# Death and Retirement: Allowing for Uncertainty

This article first appeared in the autumn 2005 issue of the Expert Witness.

Assume that a plaintiff has begun to recuperate following a serious accident. If her injuries stabilize at their current level, she will suffer a loss of earnings of $20,000 per year. If, however, she has a relapse, her losses will increase to $40,000 per year. Her doctors tell you that there is a 50 percent chance that there will be a relapse (and a 50 percent chance that there will be no relapse).

How should the annual value of her loss be calculated? If damages are set equal to $20,000 per year, there is a 50 percent chance that she will be under compensated; whereas if she is paid $40,000, there is a 50 percent chance that she will be over compensated. (And if she is paid some amount between $20,000 and $40,000, there is a 100 percent chance that she will be incorrectly compensated.)

This conundrum, in which there is uncertainty about the outcome of future events, is common in the assessment of personal injury (and fatal accident) damages. Not only is there uncertainty about the future course of the plaintiff’s injuries, as in the example above, we also face uncertainty concerning the age at which the plaintiff will retire, the plaintiff’s life expectancy, the probability that the plaintiff would have (and will be) unemployed, and a host of other factors.

The general approach that virtually all financial experts take in such cases is to calculate the average outcome that would arise from the uncertain event, if the event could be repeated a large number of times. For example, if the injury described above was to be repeated 100 times (for example, if there were 100 plaintiffs with that same injury), we would expect that the plaintiff’s injuries would remain stable in approximately 50 cases, leading to a loss of $20,000 per case. In the other 50 cases, the plaintiff would suffer a relapse and her loss would rise to $40,000. Thus, the total annual loss, across all 100 cases, would be ((50 × $20,000) + (50 × $40,000) =) $3,000,000. The average annual loss would be $30,000; which could also be calculated by multiplying 50% times $20,000 and adding 50 % times $40,000. That is, the average value of a loss can be calculated by multiplying each of the possible losses by its probability and then adding the resulting numbers together.

But, as was noted above, $30,000 is guaranteed to be the "wrong" amount in 100 percent of cases. How, then, can it be justified? One simple answer is this: if the event in question is truly uncertain, the plaintiff should be able to use the $30,000 to purchase insurance that will compensate her fully regardless of which value turns out to be her true loss – either $20,000 or $40,000. The reason for this is that if the insurer issues, say, 100 such policies, it can expect to pay out $20,000 in 50 cases and $40,000 in the other 50, for an average of $30,000. (It will have collected $3,000,000 [= 100 × $30,000] and will have paid out $3,000,000 [= 50 × $20,000 + 50 × $40,000].)

## Risk of Mortality

This type of calculation is most commonly used when dealing with the uncertainties associated with mortality. Take the extreme case in which there is a ⅓ probability that a plaintiff will live exactly one year (and then die), a ⅓ probability that he will live exactly two years, and a ⅓ probability that he will live exactly three years. If he would have earned $60,000 per year but has now been left unable to work, his loss can be calculated using the technique described above. That is, there is a ⅓ chance that he has lost one year’s income ($60,000), a ⅓ chance he has lost two years’ income ($120,000), and a ⅓ chance he has lost three years’ income ($180,000); for an average of $120,000 (=⅓ × $60,000 + ⅓ × $120,000 + ⅓ × $180,000).

Alternatively, in such cases, it is sometimes possible to
use a "rule of thumb" to estimate the loss. Given the
probabilities in the preceding example, it can be shown that,
on average, the plaintiff will live two more years before
dying. (2 = ⅓ × 1 + ⅓ × 2 + ⅓
× 3) That is, his *life expectancy* is two years.
His expected loss can then be calculated as the sum of his
losses over that life expectancy, or $120,000 (= 2 ×
$60,000). Note, however, that this approximation works best if
the losses are approximately the same in each year, as it was
here. (If the annual loss is significantly different in the
first year than, say, the third year, this approach yields a
biased estimate.)

What is clear is that it would be inappropriate to mix
together the two calculation techniques. It is not appropriate,
for example, to estimate the loss by multiplying each of the
first two years’ losses by their associated probabilities
*and* assuming that the loss continues for only two years.
That would produce an "estimated" loss of only $60,000
(=⅓ × $60,000 + ⅓ × $120,000), $60,000
less than the true loss.

## Retirement Age

The techniques described here can also be used to estimate
the effect of uncertainties about the plaintiff’s
retirement age. Assume, for example, that there was a ⅓
probability that a 63 year-old plaintiff would have worked for
exactly one year (i.e. to his 64^{th} birthday) and
then retired, a ⅓ probability he would have worked two
years, to his 65^{th} birthday, and a ⅓
probability he would have worked three years, to his
66^{th} birthday. If he would have earned $60,000 per
year while working, his loss, again, can be found from the
formula: ⅓ × $60,000 + ⅓ × $120,000 +
⅓ × $180,000 = $120,000; *or* by multiplying
the average number of years to retirement by his annual
earnings, to produce $120,000 = 2 × $60,000.

As with the mortality example, it would clearly be incorrect
to multiply each year’s earnings by the probability it
would occur *and* assume the individual would have retired
at the average age, of 65. That would produce an "estimate" of,
(⅓ × $60,000 + ⅓ × $120,000 =) $60,000,
again, only half of the correct estimate.