assume that an investor invests x in

Question

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:

         y_T = [(1 + y₁) (1 + ₁f₁) (1 + ₁f₂) (1 + ₁f₃) ..... (1 + ₁f_{T - 1})]^{1/ T} - 1

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