Yield Measure Purpose
Nominal yield Measures the coupon rate.
Current yield Measures the current income rate.
Promised yield to maturity Measures the estimated rate of return for bond held to maturity.
Promised yield to call Measures the estimated rate of return for bond held to first call date.
Realized (horizon) yield Measures the estimated rate of return for a bond likely to be sold prior to maturity. It considers specific reinvestment assumptions and an estimated sales price. It also can measure the actual rate of return on a bond during some past period of time.
Nominal and current yields are mainly descriptive and contribute little to investment decision making. The last three yields are derived from the present value model described previously.
To measure an estimated realized yield (also referred to as the horizon yield or total re- turn), a bond investor must estimate a bond’s future selling price. Following our presentation of bond yields, we present the procedure for finding these prices. We conclude with a demon- stration of valuing bonds using spot rates, which is becoming more prevalent.
18.2.1 Nominal Yield
Nominal yieldis the coupon rate of a particular issue. A bond with an 8 percent coupon has an 8 percent nominal yield. This provides a convenient way of describing the coupon charac- teristics of an issue.
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18.2.2 Current Yield
Current yieldis to bonds what dividend yield is to stocks. It is computed as:
18.2 CY = Ci=Pm
where:
CY =the current yield on a bond
Ci=the annual coupon payment of Bond i Pm=the current market price of the bond
Because this yield measures the current income from the bond as a percentage of its price, it is important to income-oriented investors (e.g., retirees) who want current cash flow from their investment portfolios. Current yield has little use for investors who are interested in total re- turn because it excludes the important capital gain or loss component.
18.2.3 Promised Yield to Maturity
Promised yield to maturityis the most widely used bond yield figure because it indicates the fully compounded rate of return promised to an investor who buys the bond at prevailing prices, if two assumptions hold true. Specifically, the promised yield to maturity will be equal to the investor’s realized yield if these assumptions are met. The first assumption is that the investor holds the bond to maturity. This assumption gives this value its shortened name,yield to maturity (YTM). The second assumption is implicit in the present value method of compu- tation. Referring to Equation 18.1, recall that it related the current market price of the bond to the present value of all cash flows as follows:
Pm=X2n
t =1
Ci=2
ð1+ i=2ịt+ Pp
ð1+ i=2ị2n
To compute theYTMfor a bond, we solve for the rateithat will equate the current price (Pm) to all cash flows from the bond to maturity. As noted, this resembles the computation of the internal rate of return (IRR) on an investment project. Because it is a present value-based computation, it implies a reinvestment rate assumption because it discounts the cash flows.
That is, the equation assumes that all interim cash flows (interest payments) are reinvested at the computed YTM. This is referred to as apromised YTM because the bond will provide this computedYTM(i.e., you willrealizethis yield)only ifyou meet its conditions:
1. You hold the bond to maturity.
2. You reinvest all the interim cash flows at the computed YTM rate.
If a bond promises an 8 percent YTM, you must reinvest coupon income at 8 percent to realize that promised return. If you spend (do not reinvest) the coupon payments or if you cannot find opportunities to reinvest these coupon payments at rates as high as its promised YTM, then the actual realized yield you earn will be less than the promised yield to maturity.
As will be demonstrated in the section on realized return, if you can reinvest cash flows at rates above the YTM, your realized (horizon) return will be greater than the promisedYTM.
The income earned on this reinvestment of the interim interest payments is referred to as interest-on-interestand is discussed in detail in Homer and Leibowitz (1972, Chapter 1).
The impact of the reinvestment assumption (i.e., the interest-on-interest earnings) on the ac- tual return from a bond varies directly with the bond’s coupon and maturity. A higher coupon and/or a longer term to maturity will increase the loss in value from failure to reinvest the cou- pon cash flow at the YTM. Put another way, a higher coupon or a longer maturity makes the reinvestment assumption more important—that is, such bonds have greaterreinvestment risk.
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Exhibit 18.3 illustrates the impact of interest-on-interest for an 8 percent, 25-year bond bought at par to yield 8 percent. If you invested $1,000 today at 8 percent for 25 years and reinvested all the coupon payments at 8 percent, you would have approximately $7,100 at the end of 25 years. We will refer to this money that you have at the end of your investment hori- zon as your ending-wealth value. To prove that you would have an ending-wealth value of
$7,100, look up the compound interest factor for 8 percent for 25 years (6.8493) or 4 percent for 50 periods (which assumes semiannual compounding and it is 7.1073). In the case of U.S.
bonds, the semiannual compounding is the appropriate procedure because almost all U.S.
bonds pay interest every six months.
