Question

Consider the following spot rate curve:

s1	s2	s3	s4	s5
0.050	0.055	0.061	0.066	0.075

(a) What is the forward interest rate that applies from period 3 to period 5? That is, what is the value of f_3,5? Assume annual compounding. (Keep your answer to 4 decimal places, e.g. 0.1234.)

(b) If the market forward rate from period 3 to period 5 is not equal to the value derived in (a), how can you create an arbitrage opportunity? (No need to key-in here.)

Answer 1

a]

The arbitrage-free forward rate is the rate that would make investing at the 3 year spot rate now, and reinvesting the proceeds after 5 years at the forward rate, the same as investing now at the 5 year spot rate. In other words, if the forward rate is arbitrage free, the investor should be indifferent between :

• investing at the 3-year spot rate now and reinvesting the proceeds at the end of 3 years at the forward rate, AND
• investing at the 5-year spot rate now

Let us say the arbitrage-free forward interest rate is f_3,5. Then :

(1 + 0.061)³ * (1 + f_3,5)² = (1 + 0.075)⁵

(1 + f_3,5)² =  (1 + 0.075)⁵ / (1 + 0.061)³

f_3,5 = [(1 + 0.075)⁵ / (1 + 0.061)³]^1/2 - 1 = 0.096

b]

If the froward rate is not equal to 0.096, there is an arbitrate opportunity

If the forward rate is more than 0.096, then :

• Sell the forward
• Buy the spot

If the forward rate is less than 0.096, then :

• Buy the forward
• Sell the spot