You hold a $50 million portfolio of par value bonds with a coupon rate of 10 percent paid annually and 15 years to maturity.How many T-bond futures contracts do you need to hedge the portfolio against an unanticipated change in the interest rate of 0.18%? Assume the market interest rate is 10 percent and that T-bond futures contracts call for delivery of an 8 percent coupon (paid annually),20-year maturity T-bond.
A)398 contracts long
B)524 contracts short
C)1048 contracts short
D)398 contracts short
E)none of the above

Question

Quizplus · Accepted Answer

To calculate the number of T-bond futures contracts needed to hedge the portfolio, we need to use the following formula:

Number of contracts = (Portfolio value / Bond value) x (Duration of portfolio / Duration of futures contract)

Portfolio value = $50 million
Bond value = $1,000 (par value)
Duration of portfolio = 15 years
Duration of futures contract = 20 years

First, we need to calculate the present value of the portfolio, assuming a 0.18% increase in interest rates:

PV = Coupon payment x [1 - (1 + r)-n / r] + Par value / (1 + r)n

Where:
Coupon payment = $100 ($1,000 x 10%)
r = 10.18%
n = 15

PV = $925.75

Next, we need to calculate the bond value for the T-bond futures contract:

Bond value = Coupon payment x [1 - (1 + r)-n / r] + Par value / (1 + r)n

Where:
Coupon payment = $80 ($1,000 x 8%)
r = 10%
n = 20

Bond value = $1,128.47

Now, we can plug in the values into the formula:

Number of contracts = ($50,000,000 / $1,128.47) x (15 / 20) = 397.53

Rounding up to the nearest whole number, we get 398 contracts. Since we are hedging against an increase in interest rates, we need to take a short position in T-bond futures contracts. Therefore, the answer is (D) 398 contracts short.

You Hold a $50 Million Portfolio of Par Value Bonds