Callable Bond Convexity

[misesorian]

I am new here and wasn't sure where else to post this question, so I apologize in advance if there was a better place to post.

Question is: Is a long position in a callable bond net long convexity?

I get that a callable has negative convexity at rates below the coupon rate (slightly less due to transaction fees).

Points to agree on. A long position in a straight bullet bond is long convexity and a short position in a call option is short convexity.

So is the combination of a long position in a straight bond and short a call (i.e. a long position in a callable bond) net long or net short convexity in the whole position?

 

[ShaktiRathore]

hi,
for rates above the coupon rate: the combination long position in a straight bond + a long position in a callable bond is positive convexity
for rates below the coupon rate: the combination long position in a straight bond + a long position in a callable bond is positive convexity+ negative convexity. the negative convexity will arise when the callable bond price will tilt towards the bond callable price so that it approaches a straight curve. if we assume it as a straight line(to be approximate) and add this straight line to positive convexity curve we will get again a positive convexity curve. So the net is positive convexity .
you have not mentioned the maturities of the bonds if they are not same its possible that net convexity might be different. If we assume the maturities are same as T where callable bond is anytime callable before time T, so suppose the net maturity of callable bond reduces to T' where T'<T
convexity of bond is proportional to T^2
so convexity of straight bond is proportional to T^2
and that convexity of callable bond is proportional to T'^2<T^2 so convexity of straight bond dominates the convexity of callable bond so that the net convexity should be approximately synonymous to convexity of the straight bond.

thanks.

 

[misesorian]

Thanks, but I'm not talking about a position in a straight bond + a callable bond. I'm just talking about a callable bond by itself.

The straight bond is long convexity.

A callable bond is the equivalent of a straight bond and a short call option. A short call option is short convexity.

So a callable bond is the combination of a straight bond (which is long convexity) and a short call option (which is short convexity). So my question is, does the long convexity nature of the straight bond component of a callable bond outweigh the short convexity nature of the short call option?

Stated another way, is a long position in a callable bond long or short convexity?

 

[ShaktiRathore]

hi there,
The callable bond yes is in fact a combination of a straight bond and a short call option. Call option in the sense that if the bond price rises above a certain callable price(exercise price) than the bond is called back by the borrower and whatever bond gains in form of price appreciation above the callable price is forgone by the investor.
It depends on the interest rates prevailing that is the nature of callable bond depends on the level of interest rates. If the interest rate is above the cutoff yield at which the bond price equals the callable price than the callable bond behaves like a normal straight bond with positive convexity. However when the interest falls below this cutoff yield the price rises above the callable price at which the bond is call back at the callable price so that the curve becomes flattened showing slight negative convexity. so for interest rate below this callable yield the callable bond had negative convexity and for above this yield the bond has positive convexity.
So a callable bond is a straight bond for above call yield and below this yield the straight bond ceases to exist. So the callable bond price is only affected by positive convexity nature. Because if bond does not exist below this callable yield there is no much affect of this negative convexity and thus finally the positive convexity seems to be a major component in the end.
e.g. a callable bond callable in 3 yrs @ 1150 with current price of 900 and maturity 5 yrs. prior to 3 yrs the bond behaves like a normal straight bond. But after 3 yrs if the price falls above the callble price of 1150 the bond is called back and the bond ceases to exist so now there is no bond exist and no point of negative convexity affecting the bond price. But we see that the bond price got affected by positive convexity prior to calling or during the life of the bond. So during its life the callable bond price is only affected by positive convexity and not negative convexity. so positive convexity is dominating over the negative. In the end we conclude that the long position in a callable bond is long convexity.
thanks

 

[misesorian]

Forget everything above. Do you agree that the writer of a call option is short convexity?

