Financial Bounds for Insurance Claims Carole Bernard (University of Waterloo, WatRISQ) Steven Vanduffel (Vrije Universiteit Brussel, Belgium).
Carole Bernard
Financial Bounds for Insurance Claims
1/29
Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Background & Objectives I (“Explicit representation of Cost-efficient Strategies” with Phelim Boyle (Wilfrid Laurier University)) • Given a cdf F , there exists an explicit representation of XT?
and of ZT? such that
I XT? ∼ F and ZT? ∼ F in the real world I XT? is the cheapest strategy (= cost-efficient strategy) I ZT? is the most expensive strategy (= cost-inefficient strategy)
⇒ Price(claim)∈ c(XT? ), c(ZT? ) I Our objectives: 1 2
To propose a “market-consistent” pricing tool To find similar bounds • on prices of claims that cannot be hedged perfectly in the market. • but for which we know the cdf under the physical probability.
Carole Bernard
Financial Bounds for Insurance Claims
2/29
Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Background & Objectives I (“Explicit representation of Cost-efficient Strategies” with Phelim Boyle (Wilfrid Laurier University)) • Given a cdf F , there exists an explicit representation of XT?
and of ZT? such that
I XT? ∼ F and ZT? ∼ F in the real world I XT? is the cheapest strategy (= cost-efficient strategy) I ZT? is the most expensive strategy (= cost-inefficient strategy)
⇒ Price(claim)∈ c(XT? ), c(ZT? ) I Our objectives: 1 2
To propose a “market-consistent” pricing tool To find similar bounds • on prices of claims that cannot be hedged perfectly in the market. • but for which we know the cdf under the physical probability.
Carole Bernard
Financial Bounds for Insurance Claims
2/29
Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Some Assumptions • Consider an arbitrage-free and complete market. • Given a strategy with payoff XT at time T . There exists Q, such that its price at 0 is PX = EQ [e −rT XT ] • P (“physical measure”) and Q (“risk-neutral measure”) are two equivalent probability measures: dQ −rT ξT = e , c(XT ) =EQ [e −rT XT ] = EP [ξT XT ]. dP T
Carole Bernard
Financial Bounds for Insurance Claims
3/29
Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
• Given a strategy with payoff XT at time T , and initial price at time 0 c(X ) = E [ξT XT ] • F : XT ’s distribution under the physical measure P. The distributional price is defined as PD(F ) =
min
{Y | Y ∼F }
{E [ξT Y ]} =
min
{Y | Y ∼F }
c(Y )
(lower bound on the price of a financial claim with cdf F )
⇒ Example of X ∼ Y with different costs in a binomial tree.
Carole Bernard
Financial Bounds for Insurance Claims
4/29
Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
• Given a strategy with payoff XT at time T , and initial price at time 0 c(X ) = E [ξT XT ] • F : XT ’s distribution under the physical measure P. The distributional price is defined as PD(F ) =
min
{Y | Y ∼F }
{E [ξT Y ]} =
min
{Y | Y ∼F }
c(Y )
(lower bound on the price of a financial claim with cdf F )
⇒ Example of X ∼ Y with different costs in a binomial tree.
Carole Bernard
Financial Bounds for Insurance Claims
4/29
Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
A simple illustration for X2 , a payoff at T = 2 Real-world probabilities=p = 21 and risk neutral probabilities=q = 14 . p
p
S 6 1 = 32
S 6 2 = 64
(
p
(
1 16
X2 = 1
1 2
6 16
X2 = 2
1 4
9 16
X2 = 3
1−p
S0 = 16 1−p
1 4
S 6 2 = 16
S1 = 8 1−p
(
S2 = 4
U(1) + U(3) U(2) 3 + , PD = Cheapest = 4 2 2 1 6 9 = Price of X2 = + 2+ 3 , Efficiency cost = PX2 − PD 16 16 16 E [U(X2 )] =
PX2
Carole Bernard
Financial Bounds for Insurance Claims
5/29
Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Y2 , a payoff at T = 2 distributed as X2 Real-world probabilities=p = 12 and risk neutral probabilities=q = 14 . p
p
S 6 1 = 32
S 6 2 = 64
(
p
(
1 16
Y2 = 3
1 2
6 16
Y2 = 2
1 4
9 16
Y2 = 1
1−p
S0 = 16 1−p
1 4
S 6 2 = 16
S1 = 8 1−p
(
S2 = 4
