The Offset Rule:
R.A.L. Carter and John P.
The University of Western Ontario,
London, Ontario. The authors wish to thank Professor Robbie Jones
for his suggestions concerning holding-period rates of return.
The Offset Rule:
In civil cases involving personal
damages, courts are concerned that a lump sum award adequately
compensate the victim for the loss of an expected future flow of
income. To convert this expected flow into a present stock, the
courts implicitly or explicitly take the present discounted value
of the expected nominal dollar value of the loss in each future
time period during which the victim expects to suffer the loss.
The process can become quite complex and quite costly, as
plaintiffs and defendants contest the underlying assumptions
about expected real rates of interest, expected rates of
inflation, and the expected growth in labour productivity, which
are just a few of the variables usually introduced in personal
We have previously provided strong
evidence for The United States and Canada(2)
that the cost and complexity of adjudicating disagreements about
the sizes of these variables can probably be avoided. This
evidence supports the underlying assumptions of what has come to
be known as "the offset rule", which asserts that the
appropriate award, ceteris paribus, is simply the amount
of the loss in the present year multiplied by the number of years
over which the loss is expected to occur. There is no reason to
engage in costly litigation concerning these variables because
expected nominal aggregate growth in labour productivity tends to
"offset" the nominal rate of interest.
In this paper, we test the underlying
assumptions of the offset rule using data from eleven additional
countries.(3) We also incorporate a more formal test for
non-stationarity in the context of seasonal data than we used in
our previous work. In general, we find that the results for these
countries are consistent with our earlier results for Canada and
The United States: the quarterly average rates of growth of
nominal labour income are similar to the average holding period
rates of return on long-term securities; i.e. the results are
consistent with the underlying assumptions of the offset rule.(4)
The Offset Rule: an algebraic
The full development of the arguments
underlying the offset rule is provided in our earlier work. Here
we provide just a summary [note to the editor and reviewers: this
summary can be condensed at your discretions]. Suppose that
someone suffers a personal injury at time t
= 0. Suppose further that the amount of loss the
person expects to suffer during the next year is $D0.
Let us suppose, in addition, that the loss is expected to
continue with certainty for exactly N
years. The issue before the court is how to determine AN,
the dollar value of the award that must be paid today to
compensate the plaintiff for all the expected future losses.
We begin the development of the
algebraic model with two additional assumptions (which we relax
as the development proceeds). First, we assume that the relevant
rates of growth, interest, and inflation are constant over the
entire N-year period. Second, we assume
that there are no life-cycle effects which would cause the amount
$D to vary for productivity reasons
during the N years. The goal of this
exercise is to determine an expression for AN,
which is the sum over N years of the
present values of the expected losses in each future year, T
The summation in equation (1) is
composed of a series of annual amounts, each of which can be
This expression says that it is
reasonable to expect that in each year, T,
the present value of the loss, PVT,
will increase beyond D0
over time for two reasons. First, it will grow due to expected
real increases in labour productivity, w.
Second, it will also grow due to expected wage inflation, p.
At the same time, though, the amount that will be lost in year T
must be discounted by the nominal interest rate. In equation (2),
we have split out the expected nominal interest rate, r,
into its two components, the expected real rate of interest, i,
and the expected rate of price inflation, g,
according to the well-known Fisher equation,(5)
(3) (1+r) = (1+i)(1+g).
Substituting equation (2) into equation
(1), the amount the courts should award is given by
then the term following the summation
sign in equation (4) equals 1 for each year, and the summation
simply equals N. Hence, because of the
offsetting numerator and denominator, equation (4) can be
(6) AN =
N × D0 .
In other words, discounting is
unnecessary, determining the expected rate of inflation is
unnecessary, testimony about the real rate of interest is
unnecessary, and studies of the overall average rate of growth of
labour income are unnecessary. The offset rule eliminates the
need for many costly portions of litigation by allowing the court
simply to multiply the amount of the loss in the initial year
times the number of years that the plaintiff is expected to
suffer the loss.
