FAIR VALUATION OF INSURANCE LIABILITIESjanroman.dhis.org/finance/Deflators/Deflators PPT.pdf ·...

FAIR VALUATION OF INSURANCE LIABILITIES

Pierre DEVOLDERUniversité Catholique de Louvain

03/ 09/2004

Pierre Devolder 09/2004 2

Fair value of insurance liabilities

1. INTRODUCTION TO FAIR VALUE2. RISK NEUTRAL PRICING AND DEFLATORS3. EXAMPLES : THE BINOMIAL and THE

BLACK/ SCHOLES CASES4. FAIR VALUE OF LIFE INSURANCE

PARTICIPATING CONTRACTS5. FAIR VALUE OF VARIABLE ANNUITIES6. CONCLUSION


1. Introduction to FAIR VALUE

International norms IAS / IFRS for all financial institutionsin Europe soon ( …)

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

General principle of Fair valuation of elements for assets as well as for liabilities: market values instead of historical values



Basic principle : from an historical or statutory accountingpoint of view to fair value bases

Fair value : price at which an instrument would be tradedif a liquid market existed for this instrument

ASSETS : market values

LIABILITIES : ???If no market value : principle of estimation of futurecash flows properly discounted and taking into accountthe different kinds of risk



Need to develop good models of valuation especially foractuarial liabilities where there is no market price

Consistency between modern financial pricing theory and classical actuarial models

Important example : guarantees in life insurance andpension:one of the most challenging risk nowaday

Even if for competition reasons methods of pricing could remain very classical , fair valuation will require new insightstaking into account modern finance


2. Risk neutral pricing and Deflators

Purpose :

- introduction to the modern financial paradigmof risk neutral pricing

- link with deflator methodology

- link with classical actuarial principle of discounting

- development in a simple discrete market model



Paradigm of risk neutral pricing:

Purpose: to compute the present price of a futurestochastic cash flow correlated with financial marketin an uncertain environment

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

))T(L(E)i1(

1)0(L T+= NO!!!



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 butthe expectation of the cash flows must be doneusing another probability measure than the real one

Change of discounting factor : deflators method:You can use the real probability measure but the discounting factor has to be changed and becomesstochastic



Risk neutral pricing :

))T(L(E)r1(

1)0(L QT+=

Q= risk neutral measure

Deflator method :

))T(L)T(D(E)0(L =

D(T) =deflator = stochastic discounting


2. Risk neutral pricing and DeflatorsSingle period market model :

one riskless asset :

rateriskfreerwithrSS =+= )1()0()1( 00

d risky assets defined on a probability space :

NjPpwith jjN ,...,1)(,...,, 21 ===Ω ωωωω

diSSSS Niiii ,...,1)),1(),...,,1(),,1(()1( 21 == ωωω

classical assumption of absence of arbitrage opportunities



STATE PRICE : Basic property :if the market is without arbitrage there existsa random variable ψ such that for any asset :

∑=

⟩Ψ=ΨΨ=N

jjjjiji withSS

10)(),1()0( ωω

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

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

∑= +

=Ψ=ΨN

1jj r1

1



RISK NEUTRAL MEASURE

Definition : artificial probability measure given by:

jj

jj )r1()(Qq Ψ+=Ψ

Ψ=ω=ω=

Properties :

∑=

=≤≤N

1jjj 1qand1q0

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

1°)

∑=

ω+

=N

1jjiji ),1(Sq

)r1(1)0(S



DEFLATOR : random variable D defined by :

j

jjj p

DDΨ

==)(ω

Properties :

∑= +

==N

1jjj r1

1)D(EDp.i

( expected value of the deflator = classical discount )

∑=

ω=N

1jjijji ),1(SDp)0(Sii.



∑=

Ψ=N

1jj )j,1(X)0(X:visionpriceState.i

Generalization : if X is a financial instrument on this market (replicable by the underlying assets) and giving for scenario ja cash flow X(1,j) then the initial value of this instrument can be written as :

))1(X(Er1

1)j,1(Xqr1

1)0(X:visionneutralRisk.ii Q

N

1jj +

=+

= ∑=

))1(XD(E)j,1(XDp)0(X:visionDeflator.iiiN

1jjj ==∑

=



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

rateriskfreerwithrStS t =+= )1()0()( 00

Risky assets :

ditStStStS Niiii ,...,1)),(),...,,(),,(()( 21 == ωωω

STATE PRICE :

