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''0 1253 CeP¥^
STX
BEBR'
FACULTY WORKING
PAPER NO. 1253
Estimating Pecuniary Damages in Lawsuits With
a Reasonable Degree of Economic Certainty
William R. Bryan
Charles M. Linke
College of Commerce and Business Administration
Bureau of Economic and Business Research
University of Illinois, UrbanaChampaign
BEBR
FACULTY WORKING PAPER NO. 1253
College of Commerce and Business Administration
University of Illinois at Urbana— Champaign
May 1986
Estimating Pecuniary Damages in Lawsuits With
a Reasonable Degree of Economic Certainty
William R. Bryan, Professor
Department of Finance
Charles M. Linek, Professor
Department of Finance
ABSTRACT
The determination of pecuniary damages in personal injury and death
actions often requires actuaries and economists to make estimates that
encompass expected future experience over extended periods of time.
Moreover, such experts must be prepared to assert that their estimates
are made with a reasonable degree of economic certainty.
An anomoly plays an integral role in the estimation process.
Namely, although neither the actual levels of future interest rates nor
future wage growth rates can be predicted with reasonable certainty,
the present value of magnitudes emerging from them can be. It is impor
tant to understand why that is so.
This paper provides empirical evidence that the present value of a
worker's future earnings can be estimated with a reasonable degree of
economic certainty. Such certainty emerges by virtue of the predictable
covariance between the rates of growth of workers ' earnings and interest
rates reflecting the time value of money.
I. INTRODUCTION
Typically, the measurement of pecuniary damages is an important
aspect of lawsuits relating to wrongful death or disability, and employ
ment termination. Expert economic testimony is frequently placed into
evidence to assist the court in establishing a trialdate or present
value of a future stream of payments. The question of whether econo
mists and actuaries can make reliable present value estimates of pecu
niary damages in personal injury and death actions has been a source of
frequent debate in this Journal [5, 8, 9, 10, 13, 15, 16, 22, 23, 26].
Four variables determine the trial date value of a future payments
stream: (1) the beginning payment level; (2) the time period over which
payments will extend; (3) the anticipated annual rates of growth in the
payments; and (4) the investment rates of return at which the expected
stream of payments is to be discounted to its present value. The first
two variables are readily subject to estimation in most lawsuits. How
ever, economists and actuaries cannot predict with a reasonable degree
of economic certainty the yearbyyear future investment returns (r)
or the annual growth rates (g) of workers' earnings. Even so, because
the valuation process revolves around the predictable average differen
tial between r and g rather than their actual levels, economists can
make reliable present value estimates.
The purpose of this article is to provide empirical evidence that
the present value of the average worker's future stream of earnings can
be estimated with a reasonable degree of economic certainty." Section II
explains why the (rg) differential is of controlling importance in
2
determining the present value of future earnings streams. Section III
examines the role of the covariance between discount rates and growth
rates in specifying the (rg) differential. Section IV provides a
simulation analysis of the estimation of a worker's earnings using the
(rg) differential technique. The Implications of these results are
discussed in Section V. Concluding remarks are presented in Section VI.
II. THE (rg) DIFFERENTIAL AND PRESENT VALUE ESTIMATES
The measurement of pecuniary damages in personal injury and death
actions often requires actuaries and economists to make projections over
3
long time periods. It is important to understand how that can be done,
in light of the fact that actuaries and economists cannot predict with
confidence what the actual level of interest rates and the actual wage
growth rates will be over an extended period in the future. Fortunately,
it is neither necessary or important to project the time path of discount
rates in estimating the present value of a future earnings stream. The
purposes of the court are sufficiently served with testimony relating to
the (rg) differential — that is, to the difference between the discount
4
rate (r) and the earnings growth rate (g).
This point is illustrated in Table I. The present values associated
with specific differentials are contained in the diagonal vectors.
Although there are great differences in the magnitudes of the income
streams corresponding to alternative growth rates in income (see bottom
tier of Table I), there is virtually no difference in the present values
consistent with any specific differential. For example, the present
value of a 25year earnings stream that has an initial $1000 value is
3
$22,023 if a one percent (rg) differential occurs at a r = 1.0 percent
and g = 0.0 percent; the present value is $22,202 if the one percent
differential occurs at r = 8.0 percent and g = 7.0 percent.
Table I about here
A brief review of present value logic reveals why an economist or
an actuary should be more interested in predicting the average (rg)
differential than with predicting the expected average levels of r and
g separately. The present value of a worker's future earnings can be
represented as
n t t
PV = Z [(E ) n (l+g^)]/[ n (1+r^)] (1)
o , o , a  a
t=l a=l a=l
where
PV = present value at t=0 of a stream of payments to replace a
worker's future earnings;
g = rate of growth in earnings in period a (a = l ,2 ,3 , . . . ,n) ;
t
E = E TT (1+g ) = earnings in period t (t = l , 2 , 3 , . . . ,n) ;
a = l
r = discount rate in period a (a=l, 2,3, . . . ,n) ; and
H = multiplication operator.
