Pricing insurance contracts: an incomplete market approach ACFI seminar, 2 November 2006 Antoon Pelsser University of Amsterdam & ING Group
Jan 11, 2016
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
3
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”
4
Outline Recap of Arbitrage-free pricing Utility Functions & Optimal Wealth Principle of Equivalent Utility Pricing Insurance Contracts Optimal Wealth with Shortfall Constraint
5
Outline Recap of Arbitrage-free pricing Utility Functions & Optimal Wealth Principle of Equivalent Utility Pricing Insurance Contracts Optimal Wealth with Shortfall Constraint
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
7
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%)
8
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!
9
Outline Recap of Arbitrage-free pricing Utility Functions & Optimal Wealth Principle of Equivalent Utility Pricing Insurance Contracts Optimal Wealth with Shortfall Constraint
10
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
11
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
12
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
13
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
14
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.
15
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)
17
Outline Recap of Arbitrage-free pricing Utility Functions & Optimal Wealth Principle of Equivalent Utility Pricing Insurance Contracts Optimal Wealth with Shortfall Constraint
18
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
19
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
20
Outline Recap of Arbitrage-free pricing Utility Functions & Optimal Wealth Principle of Equivalent Utility Pricing Insurance Contracts Optimal Wealth with Shortfall Constraint
21
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
22
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
23
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)
24
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
25
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
26
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
28
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
30
Outline Recap of Arbitrage-free pricing Utility Functions & Optimal Wealth Principle of Equivalent Utility Pricing Insurance Contracts Optimal Wealth with Shortfall Constraint
31
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
32
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
33
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
34
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
35
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
36
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”