In this module, we're going to introduce you to interest rates, present values, and how present values are implied by the no arbitrage principles, and also, introduce you to the idea of fixed income instruments. An amount A, invested for N periods at simple interest rate of R per period, is going to be worth A times 1 plus n r at maturity. In every period, you'd get an interest R times A. So over n periods, you're going to get an interest of n time a times r, and you get back your initial investment and therefore the total amount is going to be a, times one, plus n times r. An amount a, invested for n periods at the compound interest of r per period, is going to be worth, a, times one plus r raised to the power of n, at maturity. And the way to think about it is what is happening at time zero you invest A amount. One period later this amount A has now become 1 plus R times A. This total amount is now invested for another period. So it becomes A times one plus R squared, and so on. So after N periods, it becomes one plus R to the power N times A, or A times one plus R to the power N. Interest rates are typically quoted on an annual basis, even if the compounding period is less than one year. If there are n compounding periods in a year and the interest rate is r per year or per annum, then an amount a, invested for y years, is going to yield a times 1 plus r over n. So R over N, this is the interest per period and the total number of periods in Y years is going to be Y times N so this is the total number of periods. And the amount that you end up getting is A times one plus the interest per period raised to the power the total number of periods. Continuous compounding corresponds to the situation where the length of the compounding period goes to zero, therefore an amount A invested for Y years is now going to be worth, the number of periods of N going to infinity of A times one plus R over N raised to the power YN. And if you take the limit of N going to infinity, that expression solves as A times E raised to the power of Y at maturity. The number of. The compounding period is going to zero, therefore the number of compounding periods is going to infinity. And you end up getting that the expression simplifies to A times E raised to the power RY. For those of you who are [INAUDIBLE] differential equations, you can find out that basically what's going to happen is dA dt. Is going to be R times A, and if you solve this differential equation you'll get exactly the same expression that we have on that slide. Next, what we want to use is this idea of compounding to calculate present value. Remember, in the last module, we showed how the present value for a very simple bond, which pays A dollars in one year, you can calculate that using a no-arbitrage argument. We're going to do the same thing here using an interest rate R, and then we're going to expand that to the idea of borrowing rates and lending rates being different. So we want to compute the price p for a contract that pays c0, c1, c2, and so on cn at times 0, 1, 2, 3, 4, n and so on. If ck is greater than zero, that's a cash inflow, if ck is less than zero, that's a cash outflow. The present value, assuming an interest rate R per period, can be written as C0 plus C1 divided by one plus R, C2 divided by one plus R squared, and CN divided by one plus R N, capital N. In terms of an expression, this is the expression for the present value. We are going to argue that this present value is in fact the arbitrage free price b for this contract. Here's the argument. We're going to assume that we can borrow and lend at unlimited amounts at the rate r. And the portfolio vehicle to construct is false. You're going to buy the contract. If you buy the contract, what happens? At time t equal to 0, you have to pay an amount minus b, and you receive and amount c 0. At time t equal to 1, you pay, you'd receive c 1, time t equal to 2, you'd receive c 2, and time equal to k, some general k, you'd receive c k and at time t equal to capital T, you'd receive c T. I'm going to borrow c1 divided by 1 plus r amount for up to time 1. So at time t equal to 0, this is the amount that comes into my pocket. At time t equal to 1, I have to pay out, I have to return the amount that I borrowed. And how much do I have to return? It's going to be the amount that I borrowed times 1 plus r. And which cancels out the divide by 1 plus r you end up getting exactly minus c1. Similarly I'm going to borrow c2 divided by 1 plus r squared up to time 2. I will get c2 divided by 1 plus r squared into my pocket at time t equal to 0 and I have to pay out an amount c2 at time t equal to 2. Generically for a general time k, I'm going to borrow ck divided by 1 plus r to the power k. I received that amount at time t equal to 0 and then I have to pay out minus ck at time t equal to k. And what you notice by doing this construction, is that the cash flows at all future times cancel out exactly. So, the portfolio's cash flow is for all time k equal to 1, 2, 3 up to capital T turns out to be 0. And the weak no-arbitrage condition then tells me that the price that I would have had to pay for constructing this portfolio at time t equal to 0 must be greater or equal to 0. All future cash flows are greater than or equal to 0. Therefore the price at time t equal to 0 must be greater than or equal to 0. What is the price at time t equal to 0? It's the negative of the cash flow. How much did I have to pay? It's the negative of the amount that I received. So the price of the portfolio is going to be, minus the summation from k equal to 0 to capital T, c k over 1 plus r, to the power k. That must be greater than or equal to 0. That gives me a lower bound, and the price must be greater than or equal to this expression, which is exactly the present value calculation. In order to get an upper bound of the price, I'm going to reverse the portfolio. I'm going to sell the contract. And if I sell the contract, what happens? I receive an amount, p, at time t equal to zero, but now I'm responsible for the cash flows associated with the contract. I have to be, paying the cash flows to the buyer. So I have to pay out C0. I have to pay out C1, C2, all the way up to minus CT. The negative is in front. Because now instead of receiving these cash flows, I have