In estimating the present value of future earnings using an average
growth rate (g) and an average discount rate (r), equation (1) converts
into
" t t
PV = E [ Z (l+g)V(l+r)^] (2)
° ° t=l
= E [(l((l+g)"/(l+r)"))/(((l+r)/(l+g))l)]. (3)
o
4
If in the interest of expositional convenience, we let n >• «>, then the
summation term in equation (2) converts into the following geometric
expansion
PV = P^ —^ (4)
o o 1  R
where
A = the initial term in the expansion; and
R = the ratio between successive terms in the expansion.
It turns out that the initial term, A, is given by (l+g)/(l+r), and
the ratio between successive terms is also given by (l+g)/(l+r). Hence,
PV = P [(l+g)/(l+r)]/[l(l+g)/(l+r)]. (5)
o o
Letting k equal the difference between r and g (i.e., k=(rg)), substi
tuting (rk) for g, and taking derivatives of (5) with respect to r and
k , we get
3PV
^ = d/k) = [l/(rg)] (6)
and
8PV „ „
yj^ = [(l+r)/k^] = [(l+r)/(rg)^]. (7)
It is clear from inspection that (7) is [(l+r)/(rg) ] times larger
than (6). For this reason, an analyst should be more interested in the
(rg) differential than in the actual likely levels of either r or g.
Thus, given the initial level of income and the length of the time
period, all that is necessary in order to estimate the present value
of a worker's anticipated future earnings is knowledge of the average
5
differential (rg). The selection of an appropriate differential must
be based upon wage growth and discount rates that are consistent with
theoretical considerations and bear some relation to experience.
III. COVARIANCE BETWEEN GROWTH RATE AND DISCOUNT RATE
Economic theory indicates that there is a strong covariance between
rates of growth in the annual earnings of workers and interest rates.
They have common determinants. Growth theory does not, however, specify
the steadystate equilibrium values of r and g, or the (rg) differen
tial. The differential is, ultimately, an empirical issue.
The determination of the growth in wages and the general level of
interest rates are often summarized as follows:
annual wage annual rate of annual rate of
growth rate = increase in labor + change In the
productivity price level
annual time annual real rate annual rate of
value of = of interest (or + change in the
money productivity of price level
capital)
Having said this much, however, it is important to move rapidly
toward qualifying statements. First, it is important to recognize that
there is not a single wage rate. At any point in time there is a col
lection of wage levels and a collection of growth rates in wages. In a
market system, such differences (in wage levels and in growth rates)
are the means by which resources are allocated. Even so, as markets
accomplish their resource allocation objectives there is a tendency for
growth rates attaching to specific occupations to regress toward an
economywide mean.
6
Second, it is important to recognize that there is no single unam
biguous rate summarizing the return on capital. At any time there is a
constellation of interest rates, with rates of return within the con
stellation reflecting differences in various aspects of risk. Moreover,
through time there are changes in the general level of the constellation
(i.e. , the level of interest rates) as well as shifts in the relation
ship among rates of return within the constellation (i.e., the terra
structure of interest rates).
Empirical studies show that the longerrun annual increase in labor
productivity in this country has averaged 2.00 to 3.50 percent, and
that the real rate of interest is in the 2.00 to 4.00 percent range.
Thus, over protracted stretches of time the growth in money earnings and
the time value of money can be expected to display a high covariance.
It is our view that the oneyear U.S. Treasury yield is the most
useful proxy for the time value of money. In this study, we measure
the oneyear rate (r ) with the oneyear U.S. Treasury securities con
stant maturities data. The use of shorter maturities is questionable
because their yields are influenced importantly by money market pres
sures. The use of longerterm securities as a measure of the time value
of money is questionable by virtue of the fact that their yields embody
a premium for interest rate risk. Investing in one year maturities,
whose yields covary with expected changes in the price level, avoids the
substantial risk of investing in long maturity instruments to meet an
annual payment need that will also covary with the changing price level.
Stated differently, the duration of oneyear U.S. securities is consis
tent with the time horizon of workers' annual compensation reviews.
7
Data are presented in Table II that depict the relationships
between wage growth and the contemporaneous rates of return on equities
9
(r ), oneyear (r ) and twenty year (r„ ) U.S. securities. The data
show that during the 19531984 period the (r g) differential averaged
0.56 percent. The 6.07 variance of the (r g) differential was sub
stantially lower than the 10.68 variance of r alone because of the
positive covariance between r and g. The variance of the (r g)
differential is
Var(r^g) = Var r^ + Var g  2 Gov (r^^.g). (8)
A large and positive covariance between r and g obviously contributes
to the relative stability of the (rg) differential and, therefore, to
the reliability of present value estimates of workers' future earnings.
Of course, if there were perfect covariance between r and g the variance
of the differential would be zero.
Over this same period, the differential using 20year U.S. securi
ties of constant maturity (rg) averaged .99 percent and the variance
was 6.01, somewhat smaller than the variance with oneyear securities.
