Convexity effect

[wrongsaidfred]

Hi David,

I came across a problem (not one of yours) that does not seem to make sense and I was hoping you could shed some light on the situation.

The question is:
Which of the following bonds bears the greatestprice impact if its yield declines by one percent? A bond with:

A)
30-year maturity and selling at 100.

B)
30-year maturity and selling at 70.

C)
10-year maturity and selling at 100.

D)
10-year maturity and selling at 70.

The answer is obviously one with a greater time to maturity, but I would think the bond selling at par would be effected more because it has the lower yield. From my understanding, the lower the yield, the higher the duration and convexity. The answer they gave was the discount 30 yr bond and I am having troubel seeing exactly why this is the correct answer.

Their resoning was:

The bond selling at the greatest discount will have a large price impact, a discount means that the coupon payments are low or the initial yield is low.
Not quite sure what they mean by "initial", but wouldn't the par bond have a lower yield than the discount bond?

Thanks in advance for any advice.

Mike

 

[David Harper CFA FRM]

Hi Mike,

The question is instructive for its unstated assumptions, I don't think the question has a answer. First, note the price impact is not specified in dollar or percentage (duration) terms; I can find several scenarios where they give a different answer; e.g., a $30 increase on the par bond is greater in dollar terms but less in percentage terms than $24.29 on the $70 bond (both for a 1% yield shock). That's the first thing I notice: "price impact" leads me, like you" to duration. But dollar change (i.e., DV01 * 100) is a price impact, too.

But, to your point: you are totally correct that, ceteris paribus, lower yield implies higher duration (and higher DV01). By the answer, the question is focused only on the "coupon effect:" ceteris paribus (including for a GIVEN YIELD), lower coupon also implied higher duration (however, lower coupon --> lower DV01). Because both of these are true, ceteris paribus, the answer DEPENDS and VARIES on the particular combination of coupon and yield.

That's because it is NOT TRUE that a discounted bond implies that "coupon payments are low or the initial yield is low."
And this is the fallacy that an FRM candidate does want to know: the discounted bond price only tells us that the coupon rate (c) is less than the yield (YTM).

I computed coupon/yield pairs for three 30-year bonds (annual pay coupon b/c i am lazy) with a discounted price of $70:

• c = 0% (i.e., zero coupon bond), ytm (yield) = 1.2%,
• c = 2.27%, ytm = 4%,
• c = 8%, ytm = 11.62%.

... note c < y, but even the differential is non-linear, we can only assume c>ytm, we can't even assume anything about the value of (c-ytm)​

Just as the bond priced at $100 only tells us that c = ytm and could include:

• c = 1%, ytm = 1%,
• c =10%, ytm = 10%

So in my quick calcs, i get get BOTH higher/lower price impact with respect to BOTH dollar (DV01) and percentage (%) so ... the question is interesting for the assumptions in makes!

We didn't even need to get to convexity. (As written, I would have to assume the question refers to the ACTUAL price impact and therefore convexity deserves consideration ... in which case, the question has become too much to think about). The lack of precision on convexity (i.e., are we meant to include on exclude and judge only on duration) is a third problem, IMO.

It shows why it's hard to write good questions: good questions really need to be really specific, really.

Thanks, David

 

[wrongsaidfred]

Many thanks. The one point that I did not quite follow was was:

And this is the fallacy that an FRM candidate does want to know: the discounted bond only tells us that the coupon is greater than the yield.
I thought a high coupon would lead to a higher price (because it would be a premium bond) and, hence, a lower yield. This seems to disagree with what you just wrote. Am I missing something? If we consider the same state of the world (same term structure) for two bonds with the same maturity, one with a high coupon and one with a low coupon, wouldn't the high coupon always be more expensive and have a lower yield?

I think there is a chicken/egg argument, where the term structure leads to a price and a YTM. Comparing different bonds in different states of the world does not seem like a realistic assumption. Then again, is this something that GARP would do?

I attached an Excel sheet to at least try and demonstrate wht I mean.

Thanks again.

