The Foundation
5.6 Bond Yields
130 THEFOUNDATION
by holding the bond to maturity and reinvestment income. That is, YTM implicitly assumes that the bondholder can reinvest any interest payments received from the bond at the YTM.
In Examples 5.4 through 5.6, we computed the intrinsic value of a bond by discounting the bond’s future cash flows by the required rate of return for the bond (kb). If the market properly prices the bond, intrinsic value and market value (price) are the same and the discount rate is the bond’s YTM. Thus, given the expected future cash flows for the bond, we can determine the expected return, or YTM, over the life of the bond.
To find YTM, we substitute P0 for Vb and YTM for kb in Equations 5.5 and 5.6 for annual- pay bonds, Equations 5.7 and 5.8 for semiannual-pay bonds, and Equation 5.9 for zerocoupon bonds and then solve for yield rather than intrinsic value. Solving for YTM using Equations 5.5 though 5.8 requires a trial-and-error approach. That is, we discount the future cash flows using a specific YTM and compare this result to the market price. If the computed amount is greater (less than) the market price, we select a higher (lower) discount rate and repeat the process until the computed amount equals the price. To avoid this time-consuming process, we show how to solve for YTM using a financial calculator.
Investors can decide whether to buy a bond by comparing its YTM to their required rate of return on the bond. When the YTM is greater or equal to the required return, the bond would be attractive for purchase. When the YTM is less than the required rate of return, investors would not buy the bond or might consider selling it if they already own it.
Example 5.12 Yield to Maturity: Annual-pay Bonds
Suppose a 15-year, $1,000 par value, 7 percent annual-pay bond is currently trading at a price of $1,100. What is the bond’s yield to maturity? If an investor requires an 5.5 percent return, would this bond be attractive?
Solution: We substitute P0 = $1,100, I = $70, M = $1,000, and n = 15 into a slightly modified version of Equation 5.6. Using a trial-and-error approach to solve for YTM, the YTM is about 5.97 percent.
$ ,1 = $ ( + )
$ , ( + )
=
1 00 70
1
1000
1 15 5 97
1 15
YTMt YTM YTM
t
. %
+ ⇒ =
∑
We can easily solve for the bond’s YTM by using the BA II PLUS® calculator as follows:
Inputs: 15 N; 1,100 +/− PV; 70 PMT; 1000 FV; CPT I/Y Output: 5.97 Or simply:
Thus, the YTM is about 5.97 percent. If the investor requires a 5.5 percent return, this bond would be attractive because the bond’s YTM exceeds the investor’s required rate of return.
15 N
CPT I/Y
1,100 +/− PV
70 PMT
1,000 FV (press this
key last)
Example 5.13 Yield to Maturity: Semiannual-pay Bonds
Now assume that the bond in the previous example is a semiannual-pay bond. If a 15- year, $1,000 par value, 7 percent semiannual-pay bond is currently trading at $1,100, what is the bond’s yield to maturity?
Solution: Because this is a semiannual-pay bond, we input P0 = $1,100, I/2 = $35, M = $1,000, and 2n = 30 into a slightly modified version of Equation 5.8. Using trial- and-error, we get the YTM of about 5.98 percent.
$ ,1 = $
( + / )
$ , ( + / )
=
1 00 35
1 2
1000
1 230
1 30
YTM t YTM
t
+
∑
⇒ YTM = 2.99% × 2 = 5.98%
Using the BA II PLUS® calculator, key in the following and get:
Inputs: 30 N; 1,100 +/− PV; 35 PMT; 1,000 FV; CPT I/ Y Output: 2.99 × 2 = 5.98
Or simply:
8 To be technically accurate, we should find the bond’s effective annual yield, as we discussed in Chapter 4:
(true annual) YTM = (1 + semiannual YTM)2− 1 = (1.0299)2− 1 = 0.0607 or 6.07%.
30 N
CPT I/Y
1,100 +/− PV
35 PMT
1,000 FV (press this
key last)
Because the bond involves semiannual compounding, the computed value of 2.99 percent is not the bond’s YTM. Instead, it represents the semi-annual measure of yield.
The market convention used to annualize the semiannual yield to maturity is to double the semiannual rate and to use that rate as the yield to maturity.8 The term used to describe this convention is bond-equivalent yield (BEY). Thus, the BEY for this bond would be 2 × 2.99 = 5.98 percent. Note that the BEY is less than the YTM of the annual-pay bond (5.97 percent).
132 THEFOUNDATION
Practical Financial Tip 5.7
The following relationships exist among the coupon rate, current yield, and yield to maturity.
