉 Ψ

= Ψ

Ψ

= N _{j i j j j}

i t S t with t t

S

1

0 ) , (

) ( )

, ( )

( )

0

2. Risk neutral pricing and Deflators

9 DEFLATOR :

j scenario if

to t from factor

discount p

t t

D

j j

j 0

) ( )

( = Ψ =

9Pricing : if X is a financial replicable instrument on this market generating successive stochastic cash flows :

} ;

,..., 1 );

, (

{C t

ω

ω

Then the initial price of X can be written alternatively :

∑ ∑

= =

Ψ

= T

t

N

j

j

j t

t C X

1 1

) ( )

, ( )

0

2. Risk neutral pricing and Deflators

Or with risk neutral measure:

∑

∑

= =

ω +

= T

1 t

N 1 j

j j

t q C(t, ) )

r 1 (

1 )

0 ( X

Or with deflators :

∑

∑∑

= = =

= ω

= T

1 t j

j T

1 t

N 1 j

j C(t, )D (t) E(D(t)C(t)) p

3.1. THE BINOMIAL CASE

Single period model:

9Risky asset :

with probability p u

S

S₁(1) = ₁(0)⋅

with probability 1-p d

S ⋅

= ₁(0)

Absence of arbitrage opportunities if:

u r

d < + <

< 1 0

Other form of the risky asset :

µ λ + +

+

= r

3.1. THE BINOMIAL CASE

µ

0 <

λ

volatility premium

risk =

=

µ

λ

Equations of the STATE PRICE :

1 )

1 ( )

1

( + r Ψ₁ + + r Ψ₂ =

For i=0:

1 2

1 + Ψ =

Ψ d

u

3.1. THE BINOMIAL CASE

Solution for the STATE PRICE:

) 1 ( 2 ) )( 1 ( 1 1 r d u r d r + − = − + − + = Ψ

µ

λ

µ

µ

λ

µ

Safety principle :

0

1

2 = Ψ =

Ψ if

λ

) (

0

1

2 〉 Ψ if > normal case

3.1. THE BINOMIAL CASE

Fair value in a binomial environment – single period :

If X is a financial instrument on this market with future stochastic cash flows given respectively by :

2 2

1

1) (1, )

, 1

( X and X X

X

ω

ω

Then the initial fair value of X is given by :

2 2 1

1

) 0

( = X Ψ + X Ψ

X

Or :

) r 1 ( ) X X

( 2 1 )

r 1 (

1 )

X X

( 2 1 )

0 (

X _{1 2 2 1}

+ µ

λ −

+ +

3.1. THE BINOMIAL CASE

Multiple periods model:

1

2

d

2

u

d

d u⋅ u

3.1. THE BINOMIAL CASE

Structure of STATE PRICES in multiple periods:

Assumption: financial product having successive cash flows depending only on the current situation of the market (no path dependant).

1

11

Ψ

21

Ψ

22

Ψ

Ψ

12

Ψ

State price

j scenario if

t time at

price state

3.1. THE BINOMIAL CASE

Value of the STATE PRICES:

1 2

1 1

1 − − −

− _{Ψ Ψ}

=

Ψ j j t j

t j

t C

Where j-1 = number of up (j=1,..,t+1) t-j-1 = number of down

1

−

j t

C

3.1. THE BINOMIAL CASE

Fair valuation in multiple periods – binomial :

If X is a financial instrument having successive cash flows in the tree given by :

j scenario if

t time at

flow cash

X_{t j} =

Then the initial fair value of X is given by :

∑ ∑

= +

=

Ψ

= T

t t

j

j t j t

X X

1 1

1

3.2. THE BLACK / SCHOLES CASE

Continuous extension : Black and Scholes model Riskless asset :

dt ) t ( S r )

t (

dS₀ = ₀ Risky asset :

) t ( dw )

t ( S dt

) t ( S )

t (

dS₁ = δ ₁ +σ ₁

Where:- w is a standard brownian motion

3.2. THE BLACK / SCHOLES CASE

Risk neutral measure :

) t ( * dw ) t ( S dt ) t ( S r )) t ( dw dt ) r ( )( t ( S dt ) t ( S r ) t ( dS 1 1 1 1 1 σ + = σ + − δ + = t r ) t ( w ) t ( * w : with σ − δ + =

Q = risk neutral measure ( Girsanov Theorem)

3.2. THE BLACK / SCHOLES CASE

Deflators: stochastic process D such that for i=1 (risk free asset) and for i=2 ( risky asset):

)) t ( S ) t ( D ( E ) 0 (

S_i = _i

Solution: t r 2 1 ) t ( w r rt 2 e e ) t (

D ⎟⎠

⎞ ⎜ ⎝ ⎛ σ − δ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ σ − δ − − =

Safety principle :

4. Fair value of participating

contracts

4.1. Liability side:

Life insurance contract with profit :

guaranteed interest rate + participation 4.2. Asset side :

