TWO PEDAGOGICAL ISSUES REGARDING
THE DEFINITION OF INTEREST RATES

James F. Feller, Middle Tennessee State University

This paper focuses on two unrecognized but significant issues not adequately addressed in the majority of undergraduate finance textbooks. The first issue is concerned with the number and identity of risk premiums to be included in a nominal interest rate, and whether they should be incorporated multiplicatively or additively. The second issue involves the hidden risk premium implicit in the amortization of any loan requiring periodic payments over a multi-year term. Owing to a tilt effect in interest payments, the lender's risk is partially shifted to the borrower, but the shift is not explicitly defined in the nominal interest rate. Simple examples are used to illustrate both of these pedagogical issues.

The intent of this paper is to describe and interpret two consequential issues that have an impact on the process of interest-rate determination. This process is pivotal to financial theory, and critical to the valuation of any tangible or financial asset which provides future cash flows; these cash flows, when discounted to present value, define the current wealth of the asset's owner. By extension, a society's wealth then depends on the magnitude and timing of all future cash flows accruing to that society, and upon the level of the rate used to discount these flows. Because future cash flow streams are uncertain and vary in degree of uncertainty, different rates of discount are used to determine the present value of individual investments. Although these rates are labeled differently depending on the underlying theory and particular application, all are based on the real risk-free rate of interest, defined by Brigham and Houston (1998) as "the interest rate on a riskless security during a period in which zero inflation is expected" (p. 127). This rate, known also as the pure rate (Lasher, 1997), originated in the neoclassical interest rate theory of Irving Fisher, whose ideas were summarized by Lutz (1968). Both Marshall (1968) and Fisher considered the pure rate to be the rate at which the supply and demand for money would reach equilibrium; with zero risk and zero inflation expected, savers would give up the current use of their money only if appropriately rewarded on the demand side. Lasher (1997) related the pure rate with the earning power of money, and Kellison (1991) considered it a base rate analogous to the long-term growth rate of overall economic productivity. Lee, Finnerty, and Norton (1997) also related the pure rate to the economy's long-run growth rate while emphasizing the rate's role in equilibrating the supply of and demand for capital investment funds.

The nominal risk-free rate results when the real risk-free rate is adjusted for expected inflation. When "risk-free" is interpreted in the narrow sense, the only risk considered is default risk, and this rate is measured objectively as the discount rate on a short-term U.S. Treasury Bill. All other discount rates used to discount the expected cash flows of all existing investments are derived from the nominal risk-free rate. They result from an upward adjustment of this rate to account for risk factors other than inflation.

The first issue examined in this paper has two aspects: how many risk factors are included in the risk adjustment process, and how are they incorporated into the resulting nominal rate? The intent is not to answer these questions, but rather to demonstrate that large differences in wealth estimates result when different assumptions are applied to the discount rate computation.

