Retiring the Historical Net Discount Rate
Abstract
The methodology of using a net discount rate (“NDR”) to calculate the present value of future labor earnings is now over 50 years old. First found in public health economics, the earnings NDR later reached forensic economics where it was adapted to other calculations such as the present value of medical costs. At first, some forensic economists embraced the historically determined wage NDR arguing that it was simple, fair, and 0% since annual wage growth and interest rates had been fairly equal to that time. The 0% wage NDR did not sit well with some other forensic economists, and so a discourse began in the literature regarding its numeric value. Largely missing from the debate has been an analysis of the historically determined NDR, as it has been constructed and used, as a legitimate present value methodology. Building upon the initial criticisms of the NDR by Jones (1985, 1986) and the recent work of Foster (2015), this article examines the historical wage NDR method and concludes that, for several reasons, it is theoretically faulty, empirically inconsistent, and not relevant to forensic economics.
I. Introduction
The methodology of using a net discount rate (“NDR”) to calculate the present value of future labor earnings is now over 50 years old. First found in public health economics, the earnings NDR later reached forensic economics where it was adapted to other calculations such as the present value of medical costs. At first, some forensic economists embraced the historically determined wage NDR arguing that it was simple, fair, and 0% since annual wage growth and interest rates had been fairly equal to that time. The 0% wage NDR did not sit well with some other forensic economists, and so a discourse began in the literature regarding its numeric value. Largely missing from the debate has been an analysis of the historically determined NDR, as it has been constructed and used, as a legitimate present value methodology. Building upon the initial criticisms of the NDR by Jones (1985, 1986) and the recent work of Foster (2015), this article examines the historical wage NDR method and concludes that, for several reasons, it is theoretically faulty, empirically inconsistent, and not relevant to forensic economics.
This article begins with a historical retrospective on present value damage calculations which shows that the historical wage NDR emerged in forensic economics at the end of a long past period when inflation and wages had been persistently outpacing interest rates. Since its inception, the wage NDR has swung wide both in value and statistical significance. While the wage NDR is supposed to represent an average past annual constant rate at which returns to financial investment outpaced labor earnings growth, this article demonstrates that the wage NDR does not produce such a constant. The article points out that singular, constant wage NDRs are not a panacea to determine the present value of the varied lifetime earning situations encountered in forensic economics. Lastly, the NDR masks investment risk which could convey to the layman trier of fact the expectation that tort victims can obtain riskless financial profits unattainable in the financial market. Concluding, the article argues that the NDR is an outdated, incorrect, and unreliable tool to address the complicated present value problems found in forensic economics, and therefore, the NDR should be retired from the literature.
II. Terminology
This article contains terminology that forensic economists have become accustomed to using regarding net discount rates. Before going further, definitions are provided for clarity to the topics later presented.
The NDR method is an approach that some forensic economists use to compute the present discounted value of future money. The term “periodic” is often associated with the NDR method, but unless noted, the NDR method is nearly always periodic in one-year increments—an NDR of p% is a present value methodology which uses p% as the annual discount rate.
When a person is indifferent between $1.00×(1+d)−1 today and $1.00 in one year, d is the time-value discount rate which quantifies the person's indifference between having money today or one year later. Going a step further, we expect the same person to be indifferent between $1.00×(1+g)×(1+d)−1 today and $1.00 in one year when g is added as an annual percentage rate of increase in the money payment. The NDR or (d–g)×(1+g)−1 is simply a numeric constant, time-value discount rate, net of growth, which quantifies the indifference.1
When reducing future wage earnings to present value, a wage NDR is used; present values of future medical costs are computed using a medical cost NDR; etc.—the descriptor attached to an NDR gives the basis for the annual growth rate term. The basis for the interest rate component of an NDR is not as intuitive as that for growth. When a forensic economist uses the jargon of an n-year NDR, the NDR is periodic to one year, but the interest rate used to construct the NDR is derived from the annual yield on an n-year maturing security. For example, a 10-year wage NDR is an annual NDR constructed from a g annual rate of growth in wages and a d annual return on a 10-year security. A 10-year wage NDR should not be confused with the difference in the growth in wages over 10 years and the total return to maturity on a 10-year security.2
While a current3 NDR is created by the pronouncement of a specific forecast of a constant g and a constant d, a historical NDR is a statistic computed from time series data. The historical n-year wage NDR is often presented as the average of an annual historical time series of n-year wage NDRs. Sometimes, instead of averaging a time series of NDRs, the historical n-year wage NDR is calculated using a past average annual yield on a type of security and, over the same past period, the average annual rate of earnings growth.
III. Historical Context and the Emergence of the Net Discount Rate
A. Early Present Value Calculations
Case law dating to the early 19th century has required that future tort damages are to be awarded at their present value. Early legal decisions consistently endorsed damages that were calculated from the victim's lost annual earning level, a life expectancy table, and a table of present value factors. For example, Bowditch's “Tables of the Present Value of a Life-Annuity at Any Age, According to Dr. Wigglesworth's Bill of Mortality” (1833) gave life expectancy present value factors computed using 5% and 6% annual interest rates. Damage calculations for decades were simply made by multiplying a constant annual lost earning level by a present value factor.
Around the time of the Korean War, courts became concerned about the effects of inflation on tort damage awards.4 From 1939 to 1953, the 10-year U.S. Treasury interest rate averaged 2.1% while the Consumer Price Index increased at the average annual rate of 4.8% and total manufacturing average weekly earnings increased at the average annual rate of 8.2%.5 Because inflation and wages had been persistently outpacing interest rates, the routine of applying substantial financial annuity table discounts to earnings losses was called into question. As an example of inflation concern, in 1948 when reviewing the adequacy of an award, the Supreme Court of Washington wrote “(w)e are also keenly cognizant of the fact that the purchasing power of money is less today than it was 10, 15, or 20 years ago. This court has given utterance to that thought many times.” (Kellerher v. Porter, 1948, p. 666)
B. The Economic Analysis of the Present Value of Lifetime Earnings
Beginning in the late 1950s, the post-war generation of economists became schooled in the quantification of lifetime earnings through the publications of Mincer (1958), Schultz (1961), Weisbrod (1961 a, b), Becker (1962), and Miller (1965). At that same time, publications specific to the analysis of the present value of individuals' lifetime earnings began to appear such as Cheit's 1961 book in which he “assumed that wages would have risen at the average rate of 4 per cent per year throughout the working life expectancy of the cases studied” (p. 75) and regarding interest rates he stated that the “current yield is approximately 4 per cent. This higher and more conservative discount rate is used for discounting future earnings in this study” (p. 76). By the time of Cheit there was consistent evidence that interest rates were expected to approximately equal wage growth.
C. The First Use of the Net Discount Rate
The first use of a current “net discount” rate to compute the present value of lifetime earnings appeared in the public health economics literature when Mushkin (1962) stated:
(t)he growth in earnings and the discount rates are essentially combined in the mechanics of the estimating procedure. A 5 per cent net discount figure is used (that is, an 8 per cent discount rate, less a 3 per cent increase in average earnings per annum.. (p. 149)
Mushkin formed6 her wage NDR by assuming “that average earnings continue to rise in the future as they have in the past decades. (A composite rise in earnings of 3% per annum is assumed)” and she incorporated “an arbitrary interest rate of 8%, a rate used in recent studies of returns on educational capital” (pp. 148-9). A year later in 1963, Klarman (1965) presented a paper at a conference (which Mushkin discussed) that showed his own construction of an NDR:
(i)t is convenient, facilitating the calculations of present value, to combine two rates of change that are simultaneously operative into a single rate. For example, by coupling a discount rate of 4% with a projected extra price rise of 1.25% a year we have the equivalent of an effective net discount rate of 2.72% (1.04 ÷ 1.0125 = 1.0272). (p. 374).
