So remember we have four bonds, one issued by the US government, one issued by GE Capital, one issued by Motorola. One issued by a company owned by Trump, and we're going to take the GE Capital Bond. I remind you that the maturity we are, quote, unquote, we are in January 2002 and the bond expires in January 2007. That is five years down the down the road. The bond had an interest rate of 7.875%, which means that if we multiply that number by 100, that gives me the annual coupon. And if I split the coupon by two, that gives me how much I'm going to get at the end of each semester, that the bond is going to be paid interest for the next five years. We know the rating, which is AAA, and as we will talk about in a couple of minutes, AAA is kind of the highest rating that you can get. That means that the probability of default is not zero, but it's very close to zero. It's very, very unlikely that this issuer is going to default. This bond like most other bonds pay semi-annual coupons, which means that we're going to get the annual coupon, half of it at the end of the semester. And the other half at the end of the following semester. And that is the price that we have to pay for that bond, which is very close to 113. Now remember, the face value of the bond is 100. So at this point in time were paying more than the face value of the bond, almost 13% more than the face value of that bond. Let's think about cash flows. What cash flows do we put in our pockets? Well, per year, remember, if I multiply the coupon or interest rate by the face value, I get 7.875. That's the annual amount of money that I'm going to put in my pocket. But because this is going to be paid in two installments, each at the end of six months, then per semester I'm putting in my pocket half of that. That is 3.9 and a few more cents. And finally at maturity, I'm going to get one of those final coupons plus the face value of the bond. Now we know all the cash flows that we're going to put into our pocket. We know the cash flow is going to come out of my pocket, which is 112.799. So what we can do is that as you see that that's an Excel file that goes, we have all the semesters that we have there. Remember, the bond is five years away from maturity, zero is today, ten is the end of the tenth semester, which is the same as saying at the end of the fifth year, which is when the bond expires. If I want to buy this bond, I have a negative cash flow money that comes out of my pocket a little bit over $112. And what I'm going to put in my pocket over time is the 3.938 that the bond will be paying semester, after semester, after semester, after semester, up until the very last one when the bond expires. Where I'm going to get that final coupon plus the face value or the principle value of the bond. Now, if you want to calculate that yield to maturity, knowing the market price, knowing all the cash flows coming into your pocket, then you need to back out. Remember the expression, we need to back out the denominators. And that's the yield to maturity, which is 2.48%. Couple of important things to keep in mind, number one, you can do this in Excel very easily. And you can do this in more than one way. In fact, Excel has what is called a yield command and it gives you this number directly. I think that the best way to do, at least for the very first time, is lay out all the cash flows, and then calculate that, which is the internal rate of return. So as you see there you have the cash flows in the column. And what we are saying is give me the internal rate of return of the negative and all the positive cash flows. And that internal rate of return will give you the 2.48. There are other ways, perhaps even simpler ways to calculate the yield to maturity in Excel. But this one I like it because it gives you an idea what a yield to maturity really is. And a yield to maturity is basically, that internal rate of return that takes into account cash flows out of your pocket and cash flows into your pocket. Comment number two, and again I'll show you more numbers in just a minute. But comment number two and it's important, remember that what we're laying out there are semi-annual cash flows, which means that the rate that we're calculating is a semi-annual rate. Now, most of us tend to think much better in annual terms than in semi-annual terms. So we're going to make a slight adjustment to that rating in just a minute. If you want to check, and I do advise you that that you should do this. If you want to check whether you have really understood this, get the four bonds in the case, don't look at this table. I get the four bonds in the case, layer all the cash flows, layer all the prices that you have to pay for the those bonds. And calculate all those internal rates of returns. That will give you the YTM line, and that's basically the internal rates of return, so the yield to maturity of those bonds. Now if you look at the GE Capital, that's the 2.48 that we calculated a minute ago. You can do the same with the rest of the bonds and you would get the semi-annual yield to maturity. Now there's two ways to turn that into an annual magnitude. And I'm going to go with the easier one, which is in order to go from semi-annual to annual, you simply multiply by 2. So the annual yield to maturity is 2 times the semi-annual yield to maturity. There's a slightly different calculation that we could make which is called, and I don't want to confuse you with this. That's why I'm going to mention this in passing, but it's called the effective annual yield. And the effective annual yield takes into account that you don't have to wait until the end of the year to get the total coupon. You get half of the coupon before the end of the year, six months before the end of the year. And that means that the money that you get halfway into the year can earn interest on interest. And so if you were to calculate the effective annual yields to maturity, all those numbers would be a little higher than the annual yields to maturity that you have there. That doesn't account for the compounding of the money that you get halfway into each of the years. For all our practical purposes, those annual yields to maturities, all that we actually need and with those semi-annual yields, and even better with those annual yields. I want us to think for a little bit the numbers that we've gotten, that go between a low of 3.43 and a high of over 25%, which is on the Trump bond. Now first question that would have to come to mind is why do we have so different yields? And in finance when we think about obtaining different returns, the first thing that should come to mind is because they have different risks. In other words, the USOA, that is the US government bond has less risk than the GE Capital Bond, that has less risk than the Motorola bond, that has less risk than the Trump bond. And as a matter of fact that this is actually a good article that tries to highlight this idea. And it says that if you see a yield that is significantly higher than a peer group or tends to hang around the top of the peer group, you can bank on the fact that there is some extra risk built into that portfolio. Whether it's liquidity risk, or sector risk, or quality risk, what have you, but it's there. That's the big number at the end of the day that tells you whether or not there's risk in the portfolio, because there's no free lunch. Now, this is very important. And the reason it's very important is because we tend to think of the yield to maturity as a measure of return. Which it is, because it does indicate the return that you're going to put in your pocket if you buy the bond at the market price and you sell it at the maturity date, or you hold it until maturity, I should say. But it is also an idea of the risk that you have to bear, because going back to our numbers, there must be a reason why the Trump bond pays you over 25% when the US government pays you less than 3.5%. And that reason has to be well, because the Trump bond is a hell of a lot riskier than the US bond. So as you move from the USOA, to the GE Capital, to the Motorola, to Trump the return that you get increases. But also the risk that you stand to bear increases too.