Of course interest rates, inflation
rates, and labour growth rates cannot be expected to remain
constant over time. To take account of the possible variations of
these variables, we rewrite equation (2) as
or more simply,
A sufficient condition for equation (8)
to reduce to the offset rule in equation (6) is
The condition in equation (9) is
actually much weaker than the condition in equation (5) because
it allows the variables to change considerably from one time
period to the next, even moving differently from each other over
time. So long as the product of the terms equals one for each
time interval, T, the assumptions of
the offset rule will hold. A stronger condition would be
; t = 1, ..., T
It is this condition, on average, that
we examine in the empirical section of our study. If this
condition holds, then so will the condition in equation (9), and
hence the offset rule of equation (6) will prove useful.
Life Cycle and Life Expectancy
Because the courts cannot know with
certainty that the labour productivity of the plaintiff will grow
in a fashion that corresponds directly with the overall growth of
labour productivity in the economy, it is reasonable to
incorporate a term, l(t), into equation
(8). In particular, we let l(t)
represent the deviations over the plaintiff's life cycle
from the economy-wide increases in labour productivity.
Similarly, we allow for varying life expectancies by
incorporating into the equation the variable, x(t),
representing the probability that the plaintiff will still be
alive in year t.
If the condition in equation (10) holds,
then (11) above reduces to
As is clear from this equation,
including personalized data concerning productivity over the
plaintiff's life cycle or the plaintiff's life expectancy in a
specific case still does not require the use of interest rates,
expected inflation rates, or overall labour productivity growth
if the assumptions underlying the offset rule hold.(6)
Formulating the Econometric
The condition set out in equation (10)
says simply that a sufficient condition for the offset rule to
hold would be that for each year the expected growth in nominal
labour income, (1+wt)(1+pt),
equal the expected nominal rate of interest, rt
. But because we cannot readily observe people's expectations
about future variables, especially in an adversarial setting, we
must reformulate the model to make use of observable variables.
We begin by rewriting equation (10) in logarithmic form:
(13) log (1 + wt)
+ log (1 + pt) = log (1 + it)
+ log (1 + gt)
We cannot directly observe these ex
ante, expected variables, and so we make use of the concept
of rational expectations(7),
assuming that people's expectations will deviate from the
observed values of the variables non-systematically. These ex
post, realized values of the variables we represent with epwt
, and epgt,
respectively. Writing the ex post, realized nominal
interest rate as eprt
, and with the assumption of rational expectations, we have
(14) log (1 + eprt)
= ut + log (1 + it)
+ log (1 + gt)
is a forecasting error equal to log (1 + eprt)
- log (1 + rt), which shows the
extent to which expectations formed at time t - 1 about
log (1 + rt) turn out to be
incorrect. If expectations about it
are formed rationally, then ut
will be a random variable with mean zero, and it will be
uncorrelated with any other variables in the set of information
used to form the forecasts.
Similarly, we can link ex ante,
expected, rates of growth in nominal wage rates with ex post,
realized, growth rates:
(15) log (1 + epft)
= vt + log (1 + wt)
+ log (1 + pt) .
Again, if expectations about the growth
of nominal wages are formed rationally, then vt
is a random forecasting error with mean zero. Subtracting
equation (14) from equation (15), we obtain
(16) dt =
log(1+epft) - log(1+eprt)
= [log(1+wt) + log(1+pt)
- log(1+it) - log(1+gt)]
+ vt - ut
Under rational expectations, if the
assumptions underlying the offset rule are correct, then we
should observe that the measured, observable, variable dt
is random with mean zero.(8)
Which Interest Rate?