The relationships between the growth in wages and contemporaneous
interest rates appear to be inconsistent with our working hypothesis
that the oneyear U.S. Treasury yield is the most useful proxy for the
time value of money. On its face, portfolios consisting of 20year
U.S. securities appear to provide a less expensive means of replacing
lost wages than do portfolios consisting of oneyear U.S. securities.
Moreover, according to these data there is less variance in the
(r„„g) differential than in the (rg) differential. With equities,
the differencial (r g) averaged 6.18 percent but the variance was a
whopping 334.20.
Table II about here
Up to this point, we have been discussing relationships between the
growth in wages and contemporaneous rates of return on alternative
investment media. Such relationships have been the focus of the litera
ture relating to the measurement of pecuniary loss in lawsuits and the
annual cost of pension plans. In the remaining portions of this
article, we break away from the traditional preoccupation with growth
rates and contemporaneous investment returns. As it turns out, those
relationships are not of primary importance and can be misleading. It
is necessary to recognize that the covariance of ultimate importance
emerges from the relationship between the growth in wages and the rate
of return on portfolios dedicated toward meeting future wage payments as
they arise in the actual course of time. That is, the relationship of
chief importance is between g and the r from what we refer to as a
"dedicated" portfolio.
IV. THE (rg) DIFFERENTIAL ESTIMATION TECHNIQUE:
EVIDENCE FROM SIMULATION ANALYSIS
A desirable valuation technique would be one that generates a pre
sent value estimate as of the trial date which proves to be just suf
ficient to provide the payments stream that actually occurs. Any given
payments stream could be valued within the framework of several alterna
tive investment strategies. Presumably, the choice among investment
9
strategies will be guided by the riskcost tradeoff performance stan
dards of the court. This portion of the paper employs simulation ana
lysis to develop a set of experience which can be used to assess the
(rg) differential estimation technique.
Simulation Procedures
Simulation analysis was used to determine the present value or the
principal amount of money required at t=0 in a dedicated investment
portfolio in order to provide exactly the earnings stream of the average
U.S. worker over alternative time spans in the post Federal Reserve
Treasury Accord period (19531984). The portfolios were dedicated
in the sense that the principal and the associated realized investment
income were used to replace the actual annual earnings of the average
U.S. worker as that yearly earnings stream emerged. There were thirteen
20year time frames from 1953 to 1984, eighteen 15year periods,
12
twentythree 10year periods, and twentyeight 5year periods.
Alternative investment strategies with dedicated portfolios were
specified for each 20year, 15year, 10year and 5year period. One
strategy consisted of investing in oneyear U.S. Treasury securities
only. Other strategies consisted of investing only in, alternatively,
5year, 10year, 15year and 20year U.S. Treasury securities. A final
strategy consisted of investing entirely in common stocks. Each invest
ment strategy was made operational by means of simulation. According
to the terms of the simulations, the actual annual earnings of the
average worker were first identified for each year within a time period.
Then, given an initial principal investment and portfolio strategy,
amounts equal to the actual annual earnings of the average worker were
10
withdrawn yearbyyear from the portfolio. If the investment income
earned in any year were less than the earnings of the average worker
13
in that year, then securities were sold at prevailing market prices.
To the extent that the annual income of the portfolio exceeded the
annual earnings withdrawal, the excess was reinvested as prevailing
14
market rates. For each investment strategy, the simulation was per
formed in iterative fashion until an initial principal amount was
identified that was just sufficient to generate the annual earnings
during the time span being analyzed. That initial amount constituted
the dedicated portfolio at t=0 , or the present value needed to replace
the earnings stream that actually occurred following t=0, or in time
periods t=l,2 ,3, . . . ,n.
The simulations identified time period specific joint observations
of ex post growth rates and discount rates. Aggregate ex post income
pajnnents were known. Given initial income, it was possible to solve
for the ex post implied growth rate, g' , that created those income
payments. As indicated, the simulations found the minimum initial
investment for the dedicated portfolios for each investment medium.
Given this present value amount, along with the initial payment and the
growth rate in income, it was possible to solve for each implied ex post
return (or discount rate), r' , on each dedicated portfolio. Thus, for
each simulation it was possible to calculate the ex post differential
between the growth rate and the discount rate (r'g'). These simulation
data were summarized for each investment strategy and for each time
period.
11
Results
The simulation results are shown in Table III. Summarized there
are the means and standard deviations of the (r'g') differentials for
the alternative investment strategies across alternative loss periods.
Within investment strategies confined to U.S. securities, visual
inspection suggests that the (r'g') differential grows increasingly
negative and the standard deviation gets larger as we move from a 1year
strategy to a 20year strategy. It is also clear that the standard
deviation of the (r'g') differential associated with an investment
strategy is inversely related to the length of the loss period.