Using the same mathematics as he listed in the above quote, but with a 1.75% earnings growth rate, Klarman presented his own wage NDR of 2.21%.
As opposed to the simple discounting of future wage amounts at arbitrary interest rates,7 Klarman presented a thoughtful analysis of discounting future economic amounts by distinguishing between the private and societal rates of discount and by noting that “the individual's market calculations and collective calculations need not to coincide” (p. 372). Instead of discounting individual future earning capacities, the social NDR used in public health economics is related to the earning capacity of a population where the return to society of specific money amounts on certain dates is largely irrelevant.8 For that purpose, a growth/discount constructed NDR was seen as a reasonable tool to normatively decide perceived long-term required rate of return for current public investment.
D. The Forensic Economist Appears in Court
In his 1966 comprehensive guide to wrongful death damages, Speiser (1996) wrote that:
(i)t should be noted that at the present time there is an increasing awareness by courts and attorneys of the need for and the value of expert testimony. Such experts are being used with increasing frequency in all areas of law. Economists and statisticians have provided tables and charts and other devices which enable juries to predict with precision the loss of future earning capacity and courts are holding that such testimony is admissible. (p. 120-121)
Specific to present value expert testimony, in 1967 a federal judge in South Carolina wrote in a wrongful death opinion that “(t)he use of actuaries, economists and mathematicians as expert witnesses to provide estimates on present value of prospective earnings, the rate of inflation, anticipated wage changes, etc., has been increasing.” (Brooks v. United States, 1967, p. 624) By at least the mid-1960s, economic evidence and expert testimony had started to replace financial actuarial tables as the means to calculate the present value of earnings losses.
With the advancement of electronic computing, earnings data accumulated and the demand for it from tort litigation became so prevalent that in 1967, the U.S. Census Bureau responded with a technical paper titled “Present Value of Estimated Lifetime Earnings” which stated on page 1 that “(o)ne of the major statistical needs this report is intended to serve is the growing demand by the legal profession for information that can be used to help ascertain the pecuniary value of impaired earning capacity.” The Census report went on to say on page 1 that “(i)ncreasingly however, economists, statisticians, and similar experts are being used to prepare such information and to testify in court regarding its meaning.” The 1967 Census report contained present value of lifetime earnings estimates using inflation-free productivity growth rates of 0, 2, 3, and 4% and “four different rates of discount” of 0, 3, 4, and 5%.
While during the 1960s courts were taking notice of economic testimony, some courts took ahold of the present value calculation themselves. In 1966, New York Supreme Court Justice William B. Lawless, Jr. opined that present value calculations were an “unrealistic burden on juries” and that such mathematical calculations should be left with the judge (1965-1966, p. 1131). In the 1967 Beaulieu v. Elliott decision, the Supreme Court of Alaska removed the “burden” of discounting to present value by dictating that inflation was to be considered a full offset to interest in damage present value calculations—the birth of the sanctioned 0% NDR.
With the growing number of economists becoming expert witnesses, academic economists began publishing articles to support their methodologies. For example, in a 1969 piece based on testimony in a wrongful death case, (Har-Pen Truck Lines, Inc. v. Mills , 1967) an economist author advocated a 4.5% growth rate in earnings and a 4% discount rate (Pyun, 1969). To the mid-1970s, present value economic analyses of earnings losses were routinely published in law reviews rather than in economic journals (Leonard, 1969) (Levinson, 1973) (McConnico, 1975) (Kirby, 1975-1976).
E. Economic Forecasting and Data Comes of Age
Before 1970, the “procedures for forecasting a time series from its own current and past values were of a somewhat ad hoc nature.” (Newbold, 1975, p. 397) In a 1970 book (and earlier articles), Box and Jenkins (1970) presented detailed, strategic methods for computing forecasts of time series. Expanding on the work of Box and Jenkins, Granger and Newbold (1974) published methods of ferreting out spurious forecasting regressions and then later Sims (1980) popularized the vector autoregresson model which has been one of the most successful models used for the analysis and forecasting of time series. In 1979, the Dickey-Fuller test pushed along the analysis of the stationarity in time series to determine forecasting suitability.
The availability of data to make economic forecasts came in the 1960-1970s with the advancement of electronic computing and networks. The University of Chicago launched its CRSP database in 1964,9the Bureau of Economic Analysis was established in 1972 and began publishing macroeconomic data, and in 1978 when it launched LABSTAT, the Bureau of Labor Statistics (“BLS”) initiated direct online computer access to more than 150,000 economic time series.10
F. The Discount Function Settles in Finance
While Irving Fisher's 1930 book The Theory of Interest set the initial bar for understanding interest rates, especially the real rate, it only briefly examined interest rates of differing maturities. In 1940, Lutz (p. 36) wrote:
(i)t has long been customary in works on the theory of interest to talk about the interest rate, and to deal with the problem of the difference between rates on different maturities by adding a footnote to the effect that the author understands by the interest rate the whole ‘family' of interest rates.
From 1940 to 1960, the finance literature began mathematizing the theory of the interest rate which facilitated the empirical measurement of the opportunity cost of money across time. In 1962, Malkiel pushed the ideas of Lutz to their mathematical forms and then Buse in 1970 moved the term structure mathematics to the realm of securities that paid coupons, the most common form of market traded debt. Without labeling it as such, Buse (1970) essentially derived a bootstrap to “establish the relationship between the change in yield for a change in coupon given the set of expected rates.” (p. 811) After Buse, McCulloch (1971) formalized the estimation of zero-coupon yields within the term structure of interest rates when he “develop(ed) a technique of fitting a smooth curve, called the ‘discount function,' to observations on prices of securities with varying maturities and coupon rates.” (p. 19) McCulloch described the discount function as “(t)he most fundamental curve describing the term structure of interest rates, the one from which all others must be derived, is the discount function δ(m). It describes the present value of $1.00 repayable in m years.” (p. 19) So by the early 1970s, theory, method, and data were in place to determine the present value of economic amounts payable at any number of years in the future.
G. Return, Risk and Stocks, Bonds, Bills, and Inflation
After the settling of the discount function in finance, empirical evidence of asset return became a hot topic to facilitate estimates of the risk factors in business cash flows and also to price financial assets such as options (e.g., Black-Scholes, 1973). In his highly regarded piece on the theory of capital asset pricing, Ross (1976) stated that for any asset, W, its ex ante expected return at the time instant t is EW = rt + bWt where rt is the riskless rate of interest and bWt is some function of expected return of the asset in excess of the riskless rate. Within the confines of the centuries old law of one price (Froot, 1995), the literature on capital asset pricing demonstrated that risk was the separator of any two future identical expected cash flows not having the same current price.
In January 1976, Ibbotson and Sinquefield published their unprecedented analysis of historical rates of return (1926 to 1974) in stocks, bonds, and bills (“SBBI”). In addition to presenting inflation-adjusted annual returns on common stocks, long-term corporate bonds, long-term U.S. government bonds, and U.S. Treasury bills, SBBI contained the net return from investing in common stocks rather than bills, long-term government bonds rather than bills, and long-term corporate bonds rather than long-term government bonds. SBBI found that over the entire 1926-1974 period, long-term corporate bonds had an average inflation-adjusted return of 1.4%, long-term Treasury bonds had an average inflation-adjusted annual return of 1.0%, and bills had average 0.1% inflation-adjusted returns. From 1929 to 1974, the real inflation-free average annual growth in the average weekly earnings in total manufacturing was 2.0%.11 So, through at least 1974, the evidence clearly pointed to a historically negative, long-term, bond-based, total return wage NDR.
