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

Since the 1982 publication of the Bureau of Labor Statistics updated
worklife tables, articles have appeared in the Monthly Labor Review and
several legal journals regarding the use of such tables in liability
proceedings. As stated in these articles expert witnesses in wrongful
death and injury litigation are interested primarily in using the
increment-decrement worklife tables to find the expected number of years
an individual would have been active in the work force had an injury or
death not occurred. This expected worklife is then used to calculate
the present value of “expected” earnings lost between the date
of death or injury and the date of expected final separation from the
work force.

It will be shown here that such methods do not yield a
mathematically defensible expectation of future earnings, because the
sum of earnings over the expected worklife need not equal the sum of
expected yearly earnings over life. A model based on the
increment-decrement worklife table is developed for calculations of
expected earnings in each year of possible life. This model is then
modified to obtain the discounted present value of expected future
of expected earnings for representative individuals who die prior to age
85, and compares them with those reported by David Nelson and Kenneth
Boudreaux in past issues of the Review.

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Expected earnings

It is a simple exercise to show that the sum of earnings over
expected worklife need not equal the sum of expected yearly earnings
over life. For instance, assume that a cohort of 1,000 people are
initially active in the work force but, at the end of the first year,
400 become inactive. Similarly, in the second and third years, 300
become inactive at the end of each year. The expected worklife for this
hypothetical cohort is 1.9 years. If individual earnings in each
subsequent year are projected to be \$25,000, \$30,000, and \$35,000, then,
using current techniques, an expert witness would conclude that expected
earnings are \$52,000 (= \$25,000 + 0.9 X \$30,000), ignoring discounting
and other adjustments. But such a calculation overlooks the interaction
of the probability of being active in each year and the earnings which
are projected for the year. The true mathematical mean, or expected
earnings, 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 the
hypothetical cohort of 1,000 remains alive for all 3 years. It does not
allow for both movement into and out of the work force. These
complications affect the calculation of expected income. Using all the
information now available in the increment-decrement worklife tables,
the true mathematical expected earnings of an active individual at age x
can be derived in the following manner.

Let q.sub.x represent the probably (or more precisely, the relative
frequency) of death in the year following exact age x. Let l.sub.x
represent the number of survivors at age x. At each age, survivors can
be divided into those who are active in the work force and those who are
not. In addition, at each age, a survivor who is active may stay active
or leave the work force, while someone who is inactive may stay inactive
or move into the work force. Let the four relevant probabilities (or
relative frequencies) for work force transition be represented as
follows: I.sub.px.A = the probability that someone who is inactive at
age x will be active at age x + 1; I.sub.px.I = the probability that
someone 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 be
inactive at age x + 1; and, A.sub.px.A = the probability that someone
who is active at age x will be active at age x + 1

The above transitional probabilities are conditional on survival
from age x to age x + 1. Thus: I.sub.px.a + I.sub.px.I = 1, and
A.sub.px.I + A.sub.px.A = 1

Assuming that the probability of death and the probabilities of
transition between work force states are independent, the number of
inactive survivors at age x + 1 (that is, sup.1.l.sub.x+1) and the
number of active survivors at age x + 1 (sup.A.l.sub.x+1) can now be
defined 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, these
formulas 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+n
q.sub.x+n] where M is the number of ages remaining after x until the
cohort is extinguished.

The above formula for expected worklife is based on a cohort that
dies 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 of
age. Among these survivors there are those who are active at the start
of 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 number
of active years accumulated between ages x + n and x + n + 1. Persons
who survive the year, but move from active to inactive or from inactive
to 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 total
numbers of active years accumulated in year x + n by individuals who
live age x + n + 1, and who make midyear work force transitions from
either active to inactive or inactive to active status, respectively.
Persons who were active at the beginning of the year and die in the
interval 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 years
between years x + n and x + n + 1 for individuals who are assumed to die
at age x + n + 0.5.

Unlike the simpler mortality tables, the increment-decrement model
poses an added complication in the formulation of expected worklife
which has implications for calculating expected earnings: The values for
survivors by age and work force status depend upon the age at which one
begins the computations and the distribution of persons by work force
status at that age. In a mortality table, any arbitrary value for
l.sub.o will yield the same expectation of life for each successive age.
In the increment-decrement table, one must set either the active or
inactive population to zero at the starting age. For example, the
expected working life for persons inactive at age 16 is computed by
setting sup.I.l.sub.16 = 1,000 and sup.A.l.sub.16 = 0. The associated
sup.I.l.sub.x and sup.A.l.sub.x can then be computed from these two
initial values. If one starts at age 17, or calculates the table for
persons out of the work force at age 16, all of the sup.I.l.sub.x and
sup.A.l.sub.x values will change.