Table III about here
Over the entire range of observations, the (r g' ) differential for
e
the equities strategy averaged 3.78 percentage points while the (r'g')
differential for U.S. Treasury investment strategies generally were
negative (i.e., g' > r'). Specifically, the average return on dedicated
portfolio's comprised of 20year U.S. Treasury securities was 2.57
percentage points below the earnings growth rate, and the average return
on dedicated portfolio's comprised of 1year Treasury securities was
.17 percentage points below the earnings growth rate. Thus, in terms of
the present value cost of replacing a worker's future earnings, dedicated
equity portfolios were the least costly, dedicated portfolio consisting
of 1year securites were second, and dedicated portfolios of 20year
securites were the most expensive. But risk and return (cost) are
inversely related and, as the standard deviation data in Table III
suggest, the consistency of performance was much better with U.S.
Securities than with the eauities strategy.
12
Regression analysis was conducted with the (r'g') simulation data
in order to estimate the expected (r'g') differential for each
investment strategy. Because we were interested in controlling for
potential effects of loss periods on estimates of the (r'g')
differential, we used dummy variable techniques to control for
alternative loss periods. Also, because interest rates have generally
risen during the 19531984 period, it was plausible to believe that an
equation that did not take such a trend into account would be
misspecif ied. Hence, a trend variable was introduced. At length, the
regression model to be estimated was
3(r'g')^ = a^ + a^ D^ + a^ ^^ + 33 D3 + a^ T^
^ ^ ^2 ^ "6 ^3 ^ ^7 ^4 ^ "t (^^
where
(r'g') = the (r'g') differential on dedicated portfolio t with
investment strategy s (s=l through 6, corresponding to
portfolios consisting of 1,5,10,15 and 20year U.S.
Treasury securities, and equities),
D = 1 with 5year loss periods and at all other times,
D = 1 with 10year loss periods and at all other times ,
D3 = 1 with 15year loss periods and at all other times ,
T. = time trend (subscripts i = 1 through 4 relate to 5year, 10year,
15year and 20year loss periods, respectively),^^ and
U = the stochastic term at time t.
According to regression results with the simulation data, none of
the coefficients associated with loss period dummies — i.e., a,, a^ and
So — reaches an acceptable level of significance. This means that the
average differential does not depend upon the length of the loss period.
In regression confined to U.S. securities, the coefficients associated
13
with the trend variables, a, through a^ were not often significant.
But these coefficients were significant in tests wherein the invest
... . ^ 19
ment strategy was confined to equities.
Selected results from the regression tests are provided in the
20
Regression Summary in Table III. Shown there are intercepts and
standard deviations of the residuals around the regression hyperplane.
The intercept values shown in Table III are, of course, the means
from 20year loss periods. Estimated values of a for 1year and
18
5year securities do not differ from zero. Recall that the loss
period dummies are not statistically significant; hence, we can
interpret the intercepts with generality. Results with the regression
summaries are consistent with impressions from visual inspection.
That is, equities constitute the leastcost dedicated portfolios.
Among U.S. Treasury securities, 1year securities are the leastcost
medium; dedicated portfolios consisting of 20year instruments are the
most expensive.
But the consistency of performance was much better with debt instru
ments than with equity. Figure 1 illustrates this dimension of the
problem. Bear in mind that Figure 1 is illustrative only. Each of the
three normal curves is calculated from the means and standard deviations
of the distribution indicated. With 1year U.S. Treasury securities
the standard deviation of the residuals of the actual (r 'g') dif
ferential around the estimated (r 'g') differential was 1.07 percen
tage points; with 20year securities the standard deviation was 2.17
percentage points; and with equity it was 4.88 percentage points.
Thus, in terms of consistency in leastcost performance, 1year
14
securities ranked first; 20year securities ranked second; and equity
was least consistent.
We began with the working hypothesis that investing in oneyear
maturities, whose yields vary with expected changes in the price level,
avoids the substantial risk of investing in long maturity instruments
to provide a stream of pajnnents that will be affected by the changing
price level. As discussed earlier in Section III, the covariance
between the growth in wages and contemporaneous interest rates appeared
to be inconsistent with this hypothesis. However, the covariance of
ultimate importance is between the effective wage growth (g') and the
realized rate of return (r') on portfolios dedicated toward meeting
future wage payments as they arise in the actual course of time. The
(r'g') data shown in Table III provide support for the working
u • 21
hypothesis.
The point that emerges throughout Table III is that the leastcost
means of providing for a future payments stream of uncertain size is to
construct the dedicated portfolios with very shortterm securities.
Moreover, the standard deviation of the (r'g') differential is, with
out exception, smaller with dedicated portfolios consisting of short
term maturity securities than with dedicated portfolios consisting of
longterm securities.
The substantial stability in the differential between the growth
rate in wages and the returns on dedicated portfolios consisting of
shortterm securities results from the high covariance between growth
rates and shortterm interest rates coupled with little, if any,
offsetting capital depreciation. In contrast, the relatively high
15
covariance between contemporaneous movements in interest rates on
longterm securities and wages has served to increase the costs asso
ciated with dedicated portfolios consisting of such securities. Even
as longterm interest rates have risen along with wages, these
increases have resulted in capital losses on the dedicated portfolio.
