a unifying pricing theory for insurance and financial...
TRANSCRIPT
A Unifying Pricing Theory for Insurance and
Financial Risks: Applications for a Unified
Risk Management
Alejandro Balbas1 and Jose Garrido1,2
1Department of Business Administration
University Carlos III of Madrid, Spain2Department of Mathematics and Statistics
Concordia University, Canada
Instituto MEFF
April 24, 2002
1
Overview:
1. Introduction:
1.1 Distortion Operators, Insurance Pricing,
1.2 Choquet Pricing of Assets and Losses.
2. Wang’s Distortion Operator,
3. Applications:
3.1 Asset Pricing,
3.2 Capital Asset Pricing Model,
3.3 Black-Scholes Formula,
3.4 Deviance and Tail Index.
4. Dynamic Coherent Measures.
2
1. Introduction
Insurance and financial risks are becoming more integrated.
The actuarial literature argues for a unified pricing theory linking
both fields [Smith (1986), Cummins (1990, 1991) or Embrechts
(1996), Wang (2000), Cheuk (2001)] .
The price of an insurance risk (excluding other expenses) is called
a risk-adjusted premium. From classical expected utility theory
[Borch (1961), Buhlmann (1980), Goovaerts et al. (1984)] a
dual theory of risk has emerged in the economic literature [Yaari
(1987)].
Similarly, the price of a financial instrument (e.g. options) also
carries risk loads. But the markets and the strategies by which
insurance and financial risks are sold differ substantially.
3
We will discuss Wang’s proposal of a risk-adjustment method
that distorts the survival function of an insurance risk.
Wang (2000) shows that his (one-parameter) distortion operator
reproduces Black-Scholes theory for option pricing and extends
the Capital Asset Pricing Model (CAPM) to insurance pricing.
Major and Venter (1999), Gao and Qiu (2001) and Cheuk (2001)
consider two-parameter Wang transforms. With it Cheuk defines
a generalized Value at Risk (VaR) measure.
We characterize Wang’s transform as a “coherent measure” and
propose a dynamic (stochastic process) version. Also a tail index
for (financial or insurance) risk distributions is derived from it.
4
1.1 Distortion Operators in Insurance Pricing.
X is a non-negative loss variable, distributed as FX and with
SX = 1 − FX as its survival function.
The net insurance premium (excluding other expenses) is
E[X] =∫ ∞
0ydFX(y) =
∫ ∞
0SX(y)dy.
An insurance layer X(a,a+m] of X is defined by the payoff function
X(a,a+m] =
0 0 ≤ X < a
X − a a ≤ X < a + m
m a + m ≤ X
,
where a is a deductible and m a payment limit.
5
The survival function of this insurance layer is given by SX as
SX(a,a+m](y) =
{
SX(a + y) 0 ≤ y < m
0 m ≤ y.
It yields a net premium of
E[X(a,a+m]] =
∫ ∞
0SX(a,a+m]
(y)dy =
∫ a+m
aSX(x)dx.
Wang (1996) suggested to introduce the risk loading by first
distorting the survival function before taking expectations.
6
Distorting the survival function of any risk, Wang obtains a risk-
adjusted premium:
Hg[X] =
∫ ∞
0g[SX(x)]dx,
where g : [0,1] → [0,1] is increasing with
g(0) = 0 and g(1) = 1.
g is a distortion operator. It transforms SX into a new survival
function g ◦ SX, of the “ground-up distribution”.
For example, the risk adjusted premium of a risk layer is then
Hg[X(a,a+m]] =
∫ ∞
0g[SX(a,a+m]
]dy =
∫ a+m
ag[SX(x)]dx.
7
For general insurance pricing, g should satisfy:
• 0 < g(u) < 1, g(0) = 0 and g(1) = 1,
• g increasing and concave,
• g′(0) = +∞, where it is defined.
Wang (1996) shows that in a class of one-parameter functions,
only
g(u) = ur, u ∈ [0,1], 0 < r ≤ 1,
satisfies all of the above.
