Guaranteeing a Forward Rate

Say an investor at time t simultaneously buys one unit of a zero-coupon bond maturing at time T that is priced at P t, T) and sells P t, T) P t, T + 1) units of zero-coupon bonds maturing at T+ 1. Together these two transactions generate a zero cash flow The investor receives a cash flow equal to one unit at time Tand pays out P t, T)IP t, T+ 1) at time 7+ 1. These cash flows are identical to those that would be generated by a loan contracted at time t for the period T to T + 1 at an interest rate of P t, T) P t, T + ). Therefore P t, T) P t, T+ 1) is the forward rate. This is expressed formally in (3.25). 

Using the relationships between bond price and yield described earlier, (3.25) can be rewritten in terms of yield as shown in (3.26), which represents the rate of return earned during the forward period (T T + 1). This is illustrated in FIGURE 3.4. 

Expression (3.25) can be rewritten as (3.27), which solves for bond price in terms of the forward rates from ttoT. (See Jarrow (1996), chapter 3, for a description of this derivation.) 

Equation (3.27) states that the price of a zero-coupon bond is equal to the nominal value, here assumed to be 1, receivable at time T, after it has been discounted by the set of forward rates that apply from tio T. 

Calculating a forward rate is equivalent to estimating what interest rate will be applicable for a loan beginning at some point in the future. This process exploits the law of one price, or no arbitrage. Consider a loan that begins at time T and matures at T+ 1. The process of calculating the rate for that loan is similar to the one described above. Start with the simultaneous purchase at time t of one unit of a bond with a term of T+ 1 for price P t, 7+1) and sale of p amount of a bond with a term of T for price P t, T). The net cash position at t must be zero, sop must be 

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