In this module, we are going to introduce you to the floating rate bonds, how to use the new arbitrage principle, to price floating rate bonds, and also to the term structure of interest rates. In this module, we are going to price a floating rate bond. This is a bond for which the interest rate or the coupon payment rate that you receive is going to be dependent on the prevailing uncertain interest rate in the market. In order to price this instrument correct we have to use this idea called linear pricing. Linear pricing is the result that allows you to combine prices for different cash flows. Here is what this theorem says. Suppose there is no arbitrage in the market and suppose there are two different cash flows CA, which has a price PA and a cash flow CB which has a price BB. Then the arbitrage free price for a cash flow that pays the sum of these two cash flows. So c is a cash flow that pays CA plus CB. The price for such a cash flow must exactly be PA plus PB. The prices add up. And this is exactly what is going to be called a linear pricing result. We like this result, because now what it allows me to do is to take a complicated cash flow, split them up into simpler cash flows. Price each of those simpler cash flows separately. And then using this TRM claim that the price of the total cash flow is just going to be the sum of those prices. I'm going to show you a simple example first and then I'm going to go towards what happens at floating rate bonds. So let P denote the price of the total cash flow C which is CA plus CB. And suppose this price P is actually strictly less then PA plus PB. So cash flow C is cheap, the individual cash flow CA and CB are expensive. We're going to create an arbitrage portfolio, I'm going to create a portfolio that gives you a free lunch. But with no arbitrage, such a portfolio can not exist. And therefore we are going to get some bounds on the prices. So I'm going to purchase the cheap cash flow, C at a price P. And then I'm going to sell the cash flow, CA and CB separately. So I'm going to take a cash flow, split it up into cash flows CA and CB, and individually sell them in the market. What is the price of a such a portfolio? It's going to be P minus PA minus PB, which means it's less than 0, which means that there is a net income at time T equal to 0. The cash flows in the futures exactly cancel out. Therefore, the future cash flows are 0 and have constructed a free lunch. I've constructed for you a portfolio that somebody pays you to hold a time T equal to 0 and you have no obligations in the future. No arbitrage in the market means there is no free lunch and therefore the initial assumption that I had that P is less than equal to PA plus PB is going to be false which means that P must be greater then PA plus PB. We can reverse the argument if P is strictly greater than PA plus PB. What both of these in results together show you is that the price P must be exactly equal to PA plus PB. Note that we have to have a liquid market for buying and selling these cash flows in order to have this no arbitrage result give us a bound of the price. Once we have this no arbitrage result, we are going to see how we can apply that to a very simple example first and then, the floating rate bond next. Suppose that you have a cash flow C which pays C1, C2, C3 up to CT. I want to think of this as a portfolio of capital T different cash flows, so C super T, superscript t pays ct at time t equal, at time t, and 0 everywhere else. So the, total cash flow is now a portfolio, or a collection of cash flows of a simpler kind. So, these cash flows only pay at one time and zero otherwise. So, both the cash flows are annual and the interest, annual interest rate is r. Then, the price of this particular cash flow which is here time t is equal to 0. Here is time t equal to little t. It pays in amount CT here, and it pays 0 everywhere else. The price for this cash flow, which we have already computed before, is given by its net present value, which is going to be CT divided by 1 plus R to the power T. The price of the cash flow C, which is a sum of all these cash flows C super t, is going to be the sum of the prices, just the net present value of all the cash flows from the future. We are going to take this simple idea of taking a cash flow, split, splitting it up into simpler cash flows, pricing the simpler cash flows together, and bringing it back to get the total price for the more complex example of floating-rate bonds. So what are floating-rate bonds? These are bonds that give you, whose coupon rate is equal to the floating interest rates. What are floating interest rates? Floating refers to the fact that interest rates are random quantities, they fluctuate over time. Let RK denote the per period interest rate over the period K to K plus 1. This quantity RK becomes known only at time K. Before time K, this quantity RK is a random quantity. So, if I'm sitting at time t equal to 0, and I have an instrument that depends on this floating rate interest rate RK, I don't exactly know what the payment is going to be. The payment that I receive on that security will become clear only at the beginning of time K. One period loan issued in period K have to be, to be repaid in period K plus 1 are charged this amount RK. The cash flow of a floating rate bond is connected to these floating interest rates. The coupon payment at time K is determined by floating rate at time K minus 1. So it's going to be RK minus 1 times F. Here we are assuming that the coupons are annual coupons. If it's going to be semi-annual coupons, then it's going to be the semi-annual rate rather than the annual rate, RK minus 1. The fees value that you receive at time N is fixed F. Our goal is to compute the arbitrage free price PF for the floating rate bond. So what we are going to do is split up these cash flows into simpler cash flows. I'm going to consider a cash flow that pays RK minus 1 times F at time