This corresponds to the proportional hazards (PH) transform of
Wang (1995).
8
Despite some desirable properties, Wang’s PH transform suffers
some major drawbacks. The PH transform:
• of a lognormal is not lognormal (Black-Scholes formula for
option pricing),
• lacks flexibility yielding fast increasing risk loadings for high
layers,
• cannot be applied simultaneously to assets and liabilities. It
yields to serious inconsistencies; see the following section.
9
1.2 Choquet Pricing of Assets and Losses
Consider an asset A as a negative loss X = −A. The Choquet
integral with respect to the distortion g is then given by:
Hg[X] =
∫ 0
−∞{g[SX(x)] − 1}dx +
∫ ∞
0g[SX(x)]dx.
It has been proposed as a general pricing method for financial
markets with frictions [Chateauneuf et al. (1996)].
Definition: For any risk X and a real valued h, the payoff Y =
h(X) is a derivative of X. If h is non-decreasing, then Y is called
a comonotone derivative of the underlying X.
10
Theorem: Under the Choquet integral the price of a comonotone
derivative Y = h(X),
Hg[Y ] =∫ 0
−∞{g[SY (y)] − 1}dy +
∫ ∞
0g[SY (y)]dy,
is equivalent to the expectation of h(Xα), where Xα has the
ground-up distribution with survival function SXα= g ◦ SX.
The Choquet integral Hg provides a “risk-neutral” valuation of
comonotone derivatives, and hence of insurance risk layers.
But the above equivalence does not hold for derivatives which
are not comonotone with the underlying risk; a distortion with
a concave g always produces non-negative loadings, while the
ground-up distribution can yield negative loadings.
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Denneberg (1994) shows that for an asset A > 0
Hg[−A] = −Hg∗[A],
where g∗(u) = 1 − g(1 − u) is the dual distortion operator of g.
A loss X distorted by a concave g always yields Hg[X] ≥ E[X].
When g is concave, g∗ is convex and Hg∗[A] ≤ E[A].
For most choices g, the dual g∗ belongs to a different parametric
family. In particular, if the PH transform g(u) = ur is applied to
losses, the dual g∗(u) = 1 − (1 − u)r would apply to assets.
Symmetric treatment of assets and liabilities is possible for a new
class of distortion operators.
12
2. Wang’s Distortion Operator
Wang (2000) suggests a new distortion operator defined as:
gα(u) = Φ[Φ−1(u) + α], u ∈ [0,1],
where α ∈ R and Φ is the standard normal distribution with
density function:
φ(x) =1√2π
e−x2
2 , x ∈ R.
For α > 0, gα satisfies all the desirable properties of a distortion
operator for insurance pricing:
1. 0 < gα(u) < 1 for u ∈ [0,1], with gα(0) = limu→0+
gα(u) = 0 and
gα(1) = limu→1−
gα(u) = 1.
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2. gα is increasing and for x = Φ−1(u),
g′α(u) =∂gα(u)
∂u=
φ(x + α)
φ(x)= e−αx−x2
2 > 0, u ∈ (0,1).
3. For α > 0, gα is concave and
g′′α(u) =∂2gα(u)
∂u2= −αφ(x + α)
φ(x)2< 0, u ∈ (0,1).
4. For α > 0, g′α(u) becomes unbounded as u approaches 0.
5. The dual operator g∗α(u) = 1 − gα(1 − u) = g−α(u).
6. g∗α is convex when gα is concave, as in 3.
14
For gα the Choquet integral gives
H[X;α] =
∫ 0
−∞{gα[SX(x)] − 1}dx +
∫ ∞
0gα[SX(x)]dx,
which enjoys some very nice properties:
1. H[c;α] = c and H[X + c;α] = H[X;α]+ c, for any constant c.
2. H[bX; α] = bH[X; α] while H[−bX;α] = −bH[X;−α], for any
b > 0. In particular H[−X;α] = −H[X;−α].
