We need to calculate the total dollar duration of the bond portfolio, which is =100*7.5 - 65*12 + 120*6 = $690m. This means a 10 basis point change in interest rates changes the value of the portfolio by $690m*10bps = $0.69m, or $690,000. Therefore Choice 'b' is the correct answer.
[Here is an alternate step-by-step way of thinking about it. Let us consider changes to each position separately.a. A long position worth $100m in a bond with a modified duration of 7.5A 1% increase reduces the value of the bond by 7.5%. Therefore a 10bps = 0.1% change in interest rates will change the position by 0.75% = $100m*0.75% = $750,000 decrease.b. A short position worth $65m in a bond with a modified duration of 12A 1% increase increases the value of the position by 12% (remember this is a short position). Therefore a 10bps = 0.1% change in interest rates will change the position by 1.2% = $65m*1.2% = $780,000 increase.c. A long position worth $120m in a bond with a modified duration of 6A 1% increase reduces the value of the bond by 6%. Therefore a 10bps = 0.1% change in interest rates will change the position by 0.6% = $120m*0.6% = $720,000 decrease.The net change in the value of the porfolio is therefore:-750,000 + 780,000 - 720,000 = $690,000]
[Here is an alternate step-by-step way of thinking about it. Let us consider changes to each position separately.
a. A long position worth $100m in a bond with a modified duration of 7.5
A 1% increase reduces the value of the bond by 7.5%. Therefore a 10bps = 0.1% change in interest rates will change the position by 0.75% = $100m*0.75% = $750,000 decrease.
b. A short position worth $65m in a bond with a modified duration of 12
A 1% increase increases the value of the position by 12% (remember this is a short position). Therefore a 10bps = 0.1% change in interest rates will change the position by 1.2% = $65m*1.2% = $780,000 increase.
c. A long position worth $120m in a bond with a modified duration of 6
A 1% increase reduces the value of the bond by 6%. Therefore a 10bps = 0.1% change in interest rates will change the position by 0.6% = $120m*0.6% = $720,000 decrease.
The net change in the value of the porfolio is therefore:
-750,000 + 780,000 - 720,000 = $690,000]