Expected earnings at age x are calculated by introducing annual
earnings. Let total annual earnings in year x(y.sub.x.) be paid in two
equal biannual payments. The payments to persons changing work force
status during a year can be approximated by assuming that a person who
becomes inactive or dies is active for the first half of the year, and
that a person who is inactive and becomes active has earnings in the
last 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 for
expected 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 total
dollars earned by each of the four groups.

The above formula for calculating expected earnings involves an
assumption either that time has no value, or that productivity and
inflation gains are exactly offset by the market rate of interest. While
some expert witnesses still advocate the use of such a “total
offset method, “.sup.3 courts today will accept the discounting of
future earnings to reflect the net time value of money. The expected
earnings equation can be modified to accommodate discounting by defining
either a continuous compounding rate (r) or its annual discount rate
equivalent (d), that is, 1 + d).sup.-n = e.sup.nr.. (For instance, if
the annual rate of discount is 11 percent, its continuous compounding
equivalent is 10.44 percent.) The present value of expected earnings for
an 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.[right
brace]

Corresponding expressions for the expected worklife, expected
earnings, and present value of expected earnings for persons inactive at
age x can be derived in a similar way.

Calculation procedures

In her comment on Boudreaux’s and Nelson’s methods for
adjusting the worklife tables to estimate lost earnings, Shirley Smith notes that “frequently, economists want to look past the
lifetime-worklife expectancy figure to study the timing of the potential
earnings stream.” Here we argue that in the calculation of
expected lost earnings it is not sufficient to know the “median
number of years until final separation,” as defined by Nelson; to
adjust this figure by assuming that activity is evenly spread over the
entire period until retirement, as suggested by Boudreaux; or to know
any other single number that represents the possible length of time that
a person will be active. A true mathematical expectation of lost
earnings requires knowledge of the timing of probable activity and of
the potential (nominal or discounted) earnings during the period of
probable activity.

Because the timing of probable activity is sensitive to both the
initial work force status and the age of an individual, our development
of the true mathematical expected earnings, unlike the approaches of
Nelson and Boudreaux, emphasizes an active or inactive starting point.
To assess the consequences of this distinction, consider the example
provided by Boudreaux. A man age 30 with annual earnings of \$25,000
(using a current market interest rate of 11 percent and an annual
earnings increase of 4.5 percent) has a present value of
“expected” earnings of \$332,913, by the worklife table
estimate of 29.2 years of remaining worklife for the entire population.
Using Nelson’s 31.5 years to final separation criterion, the
present value of “expected” earnings is \$341,857.
Boudreaux’s 7.3-percentage reduction criterion drops this estimate
to \$316,901. However, our calculations show that the true mathematical
present 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 obtained
by 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 of
expected earnings is \$316,502, which compares favorably to
Boudreaux’s estimate of \$316,901. Given the ease of using
Boudreaux’s adjustment method, one might question the practical
value of using our more complicated true mathematical expectation
method.

Unfortunately, Boudreau’s approximation is close to the true
mathematical expectation only for younger men. His assumption that
inactivity is spread evenly over the entire period until retirement is
inappropriate at older ages, when the proportion that are inactive rises
rapidly. For a younger person, changes in expected earnings caused by
increasing probabilities of inactivity later in life are mitigated by
high discounting of expected earnings in distant years. For older
people, the mitigating effect of discounting is not present. Thus, for
a man age 45 with the same earnings stream used above, Nelson’s and
Boudreaux’s methods of estimating the present value of potential
earnings yield \$256,044 and \$242,217, respectively. Our mathematical
expectations are \$236,626 for an active man, \$155,310 for an inactive
man, and \$231,325 for the weighted average of active and inactive
persons.

OUR METHOD OF CALCULATION requires two modifications of the
probabilities of transition into and out of the work force at each age
must be converted to probabilities that are conditional on survival.
Second, conditional probabilities of transition between active and
inactive work force status must be added at age 76 to close the table.
The relevant probabilities of transition are provided in table 1. A
computer program for calculating the present value of expected earnings
based on these transitional probabilities is available from the authors.

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