Now, we will calculate spot rates, those are rk's. We just mentioned that the general procedure goes, so we observe prices of bonds. So from this, we can extract r's. And knowing r's, we can get the yield to maturity and not the other way around. Let's take an example. I'll put a, Three-year bond here. And again, for now, I will ignore the semiannual character of coupon payments. I will, for simplicity, state that they are annual. And this is a 5% bond, so it flows at 50, 50, and 1,050. And let's say that we have some r's here. Again, we have r1, which is 5%, we have r2, which is 6%, let's suppose that we know them, and r3 that is 7%. And if we calculated this, we would get the PV of this bond will be equal to 949.22, and that would be a standard procedure. Now the only question is how to find these r's. Because unfortunately, if we knew all the 50, 50 and 1,050 and this 949.22, we unfortunately cannot get these numbers. Well, let's talk about a very special example, that's called a zero-coupon bond. Supposedly, the bond would not have coupons like the same three-year bond, 0, 0, and 1,000. In this case, we could say, well, the PV of this bond is what? It's 1,000 divided by (1 + r3) to the 3rd power. So knowing this equation and though observing this PV on the market, we can easily extract r3. So if we had a liquid market for zero-coupon bonds for all feasible horizons of maturity, we would be all set. The problem, though, is that the market for zero-coupon bonds is liquid only for short-term maturities. Normally, zero-coupon bonds have a maturity of one year. But we can see a hint here. If we could create specific and sometimes called synthetic zero-coupon bonds for all feasible horizons, then using this equation, we would be able to extract r's for all desirable periods. Well, let's see how we can do. I will specifically flip over to analyze the following example. So calculating spot rates. We will take a look at two bonds observed in the market, again, the three years. Again, the first bond is at 4%, so cash flows out like this. And we see that the market price of this bond is, So this is all taken from the market. We do not know these rates. Well, but we also take a look at another also three-year bond. And again, remember that we are now talking about risk-less bonds, so we can easily compare them. There is no case of apples and oranges. This bond is an 8% bond, so its cash flows are 80, 80 and 1,080. And the market price of this bond, P2, is 1158.98. So I claim that from this information, I can easily find r3. How can I do that? Well, this is a classic trick that is oftentimes used in fixed-income markets. I create a synthetic bond, namely, I take a portfolio of two bonds of a kind 1. So I buy them, let's say, and I sell one bond of a kind 2. So this is my portfolio. Let's see what is the cash flow of this portfolio. Okay, and I'll put it here. You can see that from two bonds number 1, 80, 80 and 2,080, right? And then from my sold bond 2, we can see -80, -80, and here -1,080. So what's the net result? The net result is very nice, 0, 0, and then 1,000. Well, the fact that it's a 1,000 here is just the numbers I made this way. But see what we did. This portfolio is equivalent in its cash flows to a zero-coupon bond with the maturity of three years. And the nice thing is that, well, first of all, we know its principal. Well, it happened to be 1,000. It could have been something else, but it doesn't really matter. But we also know its price at zero because it's 2 times this less that. So I will put that the price of the portfolio is equal to 928.6 approximately. And that is equal to 1,000, this is the maturity, divided by (1 + r3) to the 3rd power. And solving this equation, I can get that r3 is 2.5%. Now, from these two bonds, I cannot extract r2, but I can find another set of securities that will be maturing here and I will do the same procedure. And strictly speaking, I can always find bonds that will have the same maturity and the combination that will eliminate coupons in between. Let's say if these coupons were 30 and 80, it would not be as easy, but I would have to buy, if it's 30 and 80, 8 bonds like this and subtract 3 bonds like that. And still, this portfolio would produce me the forecast of r3. So, like I said, in a market, there are sold various kinds of bonds that are equivalent to those. And we are now helped by the market, so we don't have to do these calculations all the time. But the general story is that this is a universal way to identify these rk's for any horizon k. And then we use them, this whole set, in order to calculate the PVs, and therefore, yields to maturity of all these bonds.
