Duration of a callable zero bond

[Hend Abuenein]

Hi David,
I came across a question about the duration of a 10 year zero coupon bond that is callable annually after the 6th year.
Answer said that the duration of a zero-coupon bond is always equal to its maturity, regardless of whether it is callable, explaining that if the issuer calls a zero, it will be for the purpose of retiring debt, not refunding at lower rates.
But what does the purpose of calling the bond have to do with determining its duration?
And do you agree that it is equal to its maturity? If it's called on the 7th year, how can its duration be 10 years?!

Thank you

 

[David Harper CFA FRM]

Hi Hend,

I think this is the first time this has come up, which is surprising!

No, I don't agree that the duration of a callable zero-coupon bond equals its maturity. After i record video, I will look in my library for a reference, but the answer makes no sense to me: even for a zero, the bond will get called at eventually at a lower yield. At such a low yield, the negative convexity will not give over to negative duration, but the duration is (clearly, IMO) decreasing as yields decrease(i.e. , at low yields, i don't see any other scneario except that duration and dollar duration are increasing functions of yield: if true, renders the given answer incorrect).

Re: "If it's called on the 7th year, how can its duration be 10 years?" I actually think your point here is pretty correct--although some argue that we can't characterize duration in these terms [weighted average maturity] when we introduce derivatives [i.e., the weighted maturity becomes too variant to factors that it can be difficult to characterize the bond in Mac duration terms, I think is Fabozzi's point somewhere] ... but i think the substance of your point is correct. I am 90% intuitively confident "we" are correct (you and I) and the answer is incorrect, but i will look in my library after i record for some formal support. Thanks!

 

[Hend Abuenein]

Thanks David,
About the decreasing yield, (I didn't mention this because I thought it irrelevant to my doubts) but question says yield curve is flat at 10%, and price of call would be bond's par. (no more information)

Now that issuer won't have yield-related incentive to call, does it make a difference to the right answer?

 

[David Harper CFA FRM]

I'm not sure, my intuition (visualization) could go either way: if the option's deep OTM, the bond's effectively like its noncallable counterpart; i.e., at high yield, i can imagine duration = maturity. On the other hand, i'd imagine that's an asymptote that renders the statement approximately true but technically incorrect, which the error a function of proximity to the call. I'm not confident, is why i need to find expert reference support later, because I don't want to just give you my hunch. Thanks,

 

[Hend Abuenein]

Here's the original question from Kaplan's SchweserPro 2012

A 10-year zero coupon bond is callable annually at par (its face value) starting at the beginning of year 6. Assume a flat yield curve of 10%. What is the bond duration?
A- 10 Years
B- 5 Years
C- 7.5 Years
D- Cannot be determined based on the data given.

This is Q #126323

I went for D

 

[David Harper CFA FRM]

Thanks Hend, it's much better to see the whole question (you didn't mention the strike at par at first, and in my haste, i missed your subsequent mention). It's sneaky. As for the question's intent, the key is that it strangely strikes a par, which is unrealistic: a zero will always trade at a discount, so the option here has effectively zero value. The very high 10% yield further only helps to confirm the effective worthless of the option. Then the question would have you, I suppose, lean on maybe our most important callable bond equation:

value [call option] = value [value of non-callable; i.e., option-free] - value[equivalent callable bond]

so, the question is sort of creative by setting up a worthless option, such that the Macaulay duration of this bond, although maybe not exactly 10.0, will be very close.

In regard to stepping back and answering your original question, I did locate help from my favorite book on duration. (key quote blelow)

Here is my paraphrase of their answer (combined with their academic reference) to the question: what is duration of a callable zero-coupon bond?
Answer: it varies with yield and option strike price,

• Lower bond [as strike price --> 0 and/or yield decreases] of time to call date; e.g., in the question above, lower bound would be 6.0 years, if the strike were more realistic!
• Upper bond [as strike price --> inf. and/or yield increases; i.e., as option value tends toward zero] of duration of underlying non-callable bond; e.g., in the above, 10.0 years.

"Sanjay Nawalkha page 31: We have demonstrated that many statements about duration and convexity do not hold in the simple case of a zero-coupon bond. Let us now take the more complicated example of a security with an embedded option—a callable zero-coupon bond—to illustrate how misleading these statements can be if applied more generally. Consider a $100 face value 10-year zero-coupon bond that is callable (European-style) in one year at 80 percent of its face value. Figure 2.2 plots the bond’s price, duration, and dollar duration as a function of yield. The bond price as a function of yield first steepens, and then flattens as yield increases (see Figure 2.2 Panel A). Inferring duration from the slope in Figure 2.1 implies incorrectly that duration first increases and then decreases as yield rises—whereas, the duration of the callable bond is monotonically increasing in yield (see Figure 2.2 Panel B). The correct inference is that it is dollar duration that first increases and then decreases with increasing yield (see Figure 2.2 Panel C). We conclude that inferring duration from the slope of the price-yield relationship causes substantial confusion in the case of a callable bond. -- source http://www.amazon.com/Interest-Rate-Risk-Modeling-Valuation/dp/0471427241"

• From book: http://www.amazon.com/Interest-Rate-Risk-Modeling-Valuation/dp/0471427241
• Footnote self-references Nawalkha's paper (for future reference, this is actually the most helpful, includes simulations) : http://ssrn.com/abstract=983349

 

[Hend Abuenein]

Thank you

 

[David Harper CFA FRM]

Hey Hend,

I just noticed (as i am triangulating on the handbook to write mock question...): this question is in the FRM handbook, verbatim: example 9.2, page 213.

 

[Hend Abuenein]

Hi David,

I buried Jorions' book early 2011 in the store room . (Didn't find it useful at all)
I'll have to go dig it out. Not unless you have the time to quote the question here
If not, it's ok.

