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 a simple interest rate of r per period is going to be worth A times 1 plus nr at maturity. In every period, you get an interest r times A, so over n periods, you're going to get an interest of n times A times r and you get back your initial investment. Therefore, the total amount is going to be A times 1 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 1 plus r raised to the power n at maturity. The way to think about it is what is happening is at time 0, 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. It becomes A times 1 plus r squared, and so on. After n periods, it becomes 1 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. R over n, this is the interest per period. The total number of periods in y years is going to be y times n. This is the total number of periods. The amount that you end up getting is A times 1 plus the interest per period raised to the total number of periods. Continuous compounding corresponds to the situation where the length of the compounding period goes to 0. Therefore, an amount A invested for y years is now going to be worth the number of periods n going to infinity of A times 1 plus r over n raised to the power yn. If you take the limit of n going to infinity, that expression solves as A times e raised to the power ry at maturity. The number of the compounding period is going to 0. Therefore, the number of compounding periods is going to infinity. You end up getting that the expression simplifies to A times e raised to the power ry. For those of you who are well-versed with differential equations, you can find out that basically what's going to happen is dA dt is going to be r times A. 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. We want to compute the price p for a contract that pays c_0, c_1, c_2, and so on c_N and times 0, 1, 3, 3, 4, N, and so on. If c_k is greater than 0, that's a cash inflow, if c_k is less than 0, that's a cash outflow. The present value, assuming an interest rate r per period, can be written as c_0 plus c_1 divided by 1 plus r, c_2 divided by 1 plus r squared, and c_N divided by 1 plus r N. In terms of an expression, this is the expression for the present value. Let me repeat. We're going to argue that this present value is in fact the arbitrage-free price p 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 that we are going to construct as follows, we'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 p and you receive an amount c_0. At time t equal to 1, you receive c_1, time t equal to 2, you receive c_2, at time equal to k, some general k you receive ck, and at time t equal to T, you receive c_T. I'm going to borrow c_1 divided by 1 plus r amount for up to time one. 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. How much do I have to return? It's going to be the amount that I borrowed times 1 plus r, which cancels out the divide by 1 plus r, you end up getting exactly minus c_1. Similarly, I'm going to borrow c_2 divided by 1 plus r squared up to time two. I will get c_2 divided by 1 plus r^2 into my pocket at time t equal to 0 and I have to pay out an amount c_2 at time t equal to 2. Generically, for a general time k, I'm going to borrow ck divided by 1 plus r^k. I receive that amount at time t equal to 0. Then I have to pay out minus ck at time t equal to k. What you notice by doing this construction is that the cash flows at all future times cancel out exactly. The portfolio's cash flows for all time k equal to 1, 2, 3, up to T turns out to be zero. 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 than equal to 0. All future cash flows are greater than equal to 0. Therefore, the price at time t equal to 0 must be greater than 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. The price of the portfolio is going to be minus the summation from k equal to 0 to capital T, ck over 1 plus r^k, that must be greater than equal to 0. That gives me a lower bound that the price must be greater than 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 0, but now I'm responsible for the cash flows associated with the contract. I have to be paying the cash flows to the buyer. I have to pay out c_0, I have to pay out c_1, c_2, 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 c_1 divided by 1 plus r up to time one. When I lend an amount, money goes away at time t equal to 0, but I get it back at time t equal to 1. I lend an amount c_2 divided by 1 plus r^2 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. How much comes back? It's minus c_2 divided by 1 plus r^2 times 1 plus r^2, which is exactly c_2. 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 zero or more weakly greater than equal to zero because everything cancels out. 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 0, it's going to be the sum of k equal to 0, 1 through T. ck divided by 1 plus r^k minus p. This price of the portfolio must be greater than 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_0 up through cT associated with per period. 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. We can lend at the rate r_L and borrow at the rate r_B. 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. Typically the borrowing rate is going to be strictly larger than the lending rate. 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 the 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. Let's construct those portfolios. We buy the contract and we borrow C_k over 1 plus r_B to the power 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 cashflow in year k is going to be C_k coming from the contract minus C_k over 1 plus r_B to the power k times 1 plus r_B to the power k. This is the amount of money that I have to return on the amount that I borrowed, they cancel exactly. Therefore, C_k is greater than equal to zero for all times in the future. In fact, it's exactly equal to zero. The weak no-arbitrage condition tells me that the price of this portfolio must be greater than equal to 0. What is the price for this portfolio? It's going to be P, the amount that I paid for the contract, minus C_0 minus the summation from k equals 1 through N, C_k over 1 plus r_B to the power k. That must be greater than or equal to 0. This expression here is exactly the present value for the cashflow evaluated at the borrowing rate r_B. 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, r_L and which r_L is typically strictly less than, but it is less than equal to r_B. Again, the cash-flow associated with this portfolio is going to be exactly equal to zero. In fact, I'm going to only use the fact that C_k is greater than equal to 0. 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_0, plus k going from one through N, C_k over 1 plus r_L to the power 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 is 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 r_L. 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 r_L. It's lower bounded by the present value calculated at r_B, and therefore we get a no arbitrage interval. This is an example of an incomplete market. Markets where we cannot completely hedge away the risk and therefore you don't get an exact price using a no-arbitrage argument. 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 a market power, the price would either go to the lower bound or it will go to the upper bound. The supply and demand sets the price.