∑=

⟩Ψ=ΨΨ=N

jjjjiji ttwithtStS

10),()(),()()0( ωω


2. Risk neutral pricing and DeflatorsDEFLATOR :

jscenarioiftotfromfactordiscountp

ttD

j

jj 0

)()( =

Ψ=

Pricing : if X is a financial replicable instrument on thismarket generating successive stochastic cash flows :

;,...,1);,( Ω∈= ωω TttC

Then the initial price of X can be written alternatively :

∑∑= =

Ψ=T

t

N

jjj ttCX

1 1)(),()0( ω



Or with risk neutral measure:

∑ ∑= =

ω+

=T

1t

N

1jjjt ),t(Cq

)r1(1)0(X

Or with deflators :

∑∑∑== =

=ω=T

1tjj

T

1t

N

1jj ))t(C)t(D(E)t(D),t(Cp)0(X


3.1. THE BINOMIAL CASE

Single period model:

Risky asset :with probability puSS ⋅= )0()1( 11

with probability 1-pdS ⋅= )0(1

Absence of arbitrage opportunities if:urd <+<< 10

Other form of the risky asset :µλ +++= ru 1 µλ −++= rd 1



µ0 <λ <With condition :

volatilitypremiumrisk == µλ

Equations of the STATE PRICE :

1)1()1( 21 =Ψ++Ψ+ rrFor i=0:

121 =Ψ+Ψ duFor i=1:



Solution for the STATE PRICE:

)1(2))(1(1

1 rdurdr

+−=

−+−+=Ψ

µλµ

up

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

2 rdurru

++=

−++−=Ψ

µλµdown

Safety principle :

012 =Ψ=Ψ λif

)(012 casenormalif >Ψ⟩Ψ λ



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 :

2211 ),1(),1( XXandXX == ωω

Then the initial fair value of X is given by :

2211)0( Ψ+Ψ= XXXOr :

)r1()XX(

21

)r1(1)XX(

21)0(X 1221 +µ

λ−+

++=



Multiple periods model:

1

2 d

2 u

d

d u ⋅

u

Risky asset



Structure of STATE PRICES in multiple periods:

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

1

11 Ψ

21Ψ

22Ψ

23Ψ

12 Ψ

State price

jscenarioifttimeatpricestatejt =Ψ



Value of the STATE PRICES:

12

11

1 −−−− ΨΨ=Ψ jtjjtjt C

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

1−jtCAnd: is the number of paths in the tree with

j-1 up in t periods



Fair valuation in multiple periods – binomial :

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

jscenarioifttimeatflowcashX jt =

Then the initial fair value of X is given by :

∑∑=

+

=Ψ=

T

t

t

jjtjtXX

1

1

1)0(


3.2. THE BLACK / SCHOLES CASEContinuous extension : Black and Scholes model

Riskless asset :

dt)t(Sr)t(dS 00 =

Risky asset :

)t(dw)t(Sdt)t(S)t(dS 111 σ+δ=

Where:- w is a standard brownian motion- δ is the mean return of the asset (δ>r)- σ is the volatility of this return


3.2. THE BLACK / SCHOLES CASERisk neutral measure :

)t(*dw)t(Sdt)t(Sr))t(dwdt)r()(t(Sdt)t(Sr)t(dS

11

111

σ+=σ+−δ+=

tr)t(w)t(*w:withσ−δ

+=

Q = risk neutral measure ( Girsanov Theorem) Under Q the process w*is a standard brownian motion


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 ii =

Solution:

tr21)t(wr

rt

2

ee)t(D⎟⎠⎞

⎜⎝⎛σ−δ

−⎟⎠⎞

⎜⎝⎛σ−δ

−−=

Safety principle :Lower values of « w » give higher values for deflator


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 :Fair value and equilibrium condition


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 accountthe specificities of the underlying liability

