Question

You are given the following spot rates. Determine the 1 year forward rates at the beginning of year 0 through 7. Use annual compounding. Consider the formula 8 8 (1+yo, Show that it holds for your forward rates, and explain in a sentence why it must be true 1(1+y:-1, i=1 Year n Rate yo,n 0.024 0.0245 0.026 0.027 O voo + w N - 0.0275 0.0275 0.028 0.029

Answer 1

First, we calculate the forward rates using the table given using bootstrapping

y(0,1) = 0.024

y(1,2) = ((1.0245^2)/(1.024)) -1 = 0.025

y(2,3)= ((1.026^3)/(1.024*1.025)) -1 = 0.029

y(3,4) = ((1.027^4)/(1.024*1.025*1.029)) -1 = 0.030

y(4,5) =  ((1.0275^5)/(1.024*1.025*1.029*1.030)) -1 = 0.0295

y(5,6) =  ((1.0275^6)/(1.024*1.025*1.029*1.030*1.0295)) -1 = 0.0275

y(6,7) = ((1.028^7)/(1.024*1.025*1.029*1.030*1.0295*1.0275)) -1 = 0.031

y(7,8) = ((1.029^8)/(1.024*1.025*1.029*1.030*1.0295*1.0275*1.031)) -1 = 0.036

Now, considering the formula

$(1+y_{0,8})^{8} = \coprod_{i=1}^{8} (1+y_{i-1,i})$

LHS of the formula = (1+0.029)^8 = 1.2569

RHS = 1.024*1.025*1.029*1.030*1.0295*1.0275*1.031*1.036

RHS =1.2569

LHS=RHS and hence the equation holds true for the forward rates. The forward rates are derived from the forward curve and any aberration in the rates will lead to arbitrage opportunities and hence the forward rates formula holds true.