Question

1. Suppose you manage a bond portfolio with a current value of $150,000,000 and a duration of 7.32. You need to hedge the interest rate risk of this portfolio for some reason. Today's date is Monday, December 10th, 2018 so the settlement price for a treasury bond is the 11th. You decide to use the 10-year T-note futures to hege. The cheapest to deliver bond is the 3% coupon bond with maturity date September 30, 2025 which is currently selling for a yield of 2.766% (price of 101.4766 or 1.014766 per 1$ of face value)

A. What is the duration of the CTD bond?

B. How many futures contracts do you enter into?

C. Do you go long or short these contracts?

Answer 1

a) Duration of CTD Bond can be calculated as follows:

Coupon of 3% is allotted semi-annually ,thus $1.5 every 6 months till September 2025

PV of $1 after nth period is calculated as :

PV of $1= 1 / {(1+y)^n} where y= yield to maturity= 2.766% , n= time period

Period (t)	Coupon/ Principal (C)	D= C x t	PV of $1	D x PV
1	1.5	1.5	0.992	1.488
2	1.5	3	0.978	2.935
3	1.5	4.5	0.965	4.343
4	1.5	6	0.952	5.712
5	1.5	7.5	0.939	7.043
6	1.5	9	0.926	8.337
7	1.5	10.5	0.914	9.595
8	1.5	12	0.901	10.817
9	1.5	13.5	0.889	12.004
10	1.5	15	0.877	13.157
11	1.5	16.5	0.865	14.277
12	1.5	18	0.854	15.363
13	1.5	19.5	0.842	16.418
13	100	1300	0.842	1094.54
			Total PV	1216.03

Thus, Maculay duration = Total PV/ Current value = 1216.03/ 101.48 = 11.98

b) Bond portfolio's duration is 7.32 implying an increase of 1% in interest rate would reduce the value of portfolio by 7.32 x $ 150 mn / 100 = ~$11 mn

To effectively hedge against this, we need to ensure an equal gain in value of T-Bond futures. Thus,

Investment needed in T-bond futures = $ 11 mn x 100 / 11.98 = $ 91.8 mn

Each contract being of $ 100,000, we need to purchase 91.8 mn/ 100,000 = 918 contracts = ~920 contracts

c) These contracts are used for hedging against any risk of interest rate increase on bonds in our portfolio. If interest rates rise, the price of portfolio would drop, which needs to be offset. This can be achieved if we have short position in T-Bond futures contract , the value of which would gain if interest rates increase.