What do actuaries do?
Actuaries use mathematical and business skills to define, analyze, and solve business problems involving the cost of possible future events. The following elementary article---a disguised introduction to actuarial science---illustrates this by using business ideas (how money grows with interest) and mathematical tools (intuitive probability and statistics) to examine how much money a person needs for retirement. The article appeared in print in The College Mathematics Journal, volume 29, number 4, 1998, pp. 278–283; copyright The Mathematical Association of America 1998, all rights reserved.
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Author profile:
Jim Daniel (jimdaniel@mail.utexas.edu) nine years ago naively agreed to head the actuarial program at The University of Texas at Austin, despite total ignorance of actuarial mathematics. Now an Associate of the Society of Actuaries, Jim has enjoyed his new career even more than his research in numerical analysis---the field in which he received a Ph.D. from Stanford University (1965). Hobbies include reading mystery novels about crazed killers and admiring his wife Ann and their two cats.
How much money do you (or your parents) need for retirement? $100,000 per year? A lump sum of a million dollars? As much as possible? These quick answers miss the point, partly because the question is too broadly phrased. Let's refine our question.
Different individuals will have different goals for yearly income after retirement. Let's suppose, for simplicity, that an individual wishes to receive an income of $I after taxes at the start of each year for life, starting at the moment of retirement. How much would such a person require in investments at retirement in order to provide that stream of payments?
It's clear that to receive yearly payments of $I, the retiree would need I times as much as needed for a stream of $1 yearly payments, so we can restrict our discussion to $1 payments. The amount required in investments of course depends on the yearly rate of return on the investments after taxes, and also on whether the retiree wants the $1 payments to increase over time to account for inflation. If r is the after-tax yearly rate of return, then a $1 investment at the start of a year would grow to $(1 + r) at the end of the year. But if g is the yearly inflation rate the retiree anticipates, each end-of-the-year dollar would buy only 1/(1 + g) times as much as at the start, so the real growth factor would be
(1 + r) / (1 + g) = 1 + i, where i = (r - g) / (1 + g).
Let's call i the real yearly rate of return (after expected taxes and inflation), and assume that i is nonnegative.
The original question can now be phrased more precisely: "Assuming a real yearly rate of return i, how much does a retiree need to have invested in order to provide $1 at the start of each year for life, starting at the moment of retirement?" Our analysis of this retirement problem will illustrate the kinds of modeling that actuaries perform. Actuaries are business people who use mathematical and statistical techniques from actuarial science to analyze how to provide financially now for future costs of various risks. As we shall see, the answer to our problem is: "It depends."
The answer to our question surely depends on how long the retiree lives. Let K denote the whole number of years our retiree lives after retirement. If death occurs in the first year, then K = 0; but the retiree still will have received one payment---the initial one on the day of retirement. More generally, the retiree will always receive K + 1 payments whatever the value of K, since the payments are made on the retirement anniversaries.
But K is unknown, of course. An extremely conservative approach would be to have enough invested to provide $1 per year regardless of the size of K---that is, forever. This requires a fund of $(1 + i)/ i. At the start of the first year the $1 payment would reduce the fund to $(1 + i)/ i - 1 = $1/ i, which would grow at interest for the year to exactly the original $(1 + i)(1/ i) at the start of the second year, allowing payments forever. If i = 4%, for example, this approach would require a $26 initial fund; making the $1 payment would leave $25, which at 4% interest would grow to $26 to start the whole process again the next year.
However, most people know that they won't live forever and instead figure that they only need enough invested for some specific number of payments (K + 1, in our notation). Unfortunately, K + 1 is unknown. When faced with such uncertainty, we can examine data to help us understand the various values of K that typically occur in real life. For this article, we will use data collected and analyzed by the Society of Actuaries [5]. Numerical examples will be based on typical 65-year-old female retirees, and i = 4% will be used for the rate of return.
Returning to our investment problem, and considering the data on the behavior of K, we might ask: "How much is needed on average to provide the payments?" Beware! This is not the same as asking, "How large a fund would be needed for an average person who survives an average time into the future?" Let's pause to see why not.
The average-future-lifetime mistake. Our data [5] can be used to show that if you observe the number of whole future years lived by each of 50,000 typical 65-year-old female retirees, and then average these future lifetimes over the 50,000 retirees, your average is highly likely to fall between 20.83 and 20.99 future years. Let's just use 20.9 years as the value for the average female. (The corresponding average for males, by the way, is only about 17.9 years.)
