Question

The yield on a default-free four-year zero-coupon bond is 3%; the yield on a default-free five-year zero-coupon bond is 4.5%. The bonds have a face value of $1000 and are traded in an open market. You are a money manager and know that you will have a net inflow of $100,000 four years from now, and an obligation (i.e. a net outflow) of $100,000 one year later (i.e. five years from now). Once you get your inflow, you plan to invest part of this inflow (as much as necessary) in risk-free bonds for a year, and immediately pay the rest to your investors in the form of a profit. You would like to hedge the interest-rate risk that is involved in this future bond investment, in order to be able to pre-announce your expected profit today, but also ensure that your obligation is covered.

(a) Based on today’s yields, what is the no-arbitrage yield of a one-year forward-rate agreement starting four years from now?

(b) Assuming you can obtain such a forward-rate agreement, how much of your inflow will you need to re-invest, and how much will you be able to pay to investors?

(c) Now assume that no such forward-rate agreements are being offered in the market. How can you construct one yourself (i.e. replicate one), in order to hedge your interest-rate risk? Carefully describe your strategy, and show that the resulting cash flows mirror those of the forward-rate agreement you are trying to create. Assume you can go either long or short in either bond, and you can also buy or sell fractions of bonds.

Answer 1

S5 = 4.5%, S4 = 3%,

Part (a)

F4,5 = (1 + S4)⁵ / (1 + S4)⁴ - 1 = (1+ 4.5%)⁵ / (1 + 3%)⁴ - 1= 10.72%

Part (b)

I should invest = 100,000 / (1 + F4,5) = 100,000 / (1 + 10.72%) =  90,316.57 and I will be able to pay to the investor an amount = 100,000 - 90,316.57 =  9,683.43

Part (c)

Action	Replicating Portfolio	Cash flow at t = 0	Cash flows at t = 4	Cash flows at t = 5
1	Buy 100 nos. of 5 years ZCB	(80,245)		100,000
2	Short (sell) 90.32 nos. of 3 years ZCB	80,245	(90,317)
3	Receive $ 100,000 at t= 4		100,000
4	Payout pre announced profits to your investors		(9,683)
5	Payout your obligation at the end of year 5			(100,000)
	Total Cash flows	-	-	-