Negative Duration / Convexity

[dennis_cmpe]

What does it mean when a bond has negative duration? Similarly, for negative convexity?

Let's say duration was -5 and convexity was -.05.

What does this mean for the bond's price and its relationship to yields?

 

[David Harper CFA FRM]

Your (-5) would imply that a yield increase (decrease) of 1% corresponds to an bond price increase (decrease) of 5%;
i.e., the opposite of duration (+5) in the typical long position: 5 duration implies +1% yield --> -5% price (decline)
So, typically cited examples of negative duration are (i) short position in bond or (ii) long an interest only (IO) strip [b/c yield increase --> price increase]
Some will say a short or a leverage short (put on a bond) will give negative duration.

The convexity is harder, but it's a second derivative function...which IMO is the only way to really keep it straight.
So, a plain vanilla coupon bond always has positive convexity. This means: as yield increases, the slope of the tangent (to P/Y curve; i.e., the dollar duration) is increasing. In other words, the normal positive convexity is shown by a negative dollar duration than increases as yield increases (2nd derivative = rate of change in 1st derivative)
So, a negative -.05 convexity implies the rate of change in dollar duration is negative instead of positive. Visually at this segment (e.g., low yield on bond with embedded call), the curve is concave down rather than up.
What's this really mean? typically (positive convexity) duration is always understimating bond price, but at negative convexity duration will overestimate the price price based on linear approximation. So it's matters for the direction of the error created by the linear approximation.

It's a bit more complicated than this, owing to different definitions of duration. For example, I technically don't think a short bond position has a negative duration. This is my interpretation of Jorion (and agreement with my interpretation of him), which seems to be at odds with some other sources. I could be wrong. I'd love some help....because...
As (modified) duration = (-1) * dP/dY * 1/P.
A long position on a vanilla bond is always positive = (-1)*(-dP/dY)*(1/P) = (+)
But it seems to me a short position also has a positive duration because:
(-1) * (+dP/dY) * (-1/P)
For this reason, at least mathematically, I have had a hard time generating a negative Macaulay/modified duration because this duration is "infected" by price and the price seems to cancel back to (+) !!

This may be related to why dollar (value) durations are preferred in portfolios: you can just add/subtract the dollar durations.

David

 

[harifrm1980]

HI David,

A short call has -ve duration and -ve convexity. A short bond seems to be similar. As yield increases short seller will benefit. When I calculate modified duration for a continuously compounded zero, for short sale it comes out to be exactly opposite of long bond.
for long, P=F e^(-rt); dP/dr=-tFe^(-rt)=- DP; therefore D=t.
for short, (short seller pv) Ps=-F e^(-rt); dP/dr=tFe^(-rt)=- DP; therefore D=-t.. hence -ve duration
Visually shorted bond price yield curve is a mirror image about x-axis(yield), that would mean duration and convexity are negative.
I hope you can correct me with your usual explanation.

Jorion says negative convexity is dangerous because prices go down faster and rise slower. But it seems it should be good if one is short!!

thanks,
Hari

 

[David Harper CFA FRM]

Hi Hari,

I think this is why dollar duration is preferred in portfolios, because modified durations cannot be added. It took me a while to agree with Jorion, but I think he is being precise to show the option delta as having a mirror image (for the short) but not for the bond. I think you have a slight sign error in your second formula:

* I agree with your long, so D = t for long under continuous
* But for the short, you have: dP/dr=tFe^(-rt) because dP/dr = (-t)(-Fe^(-rt)) = t*F*e^(-rt). But this = t*(-P) = -tP. Note that both the long and short result in dP/dr = -tP, given that D=dP/dr*-1/P, both result in t=D.

So, mathematically, it seems to me that you would be correct about the directional impact of the true first derivative (dollar duration) but, becuase bond price includes (infects) the modified duration, the negative (-) in the short price position (-P) is canceling the negative in the first derivative. If not for (-P), I think you would have a situation just like the option delta...hope that help..David

 

[harifrm1980]

Hi David,

Since I like the subject, I found modified duration to be confusing to use because of the compounding rates involved. But your comment about using Dollar Duration clears awkward feeling like daylight to darkness.
However I have a bone to pick with the rest. It seems to me that you are saying that impact of shorting the bond is negative dollar duration but duration will still be positive. That seems a bit contradictory, but I understand. If you consider a Macaulay's Duration, which order of magnitude wise similar to modified duration, we will see that it is positive for long position because all cash flows are income to the bond holder; and the duration is weighted average of the cashflows. By the same token, for the short seller the cashflows are identical except all out going. I think you would interpret that duration which is just weighted average of payments is same, but the direction is of dollar duration is negative for shortseller. I would think the duration relative to other long positions in a portfolio should be considered negative then.

Actually, delta calculation for option is same whether one is short or long. For short, the delta is multiplied by a negative 1, because, to the seller, increase in price of underlying stock will result in decrease in net worth. The delta of an option is same, just that sign is changed based on the position the trader is taking. It seems to be no different than bond.

I further looked in Jorion's 5th edition FRM handbook, p24. He calculated the duration of a 3 bond portfolio, consisting of one short and 2 long bonds. The effect of short bond was to decrease the effective duration. So the short bond is implied to have a negative duration.

In my calculation, I distinguished between negative present value Ps, to the shortseller, from present value of the coupon paying bond which is always positive and incorporated a negative sign as a result and came up with -ve duration.

For the FRM exam, should the short bond have an absolute of duration or a duration negated by the direction of the position?.

Thanks, Hari.

 

[David Harper CFA FRM]

Hi Hari,

I could be wrong about the negative duration, I admit to < 100% confidence ... I am more comfortable that mathematically I cannot seem to produce a negative duration (in addition to never having read about it except for IO strips, i think?); as shown above, i think you missed an additional negative (the short position) and so, merely mathematically, the short duration has a positive sign also. Again purely on mathematical terms, this is because duration is not like delta. A short option has negative delta, in a way that is easier to visualize, b/c delta is a first derivative (i.e., go short and you can flip the price/call price chart and see the tangent line flip). But duration embeds, additionally, -1/P; such that the short (-P) adds a cancelling impact. So, where i am stuck is, i cannot mathematically find the "negative duration" wherease negative delta is natural.

Re: Macaulay duration, i can't really refute that. I can only ask, if I am long with Macualay duration 10 and you are short, then we mean I am roughly "10 weighted avg. years to receipt of cashflow." I agree with you, that means, you (the short) are 10 weighted average years to payment of cashflow; i.e., to outflows" Does that mean you are -10 or just "short with 10," I don't know?

I see p24 (thanks for pointer!) but notice, he has the short bond with modified duration of (+) 29. Then, to use in a portfolio in an additive way, he is "forced" to use dollar duration where the short position gives a negative dollar duration. To me, the negative dollar duration is more useful (and the meaning of a negative modified duration eludes me).

For the FRM exam, should the short bond have an absolute of duration or a duration negated by the direction of the position? For the exam, frankly it will be as i have said: you can assume modified durations will be positive; but, importantly, the short absolutely decreases the dollar duration (just as jorion has in the example). You are, of course, correct, that a short position will lower the dollar duration of a portfolio, we don't want to lose sight of that!

Hope that is helpful...David