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# Convexity in Bonds

#### Sunil Natarajan

##### Credit Analyst
Hi David,
In FRM Handbook Ch-13, Phillipe Jorion has mentioned that bonds always have positive convexity.In options the convexity(Gamma) can be both positive and negative. Long position in an option has positive Gamma, while short position in an option has negative gamma.
I wanted to know whether bonds irrespective of long or short position always have positive convexity. I thought that long position in a bond would have positive convexity and vice versa. And also a call option in a bond creates a negative convexity for a bond holder and a put option on a bond does create a negative convexity position for an issuer.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Sunil,

It's a *truly* astute observation. There is a subtle, superficial discrepancy between option gamma and bond convexity that manifests in Ch 13 (as I look at it, I only noticed this as you've pointed it out)

Absolutely a bond can have negative convexity. Jorion means a plain bond without embedded derivatives will always have positive convexity (and, yes, IMO he also means a short plain vanilla bond will always exhibit positive convexity). To achieve negative convexity, you only need to get the price/yield curve to "bend back" (2nd derivative < 0) at low yields with a callable bond or at high yields with a puttable bond. Last year, the FRM assigned Tuckman's chapter 22 on mortgage-backed securities and MBS are good examples of classically negative convexity. At low yields, refinances are akin to callable bonds. At high yields, defaults cause prepayments which act (sort of) like puttable bonds. (Sections of the price/yield curve can be variously concave/convex.)

But in the "bond context" duration/convexity are instrument-specific; the statements hold regardless of long/short. UNLIKE Jorion's option gamma discussion, which gives rise to the superficial discrepancy.

The reason is due to the definitional (semantic) difference between gamma and convexity: gamma is the second-derivative (i.e., the slope of the tangent line), convexity is not exactly the 2nd derivative; it's is the 2nd derivative divided by bond price. (Just like duration is, contrary to what many people think, not the slope of the tangent line. It's the slope divided by, or "infected" with, bond price). Because convexity divides by price, in the short position, you do in fact (just like the option) switch to a negative slope (2nd derivative) but it is canceled by the negative price of the short position!

Put another way, if convexity were defined as gamma (as purely 2nd derivative), those charts in Jorion Ch 13 would apply to bonds, too, and a short bond would similarly have negative gamma.

I notice Jorion writes gamma is "similar to" the concept of convexity, so as usual, he is precise. The difference is subtle, in summary:

* As Jorion shows, gamma is a function of long/short
* But due to canceling effect, duration/convexity are features of the instrument (not functions of long/short). Plain vanilla will always give positive convexity, but generally adding embedded options will introduce negative convexity (for portions; again, a bond can be positively convex passing thru an "inflection point" to negatively convex)

I hope that's not too much, I think your observation is very interesting...David

Regards,
Sunil

#### fashepard

##### New Member
David,

The formula for convexity in your slides has two (change Y)^2 one in the measure and one in the adjustment. Is it correct that these can be different numbers?

Frank

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Frank,

Yes, as Fabozzi says, the measure is not meaningful per se.

I have a slide near that, which compares the two measures, one with (2) in the bottom and one without.
And then: the adjustment is either scaled or not. So, they end up in the same place.

Tuckman's approach is: exclude (2) in denominator of measure, then use (1/2) in the adjustment.
But is the same as include (2) in the measure and drop the (1/2) in adjustment

David

#### Sunil Natarajan

##### Credit Analyst
Hi David,
Holding yield constant , the bond with lower coupon would have higher duration and greater convexity.
But I read somewhere that by keeping both yield and duration constant, the bond with lower coupon has lower convexity.Convexity is a measure of dispersion of cash flows.Could you please explain the same.
If possible could you upload an excel working for this if possible.

