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Yield to Maturity

Yield to Maturity
We will talk about three things for a zero coupon bond. The face value, because there's no coupon, the interest rate 5% In the previous example, and then we figured out price. As I said, if time value of money is positive, and another cool thing about treasury bills like zero-coupon bonds, there are so many cool things about it, is that it is the most fundamental security out there. In other words, if you hold it till maturity, you are going to get $100,000, why? Because as we'll see later, we believe that any other entity except the government will pay up. The governments are unique in many ways that they tend not to default.
Having said that, I have to be careful, because default can happen to anybody. But if you were to ask anybody which is the safest security in the world, they'd probably say a zero coupon bond because if you hold it to maturity, you will get the money that you were promised. Of course, there's inflation risk, and there are instruments to take care of that, but let's stick with the basic security. So, what is yield to maturity? Remember there are three things. One is the face value, the second is the price, the third is, built in, between the price and the face value, is something called yield to maturity.
So let's talk about it a sec, and we'll see a lot in the real world about yield to maturity. And therefore I want to capture it. The return built into the pricing of a bond is called yield to maturity and so let's think about it. And then an example, suppose for simplicity, w We have a 5-year zero-coupon bond, and the simplicity is the following. Compounding is happening every year, not every six months. We'll change that in a second, we go back and forth on that small issue. But fully agree, that is nothing compared to the complexity we just did in module two. So for now, five years means five years. Five years means five periods.
If you look up it's a zero coupon bond that the face value of $100,000 and the current price is $74,726. So what happened? So imagined that Is the auctioneer, I love that, saying, I'm the government. I'm promising to pay you back $100,000, so people bid with each other and go back and forth and they decide that the price is 74,726, and it clears it.
That clearing price also determines the yield to maturity built into the bond, okay? So that's what we're trying to do right now. So let's draw a timeline as always and try to solve the problem. And I'll tell you how to do it using a function in Excel. I'm looking down to get hold of the pen. Okay so you have five years, and we are assuming five distinct compounding intervals. And you're going to be paid 100,000 over here.
This is in your think about it, what will it be? Future value or present value? It'll be future value. We also know the PV and it is $74,726, okay? So this is what we know, and I'm not putting commas simply because I'm trying to think ahead, and I know I don't put commas in Excel. How much coupon? Coupon is zero. It's a zero-coupon bond. So, the question we have is, what is the per year rate of return built-into it if I hold the bond till maturity. What is the rate of return built into it if I hold the bond til maturity? So I know this.
I know that 74,726, I put a commas, is equal to 100,000 / 1 +, I'll call it yield to maturity, power five. Remember what you are trying to calculate. You're trying to calculate that rate of return that's built into the bond pricing. Remembering what? Two important things. One is that I'm going to hold to maturity, yield to maturity. The second is what's the compounding interval. By the nature of the formula that I had wrote, what do you think is the compounding interval? The compounding interval is a year. As I said, I'll change that in a second.
Now if you look at the five, that creates the problem because it's raised to power five because of compounding, and we are quoting a number that's on a yearly basis. So now you have problem because it's a one equation, one unknown. But raise to power five is very, very tough to figure out. Turns out that there's another name for yield to maturity, we'll do later. It's called internal rate of return built into this, which makes sense again, right? Internal to the cash flows. So let's try to do it using Excel, and I'm going to do that with you. But I'll introduce a function, and that function, if you think about, and look up, please look up a function called rate.
Say =RATE(. And if you open parentheses you will see the following symbols jump at you. The first one won't be R. Why? Because you're figuring out R so this is the slight twist. The first one will be number of and I'm going to write it as m. The second number that will jump at you is PMT.
The third number that will jump at you is PV. And the fourth, you'll see (FV). And what have I done to the FV? I've put it in parenthesis. Now, the key here is that both PV and FV can't be positive, because if they are, you're assuming you have a money machine. You're getting money, you're getting money, you're getting money. One of them has to be negative in order to figure our rate of return. So be very careful. What is N? N is the number repeated, so let's figure it out. How much will you put? 5, then the comma? How much will you put for PMT?
Because it's a zero-coupon bond. How much will you put for PV? We know the number, and I'm looking here. It's 74.
726, but I put in a positive number, so what does the FV have to be? Negative, so negative is -100,000, and I was going to put a parenthesis, but so this what I'll do for each number Right? Match it and when you press enter, what will you find? You'll find 0.06 or 6%. You'll find this, okay? So, very easy to execute, but if you have Excel and the culprit here is the five. And imagine, this could be actually greater than five, can be in fractional terms, and let's talk and do a real world twist. What do I mean by that? Typically the compounding interval.
Interval is determined at six months. What do I mean by that? Is built into all pricing of government bonds, the interval for compounding is six months. So when we go to a government bond that pays coupon, it pays coupon every six months. That is just a technicality that will make your life a little bit miserable, but not too miserable, right? So what do I have to do here? As soon as I know the compounding interval is six months, I will do five years. I'll change it to 10 periods.
And that's available twist I want you to think about, okay, and when you are doing it in real time data. And at the end of this module we'll go to actually, and I'll do that even for stocks to show you how common all of this is and across the world you can do this. So I'll replace this by 10. So I'll do, rate 10, comma, coupon's a zero. Suppose the price is still the same, which it won't be, but suppose that's the price. But if we are making it realistic, -100,000. Quick question, what will this number be? And this is a quiz for you. Think about it. Let me ask you the following.
Most people will be inclined to take 6%, which they've already calculated, and divide by 2. That answer can't be right because of compounding. So the answer is, because interest earns interest, the number will be less than half of what we've calculated. And I think it'll turn out to be 2.96%. One last comment. If and when we see payment, which is coupon, we will also divide the coupon by two, because the coupons are stated annually. And so, when you take a coupon, and it's being split up every six months, you'll have to divide that by two. More on that later.
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