V, PERFORMANCE STANDARDS AND THE RISKRETURN TRADEOFF
The analysis and results presented above have a direct bearing on
issues typically before the court. Specifically, the results can be of
use in placing a trialdate value on a lost future income stream, on
prospective future medical payments, or on a range of other types of
laborintensive payments.
As suggested, if the court were simply to use the averages from the
simulation experience it is clear that equities present the leastcost
option. For example, the rate of return on dedicated equity port
folios for a 20year loss period averaged 4.36 percentage points above
the growth rate in wages. Such a growth rate/discount rate rela
tionship results in a substantially lower present value than was the
case with either 1year or longermaturity U.S. securities. A numeri
cal example may be helpful.
Suppose that the initial payments rate is $1000 and that the pay
ments will continue for 20 years. For computational convenience we can
22
assume a 0.0 percent growth rate and a 4.36 percent discount rate.
The differential related to equities implies a present value of $13,167.
By comparison, the differential related to 1year securities (.28 per
cent) implies a present value of $20,600, approximately 60 percent more
than the cost of an equities portfolio. The present value implied by
16
the differential relating to 20year maturity securities (4.36 per
cent) is $33,004, well over twice the cost of an equities portfolio.
In short, the differences in present values implied by these alter
native differentials are by no means trivial.
But there is a problem. Recall that the differentials used above
are averages. It is the very nature of measures of central tendency
that about onehalf the values in a distribution lies above the mean,
and onehalf lies below. Hence, if the court opts to use the arith
metic mean for valuation purposes it will do so knowing that roughly
onehalf the claimants will be unable to construct dedicated portfolios
that meet their needs.
And there is a problem within a problem. Recall the distributions
with alternative dedicated portfolios (see Figure 1 and Table III).
Figure 1 about here
There is a very large dispersion of experience around the average dif
ferential associated with dedicated portfolios consisting of equity,
(r 'g'). Such a result is consistent with what might be expected in
e
light of the relationship between returns and risk. That is, there is
a widelyheld view that higher returns will be related to higher risk;
conversely, lower returns will be related to lower risk. But the con
sequence of such a riskreturn tradeoff is that some claimants will
fall far short of meeting their needs. As applied to the matter at
hand such a consequence constitutes an ironic anomaly. It means that
the benefits of the increased risk accrue immediately to the defendant
(the wrongdoer) without regard to final outcomes, while the burden of
17
the Increased risk falls upon the claimant (the injured party). Such a
state of affairs reverses the riskreturn tradeoff logic undergirding
investment decisions.
The court has already faced up to at least a portion of these
issues. In Jones and Laughlin Steel Corporation v. Pfeifer [U.S.
76 L Ed 2d 768, 103 S Ct (June 15, 1983)] the court held that "... the
discount rate should not reflect the market's premium for investors who
are willing to accept some risk of default." The court appears to have
effectively limited the range of acceptable investment media to U.S.
Treasury securities. In any event, because the risk of default is
clearly present in equities, it would appear to be quite clear that
common stock returns cannot be used for valuation purposes.
But there is another basis upon which equities do not provide a
leastcost basis for loss evaluation. That basis relates to the court's
reasonable insistence upon high performance standards for dedicated
portfolios. It is reasonable to believe that the court would not be
satisfied with a decisionmaking process that results in presentvalue
amounts that are insufficient to meet the needs of roughly onehalf the
claimants. But suppose the court insists upon a process that implies
only a 5 percent error. According to Figure 1, 95 percent of the
dedicated portfolios consisting of 1year U.S. Treasury securities lie
above a (r 'g') differential of 2.04. For 20year maturity securities
a (r 'g') differential of 5.68 is required in order to include 95
percent of the sufficient dedicated portfolios. For equities, a
(r 'g') differential of 3.66 would be required. If the court were to
e
enforce an even higher standard, for example one permitting shortfalls
18
with only 2 percent of the dedicated portfolios, (r'g')s of 2.48,
6.57 and 5.66 would be required for, respectively, 1year securities,
20year securities, and equities.
In terras of numerical examples, the very highperformance dedicated
portfolio (2 percent shortfalls) consisting of 1year U.S. Treasury
securities would cost $26,308. The highperformance portfolio con
sisting of 20year securities would cost $44,032; and the portfolio
consisting of equities would cost $38,991. In short, within the con
text of speaking about highperformance dedicated portfolios, 1year
23
U.S. Treasury securities provide the leastcost investment vehicle.
VI. CONCLUDING REMARKS
It is our view that the present value of future payments streams
can be estimated with a reasonable degree of economic certainty. There
is reasonable certainty by virtue of the relative stability in differen
tials between growth rates in payments streams and returns on 1year
dedicated portfolios. Moreover, the stability in differentials does not
result from fortuitous circumstances. Rather, both wages growth rates
and interest rates are determined by a set of common factors. In par
ticular, in the short run each is importantly affected by inflation
rates. Over the long run there is a tendency towards a zero differential.
