This article first appeared in the spring 2007 issue of the Expert Witness.

## Introduction

In order to calculate the lump-sum, or discounted, value of a future stream of earnings, the financial expert must forecast the rate of interest (or discount rate) at which the plaintiff’s damages will be invested. Although most experts now base this forecast, to a large extent, on the rates of interest available on Government of Canada bonds, disputes have arisen concerning the manner in which the data concerning these bonds should be interpreted.

Fundamentally, the question comes down to one of whether the plaintiff can be assumed to take a “conservative” approach, in which she invests her damages in long-term bonds and holds those bonds to maturity; or whether she can be assumed to follow a “market-based” (or “speculative”) approach in which she buys and sells bonds as market conditions change.

Whereas Economica favours what I have called the conservative approach, some other economists employ the market-based approach. In this article, I first describe how interest rates are determined under each of these approaches and then I explain why I believe that the market-based approach is inappropriate.

## Definitions

Imagine the following scenario: the court has ruled (i) that one component of the plaintiff’s damages is a loss of $50,000 twenty years from now; and (ii) that the interest rate to be used to discount this loss is five percent. In this case, the lump sum value of the loss can be determined to be $18,844 (= $50,000/1.05^{20}).

It is possible that the court could have obtained its interest rate assumption simply by observing the rates offered on twenty-year bonds – what I will call the “posted” rates. Implicitly, in that case, it would have been assuming that the plaintiff will use his damages to purchase a twenty-year bond (or similar, long-term financial instrument) that pays five percent per annum; and that he will hold that bond until maturity. I call this the “conservative” approach.

Alternatively, however, the court might have assumed that the plaintiff will not hold his bond until maturity, but will buy and sell bonds as interest rates vary. In this approach, the effective interest rate will be the average of the rates that the plaintiff can expect to earn over the twenty year period. I call this approach the “market-based investment” approach.

Market-based investment in bonds operates in the following way: Imagine that, at a time when posted interest rates are five percent, the plaintiff has paid $18,844 to purchase a twenty-year bond worth $50,000 on maturity. Assume also that, one year later, interest rates have fallen to four percent. At that rate, it would cost $23,732 (= $50,000/1.04^{19}) to purchase a bond that paid $50,000 nineteen years in the future. The plaintiff could now sell his twenty-year bond, (which, after a year, has nineteen years left on it), for this amount – making a profit of $4,888 (= $23,732 – $18,844), or an *effective* rate of return of 25.94 percent, in one year.

Rates such as this can be earned on long-term bonds as long as the interest rates on those bonds are falling. For example, a bond that pays $10,000 ten years from now will cost $5,584 when interest rates are six percent; and a bond that pays $10,000 nine years from now will cost $6,446 when interest rates are five percent. Thus, the investor who purchases a ten-year bond at six percent will be able to sell it one year from now for a profit of $862 (= $6,446 – $5,584), or an effective return of 15.44 percent (= $862/$5,584), if interest rates fall to five percent.

Whenever interest rates are falling, the investor who actively trades long-term bonds will make effective rates of return that exceed the rates that would be obtained if those bonds were held to maturity.

Conversely, however, if interest rates are rising, the market-based trader will earn effective rates that are lower than the posted rates. For example, if the interest rate is seven percent, a bond that pays $10,000 nine years from now will cost $5,439 (= $10,000/1.07^{9}). It was seen above, however, that a bond that pays $10,000 *ten* years in the future will cost $5,584 when interest rates are six percent. Thus, an investor who purchases a ten-year bond at six percent will *lose* $145 (= $5,439 – $5,584), for an effective “return” of -2.6 percent, if interest rates fall to seven percent and he sells it one year later.

What is less clear from these examples is that, if the effective interest rates are averaged over a long period of time, they will equal the rates that investors would have obtained had they held their bonds to maturity. Over the long-run, the investor who sells multi-year bonds after one year, and replaces them each time with other multi-year bonds, will average the same rate of return as an investor who buys and holds bonds to maturity.