SBBI compared its inflation-adjusted Treasury returns to the “substantially different” 3% to 4% returns suggested at that time by the Federal Reserve Bank of St. Louis stating:
(n)ote that we compute the net returns between monthly total returns and inflation rates. The St. Louis Federal Reserve Bank measures the difference between observed high-grade long-term corporate bond yields and lagged inflation rates. Yields measure promised returns rather than realized returns. The promise extends over the entire future life of the bond so that it should not be compared with either current or lagged inflation rates. (p. 42)
The above quote from Ibbotson and Sinquefield points out the misnomer that the real returns from the investment in bonds is computed from the promised returns determined from the price of a bond at purchase less lagging inflation, which is the logic of many NDRs. So, before the historical wage NDR clearly appeared in the forensic economic literature, it was clear that return to investment is not determinable from the promised returns on bonds.
H. Forensic Economics Gets Prominent Academic Journal Attention
In 1974, forensic economics first appeared in the Journal of Risk and Insurance (“JRI”) with a comment by G. William Dick titled “Determining the Present Value of Future Income: Selecting Income Growth Rates.” For his reader's education on the role of the economist in tort litigation, Dick pointed readers to a law magazine article that he had written titled “The Economist's Role in the Trial of a Personal Injury Case” (1974, p. 731). In his comment, Dick advocated using different earnings growth rates by age, but he used a constant 4% discount rate across all ages. Six months after the Dick JRI article, another JRI comment appeared (Edwards, 1975) titled “Selecting the Discount Rate in Personal Injury and Wrongful Death Cases.” Edwards states at the first of his comment, “(o)ne of the most critical decisions that the economist must make in calculating economic loss is the selection of the appropriate discount rate” (p. 342). Edwards did not address earnings growth and he advocated using the interest rates given by commercial banks to their individual depositors as the discount rate to apply to earnings losses.
I. The JRI Studies Present Value
In September 1976, Franklin Smith had the lead regular article in the JRI titled “The Use of Inflation Factors in Determining Settlements in Personal Injury and Death Suits.” That article presented the mid-1970s state of the analysis of tort damages thuswise:
For a long time, the courts of many states have accepted or even required that settlements in personal injury and death suits be based upon the present value of future losses, determined by the use of annuities certain. The period for the annuity certain has been the life expectancy of a group of the same age and sex as the injured or deceased person, the corresponding work-life expectancy' or the period to age 65 or other commonly-used retirement age. An appropriate mortality table selected by an actuary has provided the period of life expectancy, and results based upon several interest rates close to the average rate of return expected on bank accounts or other conservative fixed-dollar investments for several decades into the future usually have been presented. (p. 369)
Smith's article was accepted in 1975 and published in 1976. During 1974-75, annual growth in the Consumer Price Index (“CPI”) averaged 10.1% and Smith noted that the legal system was having concerns about the impact of price inflation on court awards. Smith pessimistically wrote in the article:
With all the concern about inflation in recent years, it is natural that attorneys for the plaintiffs would try to introduce this element into the presentation in order to display larger numbers. Furthermore, as will appear later, the difference which an inflation factor can make is substantial. Therefore, if the attorney can get a settlement based on such results, he will improve his or her fee substantially as the fee for such cases usually runs 25 to 30 percent of the settlement amount. The typical game plan is to place an economist on the stand as an expert witness and have him or her present a history of the salary progress for the type of work engaged in by the injured or deceased person during the past 25 to 30 years which most likely will have been a period of uninterrupted increases. The economist then points to his or her professional standing to support a prediction that a depression is now a phenomenon of historical interest only and that salaries will continue to rise at the same rate observed during the past two or three decades. This rate, which shall be called the inflation factor, usually turns out to be larger than the interest rate assumed for determining present values. (p. 371)
Smith made his present value calculations with a 9% inflation rate and a 6% discount rate. He ended his article with the sentence “(t)he author hopes that this article will prompt others to present proposals for solving the issues raised in this discourse” (p. 374).
The March 1977 issue of the JRI contained three comments directed towards Dick (1974), Edwards (1975), and Smith (1976). In the first comment, Harris (1977) disagreed with Dick's age-span growth rates and wrote that “(i)t is more accurate to use annual growth rates rather than different growth rates over different age spans in determining a stream of probable future income.” (p. 122) With regards to discount rates, Harris argued that the term structure of interest rates must be accounted for because:
differences in yields due to the term structure of interest rates are more significant in selecting appropriate discount rates than are differences in yields based on the type of financial asset held. … Thus, other things equal, when yield curves slope upwards higher yields can be obtained by investing in financial instruments with longer maturities. (pp. 119-120)
In the second March 1977 JRI comment, Bell and Taub presented two approaches to discount losses to present value: the certainty equivalent approach12 and the discount rate approach:
In the former approach the discount rates are used to adjust for the time value of money while the risk of the lost earnings stream is accounted for separately. Under the latter approach discount rates are selected which adjust not only for the time value of money, but also for the risk of the lost earnings. (p. 122)
Bell and Taub describe the process for obtaining a certainty equivalent measure of earnings is to (a) calculate the growth rate of earnings by age, (b) account for expected changes in productivity and inflation, and (c) account for the probability of surviving and being employed by age. They wrote:
Once the certainty equivalent of the victim's uncertain earnings is determined for each year in the future, these certainty equivalents are discounted, using variable risk-free discount rates. Since the use of certainty equivalents has already adjusted for the uncertainty of future earnings, it is proper that the discount rate be risk-free. Discount rates under this approach are used only to account for the time value of money. (p. 125)
Bell and Taub point out that:
if the award is used to purchase long term government bonds, then periodic liquidation of bonds after interest rate changes will lead to capital losses or gains. Under such circumstances the discount rate cannot be considered risk-free. (p. 125)
To remove such risk, Bell and Taub then:
suggest that the appropriate procedure involves using a variable discount rate. If the award were used to purchase government bonds of varying maturities with no call provisions then the recipient would hold a portfolio that guarded against both interest rate risk and financial or business risk. Edwards also considered this alternative, but rejected it as being impractical for reasons which he did not state. (p. 126)
Bell and Taub's “discount rate approach” simply added a risk-premium to the discounting under the certainty approach. Bell and Taub argued that such risk-premiums may be appropriate because of the risks involved in lifetime earnings.
