CHAPTER 3
MEASURING YIELD
CHAPTER SUMMARY
In Chapter 2 we showed how to determine the price of a bond, and we described the relationship between price and yield. In this chapter we discuss various yield measures and their meaning for evaluating the relative attractiveness of a bond. We begin with an explanation of how to compute the yield on any investment.
COMPUTING THE YIELD OR INTERNAL RATE OF RETURN ON ANY INVESTMENT
The yield on any investment is the interest rate that will make the present value of the cash flows from the investment equal to the price (or cost) of the investment.
Mathematically, the yield on any investment, y, is the interest rate that satisfies the equation. P = CF11 + CF22 + CF33 + . . .+ CFNN ?1?y??1?y??1?y??1?y? where CFt = cash flow in year t, P = price of the investment, N = number of years. The yield calculated from this relationship is also called the internal rate of return. Solving for the yield (y) requires a trial-and-error (iterative) procedure. The objective is to find the yield that will make the present value of the cash flows equal to the price. Keep in mind that the yield computed is the yield for the period. That is, if the cash flows are semiannual, the yield is a semiannual yield. If the cash flows are monthly, the yield is a monthly yield. To compute the simple annual interest rate, the yield for the period is multiplied by the number of periods in the year. Special Case: Investment with Only One Future Cash Flow
When the case where there is only one future cash flow, it is not necessary to go through the time-consuming trial-and-error procedure to determine the yield. We can use the below equation.
?CFn?y = ? ? ? 1. P??1/n
Annualizing Yields
To obtain an effective annual yield associated with a periodic interest rate, the following formula is used:
effective annual yield = (1 + periodic interest rate)m – 1
where m is the frequency of payments per year. To illustrate, if interest is paid quarterly and the periodic interest rate is 8% / 4 = 2%), then we have: the effective annual yield = (1.02)4 – 1 = 1.0824 – 1 = 0.0824 or 8.24%.
We can also determine the periodic interest rate that will produce a given annual interest rate by solving the effective annual yield equation for the periodic interest rate. Solving, we find that: periodic interest rate = (1 + effective annual yield)1/m – 1. To illustrate, if the periodic quarterly interest rate that would produce an effective annual yield of 12%, then we have: periodic interest rate = (1.12)1/4 – 1 = 1.0287 – 1 = 0.0287 or 2.87%.
CONVENTIONAL YIELD MEASURES
There are several bond yield measures commonly quoted by dealers and used by portfolio managers. These are described below.
Current Yield
Current yield relates the annual coupon interest to the market price. The formula for the current yield is: current yield = annual dollar coupon interest / price. The current yield calculation takes into account only the coupon interest and no other source of return that will affect an investor’s yield. The time value of money is also ignored.
Yield to Maturity
The yield to maturity is the interest rate that will make the present value of the cash flows equal to the price (or initial investment). For a semiannual pay bond, the yield to maturity is found by first computing the periodic interest rate, y, which satisfies the relationship: CCCCMP = + + + . . .+ + 123nn?1?y??1?y??1?y??1?y??1?y? where P = price of the bond, C = semiannual coupon interest (in dollars), M = maturity value (in dollars), and n = number of periods (number of years x 2).
For a semiannual pay bond, doubling the periodic interest rate or discount rate (y) gives the yield to maturity, which understates the effective annual yield. The yield to maturity computed on the basis of this market convention is called the bond-equivalent yield.
It is much easier to compute the yield to maturity for a zero-coupon bond because we can use:
?M?y = ? ? ? 1. ?P?1/n
The yield-to-maturity calculation takes into account not only the current coupon income but also any capital gain or loss that the investor will realize by holding the bond to maturity. In addition, the yield to maturity considers the timing of the cash flows.
Yield To Call
The price at which the bond may be called is referred to as the call price. For some issues, the call price is the same regardless of when the issue is called. For other callable issues, the call price depends on when the issue is called. That is, there is a call schedule that specifies a call price for each call date.
For callable issues, the practice has been to calculate a yield to call as well as a yield to maturity. The yield to call assumes that the issuer will call the bond at some assumed call date and the call price is then the call price specified in the call schedule. Typically, investors calculate a yield to first call or yield to next call, a yield to first par call, and yield to refunding.
Mathematically, the yield to call can be expressed as follows: CCCCM*P = + + + . . .+ + 123n*n*?1?y??1?y??1?y??1?y??1?y? where M* = call price (in dollars) and n* = number of periods until the assumed call date (number of years times 2). For a semiannual pay bond, doubling the periodic interest rate (y) gives the yield to call on a bond-equivalent basis.