Liability :AssetParticipation liability linked with investmentresults


4.1. Liability sidePure 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

l x


4.1. Liability sideBonus systems :

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

Three possible schemes :Terminal bonus : bonus computed at maturityReversionary bonus : bonus computed each

year and fully integrated in the contractas additional premium

Cash bonus : bonus computed each year and paid directly to the client


4.2. Asset sideCash – 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 durationof the contract

- Stocks : stock index


4.2. Asset side Cash model :

Money market account :

ratefreerisk)t(rwithdt)t()t(r)t(d

=β=β

Risk free rate :Ornstein-Uhlenbeck process

motionbrowniandardtanswwith)t(dwdt))t(rb(a)t(dr

1

1r

=σ+−=


4.2. Asset sideBond model

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

)t(dw)T,t()T,t(Pdt)T,t()T,t(P)T,t(dP 1σ−µ=Particular case : VASICEK Model

)e1(a1)s(B:with

)tT(B)T,t()tT(B)t(r)T,t(

as

r

r

−−=

−σ=σ−λσ+=µ


4.2. Asset sideStocks model :

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

12

22

1S

woftindependanmotionbrowniandardtanswrateserestint/stocksncorrelatiowith

)))t(dw1)t(dw(dt()t(S)t(dS

==ρ

ρ−+ρσ+µ=


4.2. Asset sidePortfolio :

1stocksinproportionbondsinproportioncashinproportion

SBC

S

B

C

=α+α+α=α=α=α

2 main assumptions : - proportions remain constant

( continuous rebalancing)- self financed strategy


4.2. Asset sideBond strategy :

Assumption : at each time t zero coupon bonds of onlyone 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=NStrategy 2 : shorter maturity of the bondand successive reinvestments till end of the contract


4.2. Asset sideBond strategy

tn Ns=0 t1 t2 t3

ntreinvestmeoftstanins:t,...,t,t,0tntsreinvestmeofnumber:n,...,1,0s

n210 ==

Particular case : matched strategy : n=0


4.2. Asset sideEvolution of the portfolio :

portfoliounderlyingtheofvaluemarket)t(V =Evolution equation :

)tttfor(

)t()t(d

)t,t(P)t,t(dP

)t(S)t(dS

)t(V)t(dV

i1i

Ci

iBS

<<

ββ

α+α+α=

−


4.2. Asset side

Value at maturity of the assets :

ddistributenormally))t(Vln)t(V(lntermeachwith

))t(Vln)t(V(ln)0(Vln)N(Vln

s1s

s

n

0s1s

=−

−=−

+

=+∑

))0(Vln)N(Vln(var)N())0(Vln)N(V(lnE)N(

2 −=σ

−=µ


4.2. Asset side

Explicit form :

∫

∫∫

∫

+

++

+

−ρσσαα−−σα+σα−

−σα−ρσα+ρ−σα+

−λσα+α+α+µα=−

++

+

++

1s

s

1s

s

1s

s

1s

s

t

t1srSBS1s

22B

2B

2S

2S

t

t11srBSS

t

t2

2SS

1srB

t

tCBSs1s

du))ut(B2)ut(B(21

)u(dw))ut(B()u(dw1

du))ut(B)u(r)(()t(Vln)t(Vln


4.3. Valuation of the contractExample of a contract with single premium andterminal bonus :

Fair value at maturity = pay off of the contract

)0;)i1()N(Vmax()i1()N(FV NN +−η++=

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

)))i1(;N;V(Call)N,0(P)i1((p)0(FV NNxN +η++=


4.3. Valuation of the contractEquilibrium condition :

1))i1(;N;V(call)N,0(P)i1( NN =+η++

Consequences :1°)

)))N,0(R1(

1)N,0(Pwith(

)N,0(Ri

N+=

<

2°) depends on the investment strategy throughthe value of V(N)


4.3. Valuation of the contract3°) implicit relation for the technical rate i ;4°) explicit relation for the participation rate η:

))i1(;N;V(call)N,0(P)i1(1

N

N

++−

=η

An explicit formula of the call can be obtained in the CBS model presented before.


4.3. Valuation of the contractRisk forward neutral measure method :

measureneutralriskforwardQwhere

)0);)i1()N(V(((maxE)N,0(Pcall

N

TQN

=

+−=

In the CBS model we have :

υυ±−+−

=

Φ+−Φ=

±

−+

2

N

2/1)T,0(Pln)i1(lnN)i(D

:with))i(D()N,0(P)i1())i(D(call


4.3. Valuation of the contractWith for instance for the matched strategy :

)e1(a2)e1(

a21

aN)N(B

)e1(a1

aN)N(B

:with)N(B)1(2)N(B)1(N

aN3

aN2322

aN21

1rSBS22

r2

B2

S2

S2

−−

−

−−−+=

−−=

ρσσα−α+σα−+σα=υ


5. FAIR VALUE OF VARIABLE ANNUITIES

Purposes :

How to valuate pension annuities not in terms of technical basisbut in terms of market fair values;

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

Sensitivity of the provision with respect to financial parameters;

How to fix the technical interest rate.


