## Estimating lost future earnings using the new worklife tables Essay

Since the 1982 publication of the Bureau of Labor Statistics updatedworklife tables, articles have appeared in the Monthly Labor Review andseveral legal journals regarding the use of such tables in liabilityproceedings.

As stated in these articles expert witnesses in wrongfuldeath and injury litigation are interested primarily in using theincrement-decrement worklife tables to find the expected number of yearsan individual would have been active in the work force had an injury ordeath not occurred. This expected worklife is then used to calculatethe present value of “expected” earnings lost between the dateof death or injury and the date of expected final separation from thework force. It will be shown here that such methods do not yield amathematically defensible expectation of future earnings, because thesum of earnings over the expected worklife need not equal the sum ofexpected yearly earnings over life. A model based on theincrement-decrement worklife table is developed for calculations ofexpected earnings in each year of possible life. This model is thenmodified to obtain the discounted present value of expected futureearnings. The final section of this article presents our calculationsof expected earnings for representative individuals who die prior to age85, and compares them with those reported by David Nelson and KennethBoudreaux in past issues of the Review. Expected earnings It is a simple exercise to show that the sum of earnings overexpected worklife need not equal the sum of expected yearly earningsover life.

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For instance, assume that a cohort of 1,000 people areinitially active in the work force but, at the end of the first year,400 become inactive. Similarly, in the second and third years, 300become inactive at the end of each year. The expected worklife for thishypothetical cohort is 1.9 years. If individual earnings in eachsubsequent year are projected to be \$25,000, \$30,000, and \$35,000, then,using current techniques, an expert witness would conclude that expectedearnings are \$52,000 (= \$25,000 + 0.9 X \$30,000), ignoring discountingand other adjustments. But such a calculation overlooks the interactionof the probability of being active in each year and the earnings whichare projected for the year.

The true mathematical mean, or expectedearnings, is \$53,500 (= 0.4 X \$25,000 + 0.3 X \$55,000 + 0.3 X \$90,000). For pedagogical ease, the above example assumes that thehypothetical cohort of 1,000 remains alive for all 3 years. It does notallow for both movement into and out of the work force. Thesecomplications affect the calculation of expected income. Using all theinformation now available in the increment-decrement worklife tables,the true mathematical expected earnings of an active individual at age xcan be derived in the following manner.

Let q.sub.x represent the probably (or more precisely, the relativefrequency) of death in the year following exact age x. Let l.

sub.xrepresent the number of survivors at age x. At each age, survivors canbe divided into those who are active in the work force and those who arenot. In addition, at each age, a survivor who is active may stay activeor leave the work force, while someone who is inactive may stay inactiveor move into the work force. Let the four relevant probabilities (orrelative frequencies) for work force transition be represented asfollows: I.sub.px.

A = the probability that someone who is inactive atage x will be active at age x + 1; I.sub.px.I = the probability thatsomeone who is inactive at age x will be inactive at age x + 1;A.sub.px.

I = the probability that someone who is active at age x will beinactive at age x + 1; and, A.sub.px.A = the probability that someonewho is active at age x will be active at age x + 1 The above transitional probabilities are conditional on survivalfrom age x to age x + 1. Thus: I.sub.

px.a + I.sub.px.I = 1, andA.sub.px.

I + A.sub.px.A = 1 Assuming that the probability of death and the probabilities oftransition between work force states are independent, the number ofinactive survivors at age x + 1 (that is, sup.1.l.sub.x+1) and thenumber of active survivors at age x + 1 (sup.

A.l.sub.x+1) can now bedefined as: sup.I.l.sub.x+1 = (1 – q.

sub.x.)(.sup.I.l.sub.x.

/sup.I.P.sup.I./.sub.

x +sup.A.l.

sub.x./.sup.A.P.

sup.I./.sub.

x.); and sup.A.l.sub.x+1 = (1 -q.

sub.x.) (.sup.I.l.sub.

x./.sup.

I.P.sup.A.

/.sub.x +sup.A.l.sub.x./.

sup.A.P.sup.A./.sub.

x.) where l.sub.x = sup.

I.l.sub.x +sup.A.

l.sub.x., and l.sub.x+1 = l.

sub.x (1 – q.sub.x.) As in the published increment-decrement worklife tables, theseformulas yield: Expected worklife for persons = (1/.sup.A.

l.sub.x) * [(1- Q.sub.x+n) active at age x (sup.A.l.sub.

x+n.sup.A.p.

sup.A.sub.x+n +0.5.

sup.A.l.sub.x+n.sup.

