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# Duration question Garp practice 2008 q 15

#### john.ophof

##### dra. Ing
Hi,

I was looking at:

15. Assuming other things constant, bonds of equal maturity will still have different DV01 per USD 100
Face Value. Their DV01 per USD 100 Face Value will be in the following sequence of Highest Value to
Lowest Value:
a. Zero Coupon Bonds, Par Bonds, Premium Bonds
b. Premium Bonds, Par Bonds, Zero Coupon Bonds
c. Premium Bonds, Zero Coupon Bonds, Par Bonds,
d. Zero Coupon Bonds, Premium Bonds, Par Bonds

I thought d looking at page 133 of Market risk PDF.

John

#### skcd

##### New Member
DV01 = Sum(Face(i)*Price(i)*Duration(i)) / Sum(Face(i)*Duration(i))
So here if its single bonds then its only the Price that matters. Premium bonds are quoted above Par and ZC bonds are discounted than Par. Hence b)

#### john.ophof

##### dra. Ing
DV01 is price value of a basis point given by formula - delta P / (10000 * delta y). A zeo coupon with have the highest duration.

We can express it also like (Duration Macaulay * Price ) / 10000. So price can be offset by duration. Maybe zero is cheaper than par bond but this doesn't mean D * P is also the lowest value.

John

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
John,

FYI, I have DV01 plotted versus zero/par/premium at the member page here.
(1st page, 2nd row)

This is straightaway from Tuckman Chapter 6. But, IMO, this is a mean, nasty question because I am not sure the intuition can be found. Just as you say, John, the DV01 is impacted by both a price and duration effect, which is mixed for the non-par bond (i.e., the par bond has non changing price, so we can infer that DV01 is an increasing function with maturity because duration is) and therefore hard to find short of doing calcs. IMO, you are doing very well to be mindful of the key formula: DV01 = P*Modified Duration/10,000. Otherwise, short of memorizing Tuckman Ch 6 on this, I do not know how you intuit these results...

That said, a "more fair" question, that we should know, is the same question above but substitute "DV01" with "duration." We definitely can intuit duration vs. zero/par/premium

David

#### john.ophof

##### dra. Ing
Indeed Tuckman page 129, thx for help!

#### skcd

##### New Member
But i guess for constant maturity and face value, i don't think the answer is entirely wrong. Its correct to say Premium > Par > Discounted > Deep Discount(like ZC) in terms of DV01 given other things (maturity, face value ) are equal.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
skcd,

Agreed, the answer is correct, as Tuckman's (and my XLS) plot shows. I just mean that, short of memorizing this, how do you derive this intuitively given that DV01 is "infected" with price? If there isn't an intuition, I think it's an unfair question b/c knowing "too much" about DV01 (as John shows) leads to ambiguity. David

#### skcd

##### New Member
Thanks David. Appreciate it. I dont get the question clearly either.

#### southeuro

##### Member
ok pls let me know if my way of tackling this is wrong but here's the short-cut:

since the DV01 formula is: (1/10,000) * (1/1+periodic yield) * (sum of time-weighted present value of cash flows)... and since zero-coupon bond is the one that gets impacted the most because it has only 1 lump-sum payment in the end, it will be the one with the lowest DV01. Now, the premium bond is premium b/c the PV of its coupons are occurring relatively closer to present day (opposite of zero-coupon or compared to the par's), therefore its duration is not shortened as much as the others, so it must have the highest DV01.

2 observations: 1) I thought the explanation of duration in investopedia to be VERY helpful with the money bags being shifted around depending on maturity, coupon and yield. 2) I think Tuckman's intuitive explanation about how an increase in yield decreases duration is quite similar to what's going on here (there as well, the yield increase impacts ALL of the PV of cash flows, but it impacts the ones in farther out the most).

Am I wrong or am I wrong?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @southeuro

I totally agree with your last 2 observations! Including, to me, it is (once you think about it) intuitive that an increase in yield will decrease the duration ... however, I still find the original question (posted in 2008, yikes!) very difficult. True it queries Tuckman's Figure 6.2 (Tuckman 2nd ed) but I (personally) continue to thinks it would be an unfair exam question.

Just to clarify the difference: the question concerns bonds with the same yield (e.g., 5%) and maturity (any T). Of course you are correct that "DV01 formula is: (1/10,000) * (1/1+periodic yield) * (sum of time-weighted present value of cash flows)" and here is the formula (Tuckman 6.5, 2nd Ed) to which you refer:

$$DV01=\frac{1}{10,000}\times \frac{1}{1+y/2\ }\left( \sum\limits_{t=1}^{2T}{\frac{t}{2}\frac{c/2\ }{{{\left( 1+y/2\ \right)}^{t}}}}+T\frac{100}{{{\left( 1+y/2\ \right)}^{2T}}} \right)$$

The reason it's not easy for me is that final term which contains both time (t) and coupons (c) for a given yield. It is true that this formula, once seen, makes it easy to see why DV01 is an increasing function of the coupon rate (therefore DVO1 of premium bond > DV01 of par < DV01 > discount) but i don't think it's easy because duration is a decreasing function of coupon rate. Duration and DVO1 have opposite reactions to coupon rate! In any case, thanks for your contribution, I love it!

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