pricing insurance contracts: an incomplete market approach
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Pricing insurance contracts: an incomplete market approach. ACFI seminar, 2 November 2006 Antoon Pelsser University of Amsterdam & ING Group. Arbitrage-free pricing: standard method for financial markets Assume complete market Every payoff can be replicated Every risk is hedgeable - PowerPoint PPT PresentationTRANSCRIPT
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Pricing insurance contracts: an incomplete market approach
ACFI seminar, 2 November 2006
Antoon Pelsser
University of Amsterdam & ING Group
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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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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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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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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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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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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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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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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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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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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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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
Optimal Wealth with Shortfall Constraint (3)
Less Shortfall
VaR = 0.28
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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
Optimal Wealth with Shortfall Constraint (4)
“Contingent Capital”
EC
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Optimal Wealth with Shortfall Constraint (5)
Finance EC by selling part of “upside” to policyholders: Profit-Sharing
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
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”