3. If X1, X2 are comonotone H[X1+X2;α] = H[X1;α]+H[X2;α].
Since two layers of the same risk are comonotone, then
H[X(a,b];α] + H[X(b,c];α] = H[X(a,c];α],
for any a < b < c.
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4. For any two variables X1 and X2
H[X1 + X2;α] ≤ H[X1;α] + H[X2;α], forα > 0,
≥ H[X1;α] + H[X2;α], forα < 0,
showing the benefit of diversification.
5. H[X; α] is an increasing function of α, with
min[X] ≤ H[X;α] ≤ max[X].
6. With α > 0 for losses and α < 0 for assets, H[X;α] preserves
1st and 2nd order stochastic dominance [see Rothschild and
Stiglitz (1971)].
7. For α > 0, g′α(u) goes unbounded as u approaches 0. For
example, if X ∼ Bernoulli(θ) then
limθ→0
H[X;α]
E[X]= lim
θ→0
Φ[Φ−1(θ) + α]
θ= g′α(0) = +∞.
16
If X ∼ Normal(µ, σ2) then the ground-up survival function SXα=
gα ◦ SX defines a new Xα ∼ Normal(µ + ασ, σ2), that is
H[X;α] = E[Xα] = E[X] + ασ[X] ,
which is the usual standard deviation principle.
If Y ∼ Lognormal(µ, σ2) then the ground-up survival function
SYα= gα ◦ SY defines a new Yα ∼ Lognormal(µ + ασ, σ2).
gα can be applied to any distribution, but closed forms other
than Bernoulli, normal and lognormal do not exist. Numerical
evaluations are simple (e.g. in Excel NORMDIST, NORMINV).
The generalization to multivariate variables is possible, simply
applying the distortion to the joint distribution function.
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3. Applications
3.1 Asset Pricing
Consider asset i with current price Ai(0) and future price Ai(1)
after one period.
Ri = Ai(1)Ai(0)
− 1 is the annual return, assumed normal with mean
E[Ri] and standard deviation σ[Ri].
Now, if assets can be priced by applying H[·;−αi] to the present
value of the future asset price at some time, then
Ai(0) = H[Ai(1)
(1 + rf);−αi] = H[
Ai(0)(1 + Ri)
(1 + rf);−αi],
where rf is the risk-free return compounded annually.
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It follows that the risk-adjusted rate of return must coincide with
the risk-free rate and hence
H[Ri;−αi] = E[Ri] − αiσ[Ri] = rf ,
which implies [Wang (2000)]:
αi =E[Ri] − rf
σ[Ri].
A similar implied α applies to an asset portfolio or, by extension,
to the market portfolio M . Under similar assumptions
αM =E[RM ] − rf
σ[RM ],
which is called the market price of risk (Cummins, 1990) in the
expression for the capital market line (CML) in the CAPM.
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The idea can be generalized to T periods. For asset i let Rit bethe annual return in period t = 1, . . . , T and Ai(t) the price attime t. Then
Rit = lnAi(t)
Ai(t − 1).
Now assume that the Rit are iid Normal. The total T -periodreturn for asset i is
Ri(T ) = lnAi(T )
Ai(0)=
T∑
t=1
Rit,
with
E[Ri(T )] =T
∑
t=1
E[Rit] = TE[Ri]
and
V [Ri(T )] =T
∑
t=1
{σ[Rit]}2 = T{σ[Ri]}2.
20
Again this implies that
H[Ri(T );−αi] = TE[Ri] − αi
√Tσ[Ri] = Trf ,
and hence [Wang (2000)]:
αi =√
T
{
E[Ri] − rf
σ[Ri]
}
.
The idea is easily extended to the continuous time model, with
geometric Brownian motion (GBM) prices, At satisfying
dAi(t)
Ai(t)= µidt + σidW (t).
21
Let Ai(0) be the current asset price. At any future date T , the
price Ai(T ) is solution to the above equation and has a lognormal
distribution [Hull (1997)]:
lnAi(T )
Ai(0)∼ Normal[(µ − σ2
2)T, σ2T ].
