The Offset Rule:

Some Multinational Evidence


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 = 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 written as




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 rewritten as

(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

(10) (1+wt)(1+pt) = (1+it)(1+gt) ; 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 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 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+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 , eppt , epit , 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)

where ut 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 and gt 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 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 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 dt 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 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 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

(17) y4t = a + b1y1t + b2y2t + b3y3t-1 + b4y3t + c2q2 + c3q3 + c4q4 + r1y4t-1 + ... + rpy4t-p + et

where: y4t = dt - dt-4;

y1t = dt-1 + dt-2 + dt-3 + dt-4;

y2t = -dt-1 + dt-2 - dt-3 + dt-4; and

y3t = -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

H1: c1 < 0 H1: c2 < 0

(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) (.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 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 root of

where n is the sample size and (h) is the estimated autocovariance at

lag h.(13)

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


Germany .000706

(4) (.269)

Switzerland .00332

(6) (1.34)


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.