Exhibit 18.3 shows that this ending wealth of $7,100 is made up of $1,000 principal return,
$2,000 of coupon payments over the 25 years ($80 a year for 25 years), and $4,100 in interest earned on the semiannual coupon payments reinvested at 4 percent semiannually. If you never reinvested any of the coupon payments, you would have an ending-wealth value of only
$3,000. This ending-wealth value of $3,000 derived from the beginning investment of $1,000 gives you an actual (realized) yield to maturity of only 4.5 percent. That is, the rate that will discount $3,000 back to $1,000 in 25 years is 4.5 percent. Reinvesting the coupon payments at some rate between 0 and 8 percent would cause your ending-wealth position to be above
$3,000 and below $7,100; therefore, your actual realized rate of return would be somewhere between 4.5 percent and 8 percent. Alternatively, if you managed to reinvest the coupon pay- ments at rates consistently above 8 percent, your ending-wealth position would be above
$7,100, and your realized (horizon) rate of return would be above 8 percent.
Interestingly, during periods of very high interest rates, you often hear investors talk about
“locking in”high yields. These people are subject toyield illusionbecause they do not realize that attaining the high promised yield requires that they reinvest all the coupon payments at the very highpromised yields. For example, if you buy a 20-year bond with a promised yield
Exhibit 18.3T h e E f f e c t o f I n t e r e s t - o n - I n t e r e s t o n T o t a l R e a l i z e d R e t u r n
Interest-on-Interest ($4,100)
$3,000
Coupon Receipts ($2,000)
Principal ($1,000)
$7,100 B
A
$7,000
$6,000
$5,000
$4,000
$3,000
$2,000
$1,000 0
5 10 15 20 25
Years Promised yield at time of purchase: 8.00%
Realized yield over the 25-year investment horizon with no coupon reinvestment (A): 4.50%
Realized yield over the 25-year horizon with coupons reinvested at 8% (B): 8.00%
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to maturity of 15 percent, you will actually realize the promised 15 percent yield only if you are able to reinvest all the coupon payments at 15 percent over the next 20 years.
Computing the Promised Yield to MaturityThe promised yield to maturity can be com- puted by using the present value model with semiannual compounding. The present value model gives the investor an accurate result and is the technique used by investment professionals.
The present value model equation—Equation 18.1—shows the promised yield valuation model:
Pm=X2n
t =1
Ci=2
ð1+ i=2ịt+ Pp
ð1+ i=2ị2n
All variables are as described previously. This model is somewhat complex because the solu- tion requires iteration. As noted, the present value equation is a variation of the internal rate of return (IRR) calculation where we want to find the discount rate, i, that will equate the present value of the cash flows to the market price of the bond (Pm). Using the prior example of an 8 percent, 20-year bond, priced at $900, the equation gives us a semiannual promised yield to maturity of 4.545 percent, which implies an annual promised YTM of 9.09 percent.2
900=40X40
t =1
1 ð1:04545ịt
+1000 1
ð1:04545ị40
!
=40ð18:2574ị+1,000ð0:1702ị
=900
The values for 1/(1 +i) were taken from the present value interest factor tables in the appen- dix at the back of the book using interpolation. Fortunately, there are several handheld calcu- lators that will do these calculations for you.
YTM for a Zero Coupon BondIn several instances, we have discussed the existence of zero coupon bonds that only have the one cash inflow at maturity. This single cash flow means that the calculation of YTMis substantially easier, as shown by the following example.
Assume a zero coupon bond maturing in 10 years with a maturity value of $1,000 selling for $311.80. Because you are dealing with a zero coupon bond, there is only the one cash flow from the principal payment at maturity. Therefore, you simply need to determine what the discount rate is that will discount $1,000 to equal the current market price of $311.80 in 20 periods (10 years of semiannual payments). The equation is as follows:
$311:80= $1,000 ð1+ i=2ị20
You will see that i= 6 percent, which implies an annual rate of 12 percent. For future refer- ence, this yield also is referred to as the 10-year spot rate, which is the discount rate for a single cash flow to be received in 10 years.