K. And that's it. It doesn't pay at any other time. I'm going to call that particular cash flow, CK cash flow, and the price of that particular cash flow is going to be denoted by little p sub K. I'm going to have a final cash flow associated with the principal amount at time N, and I'm going to call the price of that particular cash flow as capital P. Capital P is easy to compute, it's just the net present value of the principal capital F paid at time N which is going to be 1 over 1 plus R to the power N times F and just to be clear this R down here is R sub 0 which is the interest rate Right now. Using the linear pricing theorem, we know that the price of floating rate bond, P sub F, is nothing but capital P, which is the price of the principle, plus sum of K going from 1 through N. Little p sub K, which are the prices for the instrument that just pays RK minus 1 at time K. In the rest of the slides, I'm going to try to compute for you what this PK is. Okay. I'm going to do this in two steps. So far, we've only used the no arbitrage argument to price a deterministic cash flow. Here, we are constructing the price of a contract that pays RK minus 1F at time K. RK minus 1 is not known at time T equal to 0. The price has to be computed at T equal to 0. RK minus 1 is a random quantity its exact value or its realization is only known at time K minus 1. So what I'll first do is construction a portfolio whose cash flows are deterministic, all the randomness will be gone. Once I have a deterministic cash flow I know exactly how to price them. I just have to use the net present value. But net present value using R0, which is the interest rate right now. So here's how I construct the portfolio. I buy the contract, and now I enter, I try to look at what is going to happen because of this step. I'm going to have a negative cash flow minus PK at time T equal to 0, because this is the price that I have to pay to buy the contract. At time T equal to K, I receive an amount RK minus 1 times F. Now, this is a random quantity, and I need to somehow remove the randomness from the cash flows. I'm going to borrow an alpha amount. This is unknown right now. Over the period 0 to K minus 1. So, I end up getting alpha inflow at time T equal to 0, and I have to pay out minus alpha 1 plus R0 to the power K minus 1. And notice, the amount that I have to pay out depends on the interest rate at time T equal to 0 because that is where I'm going to borrow. Next, what I'm going to do is I'm going to borrow alpha times 1 plus R0 to the power K minus 1 amount over the period K minus 1 to K. So this is the inflow that happens at time K minus 1. What happens at time K? I have to return an amount that I borrowed. This is just the principle. 1 plus RK minus 1, this is random. The amount of money that I have to return at time K is going to be exactly known only at time K minus 1. At time t equal to 0, where I'm sitting and pricing this instrument, this RK minus 1 is not known. Finally, I'm going to lend alpha for the period 0 through K. I get minus alpha here, and I get alpha times 1 plus R0 to the power K there. Since I'm lending, I have a negative cash flow and a positive cash flow at time capital K. So let, let's now look at what happens to the cash flows at time T equal to K. The net cash flow at that point, CK, is going to be RK minus 1 times F which is the cash flow associated with the contract, minus alpha, plus,1 plus RZ to the power K minus 1 times 1 plus RK minus 1, this is the cash flow associated with borrowing alpha times 1 plus R0 to the power K minus 1, over the period K minus 1 to K. Plus alpha times 1 plus R0 to the power K, which is the cash flow associated with lending alpha amounts from 0 to K. Let's rearrange this and take all the terms that correspond to RK minus 1, and put them all together. You get alpha times 1 plus R0 to power K minus 1 times negative sign, and this is coming from this term over here, and you get F from that term over there. This is random because it's multiplying a random quantity RK minus 1. This quantity here just depends on R0, and therefore it's deterministic, it is known what it's going to be at time T equal to 0. Remember, alpha was unknown. So what I'm going to do is choose my alpha, in such a way, that this particular bracket cancels out. And if you set that equal to 0 you get alpha equal to 1, over 1 plus R0 to the power K minus 1 times F. This makes sure that the random quantity is equal to 0. Now, I want to understand what happens to this cash flow. I have made the cash flow deterministic, and now I am going to use net present value to calculate what happens to the cash flows. So, the cash flows appear only at three times T equal to 0, t equal to K minus 1, T equal to K. What is the cash flow at time equal to 0? This plus alpha and this minus alpha cancel, the cash flow is minus PK, which is basically the price that you pay for the contract. At time T equal to K minus 1, this quantity and that quantity exactly cancel. And what is the net cash flow that I get at time T equal to K? That's going to be alpha R0 times 1 plus R0 to the power K minus 1 which is F times R0. And well how do I end up getting this F times R0? I'm just using the fact that alpha's going to be equal to F divided by 1 plus R0 to the power of K minus 1, it exactly cancels out and you end up getting that CK is nothing but F times R0. What is the price PK now? The price of the portfolio is going to be just CK divided by 1 plus R0 to the power K, which is FR0 divided by 1 plus R0 to the power K. R0 refers to the fact that this is the interest rate at the time T equal to 0. Recall that the price of the floating rate bond PF is going to be the component that comes from the principal payment let's call it P and the sum of all these instruments for K equal to 1 through N. I have just calculated that PK is going to be F times R0 divided by 1 plus R0 to the power of K. I substitute that. I take some terms common. I