 

[David Harper CFA FRM]

Fabozzi agrees with Shakti, emphasis mine:

The price volatility characteristic of a callable bond is important to understand. The characteristic of a callable bond that its price appreciation is less than its price decline when rates change by a large number of basis points is called negative convexity.2 But notice from Exhibit 7–7 that callable bonds do not exhibit this characteristic at every yield level. When yields are high (relative to the issue’s coupon rate), the bond exhibits the same price/yield relationship as an option-free bond and therefore at high-yield levels also has the characteristic that the gain is greater than the loss. Because market participants have referred to the shape of the price/yield relationship shown in Exhibit 7–8 as negative convexity, market participants call the relationship for an option-free bond positive convexity. Consequently, a callable bond exhibits negative convexity at low yield levels and positive convexity at high-yield levels.
.... As can be seen from the exhibits, when a bond exhibits negative convexity, as rates decline, the bond compresses in price. That is, at a certain yield level there is very little price appreciation when rates decline. When a bond enters this region, the bond is said to exhibit “price compression.” --Fabozzi, Frank J. (2011-12-16). The Handbook of Fixed Income Securities, Eighth Edition.

The way I look at it: The call option writer is short gamma (i.e., Convexity * Price, aka dollar convexity). Percentage option (%) gamma is always positive, so a short option writer is always negative position gamma, but gamma tends to zero at both deep ITM and deep OTM. So, at high yields, this option is deeply OTM (same as Shakti's point) and the underlying bond's positive convexity totally dominates; at very low yields, I *think* technically per the deep ITM option the negative convexity (i.e., negative option gamma dominating underying non-call bond positive convexity) would need to tend to zero convexity. Consistent with Fabozzi's price compression, I think; or put another way, the slope (dollar duration, not duration) at very low yields, for the callable bond, is tending both toward zero but also with very little change; i.e., very little dollar convexity or d^2P/dy^2. So, i think as you move from higher yields to lower yields, you move from the classic positive convexity region (i.e., option deeply OTM) to classic negative convexity (i.e., option near ATM and ITM), which may "reverse direction" at extremely low yields such that you'd be approaching both ~ zero duration and zero convexity (neutral) at very low yields, into the price compression.

In summary, in going from high to low yields for callable bond, it looks like:

• positive convexity at high yield; option OTM
• negative convexity at low yield, as option gamma dominates underlying non-callable bond's natural positive convexity
• tending, finally to zero convexity as yield tends to zero, as p/y is ~ flat in the extreme (rate of change of 1st derivative = 0)

 

[ShaktiRathore]

Hi,
I agree with David. My thought process was like that the slope at the cutoff yield the slope instantly changes to zero and the slope becomes zero but if the slope changes slowly and tends towards zero than for low yields than there is bound to be negative convexity. And this slope finally becomes zero i.e. a straight line parallel to x axis.

thanks

 

[misesorian]

I agree with all that has been said. I'm just not sure if it answers my question.

I'm not asking whether the callable bond has positive or negative convexity... I'm not even sure that is a reasonable question since it certainly depends on the specific level of rates relative to the coupon rate.

I'm asking if you would consider the callable bond holder to be net long or net short convexity. Not whether the position has positive or negative convexity.

In other words, let's just take a straight bond that clearly only has positive convexity. The issuer is short convexity and the investor is long convexity. A simple example of the instrument in question having positive convexity, yet one side long the convexity and the other short it.

So with the callable bond, we are all in full agreement that at yields above coupon, positive convexity and yields below coupon, negative convexity.

The question is, is the holder of the callable bond long or short convexity? It is my thinking that the holder is net long convexity and the issuer is net short convexity. I get there by just thinking about the long benefiting from positive convexity and being hurt by negative convexity (or when net convexity between the bond and option work against his overall position). And of course vice versa, the issuer benefits when convexity of the entire position "goes down" (i.e. at low yields the price doesn't rise above the call price) and such could be thought about as being short convexity. And at rates above the coupon, the price doesn't decline as fast as it otherwise would which hurts the issuer should it what to buy the instrument back in the open market (for a reason other than cheaper refinancing options).

