Government Bonds: Zero Coupon

We are going to talk about entities that issue bonds or entities that borrow. And we are focused on the most important entity, I guess, in the following sense, that it borrows a lot, and it's in every country and it's called a government. And governments borrow issuing bonds and we're gonna focus on those. They finance a ton of activity as I said, and fiscal policy is very, very common across the world. They issue bonds that are very complex, but have also been involved in market backed securities in America and other countries that are pretty complex. This is all just fine.

It's just that the level of complexity of the bonds can be so much, that at some point, technical assistance in understanding how bonds work is critical to every company, every institution, every government. So I don't want you to think that we're going to approach all that complexity. But do enough for you to understand how thing could be dramatically different in different situations. We'll start with the simplest, most traded bond in the world, probably. It's called a treasury security, and the interesting thing about a treasury security, you'll see today, is it's a zero coupon bond. What does that mean? Remember, a bond pays in two ways.

So when the issuer of the bond, the borrower says, I need money, they specify two things. What is the periodic payment, called coupon, and what is the final payment, called face value? Zero coupon bond actually does not pay any coupons, and it's issued with up to one year maturity. Remember, there's a finite life, bonds tend to have finite lives, they're contracts. And it's issued up to one year maturity. And I'll give you a very interesting episode of the beginning of, I mean, not episode, example of the beginning of something called securitization when we get to it. So what does it say? It typically says that, I'll pay you $100,000 or $1,000 or whatever.

Where I is the US government, one year from now is the maximum maturity. But actually let me just introduce the complexity I just spoke about. Because we'll use it in examples as well as I'm thinking. So we're doing this ahead of time. They are either treasury bills or strips. What is the distinction between a treasury bill and a strip? For the time being, the distinction is a strip is the treasury bill, zero coupon bond with maturity greater than one year. So think like that. It, of course, could have maturity less than one, we'll get to later. But treasury bills originally were issued with up to one year maturity.

But you can buy government bonds with greater than one year maturity, that are also zero coupon, okay?

This is a very, very interesting example. It looks very simple and it isn't pricing, but it's the most traded type of loan. So what does it have? It has face value, that's it. So let's take an example. Let's try to price the bond. The bond price will be very simple, it'll be what? It'll be equal to the present value of only one thing, and that's the face value. So I'm going to just, instead of writing it all out, I will just show you in the slide on the page, and please write it out. So what you'll see is the price of the bond is equal to face value times our old friend, present value factor.

And sometimes I say PVF, sometimes I say FPF, what is it? It is the present value factor means how much is the present value of one buck given an interest rate of r and number of periods away n? Or, in this case, it's the face value divided by 1 + r raised to power n. Remember, it's one shot. It's like the first kind of example we did in the first module, seems like we met many, many moons ago. In the first module, the first thing we considered was one shot payments gathered over multiple periods. We did the future value, we did the present value.

So this is the present value of getting one shot payment and years from now, where n can be any number. So let's do an example, suppose there is a zero coupon bond that pays $100,000 in exactly one year. And let us assume that the compounding interval is one year. You'll understand what I'm talking about in a second. Because in the real world, the compounding interval for government bonds is about six months. So let's assume, for simplicity, it's one year. What is the price or value of the bond? The two are the same. How much will it sell for today? In other words, if all other bonds of similar majority are earning about 5% and issued by the government.

The neat thing about government bonds is they're traded a lot and their information is available. So the discount rate is what? 5%. So let's try to do this. This is so simple that I'm kind of a little bit, it feels odd doing this. So let's write it down. Equals, what will the price be equal to? So you do equal sign in Excel. And we'll introduce some other Excel parameters or functions. So it'll be what? The PV function, and what will be the interest rate? 0.05. How much is n? Remember, we are saying the compounding interval is the same as the maturity of this, which is one year. And what is the PMT? 0, look how I wrote 0, it's style.

And finally, 100000 remember, don't put commas.

And when you press Enter, you should get a number 95,238.10. The nice thing about a zero coupon bond is under the assumption that interest rates are positive. And by the way, in recent times even, we have seen this not to be true. But anyways, most of the time the interest rate is positive. Right now, the interest rate on such bonds will be very low in the US, but suppose it's 5% because we want round numbers. The price of a zero coupon bond under the assumption that time value of money is inherently positive is $95,238, meaning less than $100,000. So one thing that it's also called, a zero coupon bond, is also called a discount bond, why?

Not because it's being sold at a discount, meaning it's not a sale. But the present value has to be less than face value. But that is under the assumption that interest rates are positive. So let's think about the next concept. What we're going to do next is do a little clip on what I call yield to maturity, which is related to what we just did. In our example right now, if you think about it a little bit, I gave you two numbers and I asked you to calculate the third. Which we said, was the price of the bond. Now, think about it in the real world. What will happen? Trading is happening and somebody is saying, you know what?

I'm going to give $100,000 one year from now, how much will you pay me? So it's very likely that what you know is the price and built in the price is the rate of return. So we'll talk about that in a second.