calculate out the sum of this series going from K equal 1 through N. Cancel out some terms. And lo and behold, the result that you end up getting for the price of a floating rate bond, is exactly equal to the principal or the face value F. The price PF of a floating rate bond is always to its face value. That's great, so, two things I want you to take away from this little calculation. One of them is the fact that the price of a floating rate bond is exactly equal to it's face value. But the second, is how we constructed this price. What we did was, we pushed our idea of no arbitrage one step further. We could, first we used no arbitrage condition to calculate out the price of a deterministic cash flow. Now we pushed a no arbitrage condition to calculate out what happens to the price of a stochastic cash flow, and we did it in two steps. First, we remove the stochasticity, and converted It into a deterministic cash flow, and then we use the no arbitrage condition to calculate the price of this deterministic cash flow. In this slide, we are going to introduce this idea of term structure of interest rates, and it's an idea that will come back again later on in this course. What do I mean by a term structure. What we mean is that the interest rate actually depends on the term or the duration of the loan. So far we've been assuming that today the interest rate that you'll be charged for a loan doesn't depend on whether the loan is for one period, two period, ten period, 100 periods, whatever. But in reality, that is not true. The interest rate depends on the duration. And it's typically greater as the duration increases. Why does this happen? There are many explanations for it. Investors prefer to have their funds liquid, rather than tied up. So if you want to induce them to tie up their funds for a larger period, you have to give them a higher interest rates. There are other explanations such as people have expectations that in the future rates are going to go up or go down and therefore what interest rate they'd be willing to charge for you right now might depend on their direct future expectation. There might, there's also a market segmentation story which says that different people in the market are interested in funds at different terms or at different points in time. And therefore, depending upon how the market segments, the interest rates will be different for different periods and would result in a term structure of interest rates. Let SD define the spot rate. This is the rate available today, for a loan maturing in T years. So an amount A in T years, it's present value is going to be A divided by 1 plus ST, raised to the power of T. Notice here, this S depends on T. So far, when we were doing the net present value calculation. We would have done A divided by 1 plus R to the power T. Regardless of when the time is coming, the interest rate remains the same. But now, you have two things happening. The interest rate changes, and the compounding is always there. So it's going to be 1 plus ST raised to the power T. Sometimes it's more convenient to quote the spot rates in term of the discount rate. The discount rate is nothing but 1 over 1 plus ST to the power T. You can infer spot rates by looking at bond prices. Why? Because bond prices, at least for default, the risk free bonds are going to be simply a linear combination of the coupon payments discounted at the spot rate. And if you know the prices for bonds for different maturities, you could use these prices to compute out what the spot rates are going to be. Another term associated with the term structure of interest rate is something called a forward rate. A forward rate, FUV is the interest rate coded today for lending from year U to year V. We, what, if you know all the spot rates we can compute out what the forwards rates are going to be, and the way to use them is to think about two different ways that you can go from year 0 to year V. One possibility is go, is going to be to lend $1 up to time V. In which case, this $1 lent for V years at time 0 is going to give, going to be worth 1 plus SV to the power V. The other possibility is to lend that dollar for U years and then contract it right away at time T equal to 0 to lend it for another V minus U years. So it's important to remember what we're doing here in the second level option is that we're going to lend the dollar for U years and right at time T equal to 0 I'm going to construct another contract which is going to allow me to lend at whatever amount of money that I have at time U for another V minus U years. So my initial dollar is going to be worth 1 plus SU to the power U. This entire amount is now going to be lent at the forward rate. So if I lend it, I get 1 plus FUV to the power V minus U at time V. No arbitrage condition tells me that whether I go down the blue path or whether I go up the black path, I should end up with the same amount of dollars. Which means that 1 plus SV to the power of E must equal 1 plus SU to the power U times 1 plus FUV to the power of E minus U. Using this expression you can calculate out what FUV is going to be. It's going to be 1 plus SV to the power of E divided by 1 plus SU to the power U. The whole thing raised to the power V minus U minus 1. There is another relationship between spot rates and forward rates. 1 plus ST to the power T is going to be the product of K going from 0 to T minus 1. 1 plus FK, K plus 1. So one periods. And the logic behind that is very simple. You have 0 and V. One possibility is go directly. Not V, but T. And here you get 1 plus ST to the power T, or you could hop one year at a time. Here is F01, F12, FT minus 1T. Again by no arbitrage whether you go down this path or you go down that path you should get the same amount of money which tells me that 1 plus ST to the power T must be equal to the product along the above line which is 1 plus FK, K plus 1 taking the product with K equal to 0 to T minus 1.