To indicate why it is incorrect on average merely to have enough money invested for the average person, consider first a simpler retirement plan that provides no regular yearly payments, just a single $1,000,000 payment to any retiree who survives to age 87. Since K averages 20.9, an average retiree would die between ages 85 and 86, so would not live to qualify for the payment at age 87. Thus $0 is needed for an average retiree with the average future lifetime!
Clearly, investing just enough (namely, zero) to pay the benefits for a retiree who survives the average number of years cannot be the right approach. Why not? If you start with 50,000 retirees, surely some of them will outlive the average and collect their $1,000,000$ at age 87, which a $0 initial fund could not provide. Had a fund started with $P for such a person, it would have grown with interest to $P(1 + i)^{22} by that time. Thus P would have to satisfy P(1 + i)^{22} = 1,000,000 in order to make the payment. Hence P = 1,000,000 v^{22}, where v = 1/(1 + i). That value $P is called the present value of the 22-years-later $1,000,000. If i = 4%, then P = 421,955.
The actuarial data show that about 25,866 of the 50,000 original retirees are expected to survive to age 87. Thus, on average, we need to invest
(25,866) ($1,000,000) v^{22} / 50,000
for each original retiree in order to provide the $1,000,000 to survivors aged 87. At i = 4%, this is about $218,285 per original retiree. This $218,285 average amount needed is quite different from the $0 needed for the average retiree!
Average amounts needed. The technique used above to analyze the single lump-sum payment pension can be applied to our original pension of $1 yearly for life. If we start with L_{65} = 50,000 65-year-old retirees, then we can use the actuarial data to estimate the number L_{65 + k} of those alive k years later to receive a $1 payment, for k = 0, 1,2, ... .
The amount needed in an investment at age 65 that would grow to enough to pay $1 to each of the L_{65 + k} survivors k years later is $L_{65 + k} v^k, the present value of the money needed at age 65 + k. The initial amount needed to fund all the payments for the lifetime of all the original retirees is then
L_{65 + 0} v^0 + L_{65 + 1} v^1 + L_{65 + 2} v^2 + ... .
We divide this amount by the number L_{65} of original 65-year-olds to get the average number of dollars needed per original retiree. This is called theactuarial present value (APV) of the payments for the lifetime of one 65-year-old:
APV = (L_{65 + 0}/L_{65}) v^0 + (L_{65 + 1}/L_{65}) v^1 + (L_{65 + 2}/L_{65}) v^2 + ... .
For example, for the L_{65} = 50,000 female retirees discussed previously, the first few values of L_{65 + k} are about L_{66} = 49,543, L_{67} = 49,360, and L_{68} = 48,483. Had at most four payments been promised to survivors, rather than lifelong payments, the average needed would be $(L_{65} + L_{66} v + L_{67} v^2 + L_{68} v^3)/L_{65} = $3.72 if i = 4%. With lifelong payments and i = 4%, the average investment needed turns out to be $14.25. Compare this with the $26 that we earlier saw is needed to guarantee $1 payments forever rather than for life.
Does this average investment, $14.25, settle our problem? Not entirely, because our analysis has been made from the viewpoint of an insurance company that guarantees payment for life to a group of retirees. An individual retiree who had invested only $14.25 at retirement would exhaust her account if she lived much beyond the average age for her cohort. Let's look at the problem from the individual retiree's point of view. First, we should review some probability ideas.
Lurking in the background of the preceding analysis are both probability and statistics, two fundamental tools for actuaries. Suppose that the future lifetime X of each of a large number L_{0} of newborns is assumed to have the same probability distribution for each newborn. This doesn't mean that each newborn's future lifetime is the same; it just means that they all have the same chance behavior: the probability that a newborn dies in some particular age range is the same for all of the newborns. Mathematicians usually describe the random behavior in such situations by the cumulative distribution function
F(x) = Pr[ X <= x],
the probability that X is less than or equal to x---that is, that the newborn dies by age x. Actuaries look on the bright side and describe the random behavior by the survival function
s(x) = 1 - F(x) = Pr[ X > x],
the probability that the newborn survives beyond age x. The expected number L_{x} of survivors to age x from among the L_{0} newborns, then, would be the fraction s(x) of the original L_{0} newborns:
L_{x} = s(x) L_{0}.
The values L_{x} we used intuitively earlier in this article therefore describe the distribution as well as F(x) does, since F(x) can be computed from L_{x} by
F(x) = 1 - s(x) = 1 - L_{x} / L_{0}.
Actuaries regularly collect statistics on large numbers of human lives in various categories in order to build models of survival functions s(x) that seem appropriate for those categories. Our actuarial data were prepared in this way and assembled as a table of L_x values.
