# HORIZON YIELDS AND HOLDING-PERIOD RATES OF RETURN

Suppose that you buy the 10-year zero-coupon corporate bond at 60 but you have no intention of holding it all the way to maturity. Then 5.174% (s.a.) is only a reference yield – your own realized rate of return will depend on the price at which you sell the bond. We use the term * horizon yield*, or

*for the annual rate of return when the holding period differs from the time to maturity. This can be an ex-ante yield based on a projected sale price in the future or an ex-post rate of return calculated after the fact from the actual price at the time of sale. We can even extend the idea to a holding period beyond the maturity date. Then we would need to project (or observe) the price and yield at maturity when the principal is reinvested.*

**holding-period rate of return,**A useful yardstick for assessing a horizon yield when the bond is sold prior to maturity is the * constant-yield price trajectory.* This is the path the bond price will take over the time to maturity assuming no default. Figure 2.2 shows the trajectory for the 10-year zero purchased at 60 (percent of par value). The prices for the various years are from equation 2.1. For instance, the prices at year 2 (when there are eight years remaining until maturity) and at year 7 (when there are only three years left) are 66.454 and 85.792,

**FIGURE 2.2** Constant-Yield Price Trajectory, 10-Year, Zero-Coupon Corporate Bond Priced to Yield 5.174% (s.a.)

respectively. The prices along the constant-yield trajectory also are called the * carrying values* for the bond.

When the investor is able to sell the corporate zero at a yield less than 5.174% (s.a.), the sale price will be above the trajectory and there will be a capital gain. For example, suppose that the investor sells the zero at year 2 for a price of 68 (percent of par value). At that time, the now 8-year corporate zero is trading at 4.879% (s.a.).

The investor's realized 2-year holding-period rate of return turns out to be 6.357% (s.a.), which is greater than the original yield of 5.174% because the price is above the constant-yield trajectory.

Notice that we set FV equal to 68 – the redemption value of 100 is irrelevant here because the bond is sold for 68.

How much is the capital gain if the bond that is purchased at 60 is sold in year 2 at 68? Unlike equity, it should not be 8, the difference between the sale and purchase price. The movement along the constant-yield price trajectory shown in Figure 2.2 represents interest earned. So, interest income is 3.145 for the first year (= 63.145 – 60) and 3.309 for the second year (= 66.454-63.145). The key point is that, in principle, interest income is the change in price associated with the * passage of time.* Capital gains and losses are the changes in price related to

*– for bonds that means a change in the yield and a price above or below the carrying value. We see in Chapter 4 when we get into bond taxation how well these economic principles hold up in practice.*

**changes in value**Now suppose that the investor does not sell after two years and instead holds on to the bond for seven years. At that time, the now 3-year corporate zero is sold for 83 because it is being priced to yield 6.308% (s.a.).

The investor's realized 7-year holding-period rate of return is 4.690% (s.a.), which is less than 5.174% because the sale price is below the trajectory.

In this case, there is a capital loss even though the investor buys at 60 and sells later at the much higher price of 83. The relevant comparison is between 83 and 85.792, the carrying value on the constant-yield price trajectory.

There in an important investment lesson in these scenarios. Suppose the buyer of the 10-year corporate zero actually has an investment horizon of 10 years and plans to hold the bond to maturity. Then the * unrealized* gain in year 2 caused by the lower yield at that time, as well as the

*loss in year 7 caused by the higher yield, has no impact on the*

**unrealized***total return. The bull market prevailing in year 2 and the bear market in year 7 are irrelevant news stories to this investor. The bond buyer achieves a locked-in yield to maturity of 5.174% (s.a.) regardless of the path that the price takes to its destination of par value – before taxes and inflation and assuming no default.*

**realized**