Thanks anyway

 

[David Harper CFA FRM]

Hi Hend, the FRM handbook question is exactly the same, verbatim, as yours above, with the same answers except a,b,c a are shuffled, so it's weird they didn't properly attribute.

Answer given is: 10 years "because this is a zero-coupon bond, it will always trade below par, and the call should never be exercised. Hence its duration is the maturity, 10 years."

 

[Hend Abuenein]

Hi,
Oooh ...now I get it. Truly weird that they didn't attribute it. I got it from their QPro test management software, nothing printed by them. In the end of every book they have a mock exam made up of some 25 GARP practice questions, but they attribute each to its year.

On a side note: remember when I asked you last year how come Schweser say their book-end exams are REAL GARP exam questions when GARP says they never let out any of their questions?
Well, in this year's books, Schweser corrected that to :"GARP previous Practice Exams questions".

Looking forward to working on YOUR mini mock exams soon

Thanks.

 

[David Harper CFA FRM]

Hi Hend,

Ohhhhh ... well, we have those too (except we, you know, attribute because that's how we roll)

re mock, I will only have one each (25 Q P1 and 20 Q P2) ready for this May, fwiw. Although I am accustomed by habit to writing 3 questions per day, writing good mock exams questions (non AIM based) is actually quite time consuming, it ruined my planned weekend (I missed my cousin's confirmation), so i'm a little cranky The forum support is such an unknown variable, that I should be more careful about my promises ... thanks,

 

[Hend Abuenein]

YOU NEED STAFF! Pronto!

 

[David Harper CFA FRM]

Ha, well, a small business can't just hire staff. The key economic problem, not unique to me (shared by Schweser, I know them) is stealing, if all the thiefs paid, I could hire a person. But really qualified people are really expensive, often too expensive. I did outsource some of the notes but then we produced a regrettable (horrifying, to me really) number of errors on those portions. (I am amazed at how difficult it is to find somebody who can control the error rate. Most people make errors, me included. Some people make a lot of errors. To write an incorrect question can be quick. To write a correct question can take a surprisingly long, many many times i have gotten stuck in technical details for a half day, for a single question!)

Forum support is a massive spend of my time, of course (a lot think i should limit; or at least limit certain users ... well, let's just say i've argued with my advisors about the forum, a lot). But the thing is, the business is to help people learn, so i feel my current approach to the forum is consistent with the mission (if not always the economics). More than you asked! Thanks,

 

[Aleksander Hansen]

Hiring is definitely easier said than done.
The time and effort that goes into bringing on an employee is unbelievable. The Government have you fill in tons of forms, and there's insurance for this, that and the other; benefits, standards to adhere to; material about employees rights and wages, anti-discrimination and so on and so forth that you have to make available to your employees.
Single biggest job killer in my opinion is the Government, by making small businesses avoid hiring or keep staff at a minimum.
If they got out of the way, I would gladly hire 3 more people on the day!

 

[Hend Abuenein]

Hi,
I hope you've both been well.

Aleksander: I'm sorry, but I couldn't relate your reply to David's situation here.

I came across this picture 2 days ago on FB

Which I found really sad. I think it was in the Chicago anti NATO protests.
But where ever, I think it tells another story than that you are telling here.

Regards

 

[David Harper CFA FRM]

Hi Hend,

I hope you've been well (did you post how your exam went, did I just miss it?).

The picture reflects a troubling reality, here in the US a constant topic of conversation is about the question: is formal school a product worth buying? (or is it often a bubble enabled by government subsidy).

Alek's point relates to my situation in the following way: the cost to bring-in a full-time employee is much greater than the salary cost. He is so right with respect to U.S. regulations, they are basically an additional barrier to entry. I don't have data, but my hypothesis would be that regulations directly contribute to under-employment by discouraging, at the margin, the hiring of full-time employee(s) and encouraging the outsourcing to part-time independent contractors. As I understand, the "odesk economy" is thriving, and I don't doubt it at all.

 

[Aleksander Hansen]

Thanks Hend,

I'm not sure I understand what you mean by that, or how it contradicts my claim that Government tends to create a friction in the labor market... We all know where the road of good intentions lead. There are known and unknown consequences of any policy.

If this guy could do the work for David or any other business at a price both agree upon, the transaction would be mutually beneficial, and would mostsurely occur.

What I find, is that, contrary to the one guy with a cardboard poster with a statement, which may or may not be true, is that there is a lack of skilled workers. There is a very real disconnect between what recent graduates bring to the job market, and what the job market needs. Too many pursue degrees with little value in the market place, and two few are willing to work hard.

Our generation (including me) is the generation of the spoiled kids used to getting what they want, when they want it. If they can't get it, throw a tantrum, sit in a park for a month doing nothing, and vandalize the property of people who have done you no harm. Couple that with tight restrictions on the use or hiring of skilled labor from another country and you get a difficult situation.

Oh, by the way, I own a LLC so I have first had experience with federal state and local filings, laws and taxes.

That's just my view, and I respect the right of others to feel differently. I do not accept or condone however, the destruction of private property or mob rule.

 

[Hend Abuenein]

Thank you. I understand now.

I did post shortly about the exam on the first feed backs, but I really don't want to remember it. I'm just glad that it's over. For now at least.

You should have sometime off yourself

 

[Aleksander Hansen]

I should add that previous generations are no better by creating de facto giant Ponzi schemes like Medicare, Medicaid, Social Security and a bloated defense budget. It is not sustainable, but unlike Madoff who's dwarfed by these guys they are not held accountable and are merely kicking the can down the road.

I do agree though, sights like that are sad indeed.