U(3) + U(1) U(2) 3 + , PD = Cheapest = 4 2 2 X and Y have the same distribution under the physical measure 1 6 9 = Price of X2 = + 2+ 3 , Efficiency cost = PX2 − PD 16 16 16 E [U(Y2 )] =
PX2
Carole Bernard
Financial Bounds for Insurance Claims
6/29
Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
X2 , a payoff at T = 2 Real-world probabilities=p = 12 and risk neutral probabilities=q = 14 . q
q
S 6 1 = 32
S 6 2 = 64
(
q
(
1 16
X2 = 1
1 2
6 16
X2 = 2
1 4
9 16
X2 = 3
1−q
S0 = 16 1−q
1 4
S 6 2 = 16
S1 = 8 1−q
(
E [U(X2 )] =
U(1) + U(3) U(2) + 4 2
PX2 = Price of X2 = Carole Bernard
S2 = 4
,
PD = Cheapest =
1 6 9 + 2+ 3 16 16 16
=
5 2
,
1 6 9 3 3+ 2+ 1 = 16 16 16 2
Efficiency cost = PX2 − PD Financial Bounds for Insurance Claims
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Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Y2 , a payoff at T = 2 Real-world probabilities=p = 12 and risk neutral probabilities=q = 14 . q
S 6 1 = 32
q
S 6 2 = 64
(
q
(
1 16
Y2 = 3
1 2
6 16
Y2 = 2
1 4
9 16
Y2 = 1
1 6 9 3+ 2+ 1 16 16 16
1−q
S0 = 16 1−q
1 4
S 6 2 = 16
S1 = 8 1−q
(
E [U(X2 )] =
U(1) + U(3) U(2) + 4 2
PX2 = Price of X2 = Carole Bernard
S2 = 4 ,
1 6 9 + 2+ 3 16 16 16
PY2 = =
5 2
,
=
3 2
Efficiency cost = PX2 − PD Financial Bounds for Insurance Claims
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Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Minimum Price = Cost-efficient Strategy Theorem Consider the following optimization problem: min
{Z | Z ∼F }
{E [ξT Z ]}
Assume ξT is continuously distributed, then the optimal strategy is XT? = F −1 (1 − Fξ (ξT )) . Note that XT? ∼ F and XT? is a.s. unique such that PD(F ) = c(XT? ) = E ξT XT?
Carole Bernard
Financial Bounds for Insurance Claims
9/29
Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Maximum price = Least Efficient Strategy Theorem Consider the following optimization problem: max
{Z | Z ∼F }
{E [ξT Z ]}
Assume ξT is continuously distributed. The strategy ZT? that generates the same distribution as F with the highest cost can be described as follows: ZT? = F −1 (Fξ (ξT ))
Carole Bernard
Financial Bounds for Insurance Claims
10/29
Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Black and Scholes Model Under the physical measure P, dSt = µdt + σdWtP St Then ξT = e θ
σ2
−rT
dQ dP
=a
ST S0
−b
θ2
where a = e σ (µ− 2 )t−(r + 2 )t and b = µ−r . σ2 To be cost-efficient, the contract has to be a European derivative written on ST and non-decreasing w.r.t. ST (when µ > r ). In this case, XT? = F −1 (FST (ST )) Carole Bernard
Financial Bounds for Insurance Claims
11/29
Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Geometric Asian contract in Black and Scholes model Assume a strike K . The payoff of the Geometric Asian call is given by 1 RT + XT = e T 0 ln(St )dt − K which corresponds in the discrete case to
Q
n k=1 S kT n
1
n
+ −K
.
The efficient payoff that is distributed as the payoff XT is a power call option √ K + 1/ 3 ? − XT = d ST d 1− √1 S0 3 e
q 2 1 µ− σ2 T 3
1 − 2
where d := Similar result in the discrete case. Carole Bernard
.
Financial Bounds for Insurance Claims
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Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Example: Discrete Geometric Option 120 100
Payoff
80 60
Z*T
40 Y*T
20 0 40
60
80
100 120 140 160 180 200 220 240 260 Stock Price at maturity ST
With σ = 20%, µ = 9%, r = 5%, S0 = 100, T = 1 year, K = 100, n = 12. C (XT? ) = 5.77 < Price(geometric Asian) = 5.94 < C (ZT? ) = 9.03. Carole Bernard
Financial Bounds for Insurance Claims
13/29
Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Bounds on Prices • Consider a financial claim at time T with cdf F . • Denote by XT? the cheapest strategy with cdf F and by ZT?
the most expensive strategy with cdf F , ⇒ Cost(claim)∈ c(XT? ), c(ZT? ) How to use these bounds for insurance claims? 1
Let CT be a random non-negative insurance payoff (not traded) with distribution F .
