The mean excess loss function

We take a closer look at the mean excess loss function. We start with the following example:

Example
Suppose that an entity is exposed to a random loss X. An insurance policy offers protection against this loss. Under this policy, payment is made to the insured entity subject to a deductible d>0, i.e. when a loss is less than d, no payment is made to the insured entity, and when the loss exceeds d, the insured entity is reimbursed for the amount of the loss in excess of the deductible d. Consider the following two questions:

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  1. Of all the losses that are eligible to be reimbursed by the insurer, what is the average payment made by the insurer to the insured?
  2. What is the average payment made by the insurer to the insured entity?

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The two questions look similar. The difference between the two questions is subtle but important. In the first question, the average is computed over all losses that are eligible for reimbursement (i.e., the loss exceeds the deductible). This is the average amount the insurer is expected to pay in the event that a payment in excess of the deductible is required to be made. So this average is a per payment average.

In the second question, the average is calculated over all losses (regardless of sizes). When the loss does not reach the deductible, the payment is considered zero and when the loss is in excess of the deductible, the payment is X-d. Thus the average is the average amount the insurer has to pay per loss. So the second question is about a per loss average.

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The Mean Excess Loss Function
The average in the first question is called the mean excess loss function. Suppose X is the random loss and d>0. The mean excess loss variable is the conditional variable X-d \ \lvert X>d and the mean excess loss function e_X(d) is defined by:

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\displaystyle (1) \ \ \ \ \ e_X(d)=E(X-d \lvert X>d)

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In an insurance context, the mean excess loss function is the average payment in excess of a threshold given that the loss exceeds the threshold. In a mortality context, the mean excess loss function is called the mean residual life function and complete expectation of life and can be interpreted as the remaining time until death given that the life in question is alive at age d.

The mean excess loss function is computed by the following depending on whether the loss variable is continuous or discrete.

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\displaystyle (2) \ \ \ \ \ e_X(d)=\frac{\int_d^\infty (x-d) \ f_X(x) \ dx}{S_X(d)}

\displaystyle (3) \ \ \ \ \ e_X(d)=\frac{\sum \limits_{x>d} (x-d) \ P(X=x)}{S_X(d)}

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The mean excess loss function e_X(d) is defined only when the integral or the sum converges. The following is an equivalent calculation of e_X(d) that may be easier to use in some circumstances.

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\displaystyle (4) \ \ \ \ \ e_X(d)=\frac{\int_d^\infty S_X(x) \ dx}{S_X(d)}

\displaystyle (5a) \ \ \ \ \ e_X(d)=\frac{\sum \limits_{x \ge d} S_X(x) }{S_X(d)}

\displaystyle (5b) \ \ \ \ \ e_X(d)=\frac{\biggl(\sum \limits_{x>d} S_X(x)\biggr)+(w+1-d) S_X(w) }{S_X(d)}

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In both (5a) and (5b), we assume that the support of X is the set of nonnegative integers. In (5a), we assume that the deductible d is a positive integer. In (5b), the deductible d is free to be any positive number and w is the largest integer such that w \le d. The formulation (4) is obtained by using integration by parts (also see theorem 3.1 in [1]). The formulations of (5a) and (5b) are a result of applying theorem 3.2 in [1].

The mean excess loss function provides information about the tail weight of a distribution, see the previous post The Pareto distribution. Also see Example 3 below.
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The Mean in Question 2
The average that we need to compute is the mean of the following random variable. Note that (a)_+ is the function that assigns the value of a whenever a>0 and otherwise assigns the value of zero.

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\displaystyle (6) \ \ \ \ \ (X-d)_+=\left\{\begin{matrix}0&\ X<d\\{X-d}&\ X \ge d \end{matrix}\right.

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The mean E((X-d)_+) is calculated over all losses. When the loss is less than the deductible d, the insurer has no obligation to make a payment to the insured and the payment is assumed to be zero in the calculation of E[(X-d)_+]. The following is how this expected value is calculated depending on whether the loss X is continuous or discrete.

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\displaystyle (7) \ \ \ \ \ E((X-d)_+)=\int_d^\infty (x-d) \ f_X(x) \ dx

\displaystyle (8) \ \ \ \ \ E((X-d)_+)=\sum \limits_{x>d} (x-d) \ P(X=x)

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Based on the definitions, the following is how the two averages are related.

\displaystyle E[(X-d)_+]=e_X(d) \ [1-F_X(d)] \ \ \ \text{or} \ \ \ E[(X-d)_+]=e_X(d) \ S_X(d)

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The Limited Expected Value
For a given positive constant u, the limited loss variable is defined by

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\displaystyle (9) \ \ \ \ \ X \wedge u=\left\{\begin{matrix}X&\ X<u\\{u}&\ X \ge u \end{matrix}\right.

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The expected value E(X \wedge u) is called the limited expected value. In an insurance application, the u is a policy limit that sets a maximum on the benefit to be paid. The following is how the limited expected value is calculated depending on whether the loss X is continuous or discrete.

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\displaystyle (9) \ \ \ \ \ E(X \wedge u)=\int_{-\infty}^u x \ f_X(x) \ dx+u \ S_X(u)

\displaystyle (10) \ \ \ \ \ E(X \wedge u)=\biggl(\sum \limits_{x < u} x \ P(X=x)\biggr)+u \ S_X(u)

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Interestingly, we have the following relation.

