6.5 Appendix: More Exposures, Less Risk

Assume that the riskiness of two groups is under consideration by an insurer. One group is comprised of 1,000 units and the other is comprised of 4,000 units. Each group anticipates incurring 10 percent losses within a specified period, such as a year. Therefore, the first group expects to have one hundred losses, and the second group expects 400 losses. This example demonstrates a binomial distribution, one where only two possible outcomes exist: loss or no loss. The average of a binomial equals the sample size times the probability of success. Here, we will define success as a loss claim and use the following symbols:

• n = sample size
• p = probability of “success”
• q = probability of “failure” = 1 – p
• n × p = mean

For Group 1 in our sample, the mean is one hundred:

• (1,000) × (.10) = 100

For Group 2, the mean is 400:

• (4,000) × (.10) = 400

The standard deviation of a distribution is a measure of risk or dispersion. For a binomial distribution, the standard deviation is

$$\sqrt{n\times p\times \mathrm{q.}}$$

In our example, the standard deviations of Group 1 and Group 2 are 9.5 and 19, respectively.

$$\sqrt{(\mathrm{1,000})\times (.1)\times (.9)}=9.5$$ $$\sqrt{(4000)\times (.1)\times (.9)}=19$$

Thus, while the mean, or expected number of losses, quadrupled with the quadrupling of the sample size, the standard deviation only doubled. Through this illustration, you can see that the proportional deviation of actual from expected outcomes decreases with increased sample size. The relative dispersion has been reduced. The coefficient of variation (the standard deviation divided by the mean) is often used as a relative measure of risk. In the example above, Group 1 has a coefficient of variation of 9.5/100, or 0.095. Group 2 has a coefficient of variation of 19/400 = 0.0475, indicating the reduced risk.

Taking the extreme, consider an individual (n = 1) who attempts to retain the risk of loss. The person either will or will not incur a loss, and even though the probability of loss is only 10 percent, how does that person know whether he or she will be the unlucky one out of ten? Using the binomial distribution, that individual’s standard deviation (risk) is a much higher measure of risk than that of the insurer. The individual’s coefficient of variation is .3/.1 = 3, demonstrating this higher risk. More specifically, the risk is 63 times (3/.0475) that of the insurer, with 4,000 units exposed to loss.