Question

Your $100 million bond portfolio has currently $90 million invested in bonds and the remainder invested in T bills. The duration of the bond component is 5.2 years. Assume that the T bond futures contract has a duration of 7.8 years and a value of $102,500 per contract

Q) How many futures contracts should be purchased or sold to change your portfolio's duration to 6.0 years? and How many futures contracts should be purchased or sold in order to make the portfolio insensitive to changes in interest rates?

Answer 1

Weighted Average Duration of Portfolio = 6

Duration of Bond * Weigt of Bond + Duration of T Bill * Weight of T bill + Duration of T bond Futures * Weights of T bond = 6

[Note:-

Treasury Bills are like Zero coupon bond , there is only one cashflow at the end , it matures at Face Value and are issued at discount.

Thus Duration of Tbill is the Tenure of Tbill.

Since Tenure of Tbill is not mentioned, it is assumed as 1 year and thus Duration of Tbill is also 1 year.]

5.2 years * 90/(100+x) + 1*10/(100+x) + 7.8 * x/(100+x) = 6

x is assumed as the value of Tbond Futures purchased.

Why Purchase not Sell? Cause we see Duration of Original Portfolio will be any way less than 6 because both have Lower duration than 6, thus we have to purchase T bond which has higher Duration than 6.

5.2*90 + 10 + 7.8X = 600 + 6X

7.8X-6X= 600-478

1.8X = 122

x= $67.77 million Value of Tbond Futures to be purchased.

No. of Contracts of Tbond Futures to be purchased = Value of T bond Futures / Contract Size = 67.77 million/ 0.1025 million = 661.25 rounding off to 662 Contracts to be purchased.

Bond Immunization is the process by which we protect ourself from change in Maret Interest Rate.

Thus Answer to 2nd Question is same = 662 Contracts.

Happy Learning!