The general approach to defining the nominal rate appropriate to a particular investment requires an assessment of the investment's risk within some number of categories, the estimate of a risk premium relevant to each category, and the combining of the nominal risk-free rate and the risk premiums into a single rate. Keown, Petty, Scott, and Martin (1998) defined an investor's required rate of return as an additive combination of the nominal risk-free rate and a single risk premium determined by the market as a whole. Benninga and Sarig (1997) defined the risk premium as the asset's expected nominal return minus the after-tax AAA corporate bond nominal rate. Both thus derived a single risk premium from given market-determined data. Bodie and Merton (1998) approached the risk premium from both productivity and psychological aggregations, listing four fundamental determinants of rates in a market economy: (a) the productivity of capital goods, (b) the degree of uncertainty regarding their productivity, (c) the time preferences of people, and (d) the degree of risk aversion people exhibit. All of the above authors enumerated one or more risk factors, but their approaches are either too general or too abstract to be useful. Fabozzi (1996), at the other extreme, overwhelmed the reader by setting down eleven types of risk a bondholder might face. The first three risks Ñ interest rate, reinvestment, and call risk Ñ are closely interrelated. Interest rate and reinvestment risk tend to cancel each other, and call risk increases as market interest rates decline. Expected inflation, as noted previously, is adjusted for in the nominal risk-free rate determination; the adjustment, however, does not consider unexpected changes in the rate of inflation. Thus, Fabozzi designated a separate inflation risk. Exchange rate risk involves not only investment in foreign securities, but also investment in bonds of multinational corporations subject to this risk. Liquidity risk involves the bid-ask spreads of bonds, volatility risk relates to changes in interest rate variability, tax risk refers to possible changes in tax laws affecting bond returns, and event risk reflects the possibility of occurrence of natural or political catastrophes. Fabozzi also mentioned yield curve risk, but did not define it, and defined default risk as a single category, even though it might involve interest, principal, or sinking fund payments, or some combination. Finally, he defined risk risk in terms of an unawareness of risk owing to one's not understanding the attributes of a security, such as a junk bond or a complex derivative. Emery, Finnerty, and Stowe (1998) listed six risk factors: (a) interest rate risk, modified by maturity; (b) default risk; (c) call, or early payment risk; (d) inflation risk; (e) currency risk; and (f) marketability, or liquidity risk. Levy (1997) set down nine categories of risk relevant to security investment, including political and regulatory risk and business risk; neither of these is mentioned in the lists of risks surveyed above. Levy also pointed out that not all risks apply to a given security class. For example, common stocks are not subject to default risk. Jones (1998) named eight traditional sources of risk which relate to variability in investment returns. In addition to any sources mentioned previously, he included financial risk, which refers to the use of debt by companies, and country risk, which relates to the viability of a given foreign economy. Jones then separated total risk into two categories, general and specific, but did not define their components with regard to the eight specific risk categories he previously had noted. Van Horne (1998) included default, marketability, maturity, volatility, and taxability in his risk categories, but qualified volatility risk as a combined effect of maturity and coupon rate. Both Lasher (1997) and Brigham and Houston (1998) included four risk sources: (a) inflation, (b) default, (c) liquidity, and (d) maturity. Finally, Lee, Finnerty, and Norton (1997) and Ross, Westerfield, and Jordan (1996) followed a capital asset pricing approach by simply adding a single risk premium to the nominal risk-free rate. Any individually specified risk premiums were assumed to be impounded in this single risk premium, which results from the supply and demand functions for assets in a particular risk class.

While the above section points out the absence of agreement among textbook authors with regard to the number, identity, and definition of quantifiable risk premiums, the correct operation by which these premiums are combined into a single nominal interest rate is also disputed. The risk premiums may simply be added, or they may be multiplied to yield a geometric sum. As noted by Van Horne (1998), the relationship between the nominal and real risk-free rates was first defined by Irving Fisher in 1896. Ross, Westerfield, and Jordan (1996) demonstrated the Fisher effect both symbolically and numerically. If R is equal to the nominal risk-free rate, r is equal to the real risk-free rate, and h is the expected rate of inflation, then (1 + R) = (1 + r) x (1 + h). Ross et al. expanded the right hand side and solved for R to yield R = r + h + (r x h), where the interaction term (r x h) "represents compensation for the fact that the dollars earned on the investment are also worth less because of the inflation" (p. 248). The interaction term, then, does have economic significance, and its significance increases as the rate of inflation increases. A further transformation of the Fisher equation by Luenberger (1998) yields r = (R - h) / (1 + h) when the Ross et al. symbols defined above replace those used by Luenberger. The difference between the nominal rate and the inflation rate must be deflated to determine the real risk-free rate; the relationship, thus, is not strictly additive. Nevertheless, as Kellison (1991) stated, "since the cross product term is typically small, many people tend to ignore it and conveniently think of the nominal rate of interest as just the sum of the real rate of interest and the rate of inflationÓ (p. 299). Van Horne (1998, p. 451) noted that with moderate inflation, the cross product term is usually ignored, and Megginson (1997, p. 172) stated that for low inflation rates the additive form is "approximately correct." Brigham and Houston (1998) ignored completely the correct Fisher formulation, and showed the inflation premium as strictly additive. Fisher is neither mentioned in the text nor in the index. In one example on page 129, Brigham and Houston used the same numbers used by Marshall (1968, p. 493) in an illustration of the Fisher effect, but they failed to complete the calculation. In the next paragraph, they then showed the inflation premium as additive, and proceeded to calculate incorrectly a resulting "quoted rate of interest," using different input numbers. The solutions to Brigham and Houston's end-of-chapter problems that relate to the inflation premium are all based on the additivity formulation. Ross, Westerfield, and Jordan (1996), on the other hand, emphasized including the cross product term in the inflation adjustment. Furthermore, using the same numbers as used by Marshall, and Brigham and Houston, in their illustrations of the Fisher effect, Ross et al. correctly calculated the nominal rate. While their result of 15.5 percent is approximately the same as Brigham and Houston's 15.0 percent, discounting the same amount over an increasing number of periods using each rate would produce a significant and increasing divergence in present values. Lee, Finnerty, and Norton (1997) emphasized that the Fisher effect is multiplicative, and Bodie and Merton (1998, p. 45) pointed out that the intuitive suggestion that the real rate of interest "is simply the difference between the nominal interest rate and the rate of inflation" is not exactly correct. Keown, Petty, Scott, and Martin (1998) also set down correctly the Fisher effect; their example on page 56 and all of their relevant end-of-chapter problems are solved using the multiplicative relationship.