It is obvious when one examines typical
yield curves that interest rates differ with the maturity of the
security. Hence, it would seem, any test of the assumptions
underlying the offset rule must depend crucially on whether one
uses the interest rates on long-term or short-term financial
instruments. There is an additional potential problem that if the
tests are carried out using, say, quarterly observations of
long-term interest rates (as we do for this study), then it is
appropriate to make certain that the maturity of the security
matches the observation period; otherwise there is a possibility
that the reformulation of expectations during the observation
period will affect the forecast errors about nominal interest
rates in some systematic fashion.(9) A
convenient solution to the problem is to calculate 90-day holding
period rates of return for longer term securities.(10) This latter process has the advantage that (a)
it satisfies the concerns of those who argue that plaintiffs'
awards are typically invested in long-term, not short-term
instruments, and (b) it allows us to avoid, somewhat, the
problems with data collection, namely that the maturities for
which data are available differ from country to country. By
computing the 90-day holding-period rates of return for
longer-term securities, we are able to match the relevant term
over which the nominal interest rates hold with the quarterly
data for nominal wage growth.
The data used to calculate the dt
values are from the International Monetary Fund's International
Financial Statistics. For each country, we use the broadest
possible definition of labour income to try to reduce the bias of
including only wage earners or salaried employees. The labour
income series are all index numbers based in 1990. Unfortunately,
the underlying wage or income series differ considerably between
countries. Thus, the coverage ranges from the very narrow
(Belgium, France) to the very broad (Germany, UK). It is our
expectation that the narrowly defined series are closely
correlated with more broadly defined series for each country.
These definitions are given in Table 1.
The interest rate series use to calculate the holding period rates of return are the yields on long-term government bonds. For some countries (Australia, Denmark, Ireland, Italy, Japan, Netherlands, United Kingdom) the maturity of these bonds is shown as a single constant number through the sample period (In the case of Italy it was constant only up to the second quarter of 1991). For the rest (Belgium, France, Germany, Switzerland) it is given as a range of years above a lower bound. For Belgium, Germany and Switzerland we arbitrarily use a
Table 1: Definitions of
Australia Average weekly earnings,
excluding overtime, for full-time adult males for the pay period
ending on or near the middle of the quarter
Belgium Average hourly earnings of
workers in manufacturing, mining, quarrying, and construction.
Electricity, gas and water were included before 1976.
Denmark Hourly earnings for mainly male
workers in manufacturing and construction in firms employing 20
(6 before 1988) or more persons.
France Wages and other labour costs
established by law or contract in the manufacture of electrical
and mechanical machinery and equipment.
Germany Gross wages, including family
allowance, paid by employers in the industrial base of 1980,
Ireland Average weekly earnings by all
industrial workers in manufacturing.
Italy Contractual wages per worker in
manufacturing, mining, quarrying, utilities and construction.
Japan Monthly contract cash earnings of
regular workers in all industries.
Netherlands Hourly earnings in
Switzerland Hourly earnings of all male
workers aged 20 or more, including overtime, paid leave, bonuses
and family allowances. Reported only annually after the first
quarter of 1984.
United Kingdom Average monthly earnings
of all workers in the whole economy.
Source: IMF, "International
maturity one year greater than the given
lower bound in the holding period calculation.
Of course, data for many other countries
are given in International Financial Statistics.
However, either they refer to countries we have already
considered in earlier work (Canada and the U.S.) or they are very
sparse or of even lower quality than the data we did use.
There are two properties of the
behaviour of the dt
series that interest us. The first is whether it fluctuates
randomly around a constant median, or mean, or whether it grows
over time so that there is no fixed mean. A sufficient condition
for there to be a constant mean is that dt
be weakly stationary. Then, given that dt
is stationary, we are interested in whether its mean is zero.
Should there be any growth at all in the
series, the offset rule would not likely provide a very good
basis for generating damage awards. A non-stationary dt
would imply that either the numerator or the denominator of
equation (8) was growing or shrinking over time, and doing so
faster than the other term. It would mean that one side of
equation (10) was changing faster than the other over time. This
result is important because even if on average the sides
of equation (10) equal each other, we would not want to use the
offset rule if this equality is just a coincidence and the two
sides of the equation can reasonably be expected to diverge from
each other in the future.
