So in the previous clip we talked about valuing zero-coupon bonds, the basic building blocks. It so happens that governments and corporations issue coupon bonds, and so do individuals actually mortgages being a form of coupon bonds. So we need to know how to value coupon bonds. So let's start by writing out the cash flows on a coupon bond that might be issued by a corporation or by a sovereign government. So time zero, you don't get anything. When we're valuing the security, we always assume that the cash flows start one year later, so you get in the first year a coupon of C. You get in the second year, a coupon of C. In the third year, you get a coupon of C, in the fourth year you get a coupon of C, and so on and so forth until the maturity of the bond, when you get another coupon and the face or par value. So there we have the cash flows for a coupon bond, and we know how to value them. The price on this coupon bond is C over 1 plus r plus C over 1 plus r squared plus dot dot dot plus C plus F over 1 plus r to the t. So there are two distinct rates associated with coupon bonds. The first, is the coupon rate, which is the coupon payment divided by the face value. The next, is our old friend, yield to maturity. What's the yield to maturity? It's got the same definition as for a zero-coupon bond, but the formula for the zero-coupon bond isn't going to apply anymore. The yield to maturity is the rate such that the present value of the bonds payments equals the price. That's it. So I'm going to write out again the formula for the price of the bond. This r, and of course same r here and same r here, this r, that is my yield to maturity, because we're discounting the payments and making them equal to the price. So here's a good result for coupon bonds. The bond sells for its face value, sometimes called its par value, if and only if, YTM equals C over F. So bond sells for its face value, that means, the price equals the face value, that is true if and only if the yield to maturity equals the coupon rate. Why? Let's look at an example. I think you'll understand why it's true intuitively. So we're going to consider a bond with a four percent coupon rate and the face value of $100. So let's look at the cash flows of this bond. So you get four dollars, four dollars, four dollars, and then $104. So the price is equal to 4 over 1 plus r plus 4 over 1 plus r squared plus 4 over 1 plus r cubed plus 104 over 1 plus r to the fourth. Now, here's my claim. When this r is four percent, this price equals $100. You can prove that algebraically. But think about it for a moment intuitively. Suppose the interest rate is four percent, now somebody wants to borrow money from you and this person is going to pay you the interest every year, and then the principal back at the end. I think it makes sense that the value of that asset to you would be $100. Namely, you're getting your interest payment in the form of the coupons. So the coupons here, exactly offset the time value of money, and so in the end, you're willing to pay the face value of $100. So if P equals F, YTM equals the coupon rate. Basic results about bonds. Now in general, we can look at the present value formula and see that the price is a decreasing function of r. Makes sense, r is in the denominator. We know for the r equal to the coupon rate, the price is equal to the face value. But yields on bonds fluctuate. Sometimes, we're going to be over here, when we're over in this part of the graph, the bond is going to be here. The bond sells at a discount, or we could be over here, and then we're in this part of the graph, the bond sells at a premium. To recap, when r, which is the yield to maturity by the way, when the yield to maturity exceeds the coupon rate, the price is less than the face value, bond sells at a discount. When the yield to maturity is less than the coupon rate, price is greater than the face value, bond sells at a premium. When the yield to maturity is equal to the coupon rate, the price equals the face value, and the bond sells at par. Coupon rate face value. When do bonds sell at par? When they are first issued. Bonds are typically issued at par. So the government or the corporation will choose the coupon rate equal to the prevailing yield to maturity. Then over the course of the bonds lifetime, yields will fluctuate, and sometimes bonds will sell at a discount and sometimes they will sell at a premium. So we define the yield to maturity for a coupon bond as the rate that makes the present value of the payments equal to the price. Now, in the previous clip, we asked, is the yield to maturity a good measure of holding period return, for zero coupon bond, the answer is yes, as long as you hold the bond to maturity. What about for a coupon bond? Let's think a little bit about the holding period return on a coupon bond. So let's go back to the case where our maturity of the bond is four years, the yield to maturity is a percent let's say, the face value is $1,000. Let's make life simple for ourselves and say that the coupon rate is $80, which by the way means, or I should say the coupon payment is $80, which means the coupon rate is eight percent, and so the bond sells at par. Let's say we hold the bond to maturity. What's the holding period return? So that means we're holding this bond for four years. We're looking for V_4 over V_0_1 over t minus 1. So that's what's V_4. Well, V_0, we bought it for $1,000, t is four. So what's question mark? What's V_4? Well, wait a second. Let's think about this. What should be V_4 be? Is V_4 $1,000? No, because you've got the coupons. Well, let's put in the coupon payments. There's four of them. Well, wait a second, this is not right either. After all, some of these coupon payments were received at different times. I mean, isn't there something like the time value of money? Couldn't we have reinvested the coupon payments? So this is not as bad as this, but it's still wrong because it assumes we did not reinvest the coupons. What is the right answer? Well, it depends on what we reinvested the coupons at. Now, here's a reasonable assumption. Now, notice I said reasonable, not perfect. A reasonable assumption is that we reinvested the coupons at the yield to maturity. So the coupon that we received in year 3, we reinvested at eight percent, and that became not $80, but $86.40, the coupon we received in year 2. Well now, we have two years to reinvest this at eight percent. So this becomes $93.31. The coupon we receives in year 1, that becomes $100.78. Adding it all up, we get $1,360.49. Reinvesting at eight percent, our holding period return equals 1360.49 over 1,000_1 over 4 minus 1, we recover a percent, which was a yield to maturity. This is a general result for a coupon bond. YTM equals holding period return, if the bond is held to maturity and the coupons are reinvested at the yield to maturity. You may not be able to reinvest the coupons at the yield to maturity because yields might change. So for coupon bond, unless your need for cash exactly matches the coupons, you face reinvestment risk. If you reinvest at less than the yield to maturity, it will be no surprise, and the notes have some calculations that the holding period return is less than the yield to maturity. If you get lucky, greater than the yield to maturity, then of course the holding period return will be greater than the yield to maturity. So in the case of a coupon bond, the yield to maturity is a flawed yardstick because there's no way to guarantee that you're going to get this. This concept doesn't only apply to coupon bonds, it applies in another scenario where the yield to maturity reappears in a somewhat unexpected way. So the yield to maturity is in fact similar to the notion of the internal rate of return, and the main problem with the internal rate of return is actually the same as the problem that we see for the yield to maturity for coupon bonds.
