Here's an example which we'll look completely unrealistic now, it is. But until like 40 years ago, this was the most people could do because they didn't really work with the models of randomness. They were just trying to estimate everything and consider it to be known and deterministic. But today we will do this example in much more sophisticated way, but since have not introduce randomness so far in the course, but let's just see how much we can do just by pretending that we know everything in the future so that everything is deterministic in the future. That's an example about pricing. Leasing a goldmine for 10 years. You want to rent a goldmine for 10 years and the question is, how much is the fair price, fair rent for this goldmine if you're going to pay everything, let's say to the [inaudible] just for simplicity. Since we only know how to price deterministic, so far we have to pretend everything is know in advance. We are going to assume that we estimate and pretend that we know this isn't the exact amount that the mine will produce, let's say 10,000 ounces of gold per year. That the cost will be $200 per ounce, that the one ounce will sell for $400 for all 10 years in the future. We also assume that we know that if you don't invest in the mine, but we put our money somewhere else, then we can get a return of 10 percent per year for the next 10 years. These are huge assumptions. There's no way you could really know this in a situation like that. You cannot know that the cost will always be the same for 10 years and what it will be. You cannot know that how much the goal will sell for, for 10 years or how much you can get in other investments. But as a benchmark, kind of back of the envelope computation, this is the best you can estimate, the future payments and costs and interest rates. Let's just work with what we have. We just have to add the present value of annual profits. Now the annual profit is how much? It's 10000 ounces, each costs $200 to produce. It sells for 400, if you multiply that it's just 2 million per year. The estimate is you will be getting two million per year, each year. In my previous slide, my x is to a million, I have to discounts to a million by its discount factor for years K1-10, 10 percent is my interest rate which I'm comparing to. I divide by 1 plus 10 percent to the k. There is a formula in the last slide. This is an exercise for you can just look at the formula, the last slide. You can put in the numbers, it should look like this and if you compute all of that, it's something like $12.3 million. A rough computation, a rough estimate, if you believe these numbers, would be that a fair price for renting this mine would be something like $12.3 million. We're not going to be pricing things like this in this course. This is not a financial market, this is a project which has to do with the mining gold, but there is a so called theory, real options theory. Real options theory, which is exactly a theory for pricing projects like these. It came out of ideas or option pricing, financial markets, variety pricing, random payoffs and financial markets, but it's now used for pricing things outside of financial markets, like company project like this. It's a little bit strange history that people were first having deterministic models in which they would price things like these, examples like these, projects which are not financial markets project. Actually, first came insurance theory. That was the first theory that was pricing random payoffs. Insurance and options theory, we're pricing insurance and financial contracts. Then ideas from those theories were eventually used for something which existed before financial markets and before insurance for things like mines and whatever projects you can think out. Again, there is, there are books on real options. We will not cover that in this course, but the same ideas that we will cover in this course are used for pricing things like leasing a mine or any projects of that type. Let's apply our formula for present value of cash flow payments for a typical deterministic cash-flow in financial markets, which would be a loan with deterministic interest rate. I'm going to be looking at a loan. Think of taking a loan, let's say, for a house. It's a fixed rate loan. You'll be paying for, let's say,30 years each month, the same amount. What we want to do in this slide, I'm going to go back a couple of slides. If you think about this formula, this was for the present value of payments X that I will be paying each month. In my loan example, my bank will will give me a loan on an amount, and that's going to be the present value. Let's say, I take a $400,000 loan, that's my present value. I want to compute, and my bank wants to compute how much I should be paying them every month. I will know the present value, I want to compute X. Here, I was pretending I know X, so I compute the present value of future payments. But in the loan example, I know the present value is the amount of the loan that I've taken today. I want to compute how much it will have to pay every month. I want to invert this formula and compute X as a function of the present value. So I'm just going to divide by this factor here, and that's going to be the formula two slides from now. Let's go back to two slides from now. If you do that, the formula is like this, except instead of PV, I'm writing V here for value of the loan, I'm just calling PV, V dollars. The formula is here. If I'm taking a loan in the value of V dollars and my quoted rate is r and compounded n times a year, and I will be paying that loan for n periods, then this is the amount I will be paying every period. This is the formula for X for the amount I will be paying every period to amortize to pay off this loan. Let's use this in a couple of examples just to make it more clear. Here's this example, $400,000 loan. It's a 30-year loan, you will be paying it off for 30 years. Let's say the annual rate is 8 percent. This is in United States, very high now. Wouldn't be this high, but we were writing our book from, which I'm taking this example, we were writing when the rates were high, more than 10 years ago. As usual with the house loans, this is compounded monthly. The question is to use the previous formula to find the amount X of your monthly payments on this loan. n is 12. Small n is 12 because there's 12 months in a year. There's 12 times compounding in a year, which means that n, the number of periods for 30 years is 30 times times 12. Each month is a period, so 30 times times 12 is the number of periods, 360 periods. r is 8 percent. r over n is 8 percent over 12, it's 0.067, and the value of the loan is 400. I have everything plug-in to my previous formula. This formula V is 400,000, r is 8 percent over 12, n is 360. I can compute this. If I compute this, if I did my computations right, you get something like about $3,000, $2,946 each month to be paid. That's simple enough. I