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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 =
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 following equation.
.
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 maturityis 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:
P =
where P = price of the bond, C = semiannualcoupon 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:
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