An examination of the plots of dt
for each country, shown in Figures 1 to 12, does not reveal any
growth in the variable.(11)
However, this graphical procedure has the disadvantage of being
rather subjective. What some observers conclude is a long run
upward trend might appear to others as just a temporary upward
disturbance. Also, it may be quite difficult to detect moderate
systematic growth in a series that has large seasonal and
An alternative and somewhat less
subjective procedure is to examine the autocorrelations for signs
of non-stationarity. In Table 2 we show the sample
autocorrelations for each country's dt.
The number appearing in parentheses under a country's name is the
maturity of its government bonds whose interest rate we used.
Please note, however, that regardless of the maturity of the
security, we used the 90-day holding-period rate of return for
all the empirical work in this study. The horizontal line part
way down the table marks the division between countries for which
a single, fixed maturity was provided (the value for Italy's
bonds is the average of the range of maturities 15 to 20 years)
and those for which we assigned a maturity as discussed above. We
used two different maturity values for France; hence there are
two sets of results for France.
A more formal test for stationarity
using seasonal data is provided by Hylleberg, Engle, Granger, and
Yoo.(12) This test allows for the presence of unit roots
at frequencies of a year, a half year, or a quarter of a year.
The test is based on the regression
= a + b1y1t + b2y2t
+ b3y3t-1 + b4y3t
+ c2q2 + c3q3 + c4q4 + r1y4t-1
+ ... + rpy4t-p
where: y4t = dt - dt-4;
y1t = dt-1 + dt-2 + dt-3 + dt-4;
y2t = -dt-1 + dt-2 - dt-3 + dt-4; and
-dt-1 + dt-3.
We choose p, the number of lagged terms y4t-1 to y4t-p added to equation (17), so that the residuals are free of auto correlation. The series dt will have no unit roots at any freqency if b1 < 0, b2 < 0, and b3 and b4 are not both zero. Thus to test for the presence of unit roots in our quarterly series, we use least-squares results from equation (17) to test the following hypotheses:
(i) H0: c1 = 0 (ii) H0: c2 = 0
< 0 H1: c2
(iii) H0: c3 = c4 = 0
H1: either c3 0, or c4 0, or both 0.
Table2: Sample Autocorrelations
(Standard Errors in Parentheses)
Country 1 2 3 4 5 6 7
Australia .316 .134 .240 .186 .198 .0487 .0502
(15) (.0874) (.0957) (.0971) (.102)
(.104) (.107) (.107)
Denmark .0428 .411 -.0133 .307 -.0765 .157 .0104
(5) (.0874) (.0875) (.101) (.101) (.108)
Ireland .0482 .148 -.0691 .215 -.201 .0978 -0.120
(15) (.0825) (.0827) (.0845) (.0848)
(.0885) (.0915) (.0922)
Italy .407 .286 .219 .176 .172 .0982 .123
(17.5) (.135) (.156) (.165) (.170)
(.173) (.176) (.177)
Japan .151 -.00515 .187 .460 .0318 -.112 .0208
(7) (.0945) (.0966) (.0966) (.0998)
(.117) (.117) (.118)
Netherlands .161 .121 .190 .187 .0317 .109 -.00594
(10) (.0917) (.0940) (.0953) (.0984)
(.101) (.101) (.102)
U K .0478 .00458 .0765 -.0138 -.0730 .168 .0209
(20) (.0891) (.0893) (.0893) (.0898)
(.0898) (.0903) (.0927)
Belgium .147 .459 .102 .510 -.00723 .278 .0104
(6) (.0816) (.0834) (.0988) (.0995)
(.116) (.116) (.120)
France .392 .325 .418 .176 .178 .206 .232
(6) (.0816) (.0933) (.101) (.112) (.113)
(12) .358 .248 .358 .0757 .0723 .126 .127
(.0816) (.0915) (.0959) (.104) (.105)
Germany .0495 .0195 .205 .155 .0226 .0195 .00872
(4) (.0874) (.0876) (.0876) (.0912)
(.0932) (.0932) (.0932)
Switzerland .285 .167 .134 .399 -.0498 -.174 -.242
(6) (.113) (.121) (.124) (.126) (.141) (.141) (.144)
Tests (i) and (ii) are very much like
standard one-tailed t-tests (large
negative t-ratios induce rejection of H0),
except that one uses the table of (estimated) critical values
provided by Hylleberg et. al. instead of the usual t
or normal tables. Similarly, test (iii) resembles a standard F-test
(large positive values of the F-ratio
induce rejection of H0),
except one uses the tables in Hylleberg et. al. instead of the
usual F table. If we reject all of null
hypotheses (i), (ii) and (iii) above we conclude that there is no
unit root at any frequency, and so the series is stationary.