If the court insists on high performance standards it may use the
standard deviation of the (r'g') differential to control for error.
One might imagine that the willingness of the court to countenance error
will vary from case to case. For example, in those cases in which
liability is clear and dependence of claimant upon the award is virtually
19
coraplete (e.g., total disability), the court may wish to hold potential
shortfalls to a minimum. To accomplish this, the court could adjust the
mean value by, say, two standard deviations. In other instances the
court might be willing to countenance only average performance.
Perhaps the strongest conclusion from, this study is that, for valua
tion purposes, we may limit our attention to returns on dedicated port
folios consisting of 1year securities. In part, this conclusion is
based on priors, specifically, on the belief that there is no basis in
logic for a claimant to carry the burden of increased risk without an
entitlement to the benefits associated with risktaking. But the con
clusion is also based on the empirical finding that portfolios consisting
of 1year securities are leastcost portfolios. Their advantage with
high performance portfolios is especially pronounced. This advantage
reflects the low variance in the (r'g') differential. In turn, that
low variance reflects the high covariance between the wages growth rate
and the return on portfolios consisting of 1year treasury securities.
Our study suggests that there is virtually no basis for a belief that
the appropriate discount rate is greater than the growth rate. Rather,
in rough terms, our results are consistent with the presumption that
the average differential is zero.
20
BIBLIOGRAPHY
1. Brealy, R. A. and S. Schaefer, "Term Structure and Uncertain
Inflation," Journal of Finance , Vol. 32 (May 1977), pp. 277290.
2. Brody, Michael T. , "Inflation, Productivity, and the Total Offset
Method of Calculating Damages for Lost Future Earnings," The
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No. 2 (June 1975), pp. 342345.
6. Fisher, Irving, The Theory of Interest , Kelley & Millman, New York,
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Classics (New York: Augustus Kelley Publishers, 1971).
8. Franz, Wolfgang W. , "A Solution to Problems Arising From Inflation
When Determining Damages," Journal of Risk and Insurance , Vol. 45,
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9. Harris, William G. , "Inflation Risk as Determinant of the Discount
Rate in Tort Settlements," Journal of Risk and Insurance , Vol. 50,
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10. , Edward B. Bell, Allen J. Taub, and Edgar P. Hickman,
"Selecting Income Growth and Discount Rates in Wrongful Death and
Injury Cases: Comment, Additional Comment, and Further Comment,"
Journal of Risk and Insurance , Vol. 44, No. 1 (1977).
11. Henderson, Roger C. , "Periodic Payments of Bodily Injury Awards,"
The American Bar Association Journal , Vol. LXVI (June 1980), pp.
736737.
12. , "Restoring the Tort Victim to Preinjury Position,"
The American Bar Association Journal , Vol. LXVII (March 1981),
pp. 301302.
13. Hosek, VJilliam R. , "Problems in the Use of Historical Data in
Estimating Economic Loss in Wrongful Death and Injury Cases,"
Journal of Risk and Insurance , Vol. 49, No. 2 (June 1982), pp.
300308.
21
14. Ibbotsen, Roger G. and Rex A. Sinquefield, Stocks, Bonds, Bills
and Inflation: The Past and The Future (Charlottesville, Virginia:
Financial Analysts Research Foundation; 1982) and updating supplements.
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Injury Economic Losses," Journal of Risk and Insurance , Vol. 52,
No. 1 (1985), pp. 144150.
16. Linke , Charles M. , Gene Laber, and Franklin Smith, "The Use of
Inflation Factors in Determining Settlements in Personal Injury
and Death Suits: Comment, Further Comment and Author's Reply,"
Journal of Risk and Insurance , Vol. 44, No. 3 (1977).
17. Macaulay, Frederick R. , The Movement of Interest Rates, Bond
Yields and Stock Prices in the United States Since 1856 , NBER,
New York, 1938.
18. Malkiel, Burton G., The Term Structure of Interest Rates ,
Princeton University Press, Princeton, N.J. , 1966.
19. Martin, Gerald D. and Clay, William C. , Jr., How to Win Maximum
Awards for Lost Earnings , Executive Reports Corporation, Englewood
Cliffs, N.J., 1980.
20. Mennis, Edmund A. and Chester D. Clark, Understanding Corporate
Pension Plans , Financial Analysts Research Foundation,
Charlottesville, VA, 1983.
21. Miller, Herman P., "Lifetime Income and Economic Growth," The
American Economic Review , Vol. 55, No. 4 (1965).
22. Schilling, Don, "Estimating the Present Value of Future Income
Losses: An Historical Simulation 19001982," Journal of Risk
and Insurance , Vol. 52, No. 1 (1985), pp. 100116.