In the third March 1977 JRI comment, Hickman (1977) started out by stating:
Considerable debate has developed over how the appropriate interest rate should be chosen. In practice most economists appearing as expert witnesses have based their estimate of the future average interest rate on some historical average. Some have used the average yield on government bonds over a period of time. Some have used the average historical yield on long-term corporate bonds. Others have used either current or historical average rates paid by commercial banks or savings and loan associations. (p. 129)
Hickman (p. 130) stated that “(t)he cost of replacing the income stream is dependent upon current prices in the bond market.” He wrote that an investment plan can be put together from “(a) portfolio of bonds with varying maturities … so that the yearly income produced by both interest and maturing bonds would match almost exactly the estimated lost income for each year.” (p. 130) Hickman then stated that “(i)t would certainly not be surprising to see this become the accepted method” (p. 132) for computing awards because:
(f)irst of all, the results are more easily understood by the ordinary juror, with little or no training in the mathematics of finance. The juror can be shown the cost of an investment plan available today which will produce the desired income stream. A second advantage is that the method does not require an estimate of the average interest rate over the next 20 to 30 years. The investment can be made at the present time with a known yield so that there is no need to speculate about future interest rates. (p. 132)
The June 1977 issue of the JRI was silent to forensic economics but the September 1977 issue contained three more comments on “The Use of Inflation Factors in Determining Settlements in Personal Injury and Death Suits.” The first September 1977 comment was by Laber who was concerned with Smith's nominal inflation rates. Laber wrote:
If numerous 5 and 10 year periods are examined from the late 1940s to the early 1970s, average weekly earnings in the private economy increased at various rates, ranging from just over 3 percent to nearly 7 percent annually. If the annual rate of inflation over those same periods is subtracted from the rates of increase in earnings, the resulting real increases fall into a range of about 1.3 to 2.0 percent a year. These figures confirm what basic economics texts tell us: nominal rates of earnings changes vary with rates of inflation, and an underlying, more stable rate of increase reflects a rise in real earnings of workers generally. (p. 471)
Laber settles on real growth and interest rates which produced a simple13 NDR that could range from −1% to 2% as follows:
The practical application of this principle for court cases is that only real values should be used for growth rates of earnings and interest rates. Values of 1-2 percent are probably reasonable for growth of real earnings on the average, and real rates of interest have been estimated at 1-3 percent. (p. 473)
The second September 1977 JRI comment was by Linke. Like Laber, Linke criticized the nominal approach presented in the JRI by Smith and advocated a real rate of interest “between 3.0 and 3.75 percent” and growth rate in earnings of “2.9 percent” (p. 477)—a simple wage NDR of 0.1% to 0.85%. The third September 1977 JRI comment was a reply by Smith to his critics essentially thanking them for their contributions and stating that “(t)his is a complex subject, and these reviewers have added valuable theoretical background” (p. 479).
J. The NDR Appears in Law Journals
Until 1976, the economic loss literature routinely showed separate earnings growth and interest rates when computing present value. From 1964 to 1976, the 6-month Treasury bill averaged 5.5%, average weekly earnings growth averaged 5.5%, and the CPI-U averaged 5.3%. Recall that in 1967, the Alaska Supreme Court instituted the inflation-interest rate offset. In May 1976, the 0% net earnings discount rate appeared in the American Bar Association Journal with a piece by John A. Carlson, an economics professor at Purdue University. Carlson began his article with the statement:
Much of the controversy that goes on in the courtroom over the amount of damages one is entitled to for the present value of future earnings is just plain silly and unnecessary. A fair procedure for both sides, and one supported by historical data, would be to let the discount rate equal and thus offset projected economy-wide wage increases. (p. 628)
Relying on a chart of the 1950 to 1974 growth in the compensation per man-hour and the rate of interest paid on taxable government bonds, Carlson concluded that “(i)nterest rates on taxable government bonds probably reflect the current consensus on the average expected rate of increases of wages in the future” (p. 630). Despite his article being superficial, Carlson received wide citation in court cases and law reviews as the courts were looking for an easy way to handle escalating inflation when computing present value awards.
In the Montana Law Review, Formuzis and O'Donnell (1977) were critical of “Professor Carlson's offsetting technique,” stating that “Our own investigation into this relationship suggests that the rate of wage growth and the rate of interest do not change equally in the presence of inflation. They do, however, change in a predictable fashion.” (p. 299) Presenting a chart of wage growth and U.S. Treasury rates from 1955 to 1974, Formuzis and O'Donnell state that their regression analysis:
demonstrates that the rate of interest remains a constant 1.4 percentage points less than the rate of wage growth. On the basis of this analysis, the present value of lost future earnings can be ascertained by selecting a rate of discount and then setting the rate of wage growth 1.4 percentage points above the rate of discount. (p. 302)
Formuzis and O'Donnell thinly moved Carlson's 0% wage NDR to a net negative 1.4% simple discount rate as they summarized at the end of their article “(t)he method involves setting the rate of wage growth 1.4 points above the rate of discount and then calculating the present worth sum. The rate of discount should be calculated from riskless government securities with an average maturity of three years.” (p. 305)
K. The Canadian Trilogy Produces Statutory NDRs
On January 19, 1978, the Supreme Court of Canada simultaneously released three opinions (Andrews v. Grand & Toy Alberta Ltd, 1978; Thornton v. Prince George School District No. 57, 1978; and Arnold v. Teno, 1978) which have been called the “Canadian Trilogy.” These opinions were directly concerned with establishing “the correct principles of law applicable in assessing damages in cases such as this where a young person has suffered wholly incapacitating injuries and faces a lifetime of dependency on others.” (Andrews, p. 235) The Court stated that “(t)he question of ‘million dollar' awards has not arisen in Canada until recently” and it used these cases as a platform to judicially sanction a method of calculating the present value of future economic losses. With regards to the question of computing present value, the Court stated in Andrews that it needed to answer two questions “What rate of return should the Court assume the appellant will be able to obtain on his investment of the award? How should the Court recognize future inflation? Together these considerations will determine the discount rate to use in actuarially calculating the lump sum award.” After a lengthy discussion on inflation and interest rates by separately-writing justices in the Andrews and Arnold cases, the Court settled on one common present value approach “calculated by taking the current market interest rate for long-term investments and subtracting from that rate the predicted rate of long-term inflation.” (Feldthusen, 1978, p. 390) Post the Trilogy, in 1981, the British Columbia legislature passed the “Law and Equity Act” which authorizes the Chief Justice of the Supreme Court of British Columbia to make regulations prescribing the discount rate. From 1981 to April 30, 2014, the British Columbia NDR was held constant to 2.5% for future earnings losses and 3.5% for the cost of future care.14 On April 30, 2014, the earnings loss NDR was lowered to 1.5% and the cost of care NDR was lowered to 2.0%.
Differently than British Columbia, Ontario responded to the Trilogy by passing Rule 53.09 which mandates the NDR for calculating the present value of future pecuniary damages. The rule uses the same “difference between the investment rate of interest and the rate of general price inflation” as found in the Trilogy for setting the NDR. Rule 53.09 essentially states that the wage NDR is the recent long-term inflation-indexed bond less 0.5% for earnings growth.15
L. The NDR Appears in the JRI
In June 1979, a 0% wage NDR limped into the JRI with a comment by Wolfgang Franz (1978). Using little other than a citation to Carlson's American Bar Association Journal article, a piece he wrote for the plaintiff's bar Trial magazine, and three charts showing inflation, earnings growth, and bond rates, Franz concluded that “(e)mpirically, it has been established that interest rates and the rate of wage increases rise and fall together.” Franz wrote that “(o)bserving further that the rate of growth in earnings equals roughly the rate of discount, particularly when an average of the government bond rate and corporate AAA rate is used, then future earnings do not need to be forecast nor discounted.” (p. 331) Franz concluded his comment with the statement “(t)he beauty of this approach is that it is simple as well as fair.” (p. 333)
M. Inflation Skyrockets and the U.S. Supreme Court Weighs In
From 1978 to 1981, the 6-month Treasury bill averaged 10.7%, the CPI-U averaged 10.3% per year, and average weekly earnings grew at the average annual rate of 6.8% per year. The 6-month Treasury bill hit a high of 15.93% on May 21, 1981. While historically, earnings growth had consistently outpaced Treasury bill rates, in the late 1970s and early 1980s, inflation was rampant and sticky wages were not keeping up. Consequently, bill and bond rates reversed themselves to greatly exceed earnings growth—clear empirical evidence emerged against the accustomed 0% wage NDR.
On June 15, 1983, the U.S. Supreme Court released their decision in the case Jones & Laughlin Steel Corp. v. Pfeifer (1983). The Pfeifer case involved a district judge stating that he was legally bound to use a 0% wage NDR. The district judge cited decisions which held “as a matter of law that future inflation shall be presumed equal to future interest rates with these factors offsetting.” (Kaczkowski v. Bolubasz, 1980, p. 583) The U.S. Supreme Court remanded the case for reconsideration stating “(b)ut we are satisfied that whatever rate the District Court may choose to discount the estimated stream of future earnings, it must make a deliberate choice, rather than assuming that it is bound by a rule of state law.” (Pfeifer, p. 552-553) So, the Supreme Court remanded not on the economic viability of a 0% wage NDR but on the lower court's edict to the existence of a “matter of law” discount rate.