Yield To Put
If an issue is putable, it means that the bondholder can force the issuer to buy the issue at a specified price. As with a callable issue, a putable issue can have a put schedule. The schedule specifies when the issue can be put and the price, called the put price.
When an issue is putable, a yield to put is calculated. The yield to put is the interest rate that makes the present value of the cash flows to the assumed put date plus the put price on that date as set forth in the put schedule equal to the bond’s price. The formula is the same as for the yield to call, but M* is now defined as the put price and n* is the number of periods until the assumed put date. The procedure is the same as calculating yield to maturity and yield to call.
Yield To Worst
A practice in the industry is for an investor to calculate the yield to maturity, the yield to every possible call date, and the yield to every possible put date. The minimum of all of these yields is called the yield to worst.
Cash Flow Yield
Some fixed income securities involve cash flows that include interest plus principal repayment. Such securities are called amortizing securities. For amortizing securities, the cash flow each period consists of three components: (i) coupon interest, (ii) scheduled principal repayment, and (iii) prepayments. For amortizing securities, market participants calculate a cash flow yield. It is the interest rate that will make the present value of the projected cash flows equal to the market price.
Yield (Internal Rate of Return) for a Portfolio
The yield for a portfolio of bonds is not simply the average or weighted average of the yield to maturity of the individual bond issues in the portfolio. It is computed by determining the cash flows for the portfolio and determining the interest rate that will make the present value of the cash flows equal to the market value of the portfolio.
Yield Spread Measures for Floating-Rate Securities
The coupon rate for a floating-rate security changes periodically based on the coupon reset formula. This formula consists of the reference rate and the quoted margin. Since the future value for the reference rate is unknown, it is not possible to determine the cash flows. This means that a yield to maturity cannot be calculated. Instead, there are several conventional measures used as margin or spread measures cited by market participants for floaters. These include spread for life (or simple margin), adjusted simple margin, adjusted total margin, and discount margin.
The most popular of these measures is discount margin. This measure estimates the average margin over the reference rate that the investor can expect to earn over the life of the security.
POTENTIAL SOURCES OF A BOND’S DOLLAR RETURN
An investor who purchases a bond can expect to receive a dollar return from one or more of these sources: (i) the periodic coupon interest payments made by the issuer, (ii) any capital gain (or capital loss—negative dollar return) when the bond matures, is called, or is sold, and (iii) interest income generated from reinvestment of the periodic cash flows
Any measure of a bond’s potential yield should take into consideration each of these three potential sources of return. The current yield considers only the coupon interest payments. No consideration is given to any capital gain (or loss) or interest on interest. The yield to maturity takes into account coupon interest and any capital gain (or loss). It also considers the interest-on-interest component. Implicit in the yield-to-maturity computation is the assumption
that the coupon payments can be reinvested at the computed yield to maturity.
The yield to call also takes into account all three potential sources of return. In this case, the assumption is that the coupon payments can be reinvested at the yield to call. Therefore, the yield-to-call measure suffers from the same drawback as the yield to maturity in that it assumes coupon interest payments are reinvested at the computed yield to call. Also, it presupposes that the issuer will call the bond on some assumed call date.
The cash flow yield also takes into consideration all three sources as is the case with yield to maturity, but it makes two additional assumptions. First, it assumes that the periodic principal repayments are reinvested at the computed cash flow yield. Second, it assumes that the prepayments projected to obtain the cash flows are actually realized.
Determining the Interest-On-Interest Dollar Return
The interest-on-interest component can represent a substantial portion of a bond’s potential return. The coupon interest plus interest on interest can be found by using the following equation:
??1?r?n ? 1?C ?? ?r?
where r denote the semiannual reinvestment rate.
The total dollar amount of coupon interest is found by multiplying the semiannual coupon interest by the number of periods: total coupon interest = nC
The interest-on-interest component is then the difference between the coupon interest plus interest on interest and the total dollar coupon interest, as expressed by the formula
??1?r?n ? 1?interest on interest = C ???nC.
?r?
Yield To Maturity and Reinvestment Risk
The investor will realize the yield to maturity at the time of purchase only if the bond is held to maturity and the coupon payments can be reinvested at the computed yield to maturity. The risk that the investor faces is that future reinvestment rates will be less than the yield to maturity at the time the bond is purchased. This risk is referred to as reinvestment risk.
There are two characteristics of a bond that determine the importance of the interest-on- interest component and therefore the degree of reinvestment risk: maturity and coupon. For a given yield to maturity and a given coupon rate, the longer the maturity, the more dependent the bond’s total dollar return is on the interest-on-interest component in order to realize the yield to maturity at the time of purchase. In other words, the longer the maturity, the greater the reinvestment risk.
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