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 :

jscenarioforttimeatpaytoamountL j,t =

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

1 ) fixed annuity : L 2) variable annuity :

- Payment at the end of the year till death or during a fixed period of n years


5.1. Liability side

Actuarial first order bases :

ratediscounttechnicali=ttimeatyprobabilitsurvivalpxt =

Technical provision for a constant pension ( case 1):

LL jt =,

txt

n

tnxxn i

pLaLV)1(

11 +

== ∑=


5.2. Asset side

Binomial model :mixed financial strategy of the pension fundbetween riskless asset (r= riskfree rate)and risky asset (binomial model u / d)

γ : part invested in the risky asset : part invested in the riskless asset γ−1

)10( ≤≤ γ


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 rate10 ≤≤ β


5.3. Bonus schemeYearly rate of increase of the pension :

If the risky asset is up :

+−+

+−++=+ )1)1(

)1)(1((11r

ruk γγβ

)1

(11r

k+++=+ µλβγor

If the risky asset is down :

1)1)1(

)1)(1((11 =−+

+−++=+ +

rrdl γγβ


5.3. Bonus scheme

Final form of the liabilities of the variable annuity:

( ) 1, 1 +−+⋅= jtjt kLL

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

As expected THE LIABILITY DEPENDS ON TIME AND

IS STOCHASTIC


5.4. Valuation of the contract

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

( ) ⎥⎦

⎤⎢⎣

⎡Ψ= ∑∑

+

==tj

t

j

n

txtnx

LpLFV1

11,

⎥⎦

⎤⎢⎣

⎡Ψ= ∑∑

+

==tj

t

j

n

txt pL

1

11

tn

txt r

pL )1

1(1 +

= ∑=

nxraL=


5.4. Valuation of the contractComputation of the fair value of the liabilities

(variable annuity)

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

( ) ⎥⎦

⎤⎢⎣

⎡Ψ= ∑∑

+

==tj

t

jjt

n

txtnxk LpLFV

1

1,

1,

( )( )[ ]tn

txt kpL +Ψ+Ψ= ∑

=112

1

( )( ) ⎥⎦

⎤⎢⎣

⎡+ΨΨ= ∑∑

+

=

+−−−

=

1

1

11

12

1

11

t

j

jtjjt

n

txt kCpL



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

tn

t

t

xtnxk rrpLLFV ⎥

⎦

⎤⎢⎣

⎡+−

+⎥⎦⎤

⎢⎣⎡+

= ∑= )1(2

11

1)(22

1, µ

λµβγ

tn

txt i

pL )*1

1(1 +

= ∑=

nxiaL *=



Equilibrium relation :

i* = equilibrium constant discount rate given by :

))1(2

1(/))1(2

(*2222

rrri

+−

++−

−=µ

λµβγµ

λµβγ

riorif ===→ *:00 γβ

riandif <>>→ *:00 γβ


5.5. Numerical illustration

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

i=0.025Mortality: GRM 95

Risk premium : λ =0.02Volatility : µ = 0.06

n=5



0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1Fixed annuity

40%stocks

100%stocks

4000

4100

4200

4300

4400

4500

4600

4700

4800

4900

0% <12,25%stocks 20%stock 40%stocks 60%stocks 80%stocks 100%stocks



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"

Actuarial valuat ion Deflato r valuat ion B= 0 ,15B= 0 ,3 B= 0 ,5 B= 0 ,65B= 0 ,8 B=1


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

==

µλ

r=3%i=2.5%



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

==

µλ

r=3%i=2.5%



5.5. Numerical illustrationEquilibrium technical rate

0,1

0,3

0,5

0,7

0,9

100%stocks

20%stocks0

0,005

0,01

0,015

0,02

0,025

0,03

Participation rate

100%stocks 80%stocks 60%stocks 40%stocks 20%stocks 0%stocks



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

B=0.3 B=0.15 B=0

EQUILIBRIUM TECHNICAL RATE(II)


6.CONCLUSION

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

This is exactly the situation of life insurance products

One of the most important result is the way to valuate bonus and to define properly a technical interest rate in a complete ALM framework

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FAIR VALUATION OF INSURANCE LIABILITIESjanroman.dhis.org/finance/Deflators/Deflators PPT.pdf ·...

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