The modified duration will equal a Macaulay duration divided by 1, plus the current yield to maturity divided by the number of payments in a year. Once you find the duration, then you can determine the percentage change in bond prices as a result of changes in interest rates using the following:
Change in P/P * 100 = -Dmod * Change in i.
In problem 4, the first bond is a bond portfolio, so you have to do a ‘weighted average’. You see that both bonds added together have a value of 10. You also note that these are zero coupon bonds, so you have no coupon payments. That means that there is no cash flow from coupon payments, so the Macaulay duration will be the number of years to maturity. But keep in mind, you have two bonds with different maturities. One bond with a value of $4 ($4 out of $10) has a maturity of 14 years, and the other with a value of $6 has a maturity of 3 years.
$4 out of $10 is 40% and this bond matures in 14 years.
$6 out of $10 is 60% and this bond matures in 3 years.
So the ‘D’ is going to be (40%* 14)+(60%*3) = 7.4
Given this, the Modified D (Dmod) becomes ModD1 = 7.4/1.0731= 6.896 years.
In problem 4a, there is a 60 basis point change in the interest rate, which is .006. Given this, the change in price would be:
P/P -6.896 x .006 = -4.137%
In 4b, though the two portfolios have identical durations, the actual price changes will be different because the portfolios have different convexities.
In 4c, to hedge, you would do the following:
HR1 = (6.896/10.355) x [0.6 x 1.13 + 0.4 x 1.03] x (-10,000,000/109,750) = -66.14
HR2 = (6.896/10.355) x [1.01] x (-11,500,000/109,750) =70.48
Combining these two gives a net hedge ratio of 4.339. Consequently, the hedge would be to enter into a long position in 4 futures contracts.
ms.zas
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