Pricing insurance contracts: an incomplete market approach

Pricing insurance contracts: an incomplete market approach

ACFI seminar, 2 November 2006

Antoon Pelsser

University of Amsterdam & ING Group

2

Setting the stage Arbitrage-free pricing: standard method for financial

markets Assume complete market

Every payoff can be replicated Every risk is hedgeable

Insurance “markets” are incomplete Insurance risks are un-hedgeable But, financial risks are hedgeable

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Aim of This Presentation Find pricing rule for insurance contracts consistent with

arbitrage-free pricing “Market-consistent valuation” Basic workhorse: utility functions Price = BE Repl. Portfolio + MVM

A new interpretation of Profit-Sharing as “smart EC”

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Outline Recap of Arbitrage-free pricing Utility Functions & Optimal Wealth Principle of Equivalent Utility Pricing Insurance Contracts Optimal Wealth with Shortfall Constraint

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6

Arbitrage-free pricing Arb-free pricing = pricing by replication

Price of contract = price of replicating strategy Cash flows portf of bonds Option delta-hedging strategy

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Black-Scholes Economy Two assets: stock S and bond B

S0=100, B0=1

Bond price BT=erT const. interest rate r (e.g. 5%)

Stock price ST = 100 exp{ ( -½2)T + zT }

zT ~ n(0,T) (st.dev = T)

is growth rate of stock (e.g. 8%) is volatility of stock (e.g. 20%)

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Black-Scholes Economy (2) “Risk-neutral” pricing:

ƒ0 = e-rT EQ[ ƒ(ST) ]

Q: “pretend” that ST = exp{ (r -½2)T + zT}

Deflator pricing: ƒ0 = EP[ ƒ(ST) RT ]

RT = C (1/ST) (S large R small)

=(-r)/2 (market price of risk) P: ST follows “true” process

Price ƒ0 is the same!

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Utility Functions Economic agent has to make decisions under

uncertainty Choose “optimal” investment strategy for wealth W0

Decision today uncertain wealth WT

Make a trade-off between gains and losses

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Utility Functions - Examples Exponential utility:

U(W) = -e-aW / a RA = a

Power utility: U(W) = W(1-b) / (1-b) RA = b/W

-5 0 5 10 15 20 25

Wealth

Uti

lity

Exp Power

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Optimal Wealth Maximise expected utility EP[U(WT)]

By choosing optimal wealth WT

First order condition for optimum U’(WT*) = RT

Solution: WT* = (U’)-1(RT) Intuition: buy “cheap” states & spread risk Solve from budget constraint

0s.t.

)(max

WRW

WU

TT

TWT

P

P

E

E

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Optimal Wealth - Example Black-Scholes economy Exp Utility: U(W) = -e-aW / a Condition for optimal wealth:

U’(WT*) = RT

exp{-aWT*} = C ST

-

=(-r)/2 (market price of risk)

Optimal wealth: WT* = C* + /a ln(ST)

Solve C* from budget constraint

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Optimal Wealth - Example (2) Optimal investment strategy:

invest amount (e-r(T-t)/a) in stocks borrow amount (e-r(T-t)/a -Wt) in bonds usually leveraged position!

Intuition: More risk averse (a) less stocks (/a)

We cannot observe utility function We can observe “leverage” of Insurance Company Use leverage to “calibrate” utility fct.

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Optimal Wealth

-40.00

-20.00

0.00

20.00

40.00

60.00

80.00

50.00 100.00 150.00 200.00

Stock Price S(T)

We

alt

h W

*(T

)

W* Prob-RW

Optimal Wealth - Example (3)Default

for ST<75Optimal Surplus

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Optimal Wealth - Example (4) Risk-free rate r =5% Growth rate stock =8% Volatility stock =20% Hence: = (-r)/2 = 0.75 Amount stocks: e-rT/a = e-0.05T 0.75/a Calibrate a

from leverage (e.g. the or D/E ratio)

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Principle of Equivalent Utility Economic agent thinks about selling a (hedgeable)

financial risk HT

Wealth at time T = WT - HT

What price 0 should agent ask? Agent will be indifferent if expected utility is unchanged:

)(max)(max *

,

*

0** TT

WT

WHWUWU

TT

PP EE

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Principle of Equivalent Utility (2) Principle of Equivalent Utility is consistent with arbitrage-

free pricing for hedgeable claims Principle of Equivalent Utility can also be applied to

insurance claims Mixture of financial & insurance risk Mixture of hedgeable & un-hedgeable risk

Literature: Musiela & Zariphopoulou V. Young

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Pricing Insurance Contracts Assume insurance claim: HT IT,

Hedgeable risk HT

cash amount C stock-price ST

Insurance risk IT

number of policyholders alive at time T 1 / 0 if car-accident does (not) happen salary development of employee

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Pricing Insurance Contracts Apply principle of Equivalent Utility

Find price 0 via solving “right-hand” optimisation problem

)(max)(max *

,

*

0** TTT

WT

WIHWUWU

TT

PP EE

00

*

*

s.t.

