Bond investors frequently used the yield to maturity as a way of evaluating the investment. We want to look closely at this practice. We're going to ask what is the rate of return you actually earn on a bond?, and how does it relate to the yield to maturity? Well, first we're going to ask, what is the return? What does it mean to have a return? What is a return? This is going to sound abstract and philosophical, but it will get concrete. So let's say you have an investment with value v. One possible definition of return is the percent change in your investment. In that case, R is V_t. So your value at some time t, minus the value you started with, value zero divided by value zero. That's the percent change. We can rewrite this as V_t over V_0 minus one. So let's think about that definition of return as applied to our five-year bond from the previous clip. So remember, this five-year bond has a face value of $1000, and a price of $712.99. So let's say we take this bond, and we hold it to maturity. In that case, your V_t is your $1000, your V_0 is the price you paid. So R is V_t over V_0 minus one. Well, that's 1,000 over 712 minus one or 40.25 percent. Very nice. Now let's think about the three-year bond. Same face value, and keep in mind, same yield to maturity. But the price of this bond is $816.30. Well now, if we take the percent change in value, we get 1,000 over 816.30 minus one. Well, that's 22.05 percent. Wait a second, does that make sense? Should the return on the five-year bond be bigger than the three-year bond? We're missing something here. What we're missing is that it took you five years to earn this 40 percent on the five-year bond, whereas it only took you three years to earn the 22 percent. So somehow we need to adjust for the length of time that you're holding the investment. Well, there's a simple way to do that. If you remember from our discussion of effective annual rates, we do basically the same thing. We annualize the investment. Let's see how this works. So we want to find R such that one plus R_t, this t is how long you hold the investment for is equal to how much you actually earn. That's V_t over V_0. So in other words, R is equal to V_t over V_0_1 over t minus one. We will sometimes call this the holding period return. Holding period return or HPR. We might also call it the annual return depending on the setting. Note that we're calling it the holding period return because it's the return that you actually earn when you hold the investment. It's the annual return because we've annualized it. Let's return to our bond example, with our good old five-year bond. Now we're taking 1,000 over 712.99, and we're annualizing, so that gives us a holding period return of seven percent. The three-year, let's do the same thing, now we're annualizing by dividing by one- third. You will find that again, it's seven percent. Now this is more reasonable. So this by the way, as I mentioned before, these calculations are holding period return. Now you might notice something else. This holding period return is equal to the yield to maturity. Well, that's a general result. So recall our definition of yield to maturity, which says that the yield to maturity is the discount rate that makes the present value of the payments equal to the price for a bond with zero coupons. That gives us the face value over one plus the yield to maturity to the t. Let's say t is the maturity equals the price. So here this t is maturity. In other words, the YTM is F divided by P_1 over t minus one. Well, let's look at this formula. This is the same formula as for holding period return, provided that V_t is F and V_0 is P. This is equal to the holding period return if you hold the bond to maturity. Let's just plug in F is our V_t, P is our V_0, everything else is the same. So we have the following result for a zero-coupon bond, the yield to maturity equals the holding period return if the bond is held to maturity. So in this case, the yield to maturity really did measure the holding period return. Notice though, there's two provisos here, the first is that the bond is hold to maturity. The second is that the zero-coupon bond. So let's just briefly go through an example when the bond is not held to maturity, and just see what could happen.. We're going to do an example where we don't hold a bond to maturity. We have a 10-year bond, it's a zero-coupon bond. The face value of the bond is $1000. The price of the bond is $450.11. By the way, that implies that the yield to maturity on the bond is 8.31 percent. So what I'm going to ask is what happens if you purchase this 10-year bond, and sell it after one year. So after one year, the bond is a nine-year bond. Now let's just say that market conditions have changed. So here we are, one year later, market conditions have changed, and yields are higher. So now the yield to maturity on the bond is equal to 8.6 percent. That means that the bond is selling for a price of 1,000 over 1.086_9 to nine-year bond, or $475.92. So between when you bought the bond, and when you sold it, the yield rose. So what do we think this means for the holding period return? Remember holding period return is V_t over V_ 0. Now we're talking about a one-year investment, so there's no one over t minus one. So when we sold the bond, it was a nine-year bond. We sold it for $475.92. We bought the bond for $450.11, our holding period return is thus 5.7 percent. It's not the yield to maturity when we bought the bond. So this 5.7 percent is less than our yield to maturity of 8.31. This 5.7 percent is less than our yield to maturity of 8.31 percent. So our holding period return is less than our yield to maturity. So in this example, we saw that the yields rose, and our holding period return is less than the yield to maturity. If yields rise, what happens is you basically get a loss on the bond. Your holding period return is less than our yield to maturity. If yields fall, and in the notes there's an example like that, the holding period return is greater than the yield to maturity. If yields stay the same, no surprise, holding period return equals the yield to maturity. But what do we conclude from this? What we conclude is that bonds are not risk-free investments. Bonds are fixed income, that's why we call them fixed income securities, not fixed return. So when is the bond a risk-free return? Only in one situation. When your holding period matches exactly the maturity of the bond.
