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15)@\fffi1k CGJ li § C&fffiI F1f C&1 ThCC It CCCGJ April23,1 982 Are Interest RatesComparable? Many of us forget -i f we ever knew -that market traders don't always quote interest rates on comparable bases. By convention, they may quote rates on the basis of a discount or investment yield; a 360- or 365-day year; or simple or compound interest. In the special case of money-market funds, quoted rates may even represent a past-realized return rather than an expected future yield. Such discrepancies are significant indeed, especially when rates approach lofty levels. For example, a 3-month commercial paper rate of 1 6.0 percent and a bank CD rate of 1 6.7 percent, when calculated on a'comparable basis, both produce an 18.0-percent compound yield. Obviously we need directly comparable rates when considering alternative investments. But there are other reasons as well. In termstructure studies, for example, economists typically use a money-market instrument for the shortest maturity instrument and a bond for the longest maturity instrument, even though the two typically are not quoted in directly comparable terms. (In this regard, stated maturity is not normally a good measure of effective maturity, or duration, and may lead to. large term-structure biases,as shown by Joseph Bisignano in the April 3, 1981 Weekly Letter.) Economists also use the differential between the T-bill and commercial-paper rates as a measure of default risk, or the differential between the bank-CD and commercial-paper rates as a measure of bank aggressiveness. While the T-bili/commercialpaper comparison is valid, the bank-COl commercial-paper comparison is not, again because CD and commercial-paper rates are not calculated the same way. First step: annual compound interest To examine how quoted rates differ, we must first establish a single method of expressing all yields. Next, we must ferret outthe different conventions on which rates actually are quoted, andfinally, adjustthe quoted rates to a directly comparable basis. Instruments may vary in maturity from one-day Federal funds to perpetual British consols. Thus, to compare rates of return, one must first put yields on a single time basisthe year being the universally accepted denominator. But one must also decide on a comparable treatment of interest received duringthe lifeofthe instrument. Onesensible rule, especially for shorter-term moneymarket instruments, is to presume that all cash payments (either coupon- or face-value redemption) can be reinvested at the same rate of return as the initial instrument. This rule-of-thumb enables the yields from instruments with different maturities and payment streams to be compared on the basis of annual compound interest. Accepted conventions Financial-market participants quote rates on the basis of a number of conventions that puzzle the purist. Although accepted practice seems crude and often inconsistent, it resuIted from the need to make qu ick decisions when sophisticated calculators were unavailable and when historically low interest rates made fancy calculations less critical. The year thus took on 360 days in the money market because of its computational ease, while simple-interest calculations became the industry norm for short-term instruments. In the money market, which deals in securities with initial maturities of up to one year, discount securities are treated differently from interest-bearing securities. Discount securities are redeemable at face value as of a stated date; the yield is calcu lated as the difference between face value and purchase price as a percentage of face value (redemption price). In contrast, for single-payment interest-bearing securities, the yield is calculated on the basis of interest paid at maturity as a percentage of principal (purchase price). The latter method properly reflects return on investment, while discounting understates the true investment yield. Moreover, the de- ffi)©\lffi1k §@\1 l)l 11 IPrr cc11 CGJ ' Opinions expressed in this ne\vsleHer do not necessarilv reflect the views of the management of the Federal Reserve Bank of San Francisco, or of the Board of Governors of the Federal Reserve System. Money-market funds (MMF's) do not pay an ex ante promised yield as debt instruments do. Instead, they pay an ex post realized portfolio return-the result of accrued interest plus capital gains and losses. In their computations, most consumer-oriented M MF's first mark assetsto market daily (i.e., estimate unrealized capital gains/losses). Next they quote a weekly yield on the basis of the average return (including realized interest and both realized and unrealized capital gains/ losses) over a seven-day period. Finally they annualize this yield using the money-market norm of simple interest. gree of understatement rises nonlinearly with the rate of interest. Equally important, market participants use simple rather than compound interest when annualizing money-market rates. This convention is particularly puzzling because it creates difficulties for comparing instruments with different maturities. Dealers defend simple-interest calculations on the grounds ./ that compound-interest comparisons would imply reinvestment of accrued interest at the initial rate. But simple-interest comparisons are even worse, since reinvestment of accrued interest presumably occurs at a zero rate of interest. However weak the justification, dealers