Is there a short cut to computing the duration below for the 6% 10-year bond in the question below? The question doesn’t mention if the bond is semiannual or not. Do we assume semiannual? If so, then I would have to compute 20 present value calculations for each cash flow, which seems time consuming. Also “recently issued” seems key here. This seems to mean that the time to maturity is around 10 for this bond? The answer explanation also assumes that this bond is priced at par. Seems I would have to make a few assumptions in questions like these?

61. What is the best estimate of the market value of a portfolio of USD 100 million invested in recently issued 6% 10-year bonds and USD 100 million of long 10-year zero coupon bond if interest rates decline by 0.50%:

a. USD 219 million
b. USD 195 million
c. USD 209 million
d. USD 206 million

ANSWER: C

To calculate the best estimate of the market value of the portfolio if interest rates decline 0.5%, one needs to calculate the change in the market value of each bond using duration. The duration of the 10-year zero coupon bond is 10. Thus, the change in value of this bond equals 10x0.005x100,000,000, which equals 5 million dollars.

The duration of the newly issued 6% bond is 7.802 assuming that the price of the bond is par. Given a duration of 7.802, the change in the value of the bond equals 7.802x0.005x100,000,000 which equals 3.91 million.

Yes, agreed, without yield of coupon bond (or, equivalently, w/o being given the coupon bond’s price), duration cannot be computed on the coupon bond. Although you’d need 3 PV calculations, not 20 to get duration.

But, without yield, the only thing you can do is “ballpark” it via the logic given. It’s not a a great question, given it commingles Macaulay and Modified duration [notice it uses Macaulay duration to answer the sensitivity question. This is technically wrong. The zero has a Mac duration of 10 but it’s sensitivity is < 10. The .5% rate change translates into < $5 million]. So, it’s not great b/c the 206 is unlikely but there is an unlikely (high yield) scenario that could get pretty near to the 206 when you correctly used modified duration.

so, yea, you can only shortcut this one, you have to see that the zero bond adds almost $5 million and the other is going to add one-half of 6-9 duration, or +3 to +4.5 million. And then use elimination

David

Can you clarify why only 3 PV calculations are required here? I thought since the bond is 10 years, then 20 PV cash flow calculations are required to determine duration (assuming semi-annual payments).

Dennis,

I added a second sheet to the durations worksheet on the member page, with these numbers. I assumed coupon bond priced at par (please see 2nd sheet)

To get Macaulay duration *directly* you are correct, requires several PV calcs. But you don’t need to do that.

Modified duration = (100.75 - 99.26)/(2* 100* 10 bps) = 7.439 modified duration. So, you only need PV @ 6%, PV @ 6% + shock, and PV 6% - shock.

Then 7.44 * (1+6%/2) = 7.66 Macaulay duration which is what you get if you calc it out the long way.

David