A.p.sup.I.sub.

x+n +0.5.sup.I.l.sub.

x+n.sup.I.p.

sup.A.sub.x+n) + 0.5.Sup.A.l.

sub.x+nq.sub.x+n] where M is the number of ages remaining after x until thecohort is extinguished. The above formula for expected worklife is based on a cohort thatdies out over x + M + 1 years. At age x, for each of the remaining M +1 years, there are four terms over which yearly summation takes place.

The first three terms refer to persons who survive to the next year ofage. Among these survivors there are those who are active at the startof the year and stay active for a full year. For this group.

(1 –q.sub.x+n)(.sup.A.

l.sub.x+n.sup.A.p.

sup.A.sub.x+n) is the total numberof active years accumulated between ages x + n and x + n + 1. Personswho survive the year, but move from active to inactive or from inactiveto active status, are assumed to be active for one-half year.

Thus (1 -q.sub.x+n)(0.5.

sup.A.l.sub.x+n.

sup.A.P.

sup.I.sub.x+n) and (1 -q.sub.x+n)(0.5.sup.

I.l.sub.x+n.

sup.I.P.sup.A.sub.x+n) are the totalnumbers of active years accumulated in year x + n by individuals wholive age x + n + 1, and who make midyear work force transitions fromeither active to inactive or inactive to active status, respectively.Persons who were active at the beginning of the year and die in theinterval are also considered active for one-half year.

Thus,0.5.sup.A.l.sub.

x+n.q.sub.x+n is the total number of active yearsbetween years x + n and x + n + 1 for individuals who are assumed to dieat age x + n + 0.5. Unlike the simpler mortality tables, the increment-decrement modelposes an added complication in the formulation of expected worklifewhich has implications for calculating expected earnings: The values forsurvivors by age and work force status depend upon the age at which onebegins the computations and the distribution of persons by work forcestatus at that age. In a mortality table, any arbitrary value forl.

sub.o will yield the same expectation of life for each successive age.In the increment-decrement table, one must set either the active orinactive population to zero at the starting age. For example, theexpected working life for persons inactive at age 16 is computed bysetting sup.

I.l.sub.16 = 1,000 and sup.A.l.sub.

16 = 0. The associatedsup.I.l.sub.x and sup.

A.l.sub.x can then be computed from these twoinitial values. If one starts at age 17, or calculates the table forpersons out of the work force at age 16, all of the sup.I.l.sub.

x andsup.A.l.

sub.x values will change. Expected earnings at age x are calculated by introducing annualearnings.

Let total annual earnings in year x(y.sub.x.) be paid in twoequal biannual payments. The payments to persons changing work forcestatus during a year can be approximated by assuming that a person whobecomes inactive or dies is active for the first half of the year, andthat a person who is inactive and becomes active has earnings in thelast half of the year.

Under these conditions: Expected earnings for active person = (1/.sup.A.l.

sub.x.) * [(1 -q.sub.x+n.) at age x (.sup.

A.l.sub.x+n.sup.

A.P.sup.A.sub.x+n.y.sub.

x+n +0.5.sup.A.l.sub.x+n.sup.

A.P.sup.I.sub.x+n.y.sub.

x+n +0.5.sup.I.

l.sub.x+n.sup.I.

P.sup.A.sub.x+n.y.sub.x+n.

) +0.5.sup.

A.l.sub.x+n.q.sub.

x+n.y.sub.x+n.

] As does the formula for expected worklife, this formula forexpected earnings describes four groups who work (or, more precisely,are active) for different portions of the year between ages x + n and x+ n + 1. Years of work, however, are now evaluated in terms of totaldollars earned by each of the four groups. The above formula for calculating expected earnings involves anassumption either that time has no value, or that productivity andinflation gains are exactly offset by the market rate of interest. Whilesome expert witnesses still advocate the use of such a “totaloffset method, “.sup.

3 courts today will accept the discounting offuture earnings to reflect the net time value of money. The expectedearnings equation can be modified to accommodate discounting by definingeither a continuous compounding rate (r) or its annual discount rateequivalent (d), that is, 1 + d).sup.

