Once we have the spot rate curve, we can easily use it to derive the forward rates. The key idea is to satisfy the no arbitrage condition – no two investors should be able to earn a return from arbitraging between different interest periods. Let’s take an example of how this works. Let’s say an investor wants to invests his funds for two years. He is faced with two choices:
- Directly invest in a 2-year bond
- Invest in a one-year bond, and again invest the proceeds after one year in a one year bond.
Assuming the same nature of investments, the returns from both choices should be the same.
Let’s say s1 is the one-year spot rate, s2 is the two-year spot rate and 1f1 is the one year forward rate one year from now.
Assuming $1 as the initial investment, the value of investment in first choice after two years:
The value of investment in second choice after two years:
= (1+s1) (1+1f1)
If there are no arbitrage opportunities, both these values should be the same.
(1+s2)2 = (1+s1) (1+1f1)
If we have the spot rates, we can rearrange the above equation to calculate the one-year forward rate one year from now.
1f1 = (1+s2)2/(1+s1) – 1
Let’s say s1 is 6% and s2 is 6.5%. The forward rate will be:
1f1 = (1.065^2)/(1.06) – 1
1f1 = 7%
Similarly we can calculate a forward rate for any period.