# Example 8.28: should we buy snowstorm insurance?

February 28, 2011
By

(This article was first published on SAS and R, and kindly contributed to R-bloggers)

It’s been a long winter so far in New England, with many a snow storm. In this entry, we consider a simulation to complement the analytic solution for a probability problem concerning snow.

Consider a company that buys a policy to insure its revenue in the event of major snowstorms that shut down business. The policy pays nothing for the first such snowstorm of the year and \$10,000 for each one thereafter, until the end of the year. The number of major snowstorms per year that shut down business is assumed to have a Poisson distribution with mean 1.5. What is the expected amount paid to the company under this policy during a one-year period?

Let SNOW be the number of snowstorms, and pay the amount paid out by the insurance. The following chart may be useful in discerning the patttern:

`SNOW    PAY     10000*(snow-1)0       0      -100001       0       02       10000   100003       20000   20000`

The analytic solution is straightforward, but involves a truncation of the first snowstorm. Since we can assume that the random variable SNOW ~ Poisson(1.5) we know that E[SNOW] = 1.5 and E[10000*(SNOW-1)] = 10000*E[snow] – 10000 = 15000 – 10000 = 5000.

E[PAY] is equal to E[10000*(SNOW-1]) + 10000*P(SNOW=0) so the exact answer is

`10000*P(snow=0) + 15000 - 10000 =10000*exp(-1.5) + 15000 - 10000 = \$7231`

Here the advantage of simulation is that it may provide a useful check on the results, as well as a ready measure of variability. In this situation, the code is quite simple, but the approach is powerful.

R

`numsim = 1000000snow = rpois(numsim, 1.5)pay = snow - 1      # subtract onepay[snow==0] = 0    # deal with the pesky P(snow=0)sim = mean(pay*10000)analytic = 10000*(dpois(0, 3/2) + 3/2 - 1)`

Yielding the following:

`> sim[1] 7249.55> analytic[1] 7231.302`

SAS
The simulation and analytic solutions are also straightforward in SAS. Here the analytic result is only calculated once

`data snow_insurance;do i = 1 to 1000000;  nsnow = ranpoi(0, 1.5);  payout = max(nsnow -1, 0) * 10000;  output;end;analytic = 10000 * (cdf("POISSON", 0, 1.5) + 1.5 -1);output;run;proc means data=snow_insurance mean;  var payout analytic;run;`

This results in the following output:

`Variable            Mean------------------------payout           7236.96analytic         7231.30------------------------`

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