Let the annual premium for a certain risk be π and denote the losses in year i by Si and assume that they are iid.. The company has a certain initial capital w. Then the capital of the company after year i is
Xi = w + πi −
i
X
j=1
Sj.
The stochastic process {Xi} is called a random walk. From Lemma E.1 we conclude that
i) if π < IIE[S1] then Xi converges to −∞ a.s..
ii) if π = IIE[S1] then a.s.
i→∞lim Xi = − lim
i→∞
Xi = ∞ .
iii) if π > IIE[S1] then Xi converges to ∞ a.s. and there is a strictly positive proba-bility that Xi ≥ 0 for all i ∈ IIN.
As an insurer we only have a chance to survive if π > IIE[S]. Therefore the latter must hold for any reasonable premium calculation principle. Another argument will be given in Chapter 2.
1.10.1. The Expected Value Principle The most popular premium calculation principle is
π = (1 + θ)IIE[S]
where θ is a strictly positive parameter called safety loading. The premium is very easy to calculate. One only has to estimate the quantity IIE[S]. A disadvantage is that the principle is not sensible to heavy tailed distributions.
1.10.2. The Variance Principle
A way of making the premium principle more sensible to higher risks is to use the variance for determining the risk loading, i.e.
π = IIE[S] + α Var[S]
for a strictly positive parameter α.
1.10.3. The Standard Deviation Principle A principle similar to the variance principle is
π = IIE[S] + βp Var[S]
for a strictly positive parameter β. Observe that the loss can be written as π − S =p
Var[S]
β −S − IIE[S]
pVar[S]
.
The standardized loss is therefore the risk loading parameter minus a random vari-able with mean 0 and variance 1.
1.10.4. The Modified Variance Principle
A problem with the variance principle is that changing the monetary unit changes the security loading. A possibility to change this is to use
π = IIE[S] + αVar[S]
IIE[S]
for some α > 0.
1.10.5. The Principle of Zero Utility
This premium principle will be discussed in detail in Chapter 2. Denote by w the initial capital of the insurance company. The worst thing that may happen for the company is a very high accumulated sum of claims. Therefore high losses should be weighted stronger than small losses. Hence the company chooses a utility function v, which should have the following properties:
• v(0) = 0,
• v(x) is strictly increasing,
• v(x) is strictly concave.
The first property is for convenience only, the second one means that less losses are preferred and the last condition gives stronger weights for higher losses.
The premium for the next year is computed as the value π such that
v(w) = IIE[v(w + p − S)] . (1.3)
This means, the expected utility is the same whether the insurance contract is taken or not.
Lemma 1.8.
i) If a solution to (1.3) exists, then it is unique.
ii) If IIP[S = IIE[S]] < 1 then π > IIE[S].
iii) The premium is independent of w for all loss distributions if and only if v(x) = A(1 − e−αx) for some A > 0 and α > 0.
Proof. i) Let π1 > π2 be two solutions to (1.3). Then because v(x) is strictly increasing
v(w) = IIE[v(w + π1− S)] > IIE[v(w + π2− S)] = v(w) which is a contradiction.
ii) By Jensen’s inequality
v(w) = IIE[v(w + π − S)] < v(w + π − IIE[S]) and thus because v(x) is strictly increasing we get w + π − IIE[S] > w.
iii) A simple calculation shows that the premium is independent of w if v(x) = A(1 − e−αx). Assume now that the premium is independent of w. Let IIP[S = 1] = 1 − IIP[S = 0] = q and denote by π(q) be the corresponding premium. Then
qv(w + π(q) − 1) + (1 − q)v(w + π(q)) = v(w) . (1.4) Note that as a concave function v(x) is differentiable almost everywhere. Taking the derivative of (1.4) (at points where it is allowed) with respect to w yields
qv0(w + π(q) − 1) + (1 − q)v0(w + π(q)) = v0(w) . (1.5)
Varying q the function π(q) takes all values in (0, 1). Changing w and q in such a way that w + π(q) remains constant shows that v0(w) is continuous. Thus v(w) must be continuously differentiable everywhere.
