Now, I wanted to find a discount bond. Bonds typically pay coupons. This is an old word but they still say coupons. It used to be, that if you invested in a corporate bond or a government bond it would be on a piece of paper they would give you. This is like for hundreds of years. And around the exterior of the pieces of paper, were little coupons that you would clip every six months typically. And you would clip your coupons every six months and take them to a bank, and the bank would then give you the money. So, each coupon would be so much money. And then, at the end of the maturity of the bond, you could take the whole thing back and you get your amount back. So a typical bond back then, in 1900, and 1800, and 1700. Going way back. If you bought a 100-dollar bond, and it was issued for $100, and had say as a three-dollar coupon. Then, you would pay $100. You'd wait six months, you'd clip a coupon, and it would say, pay to the bearer one dollar and fifty cents. You'd go to the bank and get your one dollar and fifty cents. And you can see some of these bonds, they're framed and on display. Ones that defaulted, otherwise, the coupons would be already clipped and gone. You can see them, I think they're on displays on the fourth floor of this building. A discount bond is a bond that carries no coupon. Now, why would you buy a bond that carries no coupon? How do you get interest from it? This is also time in memorial. People have traded discount bonds for a long time, and the answer is, because you buy it for less than $100. You buy it at a discount. So, they tend to be two different kinds of bonds, the coupon bonds which are more common, tend to be sold at par. You buy it initially for $100, and you sell it for a 100, you get back at the end when it matures after so many predefined years. With a discount bond, there are no coupons but of course, you buy it at a discount. There would be no other reason to buy it. Maybe today they're selling not at a discount with a negative interest rates. But normally, they're sold at a discount. We still call them discount bonds even there's a negative interest rate than they're selling for more than $100 initially. So, if we look at the price of the discount bond, we can infer the yield to maturity from that bond. So, if someone says, I have a bond that will pay let's say, one dollar in T years, and it's compounding once a year, and the price I want is P, I can compute using this formula. What the yield to maturity is. So, I would basically take one over P, if solving this equation, I take one over P to the one over T power, and that's the yield to maturity. In a sense paying an interest rate of r, every year. Compounding once a year if I call it that. But typically, bonds pay interest rate every six months, that's the tradition. So, you might use this formula instead which has the bond compounding twice a year. If T is a number of years to maturity, and P is the price, then we will take P as the present value of the principal which I have as one dollar here, divided by one plus or over two, to the two T. So, the price today of the bond is called, the present value. If it's a one dollar principal of one dollar at time capital T. And as a general rule, P is going to be less than one. I say is a general rule because it might not hold right now which is a little puzzling. But over most of history, it's a discount. P is less than one. So, is that clear? Any questions about that? You have to specify the compounding interval. But normally, for pedagogical purposes, it's convenient to take the compounding as once a year. And we just use this formula. By the way, you could do it continuously till you could say, what does that continuously compounded yield to maturity. And that we have P equals e to the minus r times capital T. Okay. Now, we can define the present discounted value of any cash flow, not just a coupon flow which would be the case for a coupon bond, or the principal after T years for discount bond. Because we know that implicit in market prices for discount bonds, we can calculate the present discounted value of a dollar in any number of years. So, what is a dollar one year today? What is the present discounted value of that? It's one over one plus r, where r is the yield to maturity on a one year discount bond. And what is the present discounted value of a dollar in n years? It's one over one plus r to the nth power. Now, this is obvious to a banker who always thinks. When you talk money with a banker, and you're talking about money in future years, there's a little calculator going in his head, he has memorized the prices of all these discount bonds going out. And he's translating it into present value or present discounted value, PDV. Amateur's mess this up. Instead, they become vulnerable to fishes. Lots of people make mistakes. Maybe you should develop the habit of always computing the present discounted value. On the other hand, you live in a very good time for ignoring this. Because right now, interest rates are virtually zero. But it will come back. I think Nick you are right, we are going to have two percent and higher interest rates at some date in the future. I just don't know when. If you have a cash flow, x sub t, where x sub one is the money coming in in one year. x sub two is the money coming in in two years. And let's assume that the discount rate is the same for all these different maturity. This is a simplification. Then the present discounted value of the cash flow is the summation t equals one of the capital T, of the cash flow x sub t, divided by one plus r over the t. That's one of the most famous formulas in finance. So, now let's look at a conventional coupon varying bond which is issued at par. How do they issue them at par by the way? It's tradition to issue at par because you're getting your interest in the form of coupons. What they have to do is judge the market. If I want to issue with par, what coupon is the market demanding on a $100? And once I know that, I'll just pick that coupon, and I can be pretty sure that my bond will be picked up for $100 because I've got the market coupon. So, I'm going to use c for the amount of coupon. Now, this is measured in currency, if we're dealing in dollars, c is so many dollars and the prices in so many dollars. And what it is now? Now, here I have two versions. This is compounded annually, and this is the more realistic compound in every six months where t measures years. So, what you get in a simple case when you buy a coupon bond, is you get after one year, this is the annual compound after one year, I can clip a coupon for c dollars. And then, I have to wait another year, and then I can clip a coupon for c dollars again, and clip another coupon in three years for c dollars. And then at the end, I get my last coupon of c dollars, plus the principal which is a 100. So, I get 100 plus c dollars at the end. What is the present value of that if it's discounted at rate r? It turns out that's the formula for the present value. Now it's interesting to take the limit of this as t goes to infinity. If t goes to infinity, this term goes to zero, right? And this term goes to zero. So, we're left with c times one over r. So, that's the console. I think I have another slide for that. This is the more complicated formula for six-month compounding. This formula was sufficiently difficult that in the old days, and people didn't have calculators. Bankers would carry around the table, bond yield table. Also you can't solve this back. If I'm told the price of a bond, and I want to compute the yield to maturity r, I've got to solve this equation for r. And you can't. It's not algebraically possible. Unless, t is very small. So, you need a book. But now, it's probably already on your mobile phone. I think it is. I know it is. Go to Wolfram Alpha on your mobile phone and you'll get, or there must be other places. Must be hundreds of places that will solve this equation for you because it's standard. So many people think in terms of present values. And they want to know what the yield to maturity. What's the interest rate on a bond given its price.