Question

There are a number of ways to construct bond hedge ratios. One way to write a duration-based hedge ratio is as follows:

Hedge Ratio = CF_ctd X (P_b X D_b)/ (P_f X D_f ) where

CF = conversion factor for CTD bond

P_b = price of bond portfolio as percentage of par

D_b = duration of bond portfolio

P_f = price of futures contract as percentage of 100%

D_f = duration of CTD bond for futures contract

A bond portfolio manager holds a bond portfolio with a face value of $8 million that is currently worth a market value of $8.25 million. The manager is concerned about future rising interest rates and so decides to hedge with a T-Bond futures contract. The cheapest to deliver bonds have an 8 1/8% coupon and a projected duration at maturity of 12 years. Their conversion factor is 1.125 and at their current price the futures price is 98-02. The current duration of the bond portfolio is 5.25 years.

a. Based on the above data, compute the optimal hedge ratio.

b. Based on the interest rate expectations, should they take a short or long position?

c. The optimal number of contracts to hedge with is given by:

Number of contracts = HR X (Portfolio current market value/current value of futures contract)

Where each futures contract is for $100,000 face value of bonds.

Based on this, compute the optimal number of futures contracts to hold.

d.   The closing futures contract price is 89-00. Based on this, how did the futures position perform?

Answer 1

CF ctd bond = 1.125

Pb = $8.25m/$8m = 1.03125 = 103.125%

Db = 5.25 years

Pf = 98.02%

Df = 12 years

Hedge ratio = 0.518

(b) When interest rates rise, value of bond portfolio decreases, therefore we should go short on futures contract, so that loss in bond portfolio is equalled by gain in futures contract (due to short position)

(c) HR hedge ratio = 0.518

Portfolio current market value = $8.25 m = 8250000

Current Market value of 1 future contract = 98.02% * 100,000 = 98020

Number of contracts = 0.518 * 8250000/98020 = 43.598

Short 43.598 future contracts

(d) If closing futures price is 89-00

Then gain on futures position = - [(89.00 - 98.02)/100] * 43.598 * 100,000 = 393,256