Suppose now that you want to find the probability that a 65-year-old survives at least another k years. (I'll denote this by p_k for simplicity.) If nothing more is known about the 65-year-old other than that this person is a former newborn who has survived 65 years (so there is no recent health data on the person, for instance), then this probability is just the probability that a newborn survives 65+k years given that he or she already survived 65 years. That is,
p_k = s(65 + k) / s(65) = [L_{65 + k} / L_{0}] / [L_{65} / L_{0}] = L_{65 + k} / L_{65}.
Intuitively, this just says that if you divide the number of people who make it to age 65 + k by the number who make it to age 65, you get the fraction of 65-year-olds who survive k years---the intuitive meaning of the probability p_k.
Note that this quotient L_{65 + k} / L_{65}, which equals p_k, appeared in the equation for APV, the actuarial present value of the $1 payments for the lifetime of a 65-year-old. That formula for APV is therefore a sum of terms of the form p_k v^k.. The factor v^k gives the present value at age 65 of $1 at age 65 + k. Why the factor p_k? That is the probability that the payment will in fact be made, so it produces the expected value of the present value of that payment. And the APV is the sum of such terms, one for each payment that might be made. Thus, the APV is in fact the expected value of thepresent value of payments made so long as the retiree survives.
Of course, the true present value of payments made for life may well be quite different from theexpected present value of those payments---much smaller if the retiree dies soon, and much larger if the retiree lives a long time. The true present value of the K + 1 payments actually made is just
1 + v + v^2 +... + v^K = (1 - v^{K + 1}) / (1 - v).
With our example data and our usual i = 4%, this will exceed the $14.25 APV if K + 1 is at least 21; that is, if the retiree survives at least 20 whole years. The probability p_{20} = L_{85} / L_{65} of surviving 20 years turns out to be about 0.6. This means that about 60% of the retirees who start out with the APV---the average amount needed for a lifetime of payments---will run out of money before running out of life. Greater confidence in having enough money requires a greater initial fund. For example, for 99% confidence (that is, for only 1% of retirees to run out of money), the initial fund needed for an individual can be shown to be about $25.58---almost the $26 needed to guarantee payments forever. One way retirees try to protect themselves at lower cost against running out of money is to pool their risks.
Risk pooling. Although the fund needed to provide lifelong payments to an individual can vary a great deal depending on the individual's future lifetime, in large groups these variations tend to average out. Retirees who live a long time and require a large initial fund are offset by those living a short time. Large corporate pension plans and insurance companies provide the opportunity for individuals to pool their risk and thereby benefit from the more regular behavior of large groups.
For a large group of N 65-year-old retirees, whether the total initial fund is adequate to provide lifelong payments to all the retirees is governed by the sum over all the retirees of the present value of the payments to each retiree. Mathematicians have proved that the sum of a large number of independent random variables usually is well approximated by a well-understood variable, a normal random variable described by the famous "bell-shaped curve." The larger N becomes, the thinner and taller the bell becomes, indicating that values are heavily concentrated near the average.
Suppose, for example, that each of N retiring 65-year-old females will deposit the amount $P_N into a fund earning i = 4%. How large need P_N be in order that we can be 99% confident that the total fund will be able to provide lifelong payments to all N retirees? Some retirees might use more than their $P_N deposit, and some might use less. But that's OK so long as the total fund for the entire group would cover the payments to the entire group---or at least that it would do so 99% of the time such groups of retirees were observed. It is possible to show that for large N$ the needed amount per person is
P_N = 14.25 + [10.34 / N^{0.5}],
which decreases to the $14.25 APV---the average amount needed---as N increases. For a group of N = 100 retirees, for example, P_{100} = 15.29, while for N = 10,000 it is P_{10,000} = 14.35. These numbers compare rather favorably with the 25.58 needed for a single individual not in such a group to be 99% confident.
Generalizations. These ideas of actuarial science---how investments grow, effects of inflation, present value, probability, statistics, and so on---can be used to analyze a wide range of similar problems. For example: how much companies should contribute regularly to special funds to meet their future pension and health-care obligations to retirees; how high premiums need be for life or health or auto or homeowners insurance; how the costs of leasing equipment compare to those of buying; how a seat-belt law might influence future injury claims; or how a particular new disease or treatment will impact health-care costs. Actuarial science attracts talented students who enjoy its distinctive mix of probabilistic modeling and financial analysis. For further information about the field I recommend[1, 2, 3, 4, 6 ].
You may still be wondering how much you (or your parents) need for retirement. My answer is, "It depends"---but I hope you now know more about what it depends on, and how.