2
Under some conditions, it also follows that Price(CT ) > c(XT? ). but in general there is no upper bound (independent of the preferences).
Carole Bernard
Financial Bounds for Insurance Claims
14/29
Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Assumptions on Preferences Denote by XT the final wealth of the investor and V (XT ) the objective function of the agent. 1
Market participants all have a fixed investment horizon T > 0 and there is no intermediate consumption (one-period model).
2
Agents’ preferences depend only on the probability distribution of terminal wealth: “law-invariant” preferences. (if XT ∼ ZT then: V (XT ) = V (ZT ).)
3
Agents prefer “more to less”: if c is a non-negative random variable V (XT + c) > V (XT ).
4
Agents are risk-averse: E[XT ] = E[YT ] ∀d ∈ R, E[(XT − d)+ ] ≤ E[(YT − d)+ ]
Carole Bernard
⇒ V (XT ) > V (YT )
Financial Bounds for Insurance Claims
15/29
Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Bid and Ask prices for insurance claims in the absence of a financial market using “certainty equivalents” • From the viewpoint of the insured with objective function U(·) and initial wealth ω the (bid) price, p b , U[(ω − p b )e rT ] = U[ωe rT − CT ]. • From the viewpoint of the insurer with a given objective function V (·) and initial wealth ω the ask price, p a , V [(ω + p a )e rT − CT ] = V [ωe rT ].
Carole Bernard
Financial Bounds for Insurance Claims
16/29
Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Properties 1
Bid and Ask prices verify p • > e −rT E[CT ].
2
If the insurer is risk neutral (v (x) = x), then pb > pa = e −rT E[CT ]
3
In the case of exponential utility pa = pb .
4
In the case of Yaari’s theory pa = pb .
5
In general, nothing can be said. u(x) = v (x) = 1 − 1/x, both agents have same initial wealth, CT ∼ U(0, 2). Next figure
Carole Bernard
Financial Bounds for Insurance Claims
17/29
Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Properties 1
Bid and Ask prices verify p • > e −rT E[CT ].
2
If the insurer is risk neutral (v (x) = x), then pb > pa = e −rT E[CT ]
3
In the case of exponential utility pa = pb .
4
In the case of Yaari’s theory pa = pb .
5
In general, nothing can be said. u(x) = v (x) = 1 − 1/x, both agents have same initial wealth, CT ∼ U(0, 2). Next figure
Carole Bernard
Financial Bounds for Insurance Claims
17/29
Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Properties 1
Bid and Ask prices verify p • > e −rT E[CT ].
2
If the insurer is risk neutral (v (x) = x), then pb > pa = e −rT E[CT ]
3
In the case of exponential utility pa = pb .
4
In the case of Yaari’s theory pa = pb .
5
In general, nothing can be said. u(x) = v (x) = 1 − 1/x, both agents have same initial wealth, CT ∼ U(0, 2). Next figure
Carole Bernard
Financial Bounds for Insurance Claims
17/29
Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Properties 1
Bid and Ask prices verify p • > e −rT E[CT ].
2
If the insurer is risk neutral (v (x) = x), then pb > pa = e −rT E[CT ]
3
In the case of exponential utility pa = pb .
4
In the case of Yaari’s theory pa = pb .
5
In general, nothing can be said. u(x) = v (x) = 1 − 1/x, both agents have same initial wealth, CT ∼ U(0, 2). Next figure
Carole Bernard
Financial Bounds for Insurance Claims
17/29
0.974 Bid price Ask price
0.973
Prices
0.972 0.971 0.97 0.969 0.968 0.967 15
16
17 18 Initial wealth w
19
20
Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Bid and Ask prices in the presence of a financial market • From the viewpoint of the insured with objective U(·) and initial wealth ω the (bid) price, p b , follows from sup XT ∈A(ω−p b )
{U[XT ]} =
{U[XT − CT ]} .
sup XT ∈A(ω)
• From the viewpoint of the insurer with objective V (·) and initial wealth ω the ask price, p a , follows from sup XT ∈A(ω+p a )
{V [XT − CT ]} =
sup
{V [XT ]} .
XT ∈A(ω)
• In general computing explicitly p b and p a is not in reach. • (Market Consistency) If CT is hedgeable, then pb = pa = E[ξT CT ]. Carole Bernard
Financial Bounds for Insurance Claims
19/29
Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Bid and Ask prices in the presence of a financial market • From the viewpoint of the insured with objective U(·) and initial wealth ω the (bid) price, p b , follows from sup XT ∈A(ω−p b )
{U[XT ]} =
{U[XT − CT ]} .
sup XT ∈A(ω)
• From the viewpoint of the insurer with objective V (·) and initial wealth ω the ask price, p a , follows from sup XT ∈A(ω+p a )
{V [XT − CT ]} =
sup
{V [XT ]} .