to pay them out. I'm going to lend an amount, C1 divided by one plus R, up to time one. So when I lend them out, money goes away at time T equal to zero. But I get it back at time T equal to one. I lend an amount c2 divided by 1 plus r squared up to time 2. Similarly, money goes out at time t equal to 0. But it comes back now at time t equal to 2. And how much comes back? It's minus c2 divided by 1 plus r squared times 1 plus r squared, which is exactly c2. Again, if you notice, what we have done here is constructed a portfolio, such that its future cash flows are going to be equal to 0, or more weakly, greater or equal to 0, because everything cancels out. And therefore, the weak arbitrage condition tells me that the price that I paid for this particular portfolio at time t equal to 0, must be greater than equal to 0. What is the price? It's the negative of the cash flow at time t equal to zero. It's going to be the sum of k equal to 0 1 through capital T, ck divided by 1 plus r k, 1 plus r to the power k minus b. And this price of the portfolio must be greater than or equal to zero. Which means that the p must be less than equal to the present value. The two bounds together implies that the arbitrage-free price, the no-arbitrage price for a contract, which has a cash flow C zero up through C capital T associated per period. And in the market where you could borrow or lend unlimited amounts at an interest rate of R per period. Is given by the present value. Notice I put a lot of caveats here. I said you could borrow or lend at the rate r. You could borrow unlimited amounts or lend unlimited amounts at the rate r. Both of these are necessary for this price to work out. What if the lending rate is different from the borrowing rate? Now the present value calculation doesn't work and we have to construct a different arbitrage argument. Here's the arbitrage argument for borrowing and lending rate. So we can lend at the rate RL and borrow at the rate RB and typically the lending rate is going to be less than the borrowing rate. what is the lending rate for a typical investor, it's the amount of money, that the interest rate that you are given for deposits in a bank. And, the borrowing rate is the amount of interest that you have to pay for loans taken from the bank. And, typically, the borrowing rate is going to be larger than, strictly larger than, the lending rate. So, let's construct our portfolio. We had two different portfolios that we had constructed when we did the no arbitrage argument in the previous two slides. One of them gave us a lower bound, and the other one gave us the upper bound. The two bounds were the same and therefore you ended up getting an exact price. So let's construct those portfolios. We buy the contract, and we borrow c k over one plus r b to the power of k, for k years. Now notice, we are borrowing, so the borrowing rate is here. By the same argument that we had in the previous slides, the cash flow in year k is going to to be c k, coming from the contract, minus c k over 1 plus r b to the power of k, times 1 plus r b to the power of k. This is the amount of money that I have to return. On the amount that I borrowed. They cancel exactly. Therefor CK is greater than equal to zero. For all times in the future, in fact it's exactly equal to zero. The new arbitrage condition, the weak new arbitrage condition tells me that the price of this portfolio must be greater than equal to zero. What is the price for this portfolio? It's going to be P, the amount that I paid, for the contract minus C0 minus the summation from K equals 1 through capital N CK over 1 plus RB to the power of K. That must be greater than or equal to zero. This expression here, is exactly the present value for the cash flow evaluated at the borrowing rate RB. This portfolio gives me a lower bound on the price, which says the price must be greater than the present value of the cash flow evaluated at the borrowing rate r B. Now let's flip the portfolio. Sell the contract and lend c k over 1 plus r l to the power k for k years. Notice because I'm lending I'm going to have to use the lending rate here. Rl and which RL is less than typically, strictly less than but it is less than equal to RB. Again, the cash flow associated with this portfolio is going to be exactly equal to zero. In fact, and I'm going to only use the fact that CK is greater then equal to zero. The weak no-arbitrage condition now tells me that the price that I should have paid for this portfolio must be greater than equal to zero. What is the price that I paid? It's going to be minus P plus C, C zero plus K going from 1 through N, CK over 1 plus RL to the power of K. Why is this the price? Because when I sell the contract, I receive, p. When I lend, all of this money goes out of my pocket therefore the net price, for the portfolio is going to be the net outflow, which is exactly the quantity that's written out there. That must be greater than equal to zero, and you end up getting an upper bound for the price, that p must be less than equal to the present value of the cash flow, c, computed using the lending rate rl. This time around, we don't get an exact price for p. We find that the price is upper bounded, by the present value calculated RL. It's lower bounded by the present value calculated at RB and therefore we get a no arbitrage interval. This is an example of an incomplete market, markets where we can not completely hedge away the risk and therefore you don't get an exact price using a no-arbitrage argument. So how is this price P set? It's set, basically, by supply and demand, depending upon whether the buyers or the sellers, who has more of the market power. The price would either go to the lower bound, or it could go to the upper bound. So the supply and demand sets the price. Next we want to expand the ideas going from the cash flows to fixed income instruments. Fixed income instruments are securities that guarantee a fixed cash flow. They guarantee you a dollar amount. But are these instruments risk free? Not at all. There is default risk associated with them, which means that, at some point this entity that is backing these fixed income security might go bankrupt. And, therefore, the cash flow that you were really promised is not going to come through. About the only