# HORIZON YIELDS

The receipt of regular coupons, usually semiannually, makes horizon yield (or holding period return) analysis particularly important. With zero-coupon bonds in Chapter 2, we saw that selling prior to maturity has a dramatic impact on an investor's holding period rate of return. It depends on the yield at the time of sale, which indicates the buyer's required rate of return to hold the bond for the remainder of the time to maturity. With coupon bonds, horizon yield analysis includes all of that plus the interest rates at which coupon payments can be reinvested. This can be ex-ante analysis using projected future reinvestment rates, or ex-post analysis using actual realized rates.

We do not do justice to the issue of coupon reinvestment risk until Chapter 5 when we have a full sequence of forward interest rates to work with. So, for now assume a constant coupon reinvestment rate (CRR) for future cash flows. Also, we assume the investor holds the bond to maturity and there is no default. Given these assumptions, the holding period return (HPR) over the time to maturity depends only on the CRR. When the bond is sold prior to maturity, the HPR is a function also of the price (and yield) at the time of sale.

The objective here is to see the connection between the HPR and the traditional YTM (yield to maturity) statistic. Rather than write out a general expression relating these to the CRR, I'll just use the 4-year, 4% annual payment bond that is priced at 99.342 (percent of par value) to yield 4.182%. Note that this is the street convention yield – using the true yield would be really messy. The investor's total return on the bond investment obviously will depend on the CRR. That total return is the sum of the reinvested coupon payments plus the final coupon and principal. The first coupon is reinvested for three years, the second for two years, and the third for one year.

Suppose that CRR = YTM = 4.182%, so that the investor reinvests all cash flows at the original yield to maturity. The total return is 117.032 (percent of par value).

You probably can guess where this is going. Now solve for the horizon yield – the annual rate of return that connects the purchase price and the total return at maturity. It is the solution for * HPR* in this expression.

This equation shows the well-known result that the yield to maturity measures the investor's rate of return only if the coupons are reinvested at that same yield. This is a standard caveat for internal rates of return in general. We can formalize this as:

The corollaries are that:

This is the essence of coupon reinvestment risk – assuming no default, the buy-and-hold investor's rate of return depends on the rate at which coupons can be reinvested over the lifetime of the bond.