The above review of textbooks has focused on the definition of the nominal interest rate, and the contradictory approaches advocated by different authors of finance textbooks. Specifically, the number, identity, and combinatorial operation differed from textbook to textbook. Finance, at the fundamental level, is taught as though it were a precise discipline. A string of future cash flows is discounted to some definite and correct present value. This is true whether the cash flows are considered certain, or whether they are discounted as the expected values of probability distributions. The pedagogical issue involves reconciling the nominal interest rate used in the discounting with the method by which it was derived. As an example, assume a perpetual stream of $1,000,000 received at the end of each year forever. Further, as in the Brigham and Houston (1998) example on page 129, assume the nominal discount rate is 15 percent, determined additively. In this case the number of risk premiums added to the real risk-free rate is irrelevant. The resulting present value is equal to the $1,000,000 payment divided by the 15 percent rate, $6,666,667. Following this example, Brigham and Houston set the real risk-free rate equal to five percent and the expected rate of inflation to ten percent. Using their additive formulation the result would not change. However, the correct multiplicative calculation, (1.05) x (1.10), yields a discount rate of 15.5 percent and a present value of $6,451,613, which is $215,054 less than the Brigham and Houston result. Moreover, if the rate were computed assuming a multiplicative relationship, a real risk-free rate of three percent, and four risk premiums, each estimated to be three percent, the resulting rate would be 15.93 percent and the present value $6,278,485. Recall that Brigham and Houston defined four risk premiums, but incorporated them additively to arrive at the nominal rate. At the limit, assuming an indefinite number of equally weighted independent risk factors, the discount rate would be 16.18 percent, and the present value $6,179,162. Thus, this simple example demonstrates that, for the given inputs the resulting present value can lie anywhere within an absolute range of $487,505, or as much as 7.31 percent below the maximum value found using the additively derived 15 percent discount rate.

In discussing the above example, it must be reiterated that the computed present values are mathematically precise, but the assumptions are only useful in defining the range of possible solutions. Kellison (1991) pointed out that "in practice there are a large number of factors that come together in complex ways to determine rates of interest" (pp. 296-297). Further, "although putting a risk premium in the interest rate may be an appropriate method of dealing with default risk, it is not necessarily an appropriate method of dealing with other types of risk" (p. 306). Moreover, Kellison stated that the pure (risk-free real) rate "has proved to be relatively stable over many decades . . . in the range of 2% to 3% (p. 297). Van Horne (1998, p. 452), however, indicated that the relationship between changes in nominal interest rates and changes in expected inflation may not be one-to-one, and that the stability of the real rate is both theoretically and empirically controversial:

The relationship between inflation and interest rates does not lend itself to simple explanations, nor does it appear to be consistent over time. There are a host of factors affecting interest rates, one of which is inflation. Empirically, the evidence is all over the map... (p. 453)

Pennachi's study (as cited in Megginson, 1997) indicated "that real interest rates are far more volatile than inflationary expectations, and that most fluctuations in nominal yields are therefore caused by fluctuations in real required returns." Complicating the problem, Benninga and Sarig (1997) commented on the difficulty of measuring expected inflation rates; they noted that in practice historical rates are often used. Thus, given the theoretical, empirical, and practical inconsistencies, the process of determining a correct interest (discount) rate remains an unresolved issue.

