Question

The prices of the zero-coupon bonds are as follows:

Maturity	Price
1	98.860
2	96.610
3	93.188
4	89.608

Suppose that a forward contract has an ₂f₂ = 3.2%. Is this market arbitrage-free? If yes, why? If not, what can you do to exploit the arbitrage opportunity? What woulb be your profit in that case?

Answer 1

2 year spot rate = s2 = (100 / P2)^1/2 - 1 = (100 / 96.610)^1/2 - 1 = 1.74%

4 year spot rate = s4 = (100 / P4)^1/4 - 1 = (100 / 89.608)^1/4 - 1 = 2.78%

Hence 2 year forward rate at the end of year 2, 2F2 will obey the following equation:

(1 + 2F2)² = (1 + s4)⁴ / (1 + s2)² = (1 + 2.78%)⁴ / (1 + 1.74%)² = 1.0781

Hence, 2F2 = 1.0781^1/2 - 1 = 3.83%

Forward contract rate = 3.2% which is not same as no arbitrage forward rate calculated as 3.83%

Hence, the market is not arbitrage free.

In order to exploit the arbitrage opportunity:

		Cash flows at
Sl. No.	Action	t = 0	t = 2	t = 4 years
1	Short (89.608 / 96.610 = 0.9275 ) no. of 2 year ZCB	89.6080	-92.7523
2	Buy 1 no. of 4 year ZCB	-89.6080		100.0000
3	Enter into 2 year forward contract 2 (Borrow under forward contract)		92.7523	-98.7834*
	Total	0.0000	0.0000	1.2166

*92.7523 x (1 + 3.2%)² = 98.7834

Hence, there is a riskless arbitrage profit of $ 1.2166 at the end of year 4 without any initial investment.