23. Smith, Franklin C. , "The Use of Inflation Factors in Determining
Settlements in Personal Injury and Death Suits," Journal of Risk
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24. U.S. Bureau of the Census, Historical Statistics of the U.S. from
Colonial Times to 1970 , Bicentennial Ed. Part I, Washington, D.C., 1975.
25. U.S. House of Representatives, 1985 Annual Report of the Board of
Trustees of the Federal OldAge and Survivors Insurance and
Disability Insurance Trust Funds , H.R. Document 9946, 1985.
26. Vernon, Jack, "Inflation Risk as Determinant of the Discount Rate
in Tort Settlements: Comment," Journal of Risk and Insurance ,
Vol. 50, No. 3 (1985), pp. 528536.
27. Wyatt Company, 1984 Survey of Actuarial Assumptions and Funding
of Pension Plans With 1000 or More Active Participants , Washington,
DC, 1985.
D/391
22
FOOTNOTES
Other variables that may be important in a specific valuation
problem, such as the ageeducation life cycle of the payments stream,
are not discussed in this paper. Issues posed by these variables do
not produce a fundamental effect on conclusions presented here.
2
Economists and actuaries cannot predict the economic future of any
specific individual with a reasonable degree of economic certainty.
The best estimate of the economic future of any individual is the
expected economic future of the average worker in the statistical cohort
to which the worker in question belongs.
3
Similar longterm projections of wage growth and investment returns
are required for annual pension funding decisions.
4
It is interesting that most of the literature on estimating
pecuniary damages has addressed the estimation of the nominal levels of
r and/or g, which are not predictable with confidence, rather than
focusing on the more important and predictable (rg) differential. For
example, see [3, 5, 9, 10, 15, 19, 23, 26].
The present values associated with a specific differential would
show no differences if the analysis were conducted using continuous
compounding for both the wage growth rate and the discount rate.
Moreover, it makes virtually no difference whether the average
differential is created by a constant r and g set through time (e.g. ,
remaining at g = 6 percent and r = 7 percent), or whether the average
differential emerges by virtue of a succession of different sets (e.g. ,
g as a sequence of 6 percent, 8 percent, 5 percent, and 7 percent, etc.;
and r as a sequence of 7 percent, 9 percent, 6 percent and 8 percent,
etc. ) .
The derivative with respect to g is equal to but with opposite
sign of the derivative with respect to r.
g
These studies rely on alternative estimates of labor productivity
and alternative measures of interest rates.
9
Interest rate data are obtained from the Federal Reserve constant
maturity series published in various issues of the Federal Reserve
Bulletin . Equity returns are from the Ibbotsen and Sinquefield study
[14, p. 18] of 19261984 stock market returns. Wage growth rates are
calculated using the index of adjusted (for overtime and interindustry
employment shifts) hourly earnings for private nonagricultural workers
[4, p. 276]. The wage growth rate data are biased downward inasmuch
as the rapid expansion of employer provided fringe benefits (nonmoney
earnings) in the 19531984 period is not considered.
Cov(r,g) = pa o where p is the correlation coefficient between
r and g.
23
Much of the analysis reported here was also done with data relat
ing to the 19261984 period. Results with the longer data span were
generally consistent with results reported here. But the experience of
the 1930s and 1940s differed so markedly from experience since the
Accord that we feel justified in focusing our presentation on the more
recent data set.
12
The use of overlapping data results in the progressive under
weighting of experience as we move toward the beginning and the end of
the data set, and a corresponding overweighting of experience as we
move toward the middle of the set. To the extent that there are trend
dependent variables operating on the data, the results can be systemati
cally biased. This problem is further addressed in footnote 17.
13
The need to sell portfolio assets to meet an annual withdrawal
payment is not difficult to accommodate in the simulation with the one
year U.S. Treasury securities investment strategy or with the common
stock investment strategy. Investing in oneyear U.S. Treasury securi
ties causes the assets of the portfolio (principal and interest) to be
available for making a payment at the end of each year. After the
required annual withdrawal is made, the remaining balance in the port
folio is reinvested at the prevailing oneyear U.S. Treasury securities
yield. The situation is similar for the equity investment strategy
inasmuch as the annual yield is decomposed into current dividend income
and capital appreciation (loss).
Investment strategies that create portfolios composed of longer
terra U.S. Treasury securities involve added considerations. The market
value of a bond changes because of movements in market interest rates
and as a result of decreases in the time to maturity. The market values
of U.S. Treasury securities sold prior to maturity were determined by
calculating the present value of the remaining coupon interest payments
and the maturity value. The discount rate used was equal to the current
market yield for U.S. securities with a maturity equal to the remaining
time to maturity of the bonds being sold.
14
If, from timetotime, an additional purchase were required, the
maturity of those securities would be equal to the lesser of the number
of years remaining in the time period or the investment strategy maturity.
The expost growth rate, g' , was obtained by solving the equation
n n
[ I Earnings = (Earnings ) Z (1+g')^].
t=l t=l
The expost investment return or discount rate, r' , was obtained
by solving the equation.
