In this part, we will now introduce appointment contracts and interest bonuses. These two contracts now allow you to lock in interest rates for loans at future time intervals. The rate you are blocking today is what is called the outpost rate. In the case of maturity and in the case of interest rate futures, it is called the forward rate. We will get forward rates and forward payments. Forward Rate Agreement FRA on the t calendar date is indicated by a future period (T-0, T-1) with lengths that we will describe by δ, a fixed K set and a fictitious N. In the case of T-1, the holder of the advance rate agreement pays a fixed K rate on the nominal price and in turn receives the variable interest rate on face value. This is called floating, because this rate is only known in the future T-0. This advance rate agreement allows you to lock in a fixed rate for the future period (T-0, T-1) today. Suppose you know that you are going to borrow with N fictitious at the time T-0) Depending on the market conditions that prevailed at the time, you had to repay the loan with the single price L (T-0, T-1).
Assuming you don`t like the uncertainty of this interest rate cash flow today and instead want to trap a K rate that is set today and that you will pay for that loan. Keeping this forward rate deal does just that. Remember that you have to pay the fixed sentence K and you get the floating. As you can see now, suspended payments are simply cancelled. And in fact, what you pay for is the K sentence. Of course, the question that arises is what is a fair firm price K, which you will fix today in light of market information, which are all zero coupon to t bond prices. To answer this question, we are now calculating the value of this advance rate agreement and placing it at 0. We start with the payment of the advance rate agreement to T-1. Keep in mind that it is given by the variable rate difference of payments minus the fixed rate payment on the fictitious N. We are again expressing this simple course with regard to the T-1 loan price on T-0. We will then call this I-1 minus I-2 and evaluate these I-1 and I-2 components separately with discounted bonds. I-1 is a cash flow that we don`t know today, but we know it at the T-0 moment.
Therefore, the value of this I-1 cash flow at the T-1 time at the T-0 time is indicated by a simple multiplication by the T-1 borrowing price. But as I-1 is the reciproke of the T-1 loan price, we get the value at the time T-0, 1. For the time-t value to be the T-0 loan price. At the time of I-2, we become even simpler because it is a cash flow that we know at the time t. Therefore, it is time t value is given simply by multiplying with the T-1 link. That is what you get. The total time t value of the forward rate agreement is therefore indicated by the difference of these two expressions that we have here on the right, and is called V-FRA value. So we see that the value of the appointment depends on the choice of the fixed K rate. The bigger the K, the smaller the value.
From the point of view of the owner of the forward rate agreement, it may even be negative. But of course, there is a counterpart to the growth rate agreement. And for the counterparty, the value has only the opposite sign.