Duration of a bond

[girishkhare]

This question is from Phillipe Jorion's FRM handbook 5th edition page 21. This is also question 104 in FRM 2001. The question is as follows

When the maturity of a plain coupon bond increases, its duration increases

a. Indefinitely and regularly
b. Up to a certain level
c. Indefinitely and progressively
d. In a way dependent on the bond being priced above or below par

The answer given is (b) and the explanation is "With a fixed coupon, the duration goes up to a certain level of a consol with the same coupon"

I have two doubts about this. Firstly, the duration of a consol (perpetual bond) is independent of the coupon as the formula is (1+yield)/(yield). What is meant by "Duration of a consol with the same coupon"?

Secondly, as figure 1.7 on page 18 of the 5th edition of FRM handbok shows, for a par bond (when coupon is equal to yield), the duration increases with maturity and asymptotically reaches the duration of a consol. For a premium bond (when coupon is greater than yield), the duration increases with the maturity but the curve lies below the duration vs. maturity curve for a par bond.

Moreover for a discount bond (when coupon is less than yield), as maturity increases, the duration first actually goes beyond the duration of a perpetual bond and as maturity further increases, the duration asymptotically decreases and matches that of a perpetual bond.

All this is shown in figure 1.7 in FRM handbook.

Given all this, I feel that the answer to the question posed above should be (d) i.e. "In a way dependent on the bond being priced above or below par" and not (b).

Could somebody please guide me on this?

Thanks in advance

Girish

 

[David Harper CFA FRM]

Hi Girish,

Sharp observations. I mostly agree. FWIW, you may notice I don't like the FRM handbook questions. This is why. When I saw the question was from 2001, I knew there's a 50/50 chance it's got a flaw.

IMO, the problem is poorly worded and as worded, makes possible a defense of either (b) or (d). But clearly you are correct about (d). For additional context, here is Tuckman (who is, after all, assigned):

"If the discount is deep enough—that is, if the coupon is low enough relative to yield, as it is in this figure—the duration of a discount bond rises above the duration of a perpetuity. But since at some large maturity the duration of a discount bond must approach the duration of a perpetuity, the duration of the discount bond must eventually fall as maturity increases." Tuckman p. 125

My thoughts:
* Maybe the question was trying for: the coupon-bond must eventually (as maturity increases) converge to the duration of perpetuity WITH THE SAME YIELD
* On your first point, you must be correct "with the same coupon" does not makes sense. I see only two explains: 1. they meant "with the same yield" or perhaps 2. they meant DV01, as that is coupon-dependent
* The imprecision with (B), IMO, is that it could mean "it has a maximum" (is not indefinite) or it has an asymptote (the consul's duration)
* the language is too loose. Regularly? Progressively ? In a way dependent? … it's a fatally flawed question for using imprecise terms where, when it comes to function plots, they are certainly available. (i.e., as the reader we are suffering to define terms/phrases in the answers instead of trying to link the answer to the question. Bad deal)

So your question is more informed that the exam question. Trying to divine something further, I think, isn't productive.

David

P.S. if you want to understand how this can happen, it's easy: Jorion (who is one of the most precise authors I've ever read) wrote the text of the handbook. Somebody else wrote the question. If you want to arbitrate between the handbook text and a handbook question, I can tell you that, every single time (that i can remember) where we've found an error, it was in the question, not in Jorion's text.

 

[girishkhare]

Thanks a lot David.

 

[frmqiu]

thanks