In addition to defeating a legally required rate of discount, in Jones & Laughlin Steel Corp. v. Pfeifer the Supreme Court weighed in with comments on the discounting of earnings losses to present value. The U.S. Supreme Court stated that the calculation of the present value of earnings losses involved two stages:
The first stage of the calculation required an estimate of the shape of the lost stream of future income. …
The second stage of the calculation requires the selection of an appropriate discount rate. … (pp. 538-539)
In unquoted portions within the above passage, the Supreme Court stated that price inflation was embedded into the “shape of the lost stream” and the “appropriate discount rate.” By prescribing two stages in the calculation, the Supreme Court essentially gave favor to the growth/discounting methods that existed before the introduction of the wage NDR. The Court also noted a need to disregard the “old” financial actuarial table method of computing present value.
To combat the problem of price inflation, the U.S. Supreme Court in Jones & Laughlin Steel Corp. v. Pfeifer suggested two methods: “In this country, some courts have taken the same ‘real interest rate' approach as Australia.” (p. 541) or “continue relying on market interest rates. To avoid undercompensation, they have shown at least tentative willingness to permit evidence of what future price inflation will be in estimating the lost stream of future income.” (p. 543)
As could be expected during a time of rampant inflation where no clear long-term, nominal signal existed, the U.S. Supreme Court set its preferences to the real interest approach for the discount rate noting that it likely was in the 1% to 3% range. The Court was silent on a real growth rate for earnings. Presumably, the Pfeifer Court found the 1% to 3% range for the real interest rate “in O'Shea v. Riverway Towing Co., 677 F.2d 1194 (CA7 1982), [where] Judge Posner stated that the real interest rate varies between 1 and 3%” (p. 551). That same range was opined by Laber in 1977 in the JRI. If U.S. Treasury Inflation-Indexed Securities were available in 1982, one wonders if the U.S. Supreme Court would have mimicked Canada by endorsing a real discount rate measured by the current returns on inflation-indexed bonds.
N. Present Value Methodology Discourse
Once the wage NDR was introduced by some as a tool to compute the present value of earnings losses, the analysis of the appropriate methods to compute the present value of earnings losses circulated around the literature. Commentary went back and forth amongst the advocates of competing methods and data sources. During the NDR's infancy, the JRI was the only academic journal publishing forensic economics related articles. The Journal of Forensic Economics (“JFE”) did not arrive until August 1987 and the Journal of Legal Economics (“JLE”) arrived in 1991.
As an example of the early contentious communications about the wage NDR, consider the 1985 and 1986 issues of the JRI. In the March 1985 issue, on methodology veracity grounds, David Jones was very critical of Franz's 0% and Laber's 0.5% NDR, accusing them of making “hidden growth rate forecast(s)” (p. 147) “in the name of ‘simplicity'” (p. 149). Jones advocated the use of current interest rates as discount factors. Striking back in the March 1985 JRI issue, Schilling (1985) vindicated the “Alaska Method” 0% NDR by finding that in a 1900-1982 historical simulation it has “been less inaccurate than other methods” and “is much simpler to perform and easier for a jury to comprehend” (p. 114). In the September 1986 JRI issue, Brown took Jones to task stating:
Jones commits the same sin that he accuses proponents of the real discount or total offset approach of in that he incorporates ‘hidden forecasts' into his forecasting formula. More specifically he implicitly assumes that the plaintiff will be able to continually reinvest the interest earned on any award at the same rate as is earned on the original award. (p. 495)
In the same issue, Jones (1986) first addressed Brown. He agreed with Brown about the continual reinvestment of interest earned problem but he stated that the problem was “trivial” to the distortions caused by the wage NDR. Jones then responded to Schilling stating “(a)lthough it may yield the ‘best' results on average, there can be no justification for using the ‘Alaska Method' under current market conditions” (p. 500). In the same September 1986 issue, Schilling responded back to Jones by calling Jones' current interest rate assumption “fragile,” noting that “the strength of the Alaska approach is that it ignores current trends and thus avoids the risk of large errors consequent upon making a wrong choice between extrapolation and reversal” (p. 496).
The April 1989 issue of the JFE16 contained a “Controversial Issues” section on “Selecting a Discount Rate” in which 13 authors presented their separate views on the appropriate discount rate. That JFE issue produced no consensus.
O. Econometrics Takes Ahold of the NDR
The year 1991 was a turning point in the academic treatment of the historical wage NDR. To that year, the wage NDR had been determined by visually analyzing graphed time series data and/or by simply computing the average annual differences between interest rates and earnings growth during various cherry-picked historical sample periods. In 1991, Haslag, Nieswiadomy, and Slottje (1991) in the JRI and Pelaez (1991) in the JFE were the first authors to apply econometrics to the study of the wage NDR using stationarity and unit root tests. Following that initial work, there have been over 30 wage NDR empirical studies in the JRI, JFE, JLE, and the Journal of Business Valuation and Economic Loss Analysis (“JBVELA”).
In 2007, Payne in the JBVELA summarized the wage NDR econometric research to that date which had used unit root analysis. At the time this article was completed, the latest published econometric studies of the wage NDR are Schap (2014) and Foster (2015). These three sources adequately summarize the historical econometric evaluation of the NDR as follows:
- 1.
A historical NDR series is only useful to the projection of future NDRs if the historical series exhibits some type of stationarity over time (i.e., stationary around a numerical value or historical mean, trend stationary, or stationary with a structural break); otherwise, the NDR follows a mean averting, non-stationary process and future NDRs are independent of previous NDRs and are non-forecastable.
- 2.
Historical wage NDRs have largely been constructed using earnings growth measured by the annual changes in economy-wide average hourly, weekly, and annual earnings by employment sector, compensation per man hour, or the Employment Cost Index. The interest rates in the NDR have largely been Treasury bills (30-day, 91-day, 6-month, and 1-year), Treasury secondary market results (1, 3, 5 and 10-year), 20-year municipal bonds, corporate bonds (5, 10, 20, and 30 year), or 6-month commercial paper.
- 3.
Foster demonstrates that contrary to most NDR studies relying on securities with greater than one-year maturity, the conceptually correct NDR is based on rate of return over the entire maturity of the security, or the NDR requires inclusion of the consequent capital gains and losses from resale at the end of the assumed holding period; and, short holding periods are unsuitable for NDRs.
- 4.
To date, historical time periods studied for wage NDRs have been inside data periods starting and stopping within 1900 to 2013. Econometric studies of wage NDRs have found them to be stationary, non-stationary, stationary with a break(s), stationary with a trend, and mixed.
- 5.
There is wide acceptance that the wage NDR experienced a significant structural break in the late 1970s to early 1980s, so wage NDRs considering data before the early 1980s have been generally deemed irrelevant for wage NDR analysis; and, observations regarding the high net interest rates of the 1980s have been shown to be no longer relevant today.
- 6.
The published values of stationary wage NDRs, when found, range from a low of −2.02% based on the 2008 to 2012 90-day Treasury bill (Schap 2014) to a high of 3.94% based on 1981 to 2000 20-year municipal bonds.
P. Historical Summary
In the earliest years, the present value of earnings losses was calculated with financial annuity tables without incorporating inflation or earnings growth. Later references show that in the 1960s it was common to find economists opining lost earnings with specific estimates of earnings growth and interest rates. The 1970s brought a revolution in the academic production of knowledge, methods, data and computing capability to form economic forecasts. When the present value issue for forensic economic purposes first appeared for serious academic discussion, the JRI published consensus was to forecast earnings growth and then apply the current discount function measured by securities of varying maturities.
