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 + y_{6})^{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 + y_{1}) (1 + _{1}f_{1}) (1 + _{1}f_{2}) (1 + _{1}f_{3}) (1 + _{1}f_{4}) (1 + _{1}f_{5})
Since both investments must generate the same precedes an end of the investment horizon:
X (1 + y_{6})^{6 }= X (1 + y_{1}) (1 + _{1}f_{1}) (1 + _{1}f_{2}) (1 + _{1}f_{3}) (1 + _{1}f_{4}) (1 + _{1}f_{5})
Solving for 3-year spot rate,
y_{6}^{ }= [(1 + y_{1}) (1 + _{1}f_{1}) (1 + _{1}f_{2}) (1 + _{1}f_{3}) (1 + _{1}f_{4}) (1 + _{1}f_{5})]^{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 + y_{1}) (1 + _{1}f_{1}) (1 + _{1}f_{2}) (1 + _{1}f_{3}) ..... (1 + _{1}f_{T - 1})]^{1/ T} - 1
Thus, discounting at forward rates will give the same present value as discounting at spot rates.