In testing for unit roots, each country's regression was initially specified in the general seasonal form (17) without any lags of y4t on the right; i.e. with p = 0. The residuals from these regressions were examined for autocorrelation using the Breusch-Godfrey LM statistic for lags 1 to 8 inclusive and the Ljung-Box portmanteau statistic for lags 1 to 12 inclusive. If these statistics revealed residual autocorrelation, lagged values of y4t were added, and seasonal dummies with insignificant coefficients were removed. These steps succeeded in removing the serial correlation, especially as measured by the Breusch-Godfrey statistic. There was no significant residual autocorrelation, even at a significance level of 15%, left in any of these restricted equations whose results are shown in Table 3.
Table 3: Values of Unit Root Test Statistics
Country Sample est. b1 est. b2
(maturity) Size (t-ratio) (t-ratio) F-ratio
Australia 127 -.116 -.361 24.0
(15) (-3.54) (-6.23)
Denmark 121 -.0987 -.211 7.95
(5) (-2.73) (-3.75)
Ireland 142 -.117 -.159 25.8
(15) (-2.63) (-3.85)
Italy 51 -.110 -.310 12.3
(17.5) (-2.40) (-3.40)
Japan 108 -.0993 -.242 20.5
(7) (-2.85) (-4.72)
Netherlands 115 -.139 -.320 28.4
(10) (-3.88) (-6.18)
United Kingdom 122 -.224 -.307 32.7
(20) (-5.21) (-6.32)
Belgium 144 -.0869 -.360 12.0
(6) (-3.54) (-6.51)
France 140 -.085 -.553 23.0
(6) (-3.07) (-7.23)
France 140 -.112 -5.80 20.8
(12) (-3.54) (-7.36)
Germany 126 -.151 -.284 29.3
(4) (-3.87) (-5.95)
Switzerland 72 -.182 -.254 10.9
(6) (-3.48) (-3.61)
The estimated critical values used for
testing the hypotheses of interest depend upon: the sample size,
whether an intercept is included, and whether seasonal dummies
are included. They are tabulated in Hylleberg, et. al. (1990).
All the t-ratios in Table 3 are to the
left of the 5% critical value except those for b1
for Ireland, Denmark, and Japan, and those are to the left of the
10% critical value. All of the F ratios
exceed the 5% critical value. Thus we conclude that there are no
unit roots at any frequency for any country; i.e. all of the
series are weakly stationary.
Having verified that for each country, dt
is stationary, and thus has a constant mean, we now consider
whether these means are different from zero; if they are not
different from zero, there is strong support for the offset rule.