still annualize all money-market instruments on the basis of simple interest. This practice, like discounting, results in an understatement of annual yield -again, nonlinearly related to the level of rates. Yield comparisons In the money market, many of the widely traded instruments, such as T-bills, commercial paper, finance-company paper, and bankers acceptances, are quoted on a discount-simple interest-360-day basis. Atthe high interest rates prevalenttoday, rate quotes fall well below effective yields. For example, a 90-day T-bill quoted at 16.0 percent actually yields lB.O percent (see table), while the bias is even larger for a lBO-day T-bill. One can easily decompose the bias into its sou rces (see chart). For the 90-day instrument, the assumption of simple interest is the greatest source of error, followed by the effect of quoting on a discount basis. For the l BO-day instrument the roles of these two factors are reversed, which is understandable given that discounting would have a greater effect -and interest compounding a smaller effect-on the instrument of longer maturity. Finally, the 360-day-year convention imparts a small but measurable bias in both cases. The choice of a 360-day year for most money-market instruments is indefensible in today's world of $10 calculators. It also leads to an understatement of annual yield, as it presumes that no interest is earned during five (or six) days of the year. This practice results in a small bias compared with those of the other conventions cited, however. For notes and bonds (coupon-bearing instruments with initial maturities beyond a year), the accepted conventions for yield calculations come closer to a true approximation of annual compound yield. The standard "bond yield" avoids mQstof the bias present in money-market quotes; i.e., it is based on investment (not discount) yield, compound interest, and a 365-day year. However, it compounds the annual coupon payment as if it were paid in a single lump at the end of the year. Most bonds actually pay interest twice a year (half at midyear), so that the "bond yield" results in an understatement of the true annual compound-interest yield. Nonetheless, dealers customarily express money-market rates in terms of "bondequivalent yields" for comparative purposes, although not even this rate is an effective compound yield. For interest-at-maturity instruments, such as bank CD's and Eurodollar deposits, the lack of discounting results in a somewhat smaller bias. For notes and bonds, the error in quote yields is even smaller because of the use of a 365-day year. But the error still is substantial because of the failure to recognize that half of each year's interest is received at midyear. Thus, the common practice of adjusting money-market rates to the "bond equivalent yield" still results in a sizable underestimate of effective yield. 2 Does all of this really matter? As the table indicates, the difference is small at low levels of interest rates -except, of cou rse, to traders who eke out their living on basis points. At higher rates, such computations matter even to the casu a I observer. Money-market traders know the problems well, and thus apply both rule-of-thumb and exact correction factors when comparing yields. But the rest of us wou Id do well to remember that interest rates are more complex than we generally give them credit for being. Equivalency of Quoted Rates Effective Yield Type of Security 6% 1 8% 5.7% 16.0% 5.8 15.7 gO-day 5.8 16.7 1 80-day 5.8 17.0 Bond 5.9 17.2 M M F ,Weekly Return 5.8 16.6 Discount gO-day 1 80-day I Interest -at-Matu rity Jad<Beebeand Elaine Foppiano Yield Bias in Discount Securities at an 18- Percent Yield Percent 2.5 r- 360-day ....-- 2.0 I - 1.5I - to l- 0.5 I - o . - Discount gO-day 1 80-day 3 NOIlnijlbl S1G )iNb 3/Ui:3 S o:1f\jI ::1)j'Cl (18 vv I. V'r.J"J Tv tL3U:I:i \l'l\il Tj:; d A tAi\i I "1'1I 1'4 1 00b 4eln • • epe!\aN • 04ePI e!UJoJ!le::::> .• euoz!JV • e>JselV !!eMeH • .; CG> Y Jr CG> BANKIN G DATA-TWELFTH FEDERALRESERVE DISTRICT (Dollaramountsin millions) SelectedAssetsandliabilities LargeCommercialBanks Loans(gross,adjusted) andinvestments* Loans(gross,adjusted) .. total# Commercialandindustrial Realestate Loansto individuals Securities loans U.s.Treasury securities* Othersecurities* Demanddeposits- total# Demanddeposits-adjusted Savingsdeposits- total Timedeposits- total# Individuals,part.& corp. (Largenegotiable CD's) WeeklyAverages of Daily Figures Member Bani,ReservePosition Excess Reserves (+ )/Deficiency (- ) Borrowings Netfreereserves (+ )/Netborrowed( -) Amount Outstanding . Change from 4/7/82 158,027 136,952 42,846 56,574 23,261 1,992 6,279 14,796 41,502 29,033 31,824 90,020 80,793 33,100 3/31/82 94 43 177 45 31 326 94 145 983 1,017 731 -1,117 -1,180 -1,036 Weekended Changefrom yearago Dollar Percent 11,486 12,822 6,393 4,897 527 501 346 969 - (497 - . 1,713 192 14,511 14,032 4,105 Weekended 4/7/82 3/31/82 40 95 56 95 103 7 III- I- 7.8 10.3 17.5 9.5 2.3 33.6 5.2 6.1 3.5 5.6 0.6 19.2 21.0 14.2 Comparable year-agoperiod 46 2 44 * Excludes tradingaccountsecurities. # Includesitemsnotshownseparately. Editorialcommentsmaybeaddressed to theeditor (William Burke)or to the author. ... Freecopiesof this andother FederalReservepublicationscanbeobtainedbycallingor writingthe PublicInfonnationSection, FederalReserveBankof SanFrancisco,P.O.Box7702,SanFrancisco94120.Phone(415)544-2184.
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