mean, you will find obviously mortgage loan calculators online that will do this for you, but it's nice to see once in your life at least how this is computed in the simplest case with a fixed rate, fixed payments. What follows is the typical way the loan balance would be computed each month. Before the first payment, before the end of the first month, the bank would compute your balance as 400,000 plus the interest for that month, which is 0.0067 times 400,000 so that would be 402,680. Then you pay it. When you pay it, you subtract from this 402,680 what you pay, you subtract 2,946 in your balance after the first payment is 399,734. Then you add to this number, 399,734 you add the one-month interest. This means that you multiply 399,734 with 1 plus 0.0067 and that's your balance before the second payment. Then you subtract from that one, the second payment, and so on. That's the way the balance is computed. Then if you are curious, suppose that instead of taking this loan, you actually invest this at the same rate of eight percent a year, compounded 12 times a year. If you do that just for curiosity to see how much you will have, you can compute that as 400,000 times 1 plus 0.0067 to the 360, which is the number of periods, 30 years. After 30 years if you do this, you will have about 10 times more, I'd say you'll have four million and something, 4.5 million approximately. Just as a curiosity to see what the numbers are. That's this example. Another example with loans, make sure we get these formulas completely right. In US, when you take a loan for a house, there will be usually two rates listed. The mortgage interest rates and the so-called APR or the annual percentage rate. Why is there a difference? Well, there is a difference because what happens is the bank is not just giving you the loan for free, they are also charging some fees for their service. To make it easier for the loan holders to do this, you don't have to pay these fees in cash, out of your pocket. Instead, these fees are combined together with the loan amounts and then you pay the fees together with the loan along the way, a little bit each month. Maybe in this example, the bank will quote 7.8 percent interest, but then they will say," Okay, but the APR is eight percent. It's going to be higher. Which means our rate is really 7.8 percent, however, we're going to charge your fees and if we charge your fees, then to you it looks like the rate is eight percent. Because to 400,000 we will add the fees and then we will charge 7.8 percent. But to you it will look like it's equivalent to not adding the fees to paying the loan of 400,000 without fees, but with the rate eight percent." You look for the rate which is equivalent to paying 400,000 plus fees at same 7.8 percent or 400,000 without fees, let's say at eight percent. This is what the APR is. Suppose we have exactly the numbers like that. The bank is quoting 7.8 percent for the rate but for the APR on this particular mortgage, they are quoting eight percent. Let's compute the amount of fees that you are paying. You could in principle ask your bank to tell you how much in fees they are charging you. But to just practice our formulas, let's just assume that you compute the fees on your own from this information that we have. The numbers are exactly the same as in the previous example. We know what the monthly payment is, the APR is eight percent, eight is applied to 400,000 without fees. We know what the monthly payments will be, they're $2,946 from the previous slide. Once I know this, I can compute what actually the value of the loan that I am paying corresponds to. In other words, if I go back to my formulas, I know I'm paying This is 2,946, this x here. If I put 400,000 here, then r was eight percent. But now what I'm going to do, I'm going to put r 7.8 percent here. Smaller rate, that's going to give me a higher V here. I'm going to compute V. If you want, I can go to still previous slides. In this formula, I'm going to put for x 2,946 and then instead of eight percent I'm going to put 7.8 percent here for r. That's going to give me a higher value, higher than 400,000. I will see how much I'm paying in fees. Instead of eight percent put 7.8 percent, put monthly payments here, which are 2,946, and see how much we are actually paying in terms of the loan amount. If you do that, you can use the formula and see that you are actually, it's as if you are taking a loan on $407,851.10. This means that the fees that the bank is charging you is the difference between 407,851 and 400,000 so you're paying about 8,000 or $7,851 in fees. That was the second example with these type of cash-flow formulas for deterministic payoffs. Just a couple of things that are sometimes mentioned in books or in practice. Something called perpetual annuity. This is the easiest cash-flow to price. Not necessarily realistic. This is where the payments would go to infinity. You would save B, if it's an annuity, you get this like a pension thing and if you live forever, you get x every month. The present value formula becomes easier when m is infinity just becomes x/r. We're not really going to need this, I'm just mentioning in case you see it in books. Going further. Let's finish with this, this is called internal rate of return. Imagine you have a sequence of payments, X0, X1, X2, Xm, and one of them at least is negative. Some are positive, some are negative. Typically, X0 is going to be negative. Think of X0 as the initial investment into some project. In terms as a payment, it's negative because you have to pay that. Think of X1, X2, and XM as being positive. For this investment today of X0 you get X1, X2 to Xm in the future. If that's the case, then we define the internal rate of return for that project as the number r for which the sum is zero. In other words, if this is all fairly priced, if you are paying X0 today, if that's the fair price for the payments of X1, X2 to Xm tomorrow, then the total value should be zero, if you believe it's fairly priced and the value is zero then the actual rate that goes with this investment is the rate which makes it equal to zero. You add the present value of these payments. The payments that you will receive and the payment that you pay at the beginning, you add the present values. You have to find r for which this is zero. It's a non-linear equation in r. You have to solve it numerically or maybe even all the Xs are the same and becomes easier. But still, it's a non-linear equation so you find r for which this is zero and we call that the internal rate of return. Why am I defining this?, because this is going to be in the next set of slides. This is exactly how we are going to define a bond yield. For the bond, you will pay some price initially and then you will be getting coupons and your final payment at maturity. Then the yield will be defined exactly as the internal rate of return of a bond. That's it for this set of slides. Thank you.