The derivative of (1.4) with respect to q is
v(w+π(q)−1)−v(w+π(q))+π0(q)[qv0(w+π(q)−1)+(1−q)v0(w+π(q))] = 0 . (1.6) The implicit function theorem gives that π(q) is indeed differentiable. Plugging in (1.5) into (1.6) yields
v(w + π(q) − 1) − v(w + π(q)) + π0(q)v0(w) = 0 . (1.7) Note that π0(q) > 0 follows immediately. The derivative of (1.7) with respect to w is
v0(w + π(q) − 1) − v0(w + π(q)) + π0(q)v00(w) = 0
showing that v(w) is twice continuously differentiable. From the derivative of (1.7) with respect to q it follows that
π0(q)[v0(w + π(q) − 1) − v0(w + π(q))] + π00(q)v0(w) = −π0(q)2v00(w) + π00(q)v0(w) = 0 . Thus π00(q) ≤ 0 and
v00(w)
v0(w) = π00(q) π0(q)2 .
The left-hand side is independent of q and the right-hand side is independent of w.
Thus it must be constant. Hence
v00(w)
v0(w) = −α
for some α ≥ 0. α = 0 would not lead to a strictly concave function. Solving the
differential equation shows the assertion.
1.10.6. The Mean Value Principle
In contrast to the principle of zero utility it would be desirable to have a premium principle which is independent of the initial capital. The insurance company values its losses and compares it to the loss of the customer who has to pay its premium.
Because high losses are less desirable they need to have a higher value. Similarly to the utility function we choose a value function v with properties
• v(0) = 0,
• v(x) is strictly increasing and
• v(x) is strictly convex.
The premium is determined by the equation v(π) = IIE[v(S)]
or equivalently
π = v−1(IIE[v(S)]) .
Again, it follows from Jensen’s inequality that π > IIE[S] provided IIP[S = IIE[S]] < 1.
1.10.7. The Exponential Principle
Let α > 0 be a parameter. Choose the utility function v(x) = 1 − e−αx. The premium can be obtained
1 − e−αw = IIE1 − e−α(w+p−S) = 1 − e−αwe−αpIIEeαS from which it follows that
π = log(MS(α))
α . (1.8)
The same premium formula can be obtained by using the mean value principle with value function v(x) = eαx − 1. Note that the principle only can be applied if the moment generating function of S exists at the point α.
How does the premium depend on the parameter α? In order to answer this question the following lemma is proved.
Lemma 1.9. For a random variable X with moment generating function MX(r) the function log MX(r) is convex. If X is not deterministic then log MX(r) is strictly convex.
Proof. Let F (x) denote the distribution function of X. The second derivative of log MX(r) is
(log MX(r))00 = MX00(r)
MX(r)−MX0 (r) MX(r)
2
. Consider
MX00(r) MX(r) =
d2 dr2
R∞
−∞erx dF (x)
MX(r) =
R∞
−∞x2erx dF (x) MX(r)
and
MX0 (r) MX(r) =
R∞
−∞xerx dF (x) MX(r) . Because
R∞
−∞erx dF (x)
MX(r) = MX(r) MX(r) = 1 it follows that Fr(x) = (MX(r))−1Rx
−∞ery dF (y) is a distribution function. Let Z be a random variable with distribution function Fr. Then
(log MX(r))00 = Var[Z] ≥ 0 .
The variance is strictly larger than 0 if Z is not deterministic. But the latter is
equivalent to X is not deterministic.
The question how the premium depends on α can now be answered.
Lemma 1.10. Let π(α) be the premium determined by the exponential principle.
Assume that MS00(α) < ∞. The function π(α) is strictly increasing provided S is not deterministic.