Strategic asset allocation:

Cash + Bonds + Stocks 4.3. Valuation of the contract :

4. Fair value of participating

contracts

Need for a consistent ALM approach:

Double link between asset and liability in this kind of product :

Liability Asset :

Investment strategy must take into account the specificities of the underlying liability

Liability : Asset

4.1. Liability side

Pure Endowment contract : - initial age at t=0 : x

- maturity : N

- Benefit if alive at time t=N : 1

- Benefit in case of death before N : 0

- Contract with single or periodical premiums (pure premium)

- Technical parameters : -mortality table:

- guaranteed technical rate : i - participation rate on surplus: η

4.1. Liability side

Bonus systems :

general definition: percentage of the surplus (Assets – Liabilities)

Three possible schemes :

Terminal bonus : bonus computed at maturity

Reversionary bonus : bonus computed each year and fully integrated in the contract as additional premium

4.2. Asset side

Cash – Bonds- Stocks model ( CBS model):

The underlying portfolio of the insurer is supposed to be invested in three big classes of asset:

- Cash : short term position ( money account) - Bonds : zero coupon bonds with a maturity

not necessarily matched with the duration of the contract

4.2. Asset side

Cash model :

Money market account :

rate free risk ) t ( r with dt ) t ( ) t ( r ) t ( d = β = β

Risk free rate :Ornstein-Uhlenbeck process

4.2. Asset side

Bond model

P(t,T)= price at time t of a zero coupon with maturity T

General evolution equation :

) t ( dw ) T , t ( ) T , t ( P dt ) T , t ( ) T , t ( P ) T , t (

dP = µ − σ ₁

Particular case : VASICEK Model

4.2. Asset side

Stocks model :

S(t)= value at time t of a stock index

4.2. Asset side

Portfolio :

1

stocks in

proportion

bonds in

proportion

cash in

proportion

S B

C S B C

= α + α + α

= α

= α

= α

2 main assumptions :

4.2. Asset side

Bond strategy :

Assumption : at each time t zero coupon bonds of only one maturity can be held but the maturity has not

necessarily to match the duration of the liability ( price of

mismatching-long duration of life insurance contract)

Strategy 1 : matched strategy : T=N

Strategy 2 : shorter maturity of the bond

4.2. Asset side

Bond strategy

tn N

s=0 t1 t2 t3

nt reinvestme of

ts tan ins

: t ,..., t

, t , 0 t

nts reinvestme of

number :

n ,..., 1 , 0 s

n 2

1

0 =

=

4.2. Asset side

Evolution of the portfolio :

portfolio underlying the of value market ) t ( V =

Evolution equation :

4.2. Asset side

Value at maturity of the assets :

d distribute normally )) t ( V ln ) t ( V (ln term each with )) t ( V ln ) t ( V (ln ) 0 ( V ln ) N ( V ln s 1 s s n 0

s s 1

= − − = − + = +

∑

2 = −

σ

− =

4.2. Asset side

Explicit form :

4.3. Valuation of the contract

Example of a contract with single premium and terminal bonus :

Fair value at maturity = pay off of the contract

) 0 ; ) i 1 ( ) N ( V max( )

i 1 ( ) N (

FV = + N +η − + N

Initial fair value : (? In line with the single premium)

)) ) i 1 ( ; N ; V ( Call )

N , 0 ( P ) i 1 ( ( p )

0 (

FV N N

x

N + + η +

4.3. Valuation of the contract

Equilibrium condition :

1 ) ) i 1 ( ; N ; V ( call ) N , 0 ( P ) i 1

( + N + η + N =

Consequences : 1°) ) )) N , 0 ( R 1 ( 1 ) N , 0 ( P with ( ) N , 0 ( R i N + = <

4.3. Valuation of the contract

3°) implicit relation for the technical rate i ;

4°) explicit relation for the participation rate η:

) ) i 1 ( ; N ; V ( call

) N , 0 ( P ) i 1 ( 1

N N

+ +

− =

η

4.3. Valuation of the contract

Risk forward neutral measure method :

measure neutral risk forward Q where ) 0 ); ) i 1 ( ) N ( V (( (max E ) N , 0 ( P call N T Q_N = + − =

In the CBS model we have :

4.3. Valuation of the contract

With for instance for the matched strategy :

5. FAIR VALUE OF VARIABLE

ANNUITIES

Purposes :

ƒ How to valuate pension annuities not in terms of technical basis but in terms of market fair values;

ƒ Influence of reversionary bonus ( variable annuities) on the level of provision;

ƒ Sensitivity of the provision with respect to financial parameters;

5.1. Liability side

- Immediate lifetime annuity for an affiliate to a pension fund - x : initial age at time t=0

- Liability to pay: 2 cases :

j scenario for

t time at

pay to

amount L_t,j =

( possibility to increase yearly the pension depending on the financial performances – asset side)

