Relationship between spot rates and short-term forward rates, Financial Management

Assume that an investor invests $X in a 3-year zero coupon Treasury security. Three years from now, the total return received would be:

         X ( 1 + y6)6

 The other alternative available to the investor is he could buy a 6-month treasury bill and reinvest the returns every six months for three years. The 6-month forward rate would decide the future return. An investment of Rs.A would generate a return equal to

         X (1 + y1) (1 + 1f1) (1 + 1f2) (1 + 1f3) (1 + 1f4) (1 + 1f5)                                                   

Since both investments must generate the same precedes an end of the investment horizon:

         X (1 + y6)6 = X (1 + y1) (1 + 1f1) (1 + 1f2) (1 + 1f3) (1 + 1f4) (1 + 1f5)     

Solving for 3-year spot rate,

         y6 = [(1 + y1) (1 + 1f1) (1 + 1f2) (1 + 1f3) (1 + 1f4) (1 + 1f5)]1/ 6 - 1                                  

In the above equation, we see that the 3-year spot rate depends on the current 6-month spot rate and the five 6-month forward rates. Actually, the right hand side of this equation is a geometric average of the current 6-month spot rate and five    6-month forward rates. In general, the relationship between a T-period spot rate, the current 6-month spot rate, and the 6-month forward rates is as follows:

         yT = [(1 + y1) (1 + 1f1) (1 + 1f2) (1 + 1f3) ..... (1 + 1fT - 1)]1/ T - 1

Thus, discounting at forward rates will give the same present value as discounting at spot rates.

Posted Date: 9/10/2012 2:43:50 AM | Location : United States

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