Understanding the Mathematics of Personal Finance

EXPECTED VALUES

Assume that on January 1, 1900, four men whose lives we’d like to follow were born. They of course didn’t know it at the time, but their lives would end on:

First man: In 1922, when he was 22 years old.

Second man: In 1968, when he was 68 years old.

Third man: In 1974, when he was 74 years old.

Fourth man: In 1988, when he was 88 years old.

The average life span, or equivalently, the expected life span, of these four men is the average of the number of years that each man lived:

22 + 68 + 74 + 88 ^

---------------------- = 63.0.

4

Suppose we started studying these men in 1950. In 1950, there are only three of these men left alive. If we kept track of them until they were all dead, we would find that their expected life span is

Подпись: 76.7.68 + 74 + 88
3

Many people interpret results such as these as telling them that because they have lived some number of years (in this case, 50), their life expectancy is higher than it was when they were born. This interpretation is nonsense. Their life expec­tancy hasn’t changed. What has changed is that the people who, for whatever reason, were destined to die earlier have already died and are no longer being counted in the averaging process. You can see this clearly in the two average calculations above. The second calculation is the average of the life spans of the three men who hadn’t died before they were 22 years old.

Column d of the Life Table is a list of life spans just like the one used in the examples above. To find the expected life span of all men from the time they’re born, we just add up all 100,000 life spans in the numerator, put the number 100,000 in the denominator, and perform the calculation.

This procedure will work, but it certainly is a mess to set up, even on a computer spreadsheet. Fortunately, there are more efficient ways to perform this calculation. If you’re interested, look up the topic “weighted averages” on the Web or in a text. The important point to remember is that when all is said and done, the calculation is exactly the same as the two calculations I did above.

If I do perform this calculation, I find that the expected life span for all men, at birth, is 74.7 years.

What is the expected life span starting at, say, 10 years? To calculate this, do the same thing I did in the simple example. Repeat the first calculation but eliminate everybody who died before their tenth birthday. The answer comes out to be 75.4 years.

Column e of the Life Table shows this same calculation but is expressed a bit differently. Using the line x = 10-11 years (10 years old), first subtract 10 from 75.4 so that we’re talking about the number of years left to live rather than the life span. Then, add half a year so as to get a midyear value--this is more representative of people between 10 and 11 years old than a calculation taken just at the tenth birthday. Putting all of this together,

e = 75.4 -10 + 0.5 = 65.9.

Going the other way, if you want to know the average life span of a 50-year-old man, start with the value of e in the table. For x = 50-51, e = 28.8. Subtract 0.5 to get the reference back to the beginning of the year: 28.8 - 0.5 = 28.3. Then, add 50, resulting in 28.3 + 50 = 78.3 as the expected life span for 50-year-old men.

Figure 10.4 shows the expected life span versus the current age. There are two sets of data shown: the expected life spans for men and for women. Women, on the average, live longer than men. This will be an important consideration when we consider reverse mortgages in Chapter 12 because the insurance companies will have to worry about who is the second spouse to die. For now, it’s just an interesting fact.

image084

Present age

Figure 10.4 Expected age at death for men and women versus present age.

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