entity that is there that is risk free is the US government or equally stable governments that are going to guarantee you that the cash flow is going to come. Corporate fixed income securities are always open to default risk. Another risk associated with them is the inflation risk. So even if the entity that gave you the fixed income security does not default, the value of the currency might go down over the years, and as a result the fixed cash flow is now of lower value and this happens because the buying power of the currency comes down over time. In order to. Hedge against this, people have started thinking about tips, which are inflation protected securities. Even if there wasn't very heavy inflation risk in the market, there's also a market risk, meaning these securities might become less or more valuable as time goes by, and if you wanted to sell these securities in the market. The price that you might end up getting is going to fluctuate over time. and as a result you are opening yourself up to market risk. So having said all of that, let's consider some typical fixed income securities, and try to price them using a no-arbitrage condition. So one of them is the perpetuity, which says that it's going to give you a fixed amount a for all times in the future. What is the no arbitrage price for this security? If you assume that you can borrow or lend at the rate r, an unlimited amount, then the no arbitrage price for this security is going to be the sum of k going from 1 to infinity of a divided by 1 plus r to the power k. If you sum this infinite series, you end up getting that the price is nothing but A divided by R. Which is the per period interest rate. Annuity is an instrument, it's a fixed income instrument that pays an amount A for periods K equals one through N. You can write this as a difference between two perpetuities. One perpetuity that starts right now. So it pays from. So here is T equal to zero. You have a perpetuity that pays at an amount A and all these times in the future. And another perpetuity that's time, that's starting at time N plus one. Meaning here is time N. Here is time N plus 1. This is the first time you are going to get a negative amount from this perpetuity so after that, everything will cancel, so all of these will cancel and you end up getting the cash flows associated with that is A. For t equals 1 to up through n. So the difference between these two is exactly the price of the annuity. So therefore the price of the annuity is A over r, this is the price of a perpetuity that starts right now. The price of the perpetuity that starts at time n is going to be A over r, but this is going to be at time t equal to n. You have to discount this back N period, so this is the discount factor. And we discount it back, you get the price for the annuity, and this is going to be A over r, one minus one plus R to the power N. The next fixed income instrument that we are going to be interested in is a bond. A bond is characterized by five different parameters, or. There's a face value. And face value is typically a hundred or a thousand. There's a coupon rate, alpha, which is paid every six months. So every six months a bond pays you alpha times f which is a face value divided by 2. There's the maturity. This is the date of the payment of the face value in the last coupon. There is the price at which is that at which these bonds are going to be selling. In addition to this, there is also a quality rating. S and P rates them as triple A, double A, triple B, double B, triple C, double C, and so on. The quality essentially is trying to get at the default risk. A high quality bond has very low default risk. The chance they could stop paying this coupon, are going to be very low. A lower qualilty bond has a higher default risk, meaning that there's a higher chance that it would not pay the payments. So, here's a story. Every six months, a half, one year, one and a half, two and so on, it pays alpha, which is the coupon rate times F, which is the face value divided by two. And at maturity, capital T, you get the face value back, plus you get alpha F divided by two, which is the last coupon payment. Now, bonds are going to be characterized by at least four numerical quantities. The face value, the coupon rate, the maturity, and the price. And in addition there is this other quantity called quality. And since bonds differ in so many different dimensions, it's hard to compare bonds. So one quantity that has been introduced in order to be able to compare bonds of different maturities, different coupon rates, and face values, is this idea known as the yield to maturity. The yield to maturity is the annual interest rate at which the current price for the bond P is exactly equal to the present value of the coupon payments plus the face value. So you take all the coupons and you discount them at the rate lambda over 2 because that's. The coupons come every 6 months. Lambda is the annual rate of interest and therefore you have it discounted by lambda over 2 and similarly you discount the face value that appears at time capital T which is to say after 2T 6 month periods at the rate lambda would do. What this little maturity does, is that it gives you a single number to start thinking about bonds. It summarizes a face value, coupon, maturity, quality and so on. It's a number that relate, that has understandable movements with respect to quality. Lower quality means lower price, which means higher yield to maturity. Which means, essentially the intuitively to think about about it is that, cash payments in the future are going to be discounted with a higher interest rate. Why with the higher interest rate? Because I'm not certain that those cash payments are going to come, and therefore, I want to discount them very strongly. With lower quality, I am even more uncertain which means that I should have a higher yield to maturity. If the interest rate in the market were to change, then the yield to maturity would change in a similar manner. Yield to maturity, therefore, gives us a way to think about different bonds and compare them. But remember, yield to maturity is a single number. It's a very crude measure. It's trying to summarize four different numbers by a single number, so it's not going to be able to capture everything.