Calculating the Yield of a Zero Coupon Bond using Forward Rates
In a previous article, we talked about how you could compute the forward rate for a given year using zero-coupon bond yields but what if you don't know a zero-coupon bond yield and you do know the forward rate? Well, actually there's a way that you can calculate the bond yields for zero-coupon notes based on the forward rates that you have assumed that you know what the forward rates are and we've got this really nice formula.
That's gonna allow us to calculate just that so let's jump into an example and I'll show you how it works. So let's say that you didn't know the yield on a five-year zero-coupon bond but you did know the forward rates here I've got the forward rates for the next five years so you've got these different forward rates here and you can essentially just plug them into this formula above and we can go ahead and actually calculate what the yield to maturity is going to be on this bond.
So let's go ahead and start plugging in so we see here we have (1 + the forward rate) from year 1 so that's 7% so that's same as 0.07 so we'd have (1 + 0.07) is going to be that first term. So We have 1.07.
So we're gonna multiply 1.07 by the next term (1 + the forward rate) rate for year two. What's the forward rate for year two? It's 6.8%. So we're just taking (1 + the forward rate) for each of these periods. It's a five-year zero-coupon bond so we're gonna go all the way up to forward rate through year five. So we're just saying 1.07 corresponds to 7%, 1.068 corresponds to the 6.8%, 1.0624 forward rate three, 1,064 for forward rate four, and then 1.067 corresponds to the 6.7% interest for forward rate five.
So it's just that simple we go ahead and we take the product of all this and that's going to be equal to (1+ the yield) for year five what we're trying to compute then we raise that to the fifth power. So that's going to make our math a little bit interesting here, so what we want to do is get the yield of maturity just al1 by itself. Now we could subtract 1 by each side and we're gonna do that in a minute but we've got to get rid of this exponent 5.
So let me just calculate it out, I'll just give you this is 1.378 equal (1+ the yield to maturity) for year five to the fifth power. Now we have to get rid of this exponent 5. So what we're going to do is take each side and we're going to take to the 1/5 power. So we have here 1.378 we're gonna take to the 1/5 power. Now when we take the right-hand side here to the 1/5 power that (1/5 multiplied by five) that's just going to be 1. So then we're gonna be left with (1+ the YTM) for that fifth year for the zero-coupon bond which is what we're trying to calculate.
Now we can go ahead if I've just done the math for you this 1.378 which is rounded by the way to the 1/5 power and then that's going to be 1.066 and again it's rounded so if the numbers don't match up exactly what yours it's alright. So 1.066 and that's going to be equal to (1+ YTM) that we're looking for for that zero-coupon bond.
Then now we just subtract 1 from each side so that's gonna give us 0.066 is equal to our yield to maturity on a five-year zero-coupon bond and another way of expressing that 0.066 is 6.6% that's the same thing it's just our way of expressing that decimal.
So what does that mean? that means that we have used these forward rates by plugging them into our formula above. I probably should have mentioned you continue on to whatever year zero-coupon pond you're trying to calculate, we're doing a 5-year here so we go the forward rate one - five and we go on into that point but if it was a seven-year bond then we'd go all the way up to year seven. So it's easily we can take this formula and apply it no matter how many periods there are for the yield that we're trying to get on the zero-coupon bond but in this case, it's a five year zero coupon bond the yield would be 6.6%