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\displaystyle (X-d)_+ + (X \wedge d)=X \ \ \ \ \ \ \text{and} \ \ \ \ \ \ E[(X-d)_+] + E(X \wedge d)=E(X)

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The above statement indicates that purchasing a policy with a deductible d and another policy with a policy maximum d is equivalent to buying full coverage.

Another way to interpret X \wedge d is that it is the amount of loss that is eliminated by having a deductible in the insurance policy. If the insurance policy pays the loss in full, then the insurance payment is X and the expected amount the insurer is expected to pay is E(X). By having a deductible provision in the policy, the insurer is now only liable for the amount (X-d)_+ and the amount the insurer is expected to pay per loss is E[(X-d)_+]. Consequently E(X \wedge d) is the expected amount of the loss that is eliminated by the deductible provision in the policy. The following summarizes this observation.

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\displaystyle (X \wedge d)=X-(X-d)_+ \ \ \ \ \ \ \text{and} \ \ \ \ \ \ E(X \wedge d)=E(X)-E[(X-d)_+]

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Example 1
Let the loss random variable X be exponential with pdf f(x)=\alpha e^{-\alpha x}. We have E(X)=\frac{1}{\alpha}. Because of the no memory property of the exponential distribution, given that a loss exceeds the deductible, the mean payment is the same as the original mean. Thus e_X(d)=\frac{1}{\alpha}. Then the per loss average is:

\displaystyle E[(X-d)_+]=e_X(d) \ S(d) = \frac{e^{-\alpha d}}{\alpha}

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Thus, with a deductible provision in the policy, the insurer is expected to pay \displaystyle \frac{e^{-\alpha d}}{\alpha} per loss instead of \displaystyle \frac{1}{\alpha}. Thus the expected amount of loss eliminated (from the insurer’s point of view) is \displaystyle E(X \wedge d)=\frac{1-e^{-\alpha d}}{\alpha}.

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Example 2
Suppose that the loss variable has a Gamma distribution where the scale parameter is \alpha and the shape parameter is n=2. The pdf is \displaystyle g(x)=\alpha^2 \ x \ e^{-\alpha x}. The insurer’s expected payment without the deductible is E(X)=\frac{2}{\alpha}. The survival function S(x) is:

\displaystyle S(x)=e^{-\alpha x}(1+\alpha x)

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For the losses that exceed the deductible, the insurer’s expected payment is:

\displaystyle \begin{aligned}e_X(d)&=\frac{\int_d^{\infty} S(x) \ dx}{S(d)}\\&=\frac{\int_d^{\infty} e^{-\alpha x}(1+\alpha x) \ dx}{e^{-\alpha d}(1+\alpha d)} \\&=\frac{\frac{e^{-\alpha d}}{\alpha}+d e^{-\alpha d}+\frac{e^{-\alpha d}}{\alpha}}{e^{-\alpha d}(1+\alpha d)} \\&=\frac{\frac{2}{\alpha}+d}{1+\alpha d} \end{aligned}

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Then the insurer’s expected payment per loss is E[(X-d)_+]:

\displaystyle \begin{aligned}E[(X-d)_+]&=e_X(d) \ S(d) \\&=\frac{\frac{2}{\alpha}+d}{1+\alpha d} \ \ e^{-\alpha d}(1+\alpha d) \\&=e^{-\alpha d} \ \biggl(\frac{2}{\alpha}+d\biggr) \end{aligned}

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With a deductible in the policy, the following is the expected amount of loss eliminated (from the insurer’s point of view).

\displaystyle \begin{aligned}E[X \wedge d]&=E(X)-E[(X-d)_+] \\&=\frac{2}{\alpha}-e^{-\alpha d} \ \biggl(\frac{2}{\alpha}+d\biggr) \\&=\frac{2}{\alpha}\biggl(1-e^{-\alpha d}\biggr)-d e^{-\alpha d} \end{aligned}

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Example 3
Suppose the loss variable X has a Pareto distribution with the following pdf:

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\displaystyle f_X(x)=\frac{\beta \ \alpha^\beta}{(x+\alpha)^{\beta+1}} \ \ \ \ \ x>0

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If the insurance policy is to pay the full loss, then the insurer’s expected payment per loss is \displaystyle E(X)=\frac{\alpha}{\beta-1} provided that the shape parameter \beta is larger than one.

The mean excess loss function of the Pareto distribution has a linear form that is increasing (see the previous post The Pareto distribution). The following is the mean excess loss function:

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\displaystyle e_X(d)=\frac{1}{\beta-1} \ d +\frac{\alpha}{\beta-1}=\frac{1}{\beta-1} \ d +E(X)

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If the loss is modeled by such a distribution, this is an uninsurable risk! First of all, the higher the deductible, the larger the expected payment if such a large loss occurs. The expected payment for large losses is always the unmodified expected E(X) plus a component that is increasing in d.

The increasing mean excess loss function is an indication that the Pareto distribution is a heavy tailed distribution. In general, an increasing mean excess loss function is an indication of a heavy tailed distribution. On the other hand, a decreasing mean excess loss function indicates a light tailed distribution. The exponential distribution has a constant mean excess loss function and is considered a medium tailed distribution.

Reference

  1. Bowers N. L., Gerber H. U., Hickman J. C., Jones D. A., Nesbit C. J. Actuarial Mathematics, First Edition., The Society of Actuaries, Itasca, Illinois, 1986
  2. Klugman S.A., Panjer H. H., Wilmot G. E. Loss Models, From Data to Decisions, Second Edition., Wiley-Interscience, a John Wiley & Sons, Inc., New York, 2004
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