Bond selling at Relationship
Par Coupon rate = current yield = yield to maturity Discount Coupon rate < current yield < yield to maturity Premium Coupon rate > current yield > yield to maturity
Assumptions and limitations
Yield to maturity involves three embedded assumptions:
1 Investors will hold the bond to maturity.
2 Investors will receive all coupon payments in a prompt and timely fashion.
3 Investors will reinvest all coupons to maturity at a rate of return that equals the bond’s YTM.
If these assumptions do not hold, the YTM will not equal the realized yield, which is the actual yield earned by investors. For example, investors may be unable to reinvest coupon payments at the YTM due to changes in interest rates that occur over the bond’s life. This type of risk is reinvestment risk. All else equal, the reinvestment risk of a bond will increase with higher coupons because there is more interest to invest. Reinvestment risk also will increase with longer maturities because a longer period exists to reinvest interest.
A limitation of YTM and other traditional yield measures is the use of the same rate to discount all cash flows. This implies a flat yield curve, which means that the yield for all maturities is approximately equal. This condition rarely exists in practice. The most com- mon relationship is a normal or positively sloped yield curve in which the longer the maturity, the higher is the yield. Occasionally, the relationship between maturities and yields is such that the longer the maturity, the lower is the yield. Such a downward sloping yield curve is an inverted or a negatively sloped yield curve.
Yield to Call
As discussed earlier, bonds are either noncallable or callable. A noncallable bond prohibits the issuer from calling the bond for retirement before maturity. With a callable bond, the issuer has the right to retire the bond before maturity. Thus, YTM may not be an appropri- ate measure of potential return of a bond. Another yield measure, called yield to call (YTC), takes into account the potential call of a bond. As with YTM, YTC considers all three sources of potential return of owning a bond. YTC assumes that:
1 Investors will hold the bond to the assumed call date.
2 The issuer will call the bond on that date.
3 Investors will reinvest all cash flows at the YTC until the assumed call date.
Calculating the YTC requires making two adjustments to the standard YTM equation. First, analysts use the length of the investment horizon as the number of years to the first call date, not the number of years to maturity. Second, they use the bond’s call price instead of the bond’s maturity value. The call price is typically greater than the maturity value to partially compensate bondholders for exposing them to reinvestment rate risk. Often the call price decreases over time because bondholders have less exposure to reinvestment rate risk.
This is because less time remains for reinvesting cash flows before the bond matures. To calculate the YTC, solve the following equation:
V I
YTC m
CP YTC m
b t mn
t mn
= + +
∑
( / ) ( + / )=1 1 1
(5.11)
where I is the annual or semi-annual coupon payment; CP is the appropriate call price of the bond; m is the number of coupon payments per year; and n is the number of years to first call.
In practice, bond investors generally compute both YTM and YTC for deferred-call bonds that are trading at a premium. A deferred-call provision is a stipulation in a bond indenture specifying that the issuer cannot call a bond until expiration of a specified call deferment period. Thus, the bond starts out as noncallable and becomes freely callable after the call deferment period has expired.
Example 5.14 Yield to Call: Semiannual-pay Bonds
Suppose that the 15-year, $1,000 par value, 7 percent semiannual-pay bond has a deferred call. The bond now trades at $1,100 and has 5 years before the first call at a price of $1,050. What is the yield to call?
Solution: Because this is a semiannual-pay bond, we input P0= $1,100, I /2 = $35, M = $1,050, and 2n = 10 into Equation 5.11 and solve for the YTC using a trial-and- error approach to get about 5.56 percent.
$ , $ ,
. % . %
1100 35
1 2
1050
1 210 2 78 2 5 56
1 10
= + +
+ ⇒ = × =
∑
( $ / ) ( / )= YTC t YTC YTC
t
Using the BA II PLUS® calculator, we input the following:
Inputs: 10 N; 1,100 +/− PV; 35 PMT; 1,050 FV; CPT I/Y Output: 2.78 × 2 = 5.56
134 THEFOUNDATION
Thus, the YTM is 5.98 percent while the YTC is 5.56 percent. Market convention uses the lower of the two yield measures (YTM or YTC) as the more conservative measure of yield. In our example, the bond would be valued relative to its YTM.
Why? On any premium bond, the YTC will always be less than YTM. This is be- cause interest rates have decreased. If interest rates remain below the coupon rate, the issuer is likely to call the bond. The opposite situation exists for bonds trading at a discount.
Concept Check 5.6
1 What are three basic sources of return that may comprise the yield on a bond?
2 Which sources of return are included in calculating the current yield, yield to maturity, and yield to call?
3 What conditions cause a bond to sell at par value, a premium, or a discount?
4 Under what circumstances will the yield to maturity and the realized yield differ?
5 What is the meaning of reinvestment rate risk?
6 How does yield to call differ from yield to maturity?
7 For bonds selling at a premium, what is the relationship between the yield to call and the yield to maturity?