18.2.4 Promised Yield to Call
Although investors use promised YTM to value most bonds, they must estimate the return on certain callable bonds with a different measure—thepromised yield to call (YTC). Whenever
2You will recall from your corporate finance course that you start with one rate (e.g., 9 percent or 4.5 percent semi- annually) and compute the value of the stream. In this example, the value would exceed $900, so you would select a higher rate until you had a present value for the stream of cash flows of less than $900. Given the discount rates above and below the true rate, you would do further calculations or interpolate between the two rates to arrive at the correct discount rate that would give you a value of $900.
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a bond with a call feature is selling for a price above par (that is, at a premium) equal to or greater than its call price, a bond investor should consider valuing the bond in terms of YTC rather than YTM. This is because the marketplace uses the lowest, most conservative yield measure in pricing a bond. As discussed in Homer and Leibowitz (1972, chap. 4), when bonds are trading at or above a specified crossover price, which is approximately the bond’s call price plus a small premium that increases with time to call, the yield to call will provide the lowest yield measure. The crossover price is important because at this price the YTM and the YTC are equal—this is the crossover yield. When the bond rises to this price above par, the computed YTMbecomes low enough that it would be profitable for the issuer to call the bond and finance the call by selling a new bond at this prevailing market interest rate.3 Therefore, theYTC measures the promised rate of return the investor will receive from hold- ing this bond until it is retired at the first available call date, that is, at the end of the deferred call period. Note that if an issue has multiple call dates at different prices (the call price will decline for later call dates), it will be necessary to compute which of these scenarios provides thelowestyield—this is referred to as computingyield to worst. Investors must consider com- puting the YTC for their bonds after a period when numerous high-yielding, high-coupon bonds have been issued. Following such a period, interest rates will decline, bond prices will rise, and the high-coupon bonds will subsequently have a high probability of being called—
that is, their market yields will fall below the crossover yield.
Computing Promised Yield to CallAgain, the present value method assumes that you hold the bond until the first call date and that you reinvest all coupon payments at theYTCrate.
Yield to call is calculated using a variation of Equation 18.1. To compute the YTC by the present value method, we would adjust the semiannual present value equation to give:
18.3 Pm=X2nc
t =1
Ci=2
ð1+ i=2ịt + Pc
ð1+ i=2ị2nc where:
Pm=the current market price of the bond Ci=the annual coupon payment for Bond i nc =the number of years to first call date Pc=the call price of the bond
Following the present value method, we solve for i, which typically requires several computa- tions or interpolations to get the exact yield. As before, this is a promised yield that requires the two assumptions noted earlier except that rather than holding to maturity, it is assumed that you hold until the first call date.
18.2.5 Realized (Horizon) Yield
The final measure of bond yield,realized yieldorhorizon yield(i.e., the actual return over a horizon period) measures the expected rate of return of a bond that you anticipate selling prior to its maturity. In terms of the equation, the investor has a holding period (hp) or invest- ment horizon that is less than n. Realized (horizon) yield can be used to estimate rates of re- turn attainable from various trading strategies. Although it is a very useful measure, it requires several additional estimates not required by the other yield measures. First, the investor must estimate the expected future selling price of the bond at the end of the holding period. Second, this measure also requires a specific estimate of the reinvestment rate for the coupon flows
3An extensive literature on the refunding of bond issues includes studies by Boyce and Kalotay (1979), Harris (1980), Kalotay (1982a), and Finnerty (1983).
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prior to selling the bond. This technique also can be used by investors to measure their actual yields after selling bonds.
Computing Realized (Horizon) YieldThe realized yields over a horizon holding period are variations on the promised yield equations. The substitution ofPf(future selling price) andhp into the present value model (Equation 18.1) provides the following realized yield model:
18.4 Pm=X2hp
t =1
Ci=2
ð1+ i=2ịt+ Pf
ð1+ i=2ị2hp
Again, this present value model requires you to solve for thei that equates the expected cash flows from coupon payments and the estimated selling price to the current market price.
You will note from the present value realized yield formula in Equation 18.4 that the cou- pon flows are implicitly discounted at the computed realized (horizon) yield. In many cases, this is an inappropriate assumption because available market rates might be very different from the computed realized (horizon) yield. Therefore, to derive a realistic estimate of the re- alized yield, you also need to estimate your expected reinvestment rate during the investment horizon. We will demonstrate this in a subsequent subsection.
Therefore, to complete your understanding of computing estimated realized yield for alternative investment strategies, the next section considers the calculation of future bond prices. This is fol- lowed by a discussion on calculating a realized (horizon) return with different reinvestment rates.
18.3 CALCULATING FUTURE BOND PRICES