No-arbitrage with a continuous risk-free return rc implies:
Ai(0) = H[e−rcTAi(T );−αi] = e−rcTH[Ai(T );−αi],
which implies again [Wang (2000)]:
αi =(µi − rc)
√T
σi.
Notice that αi shares many properties with the volatility and is
proportional to the square root of the horizon T .
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3.2 Capital Asset Pricing Model (CAPM)
As seen, the market price of risk αM in the CML of CAPM links
the systematic risk of X to the portfolio risk.
The CAPM measures a portfolio’s risk under the constraint of
an expected return level in one compounding period and the
assumption of an efficient market.
Mixing a risk-free return rf with an efficient market portfolio,
gives a minimal risk portfolio for a given level of expected return.
Let the expected return and risk of this optimal strategy be
µp and σp, respectively. Then the following capital market line
(CML) relation holds:
µp = rf + αMσp.
23
For asset i with return Ri ∼ Normal(µi, σ2i ) and a market portfolio
return RM ∼ Normal(µM , σM), consider the security market line
(SML):
µi = rf + βi(µM − rf),
where
βi =Cov[Ri, RM ]
σ2M
= ρi,Mσi
σM
.
Since αM =µM−rf
σM, the above SML relation can be rewritten as
µi = rf + ρi,MαMσi, which in turn yields:
ρi,MαM =µi − rf
σA
= αi.
Wang (2000) generalizes CAPM to include insurance risks that
are not normally distributed, valuing them with H[·;α].
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3.3 Black-Scholes Formula
Black-Scholes option pricing formula is often seen as a stop-loss
reinsurance contract under a lognormal risk neutral price.
Wang (2000) shows how the distortion operator reproduces Black-
Scholes formula under the specific α of the CAPM.
Let A(t) be the current price on day t of an asset and K a
European call option strike price, with a right to exercise after
T periods.
The payoff of this option can be expressed as
A(t + T ;K,∞) =
{
0 if A(t + T ) ≤ K
A(t + T ) − K if A(t + T ) > K.
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The expected payoff is then
E[A(t + T ;K,∞)] =∫ ∞
0SA(t+T ;K,∞)(x)dx =
∫ ∞
KSA(t+T)(y)dy,
while the price under the distortion operator g−α is
e−rfTH[A(t + T ; K,∞);−α] = e−rfT∫ ∞
K[SA−α(t+T)(y)]dy.
It easily seen that with a GBM return of rate with mean µ and
volatility σ, the ground-up variables A−α(t) implies a lognormal
distribution
lnA′
t+T
A′t
∼ Normal[(rf − 1
2σ2)T, σ2T ], for α =
(µ − rf)√
T
σ.
Hence, the distortion and its resulting pricing formula reproduce
exactly Black-Scholes’ model.
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3.4 Deviance and Tail Index
Both, insurance and financial analysts, use indexes to measure
and compare “riskyness”, in the form of a volatility measure,
down-side risk, variance, deviance or more generally speaking, a
risk measure.
The distortion operator gα induces a risk loading H[X; α]. It is
natural to think that it can also be used to define a risk measure.
Since min[X] ≤ H[X; α] ≤ max[X], for any α ∈ R, and in particular
H[X; 0] = E(X) for α = 0, Wang (2000) defines right and left
deviance measures.
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Definition: For α > 0 fixed and any risk X, the right and left
deviance indexes are given, respectively, by
RDα[X] =1
α{H[X;α]−E(X)}, LDα[X] =
1
α{E(X)−H[X;−α]}.
Note that these deviances measures are function of α, but con-
verge to a common limit at α = 0 :
limα↘0
RDα[X] = lim−α↗0
LDα[X] =∂
∂αH[X;α]
∣
∣
∣
∣
α=0,
which measures the infinitesimal “force of distortion” at α = 0.
This motivates our definition of a deviance index.
Definition: For α > 0 fixed and any risk X, the deviance index
is given by
DI[X] =∂
∂αH[X; α]
∣
∣
∣
∣
α=0.
Note that by definition of H[X;α], this derivative always exists.
