**The Offset Rule:**

**Some
Multinational Evidence**

by

R.A.L. Carter and John P.
Palmer^{(1)}

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:**

**Some
Multinational Evidence**

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
injury trials.

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
formulation**

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*
=

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
$

The summation in equation (1) is
composed of a series of annual amounts, each of which can be
written as

This expression says that it is
reasonable to expect that in each year, * T*,
the present value of the loss,

(3) (1+*r*) = (1+*i*)(1+*g*).

Substituting equation (2) into equation
(1), the amount the courts should award is given by

If

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
rewritten as

(6) *A*_{N }*=
N × D*_{0} .

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

(10) (1+*w*_{t})(1+*p*_{t})
= (1+*i*_{t})(1+*g*_{t})
; 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
Effects**

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

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
Model**

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+w**_{t}**)(1+p**_{t}* )*,
equal the expected nominal rate of interest,

(13) log (1 +* w*_{t})
+ log (1 + *p*_{t}) = log (1 + *i*_{t})
+ log (1 + *g*_{t})

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 **epw**_{t}**
, epp**_{t}**
, epi**_{t}*
, *and

(14) log (1 +* epr*_{t})
= *u*_{t} + log (1 + *i*_{t})
+ log (1 + *g*_{t})

where **u**_{t}
is a forecasting error equal to log (1 + *epr*_{t})
- log (1 + *r*_{t}), which shows the
extent to which expectations formed at time *t* - 1 about
log (1 + *r*_{t}) turn out to be
incorrect. If expectations about **i**_{t}**
**and **g**_{t}
are formed rationally, then **u**_{t}
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 + *epf*_{t})
= *v*_{t} + log (1 + *w*_{t})
+ log (1 + *p*_{t}) .

Again, if expectations about the growth
of nominal wages are formed rationally, then *v*_{t}
is a random forecasting error with mean zero. Subtracting
equation (14) from equation (15), we obtain

(16) *d*_{t} =
log(1+*epf*_{t}) - log(1+*epr*_{t})
= [log(1+*w*_{t}) + log(1+*p*_{t})
- log(1+*i*_{t}) - log(1+*g*_{t})]
+ *v*_{t} - *u*_{t}
.

Under rational expectations, if the
assumptions underlying the offset rule are correct, then we
should observe that the measured, observable, variable **d**_{t}
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.

**Data**

The data used to calculate the **d**_{t}
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
Wage Series**

Country Definition

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,
including construction.

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
manufacturing.

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
Financial Statistics*

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 **d**_{t}
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 **d**_{t}
be weakly stationary. Then, given that **d**_{t}
is stationary, we are interested in whether its mean is zero.

Should there be any growth at all in the
**d**_{t}
series, the offset rule would not likely provide a very good
basis for generating damage awards. A non-stationary **d**_{t}
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 **d**_{t}
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
nonseasonal fluctuations.

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 **d**_{t}.
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

(17) *y4*_{t}*
= a + b1y1*_{t}* + b2y2*_{t}*
+ b3y3*_{t-1}* + b4y3*_{t}*
+ c2q2 + c3q3 + c4q4 + r*_{1}*y4*_{t-1}*
+ ... + r*_{p}*y4*_{t-p}*
+ e*_{t}

where: *y4*_{t}*
= d*_{t}* - d*_{t-4}*;*

*y1*_{t}* = d*_{t-1}*
+ d*_{t-2}* + d*_{t-3}*
+ d*_{t-4}*;*

*y2*_{t}* =
-d*_{t-1}* + d*_{t-2}*
- d*_{t-3}* + d*_{t-4}*;
*and

*y3*_{t}* =
-d*_{t-1}* + d*_{t-3}*.
*

We choose * p*,
the number of lagged terms

(i) **H**_{0}**:
****c**_{1}
= 0 (ii) **H**_{0}**:
****c**_{2}
= 0

**H**_{1}**:
****c**_{1}
< 0 **H**_{1}**: ****c**_{2}
< 0

(iii) **H**_{0}**:
****c**_{3}
= **c**_{4}
= 0

**H**_{1}**:**
either **c**_{3}*
*0, or

Table2: Sample Autocorrelations

(Standard Errors in Parentheses)

Lag

**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)
(.109) (.110)

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)
(.115) (.118)

(12) .358 .248 .358 .0757 .0723 .126 .127

(.0816) (.0915) (.0959) (.104) (.105)
(.105) (.106)

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

In testing for unit roots, each
country's regression was initially specified in the general
seasonal form (17) without any lags of **y4**_{t}
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

**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

Having verified that for each country, **d**_{t}
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
root of

where * n* is the
sample size and

lag * h*.

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
Means**

Country Est. Mean

(maturity) (t-ratio)

Australia .00157

(15) (.295)

Denmark -.00201

(5) (-.526)

Ireland .00301

(15) (.506)

Italy -.00543

(17.5) (-.563)

Japan .000812

(7) (.204)

Netherlands -.00196

(10) (-.505)

United Kingdom .000238

(20) (.397)

-----------------------------------------------------------------------

Belgium -.000782

(6) (-.254)

France .00346

(6) (1.09)

(12) .00425

(.948)

Germany .000706

(4) (.269)

Switzerland .00332

(6) (1.34)

**Conclusion**

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 *d*_{t} = *v*_{t}*
- u*_{t}, 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.