Regards,
Sunil

#### ahnnecabiles

##### New Member
Hi David,

Regarding the convexity adjustment where per Tuckman, we drop 2 in the denominator and include 1/2 in the adjustment, and the other one where we have 2 in the denominator and drop 1/2 in the adjustment, if the problem is silent, like, it gives the values of duration and convexity and we are asked to compute for the approximate price of the bond, what do we assume?

Thanks!

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@ Chinquee: it shouldn't matter, you shouldn't be asked about the convexity measure (the meaningless interim step). I suppose it would be safe to follow Tuckman and compute the convexity "metric" without a (2) in the denominator; and call that convexity. Then just make sure the (1/2) is used to estimate price change.

David

#### Daswani

##### New Member
Can the convexity adjustment formula be simplified to [ (V+) + (V-) - (2Vo) ] / (2Vo)

I don't follow why the convexity measure has the change in yield squared in the denominator and then the convexity adjustment takes it out.

Couldn't this whole thing just become ( [(V+) + (V-)] / 2Vo ) - 1

Either my math is horrible (which it definitely may be) or I'm missing something.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
They are typically different. I agree with you if the shock = ultimate yield change you are estimating. But, more typical is something like: use 10 or 20 basis points for the shock, then use 1% (100 basis points) for the convexity adjustment. So like: [V+ - V- - (2V0)]/[2*V0*0.001)^2 for Convexity measure, then [Convexity measure]*(1%)^2 * 100.

I guess you have a point if, for example, both are 100 bps, then (1%)^2 cancels, but then your convexity adjustment is only good for one yield change. I sort of see you point, maybe this is just to given the adjustment flexibility.

David

#### chirania

##### New Member
Subscriber
David, Can you please validate following observation.
For VAR adjustment for gamma(2nd order) it could be an altogether different story. A positive gamma (i.e. a long call/put and long plain vanilla bond) increases value of the derivative(i.e. option, bond) , hence reduces risk. Hence the adjustment is negative resulting in less VAR. On the other hand a negative gamma (i.e. a short call/put and short plain vanilla bond) decreases value of the derivative thereby increasing risk resulting in higher VAR. Note that for VAR we have to use delta/gamma ONLY we cannot use duration/convexity as they may not fit directly in the delta-normal 2nd order (which includes gamma) formula.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @chirania Yes, I agree with "A positive gamma (i.e. a long call/put and long plain vanilla bond) increases value of the derivative(i.e. option, bond) [david: given percentage gamma is positive, a positive position gamma implies a long position], hence reduces risk. Hence the adjustment is negative resulting in less VAR. On the other hand a negative gamma (i.e. a short call/put and short plain vanilla bond) [david: given percentage gamma is positive, a negative position gamma implies a short position] decreases value of the derivative thereby increasing risk resulting in higher VAR."

However, in regard to "Note that for VAR we have to use delta/gamma ONLY we cannot use duration/convexity as they may not fit directly in the delta-normal 2nd order (which includes gamma) formula" I do not really agree: duration/convexity in the bond context uses the same Taylor series approximation (first and second order, in your terms) that delta/gamma use in the option context; i.e., duration/convexity do utilize the Taylor fist-second order [truncated] approximation.

In mathy terms, Taylor truncated to only two terms says: df = δf/δS*δS + 0.5*δ^2f/δS^2*dS^2 + ....
In the option context, that is df = ΔS*delta + 0.5*gamma*ΔS^2
• ΔS*delta is represented by the blue line. Consequently, ΔS informs df (change in option value) obviously:
• For a long put position, risk is +ΔS with linear impact: -df = +ΔS*(-delta)
• For a short put position, risk is -ΔS with linear impact: +df = -ΔS*(-delta)
• But 0.5*gamma*ΔS^2 is always additive in the Taylor Series because (ΔS^2) is positive regardless of up/down stock price
• For a long put position, that implies the positive gamma terms mitigates the negative delta term (-df); i.e., reduces risk
• For a short put position, that implies the positive gamma terms exacerbates the postive delta term (+df); i.e., increases risk

... which includes a graphic that comports with you statement:

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