)(max*

WRW

IHWU

TT

TTTWT

P

P

E

E

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Pricing Insurance Contracts (2) Consider specific example: life insurance contract

Portfolio of N policyholders Survival probability p until time T Pay each survivor the cash amount C

Payoff at time T = Cn n is (multinomial) random variable E[n] = Np Var[n] = Np(1-p)

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Pricing Insurance Contracts (3) Solve optimisation problem:

Mixture of random variables W,R are “financial” risks n is “insurance” risk

“Adjusted” F.O. condition

00

*

*

s.t.

)(max*

WRW

CnWU

TT

TWT

P

P

E

E

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Pricing Insurance Contracts (4) Necessary condition for optimal wealth:

EPU[ ] does not affect “financial” WT

EPU[ ] only affects “insurance” risk n

Exponential utility: U’(W-Cn) = e-a(W-Cn) = e-aW eaCn

TT RCnWU )(' *PUE

T

aCnaWCnWa Reee TT

][][** PUPU EE

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Pricing Insurance Contracts (5) For large N: n n( Np , Np(1-p) )

MGF of z~n(m,V): E[etz] = exp{ t m + ½t2V }

Hence: EPU[ eaCn ] = exp{ aC Np + ½(aC)2 Np(1-p) }

Equation for optimal wealth:

TpNpCaaCNpaW Ree T

)1(2221*

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Pricing Insurance Contracts (6) Solution for optimal wealth:

Surplus W* BE MVM

Price of insurance contract: 0 = e-rT(CNp + ½aC2Np(1-p))

“Variance Principle” from actuarial lit.

)1()ln( 2211* pNpaCCNpRW TaT

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Observations on MVM MVM ½ Risk-av * Var(Unhedg. Risk)

Note: MVM for “binomial” risk! “Diversified” variance of all unhedgeable risks

Price for N+1 contracts: e-rT(C(N+1)p + ½aC2(N+1)p(1-p))

Price for extra contract: e-rT(Cp + ½aC2p(1-p))

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Observations on MVM (2) Sell 1 contract that pays C if policyholder N dies

Death benefit to N’s widow Certain payment C + (N-1) uncertain Price: e-rT(C + C(N-1)p + ½aC2(N-1)p(1-p))

Price for extra contract: e-rT(C(1-p) - ½aC2p(1-p)) BE + negative MVM! Give bonus for diversification benefit

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Optimal Wealth with Shortfall Constraint Regulator (or Risk Manager) imposes additional shortfall

constraints e.g. Insurance Company wants to maintain single-A rating

“CVaR” constraint Protect policyholder Limit Probability of WT < 0

Limit the averaged losses below WT < 0

Optimal investment strategy W* will change due to additional constraint

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Optimal Wealth with Shortfall Constraint (2)

Maximise expected utility EP[U(WT)] By choosing optimal wealth WT with CVaR constraint:

denotes a confidence level V denotes the -VaR (auxiliary decision variable)

This formulation of CVaR is due to Uryasev

0}0,max{

s.t.

)(max

11

0

,

T

TT

TVW

WVV

WRW

WUT

P

P

P

E

E

E

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Optimal Wealth with CVaR

-40.00

-20.00

0.00

20.00

40.00

60.00

80.00

50.00 100.00 150.00 200.00

W*CVaR W*-old W*+EC


Less Shortfall

VaR = 0.28

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-40.00

-20.00

0.00

20.00

40.00

60.00

80.00

50.00 100.00 150.00 200.00

W*CVaR W*-old W*+EC


“Contingent Capital”

EC

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Finance EC by selling part of “upside” to policyholders: Profit-Sharing


-40.00

-20.00

0.00

20.00

40.00

60.00

80.00

50.00 100.00 150.00 200.00

W*CVaR W*-old W*+EC

EC -/- Profit Sharing

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Conclusions Pricing insurance contracts in “incomplete” markets Principle of Equivalent utility is useful setting Decompose price as: BE Repl. Portfolio + MVM MVM can be positive or negative Profit-Sharing can be interpreted as holding “smart EC”

Documents

Pricing insurance contracts: an incomplete market approach