## Data

Table 1 reports both the posted and the effective rates of interest on Government of Canada bonds of 10+ years to maturity, for the last 25 years. It is seen in the first column of this table that posted interest rates have fallen almost continuously over the entire period: from 15.22 percent in 1981 to 4.39 percent in 2005.

Given the discussion above, therefore, we expect to see that effective interest rates over most of this period would exceed the posted rates. It is seen in the second column of Table 1 that this is what happened. Particularly noticeable is the effective rate in 2005: 15.05 percent at a time when posted rates were in the four to five percent range.

As those who argue for the use of effective rates anticipate that the average of those rates will equal the average of the posted rates over the long run, Table 2 reports average posted and effective rates, both in nominal and “real” terms, for various sub-periods in the last 50 years.

It is seen there that, whether nominal or real (i.e. net of inflation) rates are used, effective rates have exceeded posted rates in every sub-period in the last 25 years. For example, whereas the average effective nominal rate was 12.49 percent between 1981 and 2005, the average posted rate was 8.64 percent. The comparable real rates were 8.74 and 5.00 percent, respectively, a difference of over three percentage points in each case.

Only if a fifty year period is used do the effective and posted rates begin to approach one another – primarily because the 25 year period from 1956 to 1980 experienced extremely low effective rates of return (hence balancing the high rates from 1981-2005 in the calculation of the fifty-year average).

## Drawbacks to Use of the Effective Rate

A review of the information contained in Tables 1 and 2 makes it clear that there are many disadvantages to the use of the effective rate as an indicator of the rate of return that plaintiffs will be able to obtain in the future.

1. The most serious difficulty with this rate is that it has been extremely variable. As a result, one can obtain almost any estimate of the real effective rate that one wishes, simply by choosing the appropriate time period.

Those who wish to argue for a very high rate might choose the fifteen-year period 1991-2005, with an average rate of 9.39 percent, for example. Whereas those who wish to argue for a much lower rate could choose the fifty-year period 1956-2005, with an average rate of 3.71 percent – or even the 60-year period 1946-2005, with an average of 2.44 percent.

There is no sound reason for choosing any one of these periods over the other. For example, it would be difficult to justify averaging together the 1956-1980 average, of -1.31 percent, with the 1981-2005 average, of 8.74 percent, to obtain a forecast of the rate of return obtainable over the next ten years.

2. A related problem is that the extreme variability of the effective rate virtually guarantees that the rate of return the investor will actually earn will differ significantly from the average rate obtainable over the long run. Hence, it would never be prudent to advise plaintiffs to adopt the market-based investment approach on which the effective rate is based.

3. Because long-term interest rates have been falling for 25 years, they have reached levels so low that most analysts believe they will not fall further – they certainly cannot fall by more than ten percentage points, as they did between 1981 (15.22 percent) and 2005 (4.39 percent). This means that the very high effective rates that have been observed in the last 25 years will not be observed again. Indeed, it is highly likely that nominal rates will rise over the near future, causing effective rates to fall, perhaps even into negative figures. As a result, plaintiffs would not be well-advised to engage in market-based investment in bonds.

4. Even if effective rates are positive, and above posted rates, the effective rate is not an appropriate rate for determining long-term rates of return on investments.

Return to the example developed in the first section of this article: The court has found that the plaintiff will lose $50,000 twenty years from now. Hence, if the interest rate is five percent, the lump-sum value of the plaintiff’s damages amount to $18,844. If, between now and one year from now, the interest rate falls to four percent, the plaintiff will be able to sell his bond for $23,732, earning a one-year rate of return of 25.94 percent.

But the plaintiff still needs to set aside enough money to replace his future $50,000 loss. At the new four percent interest rate, this will cost him $23,732. The money he has “earned” by selling his bond now has to be spent to replace it. Thus, the high effective rate of interest is illusory.