# Calculating the Yield of a Zero Coupon Bond

In this article, we're going to talk about how to calculate the yield of maturity on a zero-coupon bond. So the yield of maturity is the rate of return that you would get if you purchased a bond at its current price and then held the bond to maturity.

So for example let's say that you bought a bond for \$95,238. The face value of the bond was \$100,000. So because it's a zero-coupon bond that we're talking about there's not going to be any periodic interest payments or anything like that it's just a simple 1-year bond where you pay  \$95,238 in a year from now and get  \$100,000 returned to you.
So that difference between the \$100,000 and \$95,238 you could just think of that as the interest that you have earned on that investment, right? So now we want to say what the rate of return is? If you buy it for \$95,238 price and you get \$100,000 a year later what was your rate of return on that investment?

### The formula of calculating Yield to Maturity:

So we can just go ahead and calculate it out we can just use some really simple algebra. If we took say our \$95,238 that we paid as our price and if we multiply that by (1 + the yield to maturity) here YTM means the rate of return so  (1 + the Rate of Return) and that's supposed to equal \$100,000.

in terms of what we've just set up in this formula that should equal \$100,000 and that'll give us our yield to maturity. So this is basically we take the investment of \$95,238 we multiply it by  (1 + the Rate of Return) and that's going to give us what we get which is \$100,000.

### Solve:

So we're trying to solve for the yield you can go ahead and just divide \$100,000 by \$95,238. So we're just dividing each side we're just doing simple math here and so that'll give us one us the yield to maturity is going to be equal to 1.05.

Now I've done some rounding here so if it's not exactly precise forgive me but we basically just divided each side by \$95,238 and so now we've got this (1 + the yield to maturity) is 1.05 and so we just subtract 1 from each side and now we have our yield is going to be 0.05 or we can just think of that as a percentage we can think of that as is 5%.

So what does this mean this means that the rate of return if we were to buy a bond today for \$95,238 it didn't have any periodic interest payments anything like that it's just a one-year bond so a year from now we get \$100,000 back we earned a rate of return of 5%.

### Zero-coupon Bond for multiple periods:

So now you might say this is a simple example, we just got one year here but what if we had multiple periods? How do we go about calculating it? So we've got a really nice formula and I'll just show it to you let's just say that we had a three-year bond. let's say that we have the same things as above, its face value is \$100,000. We're going to pay \$95,238 but this time we're not going to get that \$100,000 face value until three years later. It's a three-year bond instead of a one-year bond.

### YTM formula for multiple periods:

So now I just want to show you this formula will give us the result and give us our yield every time and no matter how many periods they are.

So n is going to be 3 and it's just a number of periods and then FV that's the face value that's going to be the \$100,000. let me actually just start plugging in some of these numbers here.

So we have the face value of \$100,000 in parentheses and then that's going to be divided by the price \$95,238 and then we're going to take that and raise it to a power write all this in here in the parenthesis we're going to raise to a power. We're going to raise it to the 1 over N power, which is in this case 3 and so we're just going to raise it to the 1/3 power, and then after we've calculated that we're going to subtract out 1.

So I'll just go ahead and skip right to the chase we're going to end up with .016. So if we're going to and we can think about that in terms of a percentage 1.6%. So when you put that into your calculator if it doesn't come out exactly like this I apologize I just did a little bit of rounding but the bottom line is that we just use our formula. We've got n is the number of periods. So let's say this was a seven-year zero-coupon bond then we would raise the fraction to the one-seventh power, right? So that's where the end comes in and then we just have the face value of the bond in our case it was \$100,000 the price of the bond right and so you just plug in those numbers and it just very simply gives you the yield to maturity.