1 ) fixed annuity : L

2) variable annuity :

5.1. Liability side

Actuarial first order bases :

rate discount

technical i =

t time at

y probabilit survival

p_x

t =

Technical provision for a constant pension ( case 1):

L L_{t, j} =

t x

t n

t n

x x

n

i p

L a

L V

) 1

( 1

1 +

=

=

∑

5.2. Asset side

Binomial model :

mixed financial strategy of the pension fund between riskless asset (r= riskfree rate)

and risky asset (binomial model u / d)

γ

: part invested in the riskless asset

γ

−

1

) 1 0

5.3. Bonus scheme

Definition of the reversionary bonus for variable < annuities

Used rule of bonus : comparison each year between the effective return of the assets and the riskfree rate; a part of this surplus is given back to the affiliate:

: participation rate

1

5.3. Bonus scheme

Yearly rate of increase of the pension :

¾If the risky asset is up :

+ − + + − + + =

+ 1)

) 1 ( ) 1 )( 1 ( ( 1 1 r r u

k

β

γ

γ

) 1 ( 1 1 r k + + + =

+

βγ

λ

µ

or

¾If the risky asset is down :

1 ) 1 ) 1 ( ) 1 )( 1 ( ( 1

1 − =

+ + − + + =

5.3. Bonus scheme

Final form of the liabilities of the variable annuity:

(

)

, 1

+ − +

⋅

= t j

j

t L k

L

Where t-j+1 is the number of times of up permitting to give a bonus.

As expected

5.4. Valuation of the contract

Computation of the fair value of the liabilities : (fixed annuity)

( )

∑

∑

x p L

5.4. Valuation of the contract

Computation of the fair value of the liabilities (variable annuity)

• Actuarial valuation : not so simple: liabilities not deterministic • Fair valuation : general formula of valuation :

( )

∑

∑

k p L

L FV 1 1 , 1 ,

(

)

(

)

[

]

n

x

t p k

L Ψ + Ψ +

=

∑

(

)

(

)

∑

∑

t p C k

5.4. Valuation of the contract

Computation of the fair value of the liabilities (variable annuity) t n t t x t n x k r r p L L FV ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + − + ⎥⎦ ⎤ ⎢⎣ ⎡ + =

∑

= 2 (1 )

1 1 1 ) ( 2 2 1 ,

_µ

λ

µ

βγ

∑

= xn

i

a

L *

5.4. Valuation of the contract

Equilibrium relation :

i* = equilibrium constant discount rate given by :

)

)

1

(

2

1

(

/

)

)

1

(

2

(

*

2 2

2 2

r

r

r

i

+

−

+

+

−

−

=

µ

λ

µ

βγ

µ

λ

µ

βγ

r i

or

if = = =

→

β

γ

r i

and

if > > <

5.5. Numerical illustration

Central scenario: u=1.1 d=0.99 r=0.03

i=0.025

Mortality: GRM 95

Risk premium : λ =0.02 Volatility : µ = 0.06

5.5. Numerical illustration

0,1 0,2 _0,3

0,4 _0,5

0,6 _0,7

0,8 _0,9 1

Fixed annuity 40%stocks

100%stocks 4000

4100 4200 4300 4400 4500 4600 4700 4800 4900

5.5. Numerical illustration

Volatility sensitivity analysis : ( 60% in risky asset)

4400 4500 4600 4700 4800 4900 5000

0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,1

"Volatility factor"

5.5. Numerical illustration

Value of the equilibrium discount rate : central scenario

1° for β = 0.5 and γ =0.6 : i*= 2.21%

2° for β = 1 and γ =0.6 : i*= 1.42%

3° for β = 0.9 and γ =0.4 : i*= 2.05%

4° for β = 1 and γ =1 : i*= 0.4%

% 6

% 2

= =

µ

λ

5.5. Numerical illustration

Value of the equilibrium discount rate : other scenario ( less volatile)

1° for β = 0.5 and γ =0.6 :

i*= 2.60% versus 2.21% 2° for β = 1 and γ =0.6 :

i*= 2.21% versus 1.42% 3° for β = 0.9 and γ =0.4 :

i*= 2.52% versus 2.05% 4° for β = 1 and γ =1 :

i*= 1.68% versus 0.4%

% 3

% 1

= =

µ

λ

5.5. Numerical illustration

Equilibrium technical rate

0,1 _0,3

0,5 _0,7

0,9

100%stocks 20%stocks 0

0,005 0,01 0,015 0,02 0,025 0,03

Participation rate

5.5. Numerical illustration

0 0,005 0,01 0,015 0,02 0,025 0,03 0,035

0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,1 "Volatility factor"

B=1 B=0.8 B=0.65 B=0.5

6.CONCLUSION

ƒ State prices , risk neutral measures and deflators are easy tools in order to valuate stochastic future cash flows correlated to future financial markets.

ƒ This is exactly the situation of life insurance products