XT ∈A(ω)
• In general computing explicitly p b and p a is not in reach. • (Market Consistency) If CT is hedgeable, then pb = pa = E[ξT CT ]. Carole Bernard
Financial Bounds for Insurance Claims
19/29
Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Lower bound • Assuming that decision makers are risk averse, Theorem Using the abusive notation p • to reflect both p a and p b , p • ≥ E[ξT .CT ]. Furthermore, the lower bound E[ξT .CT ] is the market price of the financial payoff E[CT |ξT ] • Note that
Carole Bernard
p • ≥ e −rT .E[CT ] + Cov [CT , ξT ].
Financial Bounds for Insurance Claims
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Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Comments
• Hence when the claim CT and the state-price ξT are negatively correlated we find that e −rT .E[CT ] is no longer a lower bound for p b and p a which contrasts with traditional bound stated in many actuarial textbooks on insurance pricing. • Finally, remark that the inequality essentially states that both the insured and the insurer are prepared to agree on a price for the insurance payoff CT which is larger than the price “as if CT would be a financial payoff”.
Carole Bernard
Financial Bounds for Insurance Claims
21/29
Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Comments (Cont’d): 3 cases: • CT is independent of the market,
p • ≥ e −rT .E[CT ]. • CT is positively correlated with the state-price process,
the classical lower bound e −rT E[CT ] is now strictly improved. p • ≥ e −rT .E[CT ] + Cov [CT , ξT ] > e −rT .E[CT ]. • CT is negatively correlated with the state-price process,
the lower bound is smaller p • ≥ e −rT .E[CT ] + Cov [CT , ξT ]. The best lower bound for equity-linked insurance benefits will generally be lower than e −rT E[CT ] because Cov (ST , ξT ) = E[ST ξT ] − E[ST ]E[ξT ] = e −rT (EQ [ST ] − E[ST ]), Carole Bernard
Financial Bounds for Insurance Claims
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Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Index-Linked Contract I A life insurance company wants to reinsure payments of (K − ST )+ paid to a policyholder if alive at time T . CT = (K − ST )+ 1τ >T where τ denotes the policyholder’s time of death. I A reinsurer offers full coverage. E[ξT E[CT |ξT ]] = E[ξT CT ] = p(e −rT K − S0 + Cbs (S0 , K , T )) where p = P(τ > T ) and Cbs (S0 , K , T ) is the Black Scholes call price. I u: insurer’s utility u(x) = 1 −
exp(−γx) . γ
where the absolute risk aversions γ > 0. Carole Bernard
Financial Bounds for Insurance Claims
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Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Bid Price Define k1 (.) and k2 (.) such that for a given wealth z sup E [u (XT − CT )]
k1 (z) =
XT ∈A(z)
and k2 (z) =
sup E [u (XT )] . XT ∈A(z)
To calculate explicitly k1 (z), we first observe that k1 (z) = sup E E u XT − (K − ST )+ 1τ >T |τ XT ∈A(z)
=
sup E pu XT − (K − ST )+ + (1 − p)u (XT ) XT ∈A(z)
where p = P(τ > T ) and τ is independent of XT and ST . Carole Bernard
Financial Bounds for Insurance Claims
24/29
Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Bid Price Define k1 (.) and k2 (.) such that for a given wealth z sup E [u (XT − CT )]
k1 (z) =
XT ∈A(z)
and k2 (z) =
sup E [u (XT )] . XT ∈A(z)
To calculate explicitly k1 (z), we first observe that k1 (z) = sup E E u XT − (K − ST )+ 1τ >T |τ XT ∈A(z)
=
sup E pu XT − (K − ST )+ + (1 − p)u (XT ) XT ∈A(z)
where p = P(τ > T ) and τ is independent of XT and ST . Carole Bernard
Financial Bounds for Insurance Claims
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Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Computation of k1 : Pathwise Optimization We maximize pathwise. Let ω ∈ Ω, then define φ(x) = pu x − (K − ST (ω))+ + (1 − p)u (x) − λξT (ω)x It is obvious that φ00 6 0 and therefore that φ is concave and attains its maximum at x ∗ defined by φ0 (x ∗ ) = 0. For λ > 0 and for each ω ∈ Ω, define XT∗ (λ, ω) = x ∗ . If there exists λ such that E[ξT XT∗ (λ)] = z then XT∗ (λ) is an optimal solution and k1 (z) = E[u XT∗ − (K − ST )+ 1τ >T ].