Before the skyrocketing inflation and interest rates in the late 1970s and early 1980s, the wage NDR first appeared as a 0% offset rate in courts and law reviews and then in 1979, without rigorous analysis, it appeared in the JRI with Franz who called a 0% wage NDR “simple as well as fair.” After the high interest rates of the early 1980s, the historically consistent 0% wage NDR became unreliable and the supreme courts of Canada and the U.S. responded with real interest rate analysis. During the 1980s, discussion and measurement of the wage NDR and other present value methods varied greatly under the perspective of each author/expert.
Stationarity-based NDR analysis first appeared in 1991. In a variety of studies covering periods within a hundred years of data, econometric-determined wage NDRs have been inconsistent in their value and statistical significance. They have been found to be zero, negative, positive, both stationary and non-stationary, and most recently to be trend, negative or non-stationary and/or significant or not significantly different than zero. While there has been some discussion as to why one earnings growth data series might be preferable to another, noticeably absent from the literature has been an analysis of the impact of chosen security maturity has on the value and meaning of the NDR.
IV. The NDR as an Assumed Theoretical Identity
When an economist uses an NDR constructed from his or her own explicit forward-looking forecast of constant growth and investment return, the NDR value has no underlying theory or meaning—it is simply an uninteresting combination of two independent variables making convenient the mathematics of computing present value. For example, if the forensic economist forecasts future constant annual real wage growth of 1% and a future annual constant real rate of total return on invested money of 2%, for present value calculation purposes he or she can combine the forecasts into one mathematically convenient 0.99% discount factor.
When an economist estimates a stationary, forecastable NDR based upon historical time series of growth and investment return, he or she must either (1) believe there is a theoretical linkage between annual growth and investment return which produces the stationary constant NDR, or (2) annual past growth and investment return, for unknown reasons, are correlated much like the statistically significant correlation between stork populations and births of babies (Matthews 2000). Reasonably assuming that forensic economists do not form opinions on observed correlations alone, NDR advocates must believe that there is a theoretical linkage between growth and investment return.
Recalling from the terminology section, the NDR is equal to (d–g)×(1+g)−1. The NDR is assumed to be a constant, so d = NDR×(1+g) + g. The theory of the historical wage NDR must be that period return to investment, no matter maturity, is a numeric constant to period earnings growth, or vice versa. This author admits to not knowing how economic theory could predict such a constant relationship—is d = f(g) or g = f(d) or is the NDR = f(d,g) or something else? While every NDR proponent likely has some of their own theory in mind, the following thumbnail theoretical arguments are often heard. A reduction (increase) in the risk-free interest rate is caused by capital flight from (to) corporate equities when economy is slumping (expanding). When the economy slumps (expands), there is downward (upward) pressure on wages. So, the NDR theory-espoused argument is that the risk-free interest rate and changes in real wage rates move in the same direction. Such theory is silent to the reasons for the existence of a constant NDR over any specific future present value period or as to the roll of security maturity on the NDR.
The existence of a constant wage NDR contradicts the theoretical and empirically measured responses of interest and wage markets to changes in the economic environment which have been proven inconsistent (Mankiw 2007). With the NDR, it is assumed that the overall effect of interest rates on the annual changes in wages, or vice versa, in one economic environment carry over to other environments. When forecasting the NDR as a constant based upon past values of the NDR, the implicit assumption is that the historical NDR is constant across past time (i.e., stationary and mean-reverting) so that a forecast based upon past values will be reliable. In Figure 1, the one-year wage NDR is depicted from 1964 to 2015 using one-year U.S. Treasury security interest rates by month t and the one-year change in the average weekly earnings of total private production and non-supervisory employees from month t to month t+12.17 The 1-year wage NDR, ranging from below negative 4% to above positive 10%, is not constant across time. Without going through the chore of producing figures, data tables, or statistics, the promised interest rates of securities with maturities longer than one year do not produced compelling evidence of a long term constant historical wage NDR.
Citation: Journal of Forensic Economics 26, 2; 10.5085/372.1
From Figure 1, it is evident that the wage NDR is sensitive to the economic environment during which it occurs. When estimated over lengthy past periods, the wage NDR will contain movements created by macroeconomic events unrelated to current and expected future events (Havrilesky, 1990). For example, during the DotCom bubble, there was a flight of capital to the stock market which produced rising U.S. Treasury rates—the 1-year Treasury bill averaged 7.95% in 1989. Twenty years later in 2009, with the Federal Reserve's quantitative easing, the 1-year Treasury bill averaged 0.45%. Despite the remarkable difference in macroeconomic events revolving around the interest rate, wage growth was right at 3.3% in both 1989 and 2009.
Year-to-year movements in the individual investment return and growth elements are constrained under the NDR identity. Recall that Jones (1985) accused the first wage NDR proponents of making hidden earnings growth rate forecasts; the NDR also produces hidden interest rate forecasts. Likewise, if growth is known or closely known, then by the NDR identity the investment rate is specified. Since many who use the NDR method to calculate their statistic as the average of lengthy time series, year-to-year volatility in their NDR is low which unrealistically dampens the effects of the current economic environment on either investment return or growth—it is not realistic that return and growth incrementally move together at a numeric constant rate.18 Evidence of uneven movement is depicted in Figure 2 which plots, for the last 30 years, the 10-year U.S. Treasury par yield against the annual change in the Employment Cost Index for civilian total compensation. When the interest rate has been high, the net difference between the interest rate and earnings growth has been large. When the interest rate has been low, the net difference between the interest rate of earnings growth has been small. Anyone can cherry-pick the data in Figure 2 to produce a historical wage NDR convenient to a present value purpose.
Citation: Journal of Forensic Economics 26, 2; 10.5085/372.1
V. The Fallacy of the Bond Promised Yield and Maturity
To discount future money, M, at time t to its equivalent present value, PV, at time 0, the standard formula is PV0 = Mt×(1+d)−t. Within the formula, the discount rate d must be equal to the total investment return earnable from time 0 to time t. In a forensic economic application when d is a wage NDR and M are future wages, the NDR must measure, from 0 to t, the total investment return net of total earnings growth. In forensic NDR analysis, the investment return portion of the historical NDR is most always based on some average of historical promised yields as of the date the securities were purchased. Historical promised yields are, by definition, different than the historical total investment return achieved from a security's date of purchase to the date of sale or maturity. In this section, drawing upon Foster (2015), it is demonstrated that NDRs, when constructed from fixed income yields stated as promised yields, fail to measure net total investment return, and as such, are not mathematically equivalent to function of the discount rate, d, in the standard PV0 = Mt×(1+d)−t discount formula.
Money is investable in demand and fixed-income markets. A demand investment doesn't require withdrawal/sale notice (e.g., a money market account) nor is there a cost to make the withdrawal/sale. While once advocated by Edwards (1975), demand investment interest rates have been essentially ignored in the forensic economic literature because far greater default risk-free returns can be had with fixed-income investments which are contracted exchanges of money over certain periods. There are two types of fixed-income securities: discount and coupon. Discount bonds (aka bills, zeros, or strips) are delineated by their maturity and par amounts while coupon bonds are delineated by their maturity, coupons and par amounts. Since discount bonds have no intermediate cash flows, the actual yield or total return of holding the bond to maturity is known when the bond is purchased. Conversely, because coupon bonds have intermediate cash flows, their actual yield to maturity is unknown at the time of purchase. The actual yield to maturity on a coupon bond will depend on how and at what rates the bond's coupons are reinvested to the maturity date.