Mike

 

[David Harper CFA FRM]

Hi Mike,

Yes, thank you .... I made a dumb typo. As you can see from my actual bond numbers (i.e., for the $70 bond: c = 0% (i.e., zero coupon bond) < ytm (yield) = 1.2%; c = 2.27% < ytm = 4%; and c = 8% < ytm 11.62%), I definitely should have written that a discounted bond "only tells us that the coupon is less than the yield (YTM)." As that is important, I corrected that dumb-dumb mistake above. Thanks for the catch! (while i am back, i corrected to say "coupon rate" < ytm for discounted bond)
... the intuition is: as the yield is income + appreciation [IRR], if the coupon isn't delivering enough yield, the rest needs to come from appreciation. Conversely, if the coupon delivers more than the yield, the "deduct" difference must come from expected price depreciation (as bond pulls to par)

Re: "wouldn't the high coupon always be more expensive and have a lower yield?"
Yes, of course given the same term structure the price must be higher for the higher coupon.
But, no, i don't think the yield needs to be lower. I think that is only the case if the term structure is upward sloping, as is yours; I think if you reverse your spot rates, and assume an inverted term structure, the yield on the $5 coupon bond will be higher

Because, you are absolutely, correct, there is a circularity here: the yield (YTM) is variant to the term structure; the YTM is an IRR, and is like the flat curve that "impounds" all of the spot rate (term structure) information. Your XLS illustrates this nicely: 6 zero rates characterize an upward sloping term structure of spot rates. Yet the $5 bond has yield (YTM) which is different than the $3 bond, for the same exact term structure. The term structure of spot rates is, in fact, the "objective" thing that exists in the market ... in contrast to the yield, which is a function of both the objective term structure and each different instrument
... for this reason, we really want to disconnect that final implication, from the coupon and the price, to the yield EXACTLY BECAUSE, as you illustrate, we don't have the term structure information (it gets to hairy, for example, even a parallel shift of your upward-sloping term structure by 100 bps will not bump the YTM by exactly 100 bps)

Due to this (i.e., the term structure informs the yield), the overwhelmingly common quiz assumption is "a flat term structure at x%" or "flat yield curve at x%" because, if the term structure is flat at x%, this is the special case where we can know the yield (YTM) must be x%. Under any other interesting term structure, that isn't true anymore.

(ugg, i wrote too much, now i have to worry if i made a careless mistake again )

David

 

[David Harper CFA FRM]

Mike,

Sorry to append, but since you identified the "circularity" I wanted to share the following supporting idea from Tuckman Chapter 6 (it sincerely took me three FRM seasons to get what he was saying) b/c i think you might appreciate:

Tuckman p 116 (emphasis mine):
"DV01, as defined in equation (5.2), equals the derivative of the price function with respect to the rate factor divided by 10,000. In the special case of this chapter, the rate factor is the yield of the bond under consideration. Hence,
DV01 = -1/10,000 * dP/dy (Equation 6.3)
It is important to emphasize that while equation (6.3) looks very much like equation (5.2), the derivative means something different here from what it meant there. Here the derivative is the change in the price given by equation (6.1) or (6.2) with respect to yield: The one rate that discounts all cash flows is perturbed. In Chapter 5 the derivative is the change in the price, as determined by some pricing model, with respect to any specified interest rate shift (e.g., a parallel shift in forward rates, a volatility-weighted shift in spot rates, etc.)."

What's he saying, in my paraphrase?
Duration, convexity and DV01 are single-factor models. That is, we estimate a price change based on changing (shocking) one thing. The term structure has many factors as each time horizon is a factor (six months, one year, etc). The only way to shock a term structure as one factor is to do it in parallel; e.g., all factors go up by 10 bps. We can do that to a zero/spot curve or a forward curve ... but it's easiest to do it to the yield (YTM) which is tantamount to the FLAT curve that corresponds to the non-flat term structure conditional on our bond cash flows. But ... and now i get back around to your point .... we are choosing one or the other, we are not choosing both the term structure and the yield. The yield (YTM) is flat and easy, and so our DV01 is always implicitly a yield-based DV01 (or YTM-based DV01) but it cannot simultaneously be a term structure (spot rate) DV01 b/c they are circular on each other. Although we used the zero term structure to infer the yield, we are, in a very real sense, once we've retrieved the yield, DISCARDING the term structure: a 100 bps shock to the yield (YTM) is not identical to a 100 bps parallel shock to corresponding NON-FLAT term structure.

