Question

Futures

The bond portfolio is comprised of one type of corporate BBB bond. The term is 8 years semiannual, YTM is 5%, coupon is 8%. The portfolio contains 1 million bonds.

1. Calculate the par value of the portfolio.
2. Calculate the market value of the portfolio.
3. Calculate the modified/ effective duration, using the formula (not spreadsheet).
4. If the T-bond futures contract is priced at 98 and has a duration of 4, and you wish to reduce the portfolio duration to zero, how many futures contracts do you need? Long or short?

Answer 1

1). With a par value of $100 per bond, portfolio face value is

par value per bond*number of bonds = 100*1,000,000 = 100,000,000 or 100 million

2). FV (par value) = 100; PMT (semi-annual coupon) = annual coupon rate*par value/2 = 8%*100/2 = 4; N (number of payments) = 8*2 = 12; rate (semi-annual yield) = 5%/2 = 2.5%, solve for PV.

Price per bond = 119.58

Market value of the portfolio = price per bond*number of bonds

= 119.58*1,000,000 = 119,582,504

3). Modified duration = 6.096 years (calculated using MDURATION function in excel.)

Effective duration = (P-y - P+y)/(2*P0*change in y) where P-y = price when yield decreases by 1% i.e. yield = 4%; P+y = price when yield increases by 1% i.e. yield = 6%; P0 = current price

(P-y and P+y calculated using PRICE function in excel, as shown in the table above.)

Effective duration = (127.16-112.56)/(2*119.58*0.01) = 6.102 years

4). Number of contracts to be sold = (Dt - Di)*MVp/(Df*Pf) where

Dt = effective duration to be achived = 0; Di = initial effective duration = 6.102; MVp = market value of portfolio = 119,582,504 ; Df = effective duration of the fuures contract = 4; Pf = 98*100,000 = 9,800,000 (one T-bond futures has a contract size of 100,000)

Number of contracts = (0-6.102)*119,582,504/(4*9,800,000) = -18.62 or 19 contracts to be shorted