Because investors need a uniform method of evaluating the prices trading on the multitude of coupon bonds available in the market, the fixed-income security market has settled on the “promised yield” (Sharpe 1985, p. 108). On the date of sale, the market knows the maturity of a bond (t), the number of coupons paid per year (f), the dollar amount of the coupon paid (c), and the par value of the bond (M). Based upon the supply and demand for bonds, the dollar price of the bond $Y is determined. Since M, c, and f do not change over the life of the bond, at any time t, the price of the bond can be calculated from the traded promised yield (i) or the promised yield can be calculated from the traded price of the bond, Y, using the following formula (Fabozzi 2007, p 75):
The first term of the bond pricing equation represents the present value of the stream of coupon payments at the constant annual promised yield i. The second term of the equation is the present value of the par value of the bond under the constant to maturity annual promised yield i.
If the promised yield of a $1,000 par 10-year coupon bond is 10%, then under the promised yield formula the price of the bond is divisible into the present value of its $50 coupons ($623.11) and the present value of its par value ($376.89). If the present value of future annual earnings in 10 years is $1,000, then under a 10% bond yield, the future value is $2,653.30 ($1,000 × (1 + 10% / 2)20). In order to produce a total of $2,653.30 with the purchase of a 10-year/10% coupon bond, then each coupon payment will have to be reinvested to the 10-year date at the annual percentage yield of 10.25%. Since the market interest rates available after the bond purchase date to which coupons can be reinvested19 are an unknown, the total return of the bond to maturity at the time of purchase is unknown. If, on average, coupons are reinvested at an annual yield greater (less) than 10.25%, then the bondholder will have a total return to maturity greater (less) than 10.25%.
Since the present value calculation of earnings losses seeks to measure how much money needs to be invested today to replace future earnings losses as they would have annually occurred in the future, the promised yield of bonds at their time of purchase does not provide an answer. Likewise, wage NDRs based on the averages of historical coupon bond promised yields net of past labor earnings growth are not useful to determine the present value calculation of future earnings losses.
The differences during any one year or series of years between promised yields and total return can be dramatic. For example, in Table B-25 of the Economic Report of the President, 2016 the 2011 10-year Treasury note (30-year Treasury bond) is reported with an average promised yield of 2.78% (3.71%), an average of 3.35%. An average 3.35% promised yield for 2011 does not provide the answer to what rate of return was earned on intermediate U.S. Treasury securities bought and then sold in 2011 (which is the logic of an NDR based on intermediate term securities).20 The 3.35% average rate is simply the average of promised yields found in 2011. According to Table C-5 of the 2015 Ibbotson SBBI Classic Yearbook, purchasing Intermediate Term Government Bonds at the beginning of 2011 and then selling them at the end of 2011 produced a total return of 9.5% (the price of intermediate term bonds rose dramatically from the beginning to end of 2011). The change in average weekly earnings (see above) from the beginning of 2011 to the end of 2011 was 2.6%. So, the 2011 wage NDR computed from promised yields is 0.76% while the total return wage NDR is 6.73%. The logic of the historical wage NDR is that the historical differences in annual earnings growth and promised yields are indicative of the future net differences between earnings growth and investment returns. Because nearly all historical wage NDR studies in the forensic economic literature use promised yields based on coupon-bearing securities, and since promised yields and total return is unrelated, the results from any wage NDR study are not accurate or useful to what is sought to be measured: the expected return to investment net of wage growth.
Most NDR studies hold underlying security maturity constant and use annual promised yields—the assumed security holding period is one year. For example, the NDR study might be based on historical 10-year U.S. Treasury annual promised yields. Over any holding period, increases in interest rates will reduce the market value of any held security but increase the return from the reinvestment of the coupons; and, decreases in interest rates will increase the market value of a security but decrease the return on the reinvestment of the coupons. In order for a security to be immunized from changes in interest rates after purchase, the price risk and coupon reinvestment risk must completely offset each other (Bierwag, 1977) which is a highly unlikely event. Under the reasonable assumptions that full pricing information exists in the coupon security market and investors have homogeneous expectations, picking up (or losing) return to maturity with a change in the market interest rates after purchase is unlikely since no profitable riskless arbitrage is possible. (Vasicek, 1977)21 Since the n-year NDR calculation is periodic in one-year increments, the chosen n-year maturity coupon-paying investment instrument is unable to be immunized from changes in the interest rate during the one-year holding period, so ex ante return is unknowable from the stated promised yield of the security calculated at the date of purchase.22
Weil (1970) demonstrated that the promised yield “of a bond gives little information about the rate of return to holding bonds” (p. 510). Therefore, any wage NDR that is based on coupon security promised yields is inaccurate to the net periodic investment total return net of wage growth which is the supposed purpose of NDR measurement. Foster (2015) points out that when using securities that pay coupon payments, the correct construction of the NDR requires either calculating the total return on the security to maturity or the total return on the security with a sale before maturity. Calculating the return to maturity involves determining the actual past return earnable with the reinvestment of coupon payments to maturity for which there are many strategies. Calculating the return on the security with sale before the maturity date also involves ex ante determination of a naïve investment holding period (which is nearly always one year in NDR studies) and then determining the yield lost or gained at the selling date when rolling over the net return to a new security purchase. Most NDR studies that use coupon-bearing securities ignore the price of the security one year later and assume that securities can always be rolled over to the promised yield of “next year's securities” without cost or gain.23 Again as such, most wage NDR studies incorrectly calculate the net investment return over earnings growth that they purport to be determining.
As Foster (2015) points out, when considering total return to maturity, NDR data points would be lagged by the maturity date of the underlying security (e.g., a 20-year NDR study performed in 2015 would have its last data point in 1995) and all of the information about the movements in the interest rates of new issues are uncaptured. The purpose of the present value exercise is to determine what present day dollar amount with investment can replace future economic losses as they occur—the present value of intermediate bond cash-flows are irrelevant. If total return to maturity is calculated in the NDR, then the required holding period is to maturity (coupons have to be reinvested) and there are no intermediate cash flows available because the investment has to be held to maturity to achieve the return. With a return to maturity NDR, if the NDR is computed using 2-year securities, then the first year's economic losses cannot be compensated as they occur; if the NDR is computed using 3-year securities, then the first and second year economic losses cannot be compensated as they occur, etc. If the economic loss period is 10 years and the NDR is constructed with a yield to maturity under 20-year securities, then no economic losses can be compensated as they occur.
When utilizing the naïve investment strategy of buying and selling securities at fixed periods, returns are volatile and sometimes insufficient to replace economic losses. For example, Nieswiadomy (2012) found that under 10,000 simulations of historical annual returns, portfolios consisting of entirely U.S. Treasury bills or intermediate U.S. Treasury bonds failed to produce break-even returns 51.19% of the time with bills and 51.69% of the time with intermediate bonds. Fisher and Weil found that if the investment goal is equivalent to the desire to hold riskless assets, then “unless an investor has a reasonably certain horizon (or series of horizons) at which he will consume his wealth, he should probably not invest in bonds.” (Fisher, 1971, p. 424) The volatility of the NDR with year-to-year changes in the yield curve (and differences in maturity amongst NDRs), holding other factors unchanged, is dependent upon the chosen maturity of the security used to construct the NDR: “longer term-to-maturity bonds generally suffer greater price changes than shorter maturity bonds” and “high coupon bonds are less volatile than low coupon bonds.” (Ingersoll, 1978, p. 627) So in summary, past promised yields, or past promised yields net of earnings growth, have no association to future return, net or otherwise, and are irrelevant to the present value calculations found in forensic ecoomics.
