Question

You are offered a one-year forward rate in two-years of 2.5% (i.e. you can enter into a contract today to lock in a 2.5% return on a one-year security purchased in two years). Based on the current yield curve, the implied forward rate on a one-year Treasury security purchased in two years should be 1.85%.

Treasuries (maturity)	Yield (%)
1-year	0.25%
2-year	0.61%
3-year	1.02%

How can you profit from this opportunity? Be specific (i.e. at what rates will you borrow; in which security (ies) would you invest?

Answer 1

If F is the one-year forward rate in two-years then it must satisfy the following equation:

(1 + S2)²(1 + F) = (1 + S3)³

Hence, F = (1 + S3)³ / (1 + S2)² - 1 = (1 + 1.02%)³ / (1 + 0.61%)² - 1 = 1.85% as given in the question.

Price of a 2 year Zero Coupon bond (ZCB) = FV / (1 + S2)² = 100 / (1 + 0.61%)² = 98.79

Price of a 3 year Zero Coupon bond (ZCB) = FV / (1 + S3)³ = 100 / (1 + 1.02%)² = 97.00

Arbitrage forward strategy:

		Cash flows @
Sl. No.	Action	t = 0	t = 2	t = 3
1	Borrow 97.00 for 3 years	97.00		= -97 x (1 + S3)3 = -97 x (1 + 1.02%)3 = -100.00
2	Lend 97.00 for 2 years	-97.00	= 97 x (1 + S2)2 = -97 x (1 + 0.61%)2 = 98.19
3	Enter into FRA (a contract today to lock in a 2.5% return on a one-year security purchased in two years)		-98.19	= 98.19 x (1 + 2.5%) = 100.64
	Total	0.00	-	0.64

Thus you end up making a riskless and riskfree profit of $ 0.64 at t = 3 without any initial investment. This is the profit that the question is asking.