28
In finance the down-side risk is usually measured in relation to
an underlying normal distribution of asset returns.
Wang (2000) uses his right and left deviance indexes to define
right and left tail indexes, with the normal distribution serving
as a benchmark.
Definition: For α > 0 fixed and any risk X, the right and left
tail indexes are given, respectively, by
RTIα[X] =RD2α[X]
RDα[X]and LTIα[X] =
LD2α[X]
LDα[X].
These are ment to be scaleless and differ from his deviance in-
dexes which are dependent on the parameter α.
29
Example: If X ∼ normal(µ, σ2) then H[X;α] = µ + ασ and
E(X) = µ, for α, µ ∈ R and σ > 0. It implies that
RDα[X] =µ + ασ − µ
α=
µ − µ + ασ
α= LDα[X] = σ > 0,
reproducing the standard deviation.
Note that the same result is obtained with our deviance index
DI[X] = σ,
a measure always independent of the parameter α.
Similarly
RTIα[X] = 1 = LTIα[X],
showing how normal tails are used here as a benchmark; index
values larger than 1 indicate fatter than normal tails.
30
Example: If X ∼ lognormal(µ, σ2) then H[X;α] = e(µ+ασ)+σ2
2
and E(X) = eµ+σ2
2 , for α, µ ∈ R and σ > 0. Hence
RDα[X] =e(µ+ασ)+σ2
2 − eµ+σ2
2
α= eµ+σ2
2(eασ − 1)
α,
while the left deviance index has no physical interpretation here:
LDα[X] = eµ+σ2
2(1 − e−ασ)
α.
Note that these deviances depend on the parameter α and differ
from the standard deviation:
σ[X] = eµ+σ2
2
√
(eσ2 − 1).
By contrast, our deviance index is closer in value to σ[X] :
DI[X] = σeµ+σ2
2 .
31
Finally, Wang’s tail indexes are given here by:
RTIα[X] =eασ + 1
2and LTIα[X] =
1 + e−ασ
2.
Oddly enough, here σ values need to be such that eασ > 1 for
lognormal tails to be fatter than normal.
This illustrates the undesirable dependence of Wang’s tail index
on the parameter α.
Alternative tail indexes, based on higher derivatives of H[X;α]
at α = 0, are being studied.
32
Example: For X ∼ Bernoulli(θ0), H[X;α] = θα = Φ[Φ−1(θ0)+α]
and E[X] = θ0, where α ∈ R and 0 ≤ θ0 ≤ 1. Hence
RDα[X] =Φ[Φ−1(θ0) + α] − θ0
α=
θα − θ0
α,
while again, the left deviance index has no physical interpretation:
LDα[X] =θ0 − Φ[Φ−1(θ0) − α]
α=
θ0 − θ−α
α.
These deviance measures also differ from the standard deviation
σ[X] =√
θ0(1 − θ0) and depend on the parameter α.
By contrast, our deviance index is given by:
DI[X] = φ[Φ−1(θ0)] =1√2π
e−[Φ−1(θ0)]2
2 .
33
Finally, Wang’s tail indexes
RTIα[X] =Φ[Φ−1(θ0) + 2α] − θ0
2{Φ[Φ−1(θ0) + α] − θ0}=
θ2α − θ0
2(θα − θ0)
and
LTIα[X] =θ0 − Φ[Φ−1(θ0) − 2α]
2{θ0 − Φ[Φ−1(θ0) − α]}=
θ0 − θ−2α
2(θ0 − θ−α),
depend on α in an intricate way.
34
4. Dynamic Coherent Measures
Wang’s use of distortion operators and Choquet integrals has
been embedded in the framework of coherent risk measures
[Artzner, Delbaen, Eber and Heath (1999)].
In Balbas, Garrido and Mayoral (2002) we assume a dynamic
framework which reflects the stochastic behavior of (insurance)
risks and (financial) pay-offs.
As a special application, we study the extensions of VaR (Value
at Risk) and TCE (Tail Conditional Expectation) to this dynamic
setting.
35
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