VII. The “One-Measure Fits All” NDR
Recall that the wage NDR was first presented in social program cost-benefit analysis within public health economics as a method of discounting population average lifetime earnings to present value under a normatively selected discount rate. Such analysis is not interested in individuals and their occupation, industry, education, gender, etc., but in measuring the net differential between societal required investment return and growth in economy-wide labor productivity.
While investment return is homogenous to any tort victim, future earnings growth is heterogeneous to the economic characteristics of the tort victim. There is no one wage NDR construction that can appropriately discount lifetime earnings when tort victims vary by gender, age, education, and industry/occupation. For example, between the first quarters of 2001 and 2016, the average annual growth in the Employment Cost Index for the total compensation paid to all private industry workers was 2.6% while leisure and hospitality workers compensation growth was 2.1% and hospital workers compensation growth was 3.1%. If the only wage NDR considered was “all workers” and trends in earnings by occupation continued, then the hotel clerk's earnings would be discounted with too low of a rate and a nurse's with too high of a rate.
The duration of earnings losses varies widely. Consider the example of earnings losses occurring one year in the future. While the economic science maybe dismal, it certainly contains abundant information to make an appropriate and accurate one-year forecast of earnings growth and one-year investment returns are certain and do not require a retreat 5, 10, 20, 30, or 40 years in history to construct an NDR. If economics can make accurate 1-year forecasts, what about 2-year, 3-year, 4-year, etc. forecasts? At what point in forecast length would economists have to throw in the towel to some historical wage NDR construct and what historical period is relevant to the future loss period? And then, any NDR construct would be constrained to the maturity period of the security used to estimate the NDR. These are important issues that the NDR literature has ignored.
The NDR is also unusable in many common forensic economic calculations. For example, a constant NDR is not relevant (a) when contracted future wage rates are known, (b) to some employment benefits valuations such as stock options, (c) when defined pension or disability benefits do not have a COLA tied to an economic indicator or are payable under a fixed-rate percent COLA, or (d) when making many income tax and Social Security benefit calculations. Because such forensic calculations are routine, the economist must have in his or her toolkit a method to determine appropriate nominal and/or real interest rates for computing present value. Applying an NDR for some calculations in a case and a nominal interest rate for others in the case opens a clear conflict door since the NDR calculation assumes that that any specific interest rate forecast is unreliable.
VIII. Assumed Risk and the NDR
The theory of capital asset pricing (Ross, 1977) states that for any asset, W, its ex ante expected return at the time instant t is EW = rt + bWt where rt is the riskless rate of interest and bWt is some function of expected return of the asset in excess of the riskless rate. When an asset is represented by a series of cash-flows (such as a bond), each cash-flow at time m is discounted with an interest rate applicable to time m. At the time instant t, the riskless rate of interest for m is represented by the yield on an m-maturity default-free zero coupon bond. Since no bonds are default-free, government bonds are substituted and because most government bonds pay coupons, riskless zero coupon bond spot yields are determined from the prices paid for coupon bonds.24
At any instant t, the riskless zero coupon bond spot yield curve represents the accumulation of market-clearing expectations of the present value of cash-flows across time. The market's expectation of future interest rates25 can be extrapolated from forward rates and, in a world with only anticipated returns, “if the expectations theory held and spot rates were different from each other, the yield curve would change over time in a deterministic manner” (Elton, 1990, p. 630) and all bonds would have the same rate of return. With an asset such as a stock or corporate bond, information as to bWt is heterogeneous in the market as expectations of return are firm specific. Conversely, government bonds have little asset-specific information that affect their price (Elton, 1998) and movements in their prices are determined by aggregate macroeconomic information readily available to all market participants (Elton, 1999).26 So, bWt is homogeneous to each government bond holder and the risks associated with unexpected movements in bond prices after spot time t are (a) common to all government bond holders and (b) unknowable and unavoidable at time t.
The NDR method “nets” a rate of growth from the discount rate. The simplest NDR is the price inflation NDR which is commonly referred to as the real interest rate. Since both wage growth and interest rates include price inflation, a forensic economist believing in the wage NDR must acknowledge that the price inflation NDR is imbedded and separable from the wage NDR (i.e., if the forensic economist is able to estimate a wage NDR, he or she is also able to estimate its matching price inflation NDR and real earnings growth forecast). Since the price inflation NDR sets an ex ante expectation of obtainable real returns, when the price inflation NDR is not equal to the current prevailing real interest rate, then a riskless, costless arbitrage opportunity is created.27 The economist stating the existence of any denominated NDR is thereby stating that he or she has costless ex ante insight as to bWt from the asset pricing model which, of course, is impossible.
IX. Going Forward by Backing Up
The historical NDR originated in forensic economics when theory, data, computing power, and discourse were sparse. Early NDR proponents produced graphs showing an earnings growth line consistently lying at or above an interest rate line, so they declared that complications in determining present value were unnecessary because earnings growth and interest rates offset. It was well after the great inflation that the historical NDR became subject to econometric searches for positive stationary values to contradict the long held belief of an 0% offset.
As shown, the historical wage NDR is a faulted and unreliable method to address the complicated present value problems found in forensic economics. To reset the forensic economics discounting dialogue, this paper suggests a revisit to two general economic concepts. First, forecasting the future is not a study of finding past constants. Second, investment returns are a function of term with the expected difference between the current, risk-free yield curve and total expected return being measurable risk.
Before the wage NDR rate came to forensic economics, the consensus of the 1977 JRI articles28 and six years later in Pfeifer, was to first estimate the shape of the future lost earnings stream with growth, and then second, discount the stream to present value with appropriate interest rates. The economic science of discounting to present value as presented in the finance literature demonstrates that the current, risk-free yield curve is the foundation for determining the discount to present value of any future dollar amount. Any risks not accounted for in the shape of the lost stream must be specified, estimated, and then applied as a premium ala Ross (1977) to the current risk-free yield curve. The isolation of such risks, bwt, from risk-free returns, rt, in the equation EW = rt + bWt allows all discounting perspectives to operate under one present value method. While the methodology of estimating bWt is outside the scope of this paper,29 the expected return equation conforms to the finance literature and it is transparent when compared with the NDR's hidden growth, investment, and risk forecasts as first discussed by Jones (1985, 1986).
X. Conclusion
Before the great inflation, wage growth for decades regularly equaled or exceeded the interest rates on many securities and so, for some, the 0% historical wage NDR was an easy rule-of-thumb method for computing present the value of economic damages. During the appearance course of the historical wage NDR, attention has focused almost solely on determining its numeric constant values: is the NDR 0%, is it positive, is it negative, or is it even stationary? This article has presented several overlooked features of the NDR, clearly exposing that (1) the NDR does not purport to measure what it is intended to measure, (2) it incorporates hidden forecasts and risk, (3) its value as a numeric constant has varied widely with past macroeconomic events, (4) it is a statistic that has cycled in and out of statistical significance as a forecastable time series, and (5) it is unusable in many forensic economic discounting situations.
Instead of promoting a problematic methodology, it is here argued that forensic economics should retire the historical wage NDR from its literature and reassess the present value methods of general economics and statistics: forecast an income path, adjust for path-specific risk, compute present value with the risk-free current discount function, and then discount further with any omitted risk factor including the reasoning that the historical NDR has attempted to address. Moving much of the reasoning of the NDR to an explicit term in the discount function allows viable discourse, something unattainable in the historical wage NDR's hidden forecast design.
Annual wage NDR constructed from 1-year U.S. Treasury securities and the earnings growth in average weekly earnings of all production and non-supervisory employees, 1964 to 2015
Par yield on 10-year U.S. Treasury securities and the total compensation growth for all civilian employees as measured by the Employment Cost Index, 1985 to 2015