 

[David Harper CFA FRM]

Hi misesorian,

I tend to view, for example, Fabozzi's text as tantamount to an answer (although, i could be wrong of course).

In the case of a vanilla bond, which has everywhere positive convexity, I view the concept of an investor who is "long convexity" due to fact that, where C = convexity and P = bond price, +C * +P = + dollar convexity; i.e., long due to (+). The issuer is "short convexity" due to(-P) such that -P*+C = negative dollar convexity; or, perhaps it is slightly better but to the same effect to use: -quantity * +price = -Q*+P = -V, to represent the short, so we could say a short position in a vanilla bond is short convexity due to -V * +C .

dollar durations and dollar convexity, where dollar convexity = convexity * bond value, are used to aggregate and hedge, so w.r.t. the above discussion, I'd interpret the above, for the callable bond

• at high yields, long callable bond = +Q* +P * (+C) = "long" convexity
• at low yields, long callable bond = +Q* +P * (-C) = negative dollar convexity = "short convexity"

also: I don't know what to make of your use of "net" long, i don't know what is means here. The above negative convexity of callable bond is already "net" of two components. But if net means to simplify the convexity's variance w.r.t yield, then i think it's a misunderstanding of something more basic (imo). From my perspective, the metric that matters is dollar convexity (the true second derivative, d^2P/dy^2) and our research-discussion above suggests it is variant to the yield, thanks,

 

[Turner737]

hi,

what exactly does it mean to be "long" or "short" convexity?

my interpretation...

long: ones position value is non-linear in regards to change in yield. i.e. when one is long for larger increases in yield they are in essence stopped out of how low the position value can go because of the convexity/non-linearity(the angle of the line for the highest yields approaches parallel to the x axis implying a floor price). concurrently, for larger decreases in yield there is a higher increase in price and the angle of the lowest yield points approaches parallel to the y axis implying no matter how much lower the yield goes (implying a yield floor).

per a vangaurd paper i found:

"The bond price decrease resulting from a large interest rate increase will generally be smaller than the price increase resulting from an interest rate decline of the same magnitude"

per the above conversation i am looking to understand what it means to be long vs short from the investor/issuer(seller) perspective. as in are there protections one gets at certain yield/price relationships for being long or premiums one must pay at the opposite yield/price relationships?

i get the idea of convexity, in a general sense, but i still dont feel like i "get it" where i would be comfortable speaking in a conversation about being long/short and the implications.

thanks,

Matt

 

[Mark W]

Hi Matt,

I'll try and be brief and hopefully it will help you out...

Being long convexity in a bond sense means that when rates rise (fall) a long vanilla bond holder loses less (gains more) than predicted by duration alone. This is because, loosely speaking, duration is simply a linear approximation of the relationship between price and yield.

Re. a price floor...I'm going to say this non technically as I don't have the time to be precise...but I assume you refer to the visual 'flattening' of the curve along the x-axis? This floor is really zero as if rates rose sufficiently far then a vanilla bond would be effectively worthless (try it in Excel). Most charts keep the diagram within reasonable parameters thus giving the impression of a non-zero floor...but if you zoomed out enough (expanded the yield/x-axis further out) the floor would be zero i.e. the price will tend to zero.

So convexity 'protects' you compared to duration...but that's really a bad thing to say as duration is a simple linear approximation and in reality we can get a zero price if yields rise far enough. A long bond investor always benefits from convexity per the above (we lose less than duration predicts when rates rise and gain more when rates fall) - but we pay for behaviour when we purchase the bond.

Hope this makes sense - thanks - Mark

P.S. A final point, the tending to zero of the price is slow, even putting yield to 1,000% I cannot make the bond price equal quite zero!

P.P.S. Caveat: I just realised I didn't read the above posts before posting myself - apologies!

 

[Turner737]

Very useful, and all intuitive which is how I generally like thing to be.

Thank's for clarifying and simplifying.

Matt