J. van der Hoek, M. Sherris Insurance: Mathematics and Economics 28 2001 69–82 71
so that S
T
t = exp −
Z
t
λs ds
. The expected lifetime is
E [T ] =
Z
∞
tf
T
t dt =
Z
∞
[S
T
t ] dt.
Now assume a proportional change to the hazard function so that λ
∗
t = rλt. The survival distribution function under the new hazard rate is given by
− d
dt ln [S
T
∗
t ] = rλt,
so that S
T
∗
t = exp −
Z
t
rλs ds
= [S
T
t ]
r
. The expected lifetime is now
E [T
∗
] = Z
∞
[S
T
t ]
r
dt. We note that the PH transform approach to insurance pricing considers the loss distribution in a similar manner to
the survival distribution. The PH transform distorts the probability of a claim occurring and then uses the distorted probabilities to calculate the expected value.
For X and Y non-negative random variables we have the following properties of the PH transform with gx = x
r
, where 0 r ≤ 1 Wang, 1996a:
1. E[X] ≤ H
g
[X] ≤ maxX; 2. H
g
aX + b = aH
g
X + b, a ≥ 0, b ≥ 0; 3. H
g
X + Y ≤ H
g
X + H
g
Y .
This third property is referred to as sub-additivity. H
g
X will be additive in the special case of comonotonic risks.
Risks X
1
and X
2
are comonotonic if there exists a risk Z and non-decreasing real-valued functions f and h such that X
1
= f Z and X
2
= hZ Wang and Young, 1997. The concept of comonotonic risks is an extension of perfect correlation.
In general we have the following properties using the distortion function approach: 1. If gp = p, for all p ∈ [0, 1], then H
g
[X] = E[X]. 2. If gp ≥ p, for all p ∈ [0, 1], then H
g
[X] ≥ E[X]. 3. H
g
aX + b = aH
g
X + b, a ≥ 0, b ≥ 0. 4. For X and Y comonotonic, H
g
X + Y = H
g
X + H
g
Y .
5. For concave g, H
g
[X] ≥ E[X], and H
g
X + Y ≤ H
g
X + H
g
Y .
6. For convex g, H
g
[X] ≤ E[X], and H
g
X + Y ≥ H
g
X + H
g
Y .
3. Dual theory of choice
Consider preference over risks. The symbol ≻ will denote preference or riskiness, so that X ≻ Y indicates X is preferred to Y . The use of expected utility as a risk measure is derived from five axioms. These are as follows:
72 J. van der Hoek, M. Sherris Insurance: Mathematics and Economics 28 2001 69–82
1. If risks X
1
and X
2
have the same cumulative distribution function, then X
1
and X
2
are equally risky. 2. ≻ is reflexive, transitive and connected weak order.
3. ≻ is continuous in the topology of weak convergence. 4. If S
X
≤ S
Y
, then X ≻ Y . 5. If X ≻ Y and Z is any risk then
{α, X, 1 − α, Z} ≻ {α, Y , 1 − α, Z} for all α such that 0 ≤ α ≤ 1, where {α, X, 1 − α, Z} is the probabilistic mixture with
F
{α,X,1−α,Z}
x = αF
X
x + 1 − αF
Z
x. This last axiom is referred to as the independence axiom.
The independence axiom states that if X is preferred to Y , then a lottery that pays X with probability α and Z with probability 1 − α will be preferred to a lottery that pays Y with probability α and Z with probability 1 − α.
Thus there is independence with respect to probability mixtures of uncertain outcomes. If investors conform to these axioms then they will prefer strategies that have higher expected utilities. For a
non-negative random variable X, the expected utility is given by E
[U X] = Z
∞
S
X
t dU t =
Z
1
U [S
−1 X
q ] dq.
Consider two random payments X and Y . Under the expected utility property, X is preferred to Y if E
[U X] E[U Y ], where U is assumed to be a continuous non-decreasing function. U is a concave function for a risk averse individual
and is unique up to a positive affine transformation Gerber and Pafumi, 1998. Yaari 1987 develops the following dual theory of choice where this independence axiom is replaced with the
dual independence axiom which states that if X is preferred to Y , then [pX
−1
+ 1 − pZ
−1
]
−1
is preferred to [pY
−1
+ 1 − pZ
−1
]
−1
for any 0 p 1. Equivalently, if X ≻ Y and Z is any risk, then for any 0 p 1 pS
−1 X
+ 1 − pS
−1 Z
pS
−1 Y
+ 1 − pS
−1 Z
, and this condition is equivalent to requiring that if X, Y and Z are comonotonic and X ≻ Y then 1 − αX + αZ ≻
1 − αY + αZ Wang, 1996a. In this case X is preferred to Y if and only if
H
g
X = Z
∞
g [S
X
t ] dt
Z
∞
g [S
Y
t ] dt = H
g
Y , where g is a continuous and non-decreasing function defined on the unit interval with g0 = 0 and g1 = 1. If
the investor is risk averse then g is convex. An important implication of the independence axiom and expected utility is that the preference function is linear
in the probabilities. Experimental evidence has suggested that decision making behaviour does not conform with the independence axiom Machina, 1982, 1987. The most famous of these violations of the independence axiom is
probably the Allais Paradox. This is a particular example of what is referred to as the common consequence effect. An alternative approach is to use preference functions which are not linear in the probabilities such as Yaari’s dual
theory. Other non-linear functional forms have also been suggested as referenced in Machina 1987.
J. van der Hoek, M. Sherris Insurance: Mathematics and Economics 28 2001 69–82 73
4. Investment selection