-n = e.sup.nr..

(For instance, ifthe annual rate of discount is 11 percent, its continuous compoundingequivalent is 10.44 percent.) The present value of expected earnings foran active person at age x, in continuous discounting form is:(1/.sup.A.

l.sub.x.

) * [left brace](1 – q.sub.x+n.)[0.5.

sup.A.l.

sub.x+n.sup.

A.P.sup.A sub.x+n.y.sub.x+n.

(e.sup.(n+.5)r +e.sup.(n+1)r.) + 0.

5.sup.A.l.sub.x+n.sup.A.

p.sup.I.sub.x+n.y.

sub.x+n.e.sup.(n+.5)r + 0.

5.sup.I.l.sub.x+n.sup.

I.P.sup.A.

sub.x+n.y.sub.x+n.e.sup.(n+1)r] + 0.

5.sup.A.l.sub.x+n.q.sub.

x+n.y.sub.x+n.

e.sup.(n+1)r.[rightbrace] Corresponding expressions for the expected worklife, expectedearnings, and present value of expected earnings for persons inactive atage x can be derived in a similar way. Calculation procedures In her comment on Boudreaux’s and Nelson’s methods foradjusting the worklife tables to estimate lost earnings, Shirley Smith notes that “frequently, economists want to look past thelifetime-worklife expectancy figure to study the timing of the potentialearnings stream.” Here we argue that in the calculation ofexpected lost earnings it is not sufficient to know the “mediannumber of years until final separation,” as defined by Nelson; toadjust this figure by assuming that activity is evenly spread over theentire period until retirement, as suggested by Boudreaux; or to knowany other single number that represents the possible length of time thata person will be active.

A true mathematical expectation of lostearnings requires knowledge of the timing of probable activity and ofthe potential (nominal or discounted) earnings during the period ofprobable activity. Because the timing of probable activity is sensitive to both theinitial work force status and the age of an individual, our developmentof the true mathematical expected earnings, unlike the approaches ofNelson and Boudreaux, emphasizes an active or inactive starting point.To assess the consequences of this distinction, consider the exampleprovided by Boudreaux. A man age 30 with annual earnings of \$25,000(using a current market interest rate of 11 percent and an annualearnings increase of 4.5 percent) has a present value of”expected” earnings of \$332,913, by the worklife tableestimate of 29.2 years of remaining worklife for the entire population.Using Nelson’s 31.5 years to final separation criterion, thepresent value of “expected” earnings is \$341,857.Boudreaux’s 7.3-percentage reduction criterion drops this estimateto \$316,901. However, our calculations show that the true mathematicalpresent value of expected earnings for an active man at age 30 is\$319,397, and for an inactive man at age 30 it is \$273,535. In some cases, one might wish to ignore initial work force status.A weighted average of our active and inactive estimates can be obtainedby using the proportions of men active and inactive at the initial age.In the above example of a man at age 30, this average present value ofexpected earnings is \$316,502, which compares favorably toBoudreaux’s estimate of \$316,901. Given the ease of usingBoudreaux’s adjustment method, one might question the practicalvalue of using our more complicated true mathematical expectationmethod. Unfortunately, Boudreau’s approximation is close to the truemathematical expectation only for younger men. His assumption thatinactivity is spread evenly over the entire period until retirement isinappropriate at older ages, when the proportion that are inactive risesrapidly. For a younger person, changes in expected earnings caused byincreasing probabilities of inactivity later in life are mitigated byhigh discounting of expected earnings in distant years. For olderpeople, the mitigating effect of discounting is not present. Thus, fora man age 45 with the same earnings stream used above, Nelson’s andBoudreaux’s methods of estimating the present value of potentialearnings yield \$256,044 and \$242,217, respectively. Our mathematicalexpectations are \$236,626 for an active man, \$155,310 for an inactiveman, and \$231,325 for the weighted average of active and inactivepersons. OUR METHOD OF CALCULATION requires two modifications of theincrement-decrement worklife tables published by BLS. First, theprobabilities of transition into and out of the work force at each agemust be converted to probabilities that are conditional on survival.Second, conditional probabilities of transition between active andinactive work force status must be added at age 76 to close the table.The relevant probabilities of transition are provided in table 1. Acomputer program for calculating the present value of expected earningsbased on these transitional probabilities is available from the authors.

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