Pricing Insurance Risk. Stephen J. Mildenhall

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      since xS(x)→0 as x→∞ when X has a mean. Note that dS=−dF, accounting for the sign change.

Remark 16 The expression ∫0∞S(x)dx for the mean is well known to life actuaries via the expression ex=∑tpt x for curtate future lifetime in terms of survival functions. The average future lifetime is the sum of the probability of surviving to each future age. Count the birthdays!

      Exercise 17 Figure 3.3 does not show the implicit representation. Plot it. What are the horizontal and vertical axes?

      Exercise 18 Let X be a Bernoulli random variable defined on Ω=[0,1] by X(ω)=0 for ω < 0.4 and X(ω)=1 for ω≥0.4. What are P(X=0), P(X=1), and E[X]? Plot X and its distribution and survival functions, and its Lee function. Clearly label all axes and the value of each function at any jump points. Repeat the exercise for Y defined by Y(ω)=0 if ω∈[0,0.1)∪[0.25,0.35)∪[0.5,0.6)∪[0.75,0.85) and Y(ω)=1 otherwise.

Solution. X and Y define different random variables but they have the same distribution and survival functions, and the same implicit and dual implicit representations. They are both Bernoulli(0.6) variables, with P(X=0)=0.4, P(X=1)=0.6, and mean E[X]=0.6. Figure 3.4 shows the requested plots. Strictly, the vertical segments are not part of the function graphs. The dot indicates the value of each function at jumps.

      Figure 3.4 The random variables, distribution and survival functions, and Lee diagram for two identically distributed Bernoulli random variables.

      Exercise 19 A model produces 100 equally likely events that it labels by an event identifier. The events define a sample space Ω={0,…,99} and probability Pr({ω})=1/100. The model defines two identically distributed, dependent random variable outcomes

      with sum

      1 Create the model in a spreadsheet and confirm E[X]=28 and E[Xi]=14.

      2 Plot X1, X2, and X as functions of ω=0,1,…,99.

      3 Plot the survival functions, as functions of the outcome x.

      4 Plot the Lee diagrams, as functions of probability p.

      5 Are the random variables different? The survival functions? The Lee diagrams?

      We return to this example in Chapter 15.

Solutions. Figures 3.5–3.7 show the random variables, the survival functions, and the Lee diagrams. The random variables are all distinct, but the survival function and Lee diagrams for each line are the same.

      Figure 3.5 Random variables, functions of an explicit state.

      Figure 3.6 Survival functions of the outcome.

      Figure 3.7 Lee diagrams, function of a dual implicit state.

Remark 20 The relationships illustrated in Figure 3.3 are a discrete version of the formula for integration by parts. Consider approximating Riemann sums to the integral of the survival function. Let 0=x0<x1<⋯<xn<⋯, be a fine, but not necessarily equally spaced, dissection of the positive reals. We get two equivalent representations of E[X] by

(3.2)

(3.3)

       (3.4)

       (3.5)

      using Taylor’s theorem to write S(xi−1)−S(xi)=S(xi−(xi−xi−1))−S(xi)=−S′(xi′)(xi−xi−1)=f(xi′)(xi−xi−1), for some xi−1≤xi′≤xi.

      Exercise 21 Confirm the change in indexing between Eq. 3.2 and Eq. 3.3 is correct by looking at panels (d) and (e).

      Technical Remark 22. In addition to the outcome-probability and survival function forms, there is a third, dual implicit outcome expression

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