The second issue examined in this paper is concerned with the shifting of risk from lender to borrower that occurs when a loan is amortized over a term of multiple years. The required periodic payments include both interest and principal components determined mathematically in the discounting process. Although the periodic payment remains constant over the whole term of the loan, the interest component comprises a very high percentage of the payment during the early periods, but decreases exponentially over the loan's term. Contrarily, the repayment of principal is at a very low rate early in the loan's term, but this rate of repayment increases with time. Such loans generally are collateralized by some asset on which the lender possesses a first lien. If, for example, the loan were a home mortgage loan, the slow amortization of loan principal would indicate a commensurately slow transfer of equity to the borrower. Given the mathematics of discounting, a mortgage loan is treated as if it were a simple interest loan automatically renewed each year, rather than as an integral loan for a single term of multiple years. Von Mises (1996) pointed out that:

The custom of computing interest pro anno is merely commercial usage and a convenient rule of reckoning. It does not affect the height of the interest rates as determined by the market ... . It is customary to stipulate a uniform rate of interest for the whole duration of the loan contract and to apply a uniform rate in computing compound interest ... . The terms of a loan are not independent of the stipulated duration of the loan. (pp. 536-537)

As the above quotation indicates, custom and convenience support the stating of interest rates on an annualized basis. Custom likely derives from the financing of agriculture, which generally follows an annual crop cycle, and convenience from the simplicity of stating a single annual rate, rather than multiple rates with each applicable to each potential loan term.

The nonlinear amortization of principal Ñ and slow transfer of equity to the borower Ñ represents a shift in risk from borrower to lender. In absolute terms the lender receives the greater portion of interest early in the loan term while retaining the greater portion of implied equity in the loan collateral. The contrary is true for the borrower, and the disparity is much greater when the present values of the interest and principal portions of the total loan payment are considered. Feller and Rogers (1985) mathematically derived a risk premium to reflect the risk shift by equating the durations of the interest and principal repayment portions of a loan's total periodic payment. The approach taken here is less complex and only to illustrate the issue. Assume a 30-year mortgage loan of $200,000, with payments at the end of each of the next 360 months based on monthly discounting with a nominal rate of nine percent. The monthly payment is computed to be $1,609.25. Referring to Table 1, it is apparent that the

interest portion of the payment is very high and the principal repayment very low during the first months of the 30-year term. The converse is true for the later months. Table 2 shows that on an undiscounted basis the payments sum to $579,328, or nearly 290 percent of the present value of $200,000; the principal repayment, in undiscounted terms, is equal to the present value of total payments, as must be the

case. Interest comprises 65.48 percent of the undiscounted payment total, and principal repayments 34.52 percent. Intuitively, one might expect the same relative percentages to hold for the present values, but interest accounts for 80.48 percent of the $200,000 present value, while principal repayments account for only 19.52 percent. This phenomenon is attributable to the tilt effect induced by the tilting of interest payments relative to the constant total mortgage payments. Bierwag (1987, p. 39) discussed this effect in terms of graduated payment mortgages. He also had noted earlier (pp.8-13) the equivalence of the interest rate with the internal rate of return of an investment. Assuming the present value of $200,000 and monthly discounting, the calculated internal rate of return of the interest payments is 0.52 percent, while that of the principal repayments is zero. The sum of these internal rates of return is substantially below 0.75 percent, the nominal rate adjusted for monthly discounting. The missing difference is [(1.0075) / (1.0052)] - 1 = 0.0022. When annualized, the internal rate of return, 6.30 percent for interest and 2.70 percent for principal repayment, do sum to 9.0 percent, the originally stated nominal rate of interest. The 2.70 percent represents a hidden risk premium that results from the tilt effect, and thus is an additional but overloooked cost to the borrower.

From a pedagogical viewpoint, the tilt effect helps explain to incredulous beginning finance students that a quoted rate of nine percent on a 30-year mortgage loan works out to a total interest payment of nearly twice the initial principal value, or, in absolute dollars, an interest rate of 190 percent over the whole term. The apparent usury can be rationalized by pointing out that an inflation rate of 3.61 percent applied to the home's nominal value would compensate for the large total interest payment over the thirty years. However, should nominal values of homes decline, it is likely that a larger-than-expected number of recent home purchasers would default on their loans. They would lose very little in terms of equity, and in a time of financial exigency would have little incentive to continue paying for a depreciating home. Thus, the hidden risk premium, meant to protect the lender, could potentially backfire, leading to economic disaster for both lender and borrower.

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