An unbiased estimator of the population
mean of a weakly stationary series is the sample average. This
estimator has the advantages of extreme simplicity and of being
independent of the particular structure of the time series
process generating the data. In contrast, estimators of the
population mean derived from ARMA models are generally biased and
are consistent only if the ARMA specification corresponds to the
data generating process. Even if this condition is met their
asymptotic variance is no smaller than that of the sample
average. To account for the autocorrelation in the data the
standard error of the sample average is estimated by the square
where n is the sample size and (h) is the estimated autocovariance at
The results of these calculations are
shown in Table 4. Ten of the twelve t-ratios are below one and
the remaining two are less than 1.5. Thus for every country we
are unable to reject the null hypothesis that the population mean
is zero for any (asymptotic)
significance level below 18%.
Table 4: Estimates of Population
Country Est. Mean
United Kingdom .000238
With this study, the assumptions underlying the offset rule have now been confirmed for thirteen different countries. We cannot, for any one of these countries, reject the hypothesis that the nominal growth rate of labour income is, on average, equal to the nominal holding period rate of return.
These results confirm that it is appropriate for the courts in these countries to use the offset rule as a starting point for the calculation of damage awards. It is, in fact, more appropriate to use the offset rule as a starting point than it is to use a "net" discount rate of 2% or 2.5%, as many jurisdictions do to approximate the real interest rate.
We do not deny that there are personal differences from case to case that might merit inclusion in the calculations of damage awards. Rather our conclusion from these studies is that the courts should be freed to spend more time considering these personal differences; the courts should not have to consider the economy-wide aggregates that make up nominal wage-rate growth or the nominal rate of interest.
2. "Real Rates, Expected Rates, and Damage Awards," Journal of Legal Studies, vol XX, (June 1991), 439 - 62; "Simple Calculations to Reduce Litigation Costs in Personal Injury Cases: Additional Empirical Support for the Offset Rule," Osgoode Hall Law Journal, vol 32 number 2 (summer 1994), 197 - 223.
3. Australia, Belgium, Denmark, France, Germany, Ireland, Italy, Japan, the Netherlands, Switzerland, and the United Kingdom.
4. Also known as the total offset rule.
5. I. Fisher, "Appreciation and Interest," (1896) 2, Publications of the American Economic Association (3d) 341.
6. Probabilities of unemployment can be included the same way. The interested reader is referred to our earlier work, supra n 2,3, where we develop these equations in greater detail.
7. John F. Muth, "Rational Expectations and the Theory of Price Movements," 29 Econometrica 315 (1961). See also Frederic S. Mishkin, "The Real Interest Rate: A Multi-country Empirical Study," 17 Canadian Journal of Economics 283 (1984).
8. If equation (13) holds, then equation (16) reduces to dt = vt - ut, which is a random variable with mean zero.
9. See, for example, Mishkin, op. cit.
10. See R. J. Shiller, "The Volatility of Long-Term Interest Rates and Expectations Models of the Term Structure," 87 Journal of Political Economy 1190 (1970); R. J. Shiller, J.Y. Campbell, and K.L. Schoenholtz, "Forward Rages and Future Policy: Interpreting the Term Structure of Interest Rates," 1 Brookings Papers on Economic Activity 173 (1983). The computations are fairly simple but also fairly drawn out. The reader is referred to our earlier work and to that of Shiller and of Shiller et al. for more details. As in our previous studies, when appropriate in the formulae, we approximated the ex post nominal rate of return for securities with maturities of, say, 9 years with the returns on 10-year securities.
11. The means and standard deviations shown in the figures are simple sample means and standard deviations. Because of autocorrelation, the correct standard error to use for hypothesis testing is provided infra, equation (18)
12. Hylleberg, S., R.F. Engle, C.W.J. Granger and B.S. Yoo, (1990) "Seasonal Integration and Cointegration", Journal of Econometrics, 44, pp 215-238.
13. see Brockwell, P.J. and R.A. Davis (1987), Time Series: Theory and Methods. This formula was incorrectly stated in our earlier paper, Carter and Palmer (1991), although the calculations were done correctly.