Question

The following one-year "forward" rates exist for each of the years 2012-2016:

            1/1/12 - 31/12/12   20.00%

1/1/13 - 31/12/13   25.00%

1/1/14 - 31/12/14   20.00%

1/1/15 - 31/12/15   20.00%

1/1/16 - 31/12/16   11.11%

Calculate the 1, 2, 3, 4, and 5 year spot rates as at 1/1/12. Accuracy to two decimal places is required.

Help to explain what kind of formula should we approach for this question. THanks!

Answer 1

Following forward rates are given

f₁ = 20%, f₂ = 25%, f₃ = 20%, f₄ = 20%, f₅ = 11%

1-year spot rate

1 year spot rate will the same as 1 year forward rate. So, S₁ = 20%

2-year spot rate

(1+S₂)² = (1+f₁)(1+f₂)

(1+S₂) = (1.2*1.25)^1/2 = 1.224745

Therefore, S₂ = 22.47%

3-year spot rate

(1+S₃)³ = (1+f₁)(1+f₂)(1+f₃)

(1+S₃) = (1.2*1.25*1.2)^1/3 = 1.21644

Therefore, S₃ = 21.64%

4-year spot rate

(1+S₄)² = (1+f₁)(1+f₂)(1+f₃)(1+f₄)

(1+S₄) = (1.2*1.25*1.2*1.2)^1/4 = 1.212309

Therefore, S₄ = 21.23%

5-year spot rate

(1+S₅)⁵ = (1+f₁)(1+f₂)(1+f₃)(1+f₄)(1+f₅)

(1+S₅) = (1.2*1.25*1.20*1.20*1.1111)^1/5 = 1.191356

Therefore, S₅ = 19.14%

Answer

S₁ = 20%, S₂ = 22.47%, S₃ = 21.64%, S₄ = 21.23%, S₅ = 19.14%