What's the point? Our duration and DV01 are yield-based, not term structure based. A 100 bps shift, in using these, is not exactly the same thing as a 100 bps shift in the term structure!
(e.g., if we shock yield by +10 bps --> for a non flat term structure, the corresponding shift will be a little more or less. Or, if we shock the term structure by +10 bps --> the corresponding yield shift is a little more or less)
(sorry for length, it is my "Exhibit B" in support of the circularity idea)

Thanks, David

 

[wrongsaidfred]

This was extremely helpful.

Thank you for all of your help.

Just curious, and please forgive me if this opens a can of worms, but are we supposed to assume a flat term structure if nothing else is stated or can we only assume this if it is explicitly stated? I understand the YTM is implicitly a flat curve because it is constant, but I have seen old questions that included phrases like "rates drop" or "yields increase" and was not 100% sure exactly what we could infer from these types of statements. I appologize if this question is a bit vague.

My sincere thanks for all of your help this evening,

Mike

 

[David Harper CFA FRM]

Hi Mike,

The exam tends to use yield-to-maturity ("yield"). Even if not specified, the most common is (eg) "a drop of 1% in the yield," which means the implicitly flat yield-to-maturity. If spot rates or forward rates are used, they tend to be labelled as such. (It would be great if "yield" only referred to YTM, but it is sometimes un-strictly applied to a spot or forward rate). Because the exam can't make questions too complex, spot rates might be invoked to ask you to price a short maturity bond with a few rates, but multiple rates will betray it's not a yield!

But if there is only one yield given, best default assumption is "yield to maturity." Otherwise, you'll see the kind of assumption that John Hull makes in his questions; e.g., "a flat term structure of 4%" which, less precise authors will convey as "a flat yield of 4%" ... this is the most common, and as discussed, mostly moots the issue; i.e., the flat zero/spot rate term structure is the special case where a parallel shift in the yield of X bps is equal to a parallel shift in the zero term structure. I hope that helps, thanks for exploring this with me!

David

 

[wrongsaidfred]

It certainly does.

Thanks again,
Mike

 

[bhar]

Nice discussion - however from exam perspective - to keep things simple - Discounted bond < Par Bond< Premium Bond.

 

[David Harper CFA FRM]

I agree that, by obvious definition, Price(Discounted bond) < Price(Par Bond) < Price(Premium Bond) but I am unclear on what other simple rule this divined from above?

 

[BioNerd]

when i first look at this, i would think A is a coupon paying bond, so D <30; B is a zero-coupon bond, so D=30. To me that's what the question is trying to tell. So i would pick B without more thinking.

 

[David Harper CFA FRM]

BioNerd,

I agree, that's what the question is looking for. But Mike's original observation, in my opinion, exploits a two-part imprecision in the question. For example, to borrow from mine above, we can have:

• $100 price bond, at 3% yield, 30-year term, therefore coupon = 3.0%
• $70 price bond, at 4% yield, 30-year term, with coupon at 2.27% (annual pay); i.e., =-PV(4%,30,2.27%*$100,100) = $70

At 1% decline in yield gives:

• $100 bond increases to $122.40, a +$22.40 or +22.4% change; i.e., =-PV(2%,30,$3, 100) = $122.40
• $70 bond increases to $85.60, a +15.60 or +22.3% change; =-PV(3%, 30, $2.27, 100) = $85.60; i.e., less in both dollar and percentage terms!

I think it's avoided if the question inserts two qualifications: "assuming the bonds have identical original yields ... and price impact as a percentage"

I hope that's interesting!