" t t
Present Value ^^ = ^ [(Earnings ^^)(l+s') /(1+r') ]
t = l
using the previously calculated g' value and the (t=0) present value.
24
For 20year loss periods, t = 1 through 13; for 15year loss
periods, t = 1 through 18; for 10year loss periods, t = 1 through 23;
and for 5year loss periods, t = 1 through 28. In fact, however, the
trend variables are expressed as differences from arithmetic means.
This is done in order to remove the effects of estimation from the
intercepts.
18
The finding that the (r^g) and (r5g) differentials do not
differ from zero provide support for use of the socalled "Alaska
Method" which employs a total offset (r=g) logic.
19
Because there is autocorrelation in the residuals, the normal
tests of significance cannot be trusted. Such a state of affairs is
frequently a barrier to useful research because the tstatistic is
biased towards enlarged departures from zero, and, therefore,
unwarranted findings of statistical significance. In this instance,
however, the presence of autocorrelation actually serves to strengthen
our findings. As it turns out, our research interests are satisfied if
coefficient estimates are not statistically significant. Because
serial correlation contributes to an unwarranted finding of statistical
significance, our assertions relating to a lack of statistical signifi
cance are strengthened.
20
Regression results are available upon request from the authors.
21
The wage growth and risky investment (discount) return assumption
used in pension planning by the largest U.S. corporations also provide
support for the reasonableness of this (r'g') differential hypothesis
[20, pp. 3537; 25, pp. 2728]. A recent study [27] by the Wyatt
Company, a prominent national actuarial firm, of the 1984 actuarial
assumptions of 961 large pension plans covering 1000 or more active
participants revealed the average wage growth rate used was 5.9 percent
and the average investment (discount) rate was 7.2 percent. Moody's
Financial Services, a prominent national investment analysis firm,
conducted a similar survey in 1979 which revealed the Aaa, Aa, A and Baa
rated U.S. corporations employed an average 5.15 percent wage growth
rate and an average 6.45 percent investment rate in pension planning.
It is noteworthy that the (rg) differential was 1.3 percent in both
1979 and 1984 even though the yield on longterm U.S. securities was
three percentage points higher in 1984 than in 1979, and the wage growth
rate in the U.S. economy was approximately 2.5 percentage points lower
in 1984 than in 1979. These (rg) pension assumptions data support the
conclusion expressed in this paper that actuaries and economists faced
with estimating the present value of a stream of future payments rely
upon the predictable (rg) differential rather than upon speculative
estimates of the average r and average g that might prevail during the
time period in question.
22
As shown in Table I, the nominal levels of the growth rate and
the discount rate are of little importance.
23
The specific calculations, based upon the assumption of a normal
distribution, are illustrative only.
Table I
Present Value of a Growing 25Year Earnings Stream:
Alternative Growth Rates and Alternative Discount Rates
(Earnings = $1000)
Rates of Growth in
Earning
s
Discount
0.0%
1.0%
2.0%
3.0%
4.0%
5.0%
6.0%
7.0%
8.0%
Rates
0.0%
$25,000
$28,526
$32,671
$37,553
$43,312
$50,113
$58,156
$67,676
$78,954
1.0%
22,023
25,000
28,488
32,582
37,396
43,065
49,749
57,640
66,963
2.0%
19,523
22,050
25,000
28,451
32,495
37,243
42,825
49,395
57,138
3.0%
17,413
19,569
22,077
25,000
28,415
32,410
37,094
42,591
49,051
4.0%
15,622
17,471
19,614
22,103
25,000
28,379
32,328
36,948
42,362
5.0%
14,094
15,688
17,528
19,658
22,128
25,000
28,345
32,246
36,805
6.0%
12,783
14,164
15,752
17,584
19,701
22,153
25,000
28,311
32,167
7.0%
11,654
12,856
14,233
15,816
17,639
19,744
22,178
25,000
28,277
8.0%
10,674
11,727
12,928
14,302
15,879
17,694
19,785
22,202
25,000
Earnings in
Year 25 $ 1,000 $ 1,282 $ 1,641 $ 2,094 $ 2,666 $ 3,386 $ 4,292 $ 5,427 $ 6,848
Cumulative
Earnings 25,000 28,526 32,671 37,553 43,312 50,113 58,156 67,676 78,954
Table II
Relationships Between Wage Growth and
Contemporaneous Rates of Return: 19531984
Alternative (rg) Differential
Investments Mean Variance Cov(r,g)* Mean Variance
(%) (%)
U.S. Securities
1 Year (r^)
6.07
10.68
4.17
.56
6.07
20 Year (r^^)
6.50
9.35
3.54
.99
6.01
Equities (r )
e
11.68
317.51
6.48
6.18
334.20
*The mean and variance of the annual wage growth rates (g) are 5.51
percent and 3.73, respectively.
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Figure 1
Distributions of (r'g') Differential for
1Year, 20Year and Equities Investment Strategies
Percent
10
9 
8 
7 
6
5 
4 
3 
2 
1 
i
1Year
20Year
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