Carole Bernard
Financial Bounds for Insurance Claims
25/29
Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Illustration Next slide illustrates how to calculate explicitly bid prices. Recall that for a given wealth z sup E [u (XT − CT )]
k1 (z) =
XT ∈A(z)
and k2 (z) =
sup E [u (XT )] . XT ∈A(z)
Parameters are r = 2%, σ = 0.2, µ = 4%, S0 = 10, T = 1, K = 12, γ = 0.2, p = 0.7.
Carole Bernard
Financial Bounds for Insurance Claims
26/29
1 k1(z)
0.9
k (z)
Optimal Expected Utility
2
0.8 0.7
pb
k2(z0−pb) k1(z0)
p
0.6
b
p
b
0.5 p 0.4
b
0.3 0.2 3
4
5 z −p 0
b
6
z0
7 8 Initial Wealth z
Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Bid and ask prices with respect to survival probability p 2.5
Bid Price
2
1.5
1 Lower bound Bid Price when γ=0.5 Bid Price when γ=0.2 Bid Price when γ=0.05
0.5
0
Carole Bernard
0.2
0.4 0.6 Survival probability p
0.8
Financial Bounds for Insurance Claims
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Conclusion • Market consistent pricing of insurance claims • Preference-free bounds on prices of financial and insurance
claims • These bounds correspond to prices of some financial payoffs
that we give explicitly • These bounds are robust in the sense that they are derived
under rather mild assumptions
Carole Bernard
Financial Bounds for Insurance Claims
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Introduction
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Example
Conclusions
Thanks!
Carole Bernard
Financial Bounds for Insurance Claims
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Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Additional Material
Carole Bernard
Financial Bounds for Insurance Claims
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Example
Conclusions
Put option in Black and Scholes model Assume a strike K . The payoff of the put is given by LT = (K − ST )+ . The payoff that has the lowest cost and is distributed such as the put option is given by YT? = FL−1 (FST (ST )) = K −
S02 e
2 2 µ− σ2 T
ST
+ .
This type of power option “dominates” the put option.
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Financial Bounds for Insurance Claims
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Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Cost-efficient payoff of a put cost efficient payoff that gives same payoff distrib as the put option 100
80 Put option
Payoff
60
Y* Best one
40
20
0 0
100
200
300
400
500
ST
With σ = 20%, µ = 9%, r = 5%, S0 = 100, T = 1 year, K = 100. Distributional price of the put = 3.14 Price of the put = 5.57 Efficiency loss for the put = 5.57-3.14= 2.43 Carole Bernard
Financial Bounds for Insurance Claims
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Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Toy example Equity-Linked Insurance
Simplest possible insurance claim that pays at time T = 1 a payoff C1 distributed as a Bernoulli with parameter p = 0.001. P(C1 = 1) = p and P(C1 = 0) = 1 − p. 3 cases: First, the insurance claim C is linked to the death of a specific individual, then E[C1 |ξ1 ] = E[C1 ]. Bid and ask prices p • satisfy p • ≥ E[ξ1 E[C1 |ξ1 ]] = e −r E[C1 ] = e −r P(death).
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Financial Bounds for Insurance Claims
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Introduction
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Bounds
Example
Conclusions
Second, C1 pays 1 if a designated person dies and the risky asset in the market is higher than a value H or equivalently {ξ1 < L} = {S1 > H}) and E[C1 |ξ1 ] = E[1death 1ξ1 H), e −r E[C1 ] = e −r P(death)P(S1 > H) > e −r P(death)Q(S1 > H). Third, C1 pays 1 if a designated person dies and the risky asset in the market is lower than a value H. Then, Cov (C1 , ξ1 ) > 0 and p • ≥ E[ξ1 E[C1 |ξ1 ]] = P(death).Q(S1 < H) > e −r E[C1 ]. Carole Bernard
Financial Bounds for Insurance Claims
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Introduction
Cost-Efficiency
Example
Bounds
Example
Conclusions
Corollary: Optimal Investment (key) Corollary Denote by V (·) the objective function and given an initial wealth w ∈ R+ it holds that sup XT ∈A(w )
V (XT ) =
sup
V (XT ),
(1)
XT ∈Aξ (w )
where I A(w ) is the set of random wealths XT that can be generated at maturity T > 0 with an initial wealth w , I Aξ (w ) is the subset of random wealths that are almost surely anti-comonotonic with ξT (in other words which are almost surely a non-increasing function of ξT ).
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Financial Bounds for Insurance Claims
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