In this part we will now introduce forward rate agreements and interest rate futures. Both contracts allow you to lock in rates today, for loans on future time intervals. The rate that you lock in today is what is called forward rate. In case of the forward rate agreement and in case of the interest rate futures it is called the futures rate. We will relate futures rates and forward rates. Forward rate agreement FRA at calendar date t is specified by a future period (T-0, T-1), with lengths that we denote by δ, a fixed rate K, and a notional N. At T-1 the holder of the forward rate agreement pays a fixed rate K on the notional, and in turn receives the floating rate on the notional. This is called floating because this rate is only known at the future time point T-0. This forward rate agreement allows you to lock in a fixed rate today over the future period (T-0, T-1). Suppose you know you are going to take a loan with notional N at time T-0) According to the then prevailing market conditions you would have to pay back the loan along with the simple rate L(T-0, T-1). Suppose you dislike the uncertainty of this interest cash flow today and instead you want to lock in a rate K that is fixed today which you are going to pay on this loan. Holding this forward rate agreement does exactly that. Remember that you will have to pay the fixed rate K and you will receive the floating. As you now can see, the floating payments simply cancel off. And what you are in fact paying is the fixed rate K. Of course the question arises, as to what is a fair fixed rate K that you will fix today given the market information that is all the zero coupon bond prices at t. To answer this question we now compute the value of this forward rate agreement and set it equal to 0. We start by the pay off of the forward rate agreement at T-1. Remember it is given by the difference of the floating rate payments minus the fixed rate payment on the notional N. We now re-express this simple rate in terms of the T-1 bond price at T-0. We will then call this I-1 minus I-2 and we will value these components I-1 and I-2 separately using discount bonds. I-1 is a cash flow that we do not know today at time t but we know it at time T-0. Hence the value of this cash flow I-1 at time T-1 at time T-0 is given by simply multiplying it with the T-1 bond price. But since I-1 is the reciprocal of the T-1 bond price we get as value at time T-0, 1. So that the time t value is the T-0 bond price. At the time t value of I-2, we get even simpler because this is a cash flow that we know at time t. Hence it's time t value is simply given by multiplying it with the T-1 bond. This is what we get. So the total time t value of the forward rate agreement therefore is given by the difference of these 2 expressions that we have here on the right hand side, and be called the value V-FRA. So we see that the value of the forward rate agreement depends on the choice of the fixed rate K. The larger K is, the smaller the value. It can even be negative from the point of view of the holder of the forward rate agreement. But of course there is a counter party to the forward rate agreement. And for the counter party, the value will just have the opposite sign. So it's a reasonable choice to choose K the fixed rate such that the value is 0. Simple algebra shows that the rate K which renders this value equal to 0 is given by this expression here. We call it the simple forward rate prevailing at time t. When the time t coincides with the initial date T-0 of the loan period we get back the simple rate L. Which we now sometimes also call simple spot rate just to highlight the difference to the forward rate. It's further an easy exercise to see that we can re-express The time t value of the forward rate agreement in terms of the forward rate as follows. Now this expression nicely shows again that there is a fair choice of the fixed rate K namely the forward rate which renders this value of the forward rate agreement 0. If K is chosen too large the value will become negative. If K is chosen too small the value becomes positive. Now think of the following situation that K is yesterday's forward rate. Now yesterday's forward rate in general will be different from today's forward rate. That means if you entered a forward rate agreement yesterday, yesterday's prevailing forward rate seen from today's point of view the value may have become non zero. It may have become positive incase the interest rate went up. But it may also have become negative in case the interest rate had gone down. If we let the length of the lending period converge at 0, that is, if we let T-1 converge at T-0 in the limit, we obtain what we call the instantaneous forward rate with maturity T-0. Elementary calculus shows you that this limit is given by minus the logarithmic derivative of the T bond price with respect to maturity. Because the terminal value of the T bod is 1, this relation is equivalent to this formula. The nice thing about instantaneous forward rates is that knowing all the instantaneous forward rates is equivalent to knowing all the T-bond prices. As before, for varying maturities we obtain a term structure of rates. In this case, it's the term structure of forward rates prevailing at calendar date t. This is also called the forward curve. It's easy to see that forward rates are related to yields. In this way that means that the yield is the time average of forward rates. It also follows from this relation that the short ends of the forward curve and of the yield curve coincide and are equal to the short rate. Similar to a forward rate agreement, an interest rate futures contract allows the management of exposure to the futures spot interest rate prevailing over a future period (T-0, T-1). In contrast to forward rate agreements however interest rate futures are daily marked to market or resettled. To understand what this means recall that the value of a forward rate agreement may change in time. At time t when you enter the forward rate agreement, you will do it typically at conditions which render its value equal to 0. But already the next day, economic conditions may have changed. And the value of the forward rate agreement turns out to your favor to become positive. This value may of course also become negative. Eventually, it will converge to the pay off of the forward rate agreement. Now instead of sitting, and waiting, and seeing the exposure to your counter party getting a positive value or negative value; market participants prefer to settle any change in value immediately. This is done by marking to market. In this example it would mean on the day after today you would settle this increase of value by receiving a cash flow from your counter party. Making the exposure to your counter party again 0. While the next day you may have to return a part of that to your counter party and so on and so forth. Formally, the marking to market of interest rate futures contracts works as follows. At any calendar date t a so called futures price is quoted. This futures price is related to the futures rate in this way. This expression looks different than the value of a forward rate agreement but in fact, it reacts the same way to change in interest rate environment. The day after today one determines the futures price And the difference of the futures price on ( t + Δt) minus the futures price from yesterday is the cash flow to the holder of the futures contract. Recall this cash flow can be positive or negative. Futures rate here is chosen such that the following 2 conditions hold. At the settlement date, or delivery date of the futures contract T-0. The futures rate equals the underlined spot rate. At any earlier date the present value of the cash flow in the future, from holding the futures contract has to be equal to 0. These rules fix the futures rate. But there is no direct functional relationship between futures rates and zero coupon bonds as is the case for forward rates. As a consequence, futures rates are generally different from forward rates. The difference between futures rates and minus forward rates is called convexity adjustment. This difference is typically positive. As a market example, I mentioned here the Eurodollar futures contract which is tied to the LIBOR. It was introduced by the Chicago Mercantile Exchange in 1981 and has been traded since then on that exchange. It is designed to protect the owner from fluctuations in future 3-month LIBOR rates. These futures contracts are highly standardized. Their settlement or delivery months are in March, June, September and December. This is in contrast to forward rate agreements which are traded over the counter.