Proof. Take the derivative of (1.8) π0(α) = MS0(α)
αMS(α) − log(MS(α))
α2 = 1
α
MS0(α)
MS(α) − π(α) . Furthermore
d
dα(α2π0(α)) = MS0(α)
MS(α) − π(α) + αMS00(α)
MS(α) −MS0(α) MS(α)
2
− απ0(α)
= αMS00(α)
MS(α) −MS0(α) MS(α)
2 . From Lemma 1.9 it follows that
d
dα(α2π0(α)) T 0 iff α T 0 .
Thus the function α2π0(α) has a minimum in 0. Because its value at 0 vanishes it fol-lows that π0(α) > 0. π0(0) = 12Var[S] > 0 can be verified directly with L’Hospital’s
rule.
1.10.8. The Esscher Principle
A simple idea for pricing is to use a distribution ˜F (x) which is related to the dis-tribution function of the risk F (x), but does give more weight to larger losses. If exponential moments exist, a natural choice would be to consider the exponential class
Fα(x) = (MS(α))−1 Z x
−∞
ery dFS(y) . This yields the Esscher premium principle
π = IIE[SeαS] MS(α) ,
where α > 0. In order that the principle can be used we have to show that π > IIE[S]
if Var[S] > 0. Indeed, π = (log MS(r))0|r=α and log MS(r) is a strictly convex function by Lemma 1.9. Thus (log MS(r))0 is strictly increasing, in particular π >
(log MS(r))0|r=0 = IIE[S].
1.10.9. The Proportional Hazard Transform Principle
A disadvantage of the Esscher principle is, that MS(α) has to exist. We therefore look for a possibility to increase the weight for large claims, such that no exponential moments need to exist. A possibility is to change the distribution function FS(x) to Fr(x) = 1 − (1 − FS(x))r, where r ∈ (0, 1). Then Fr(x) < FS(x) for all x such that FS(x) ∈ (0, 1). The proportional hazard transform principle is
π = Z ∞
0
(1 − FS(x))rdx − Z 0
−∞
1 − (1 − FS(x))rdx .
Note that clearly π > IIE[S]. Because dFr(x) = r(1 − FS(x))r−1 dFS(x) it follows that the distribution Fr(x) and FS(x) are equivalent. A sufficient condition for π to be finite would be that IIE[max{S, 0}α] < ∞ for some α > 1/r.
1.10.10. The Percentage Principle
The company wants to keep the risk that the accumulated sum of claims exceeds the premium income small. They choose a parameter ε and the premium
π = inf{x > 0 : IIP[S > x] ≤ ε} .
If the claims are bounded it would be possible to choose ε = 0. This principle is called maximal loss principle. It is clear that the latter cannot be of interest even though there had been villains using this principle.
1.10.11. Desirable Properties
• Simplicity: The premium calculation principle should be easy to calculate. This is the case for the principles 1.10.1, 1.10.2, 1.10.3 and 1.10.4.
• Consistency: If a deterministic constant is added to the risk then the premium should increase by the same constant. This is the case for the principles 1.10.2, 1.10.3, 1.10.5, 1.10.7, 1.10.8, 1.10.9 and 1.10.10. The property only holds for the mean value principle if it has an exponential value function (exponential principle) or a linear value function. This can be seen by a proof similar to the proof of Lemma 1.8 iii).
• Additivity: If a portfolio consists of two independent risks then its premium should be the sum of the individual premia. This is the case for the principles 1.10.1, 1.10.2, 1.10.7 and 1.10.8. For 1.10.5 and 1.10.6 it is only the case if they coincide with the exponential principle or if π = IIE[S]. A proof can be found in [43, p.75].
Bibliographical Remarks
For further literature on the individual model (section 1.6) see also [43, p.51]. A reference to why reinsurance is necessary is [73]. The Panjer recursion is originally described in [64], see also [65] where also a more general algorithm can be found.
Section 1.10 follows section 5 of [43].