A contradictory concept on Convexity of Bonds?

[monsieuruzairo3]

Dear David,

I read in Hull that higher the coupon payments, greater is the convexity of Bonds as it immunizes against movement in the market yields. Correspondingly bonds with payments centered around a single time (like Zero coupon bonds) have lower convexity.
Key insight: Zero coupon bonds have lower convexity
An article http://www.investopedia.com/university/advancedbond/advancedbond6.asp
tells that zero coupon bond has the highest convexity while also stating "A bond with greater convexity is less affected by interest rates than a bond with less convexity".

I believe something is amiss here. Can you clear the confusion of this?

Best
Uzi

 

[David Harper CFA FRM]

Hi Uzi,

I extracted the Hull below. Frankly, I find some of text of both (investopedia and Hull) to be imprecise, but this is the nature of qualitative, comparative statements about duration/convexity. It can be hard to describe in words, whereas the math is exact.

In my opinion, best lens through which to view convexity is that convexity is the weighted average of maturity-squares (M^2) of the bond; analogous to duration as the weighted average maturity of the bond. I attached my copy of Tuckman 6.3 if you want to see this in action: https://www.dropbox.com/s/6zvzhj6pmhql1aa/Tuckman 6.3.xlsx ; as this is semi-annual, each maturity, T, is effectively squared with T*(T+0.5), in continuous compounding, it would be T^2. This is, to me, the best conceptual lens, because we can see, for example, the (continuously compounded) convexity of a 10 year zero-bond is 10^2; of a 30 year zero-coupon bond is 30^2. Hence, the most basic idea that convexity increases with the square of maturity.

Then, further, a coupon bond can be treated (decomposed into) a portfolio of zero coupon bonds; with the portfolio's convexity simply the weight of components.

Given that, for example:

• Investopedia's graph is good: higher convexity is associated with higher curvature. But we would not generally say, out of context, like the author does, "A bond with greater convexity is less affected by interest rates than a bond with less convexity." He's means the higher convexity bond will decrease less if yields rise (see graph), but we would not say it that way due to the confusion
• Hull probably means, what Tuckman says, that a barbell portfolio has greater convexity than a bullet portfolio with equivalent duration; e.g., barbell = 2-year and 30-year bonds, with same portfolio duration as a bullet portfolio of 9-year bonds. Because the 30-year disproportionately impacts convexity; ie., 30^2. So, this is true simply because the barbell requires a long-maturity bond component that has greater convexity. So, I don't see a contradiction if i interpret these two statements, which without context, in my opinion, each is imprecise and even misleading.

I hope that helps, here is Hull:

"The convexity of a bond portfolio tends to be greatest when the portfolio provides payments evenly over a long period of time. It is least when the payments are concentrated around one particular point in time. By choosing a portfolio of assets and liabilities with a net duration of zero and a net convexity of zero, a financial institution can make itself immune to relatively large parallel shifts in the zero curve. However, it is still exposed to nonparallel shifts." -- Hull

 

[ShaktiRathore]

I can let you understand the relation of T^2 as David explains that convexity varies with square of maturity,
Price of Z coupon bond with maturity T=P=F/(1+y)^T
Duration= D= (1/P)*(dP/dy)=(1/P)*(-T)*(F/(1+y)^T+1)....duration formula
convexity= dD/dy=(1/P)*(d^2P/dy^2)=(1/P)*(T)*(T+1)*(F/(1+y)^T+2)...convexity id rate of change of duration
convexity=((1+y)^T/F)*(T)*(T+1)*(F/(1+y)^T+2)...putting Price of bond
convexity=(T)*(T+1)*(1/(1+y)^2)
from above we see that convexity of Zero coupon bond varies directly as T(T+1) ~T^2 to be approximate. Thus convexity increases with square of maturity as a matter of fact presented by David.
Barbell has more distributed payments at extremes while bullet has one single point of pay. As David said the Barbell has greater convexity than a bullet one as is evident COnvexity of bullet=30(30+1)~930 whereas of bullet pay of maturity 9 yrs is 9(9+1)~90 so barbell has greater overall convexity.
Graph explains that high convexity is less affected by interest rates high convexity is associated with left part of the graph explains the equation price change= -duration*yield+convexity*yield^2 for a given duration a high convexity nullifies the negative effect of increase in yield the higher the convexity the more nullification from positive side of convexity.

thanks

 

[monsieuruzairo3]

Thanks now lets suppose we have a 10 year zero coupon bond and a 10 year coupon paying bond(semi-annual payments). Now we know that zero coupon bond has higher interest rate risk as it is more affected by rate movements, but we say that Zero coupon HAS HIGHER CONVEXITY according to which Zero coupon bond should be LESS AFFECTED by interest rate movements. How do you explain the contradiction. Hope my point is clear.

 

[David Harper CFA FRM]

monsieuruzairo3

While the first statement is specific and true if we make assumptions (" Zero coupon HAS HIGHER CONVEXITY" ... i.e., compared to an coupon-bearing bond with the same maturity and assuming same theoretical zero rate curve for discounting), as I mentioned above, the second statement is ambigious ("Zero coupon bond should be LESS AFFECTED by interest rate movements."); in my opinion, strictly speaking, it returns a FALSE (is how i reconcile!).

If you look at the investopedia graph, which is accurate enough (your link at http://www.investopedia.com/university/advancedbond/advancedbond6.asp ), what the author means, I think, is:

• If you keep in mind that higher convexity --> more curvature (relative to the linear tangent), when the yield increases, the bond price goes down less (as the convexity is higher) or "is less affected"
• However, this is not true for a yield decreases: when the yield decreases, the price is higher than otherwise (with more convexity, relative to linear tangent); i.e., the bond price goes up more or "is more affected"

So, we might say "with greater convexity, the price is less affected for a yield increase, and more affected for a yield decrease" but I would not say that or even try to defend that statement, that's merely (to me) an improvement. "Affected" is imprecise in this context, I am not sure it helps even with a comprehension of convexity, thanks,

 

[ShaktiRathore]

To prove it mathematically that zero coupon bond has less sensitivity to interest rate than coupon bearing bond,
convexity of Z bond=C1 and convexity of coupon bond of same maturity and same discount curve=C2
we know that C1>C2...1
Duration of Z coupon bond=D1 and duration of coupon bond of same maturity and same discount curve=D2,D1>D2...2
now for a change in interest rate two cases:
i) change in price of coupon bond when interest rate shift upwards by +i= -D2*i+.5*C2*i^2
lets put numbers to see results, D2=5,C2=100,i=2%=>change in coupon bond price=-5*.02+.5*100*.02^2=-.1+.02=-.08=-8%
change in price of Z bond, D1=6,C1=150=>change in Z bond price=-6*.02+.5*150*.02^2=-.12+.03=-.09=-9%
ii) change in price of coupon bond when interest rate shift downwards by -i= -D2*i+.5*C2*i^2
lets put numbers to see results, D2=5,C2=100,i=-2%=>change in coupon bond price=-5*(-.02)+.5*100*(-.02)^2=.1+.02=.12=12%
change in price of Z bond, D1=6,C2=150=>change in Z bond price=-6*(-.02)+.5*150*(-.02)^2=.12+.03=.15=15%
we see from above two results that validates what David has said above,
when the yield increases, the bond price goes down less (as the convexity is higher); i.e., "is less affected": that is when yield increases by 2% the Z Bond price goes down less by 9% as compared to when the yield decreases i.e. goes down by 2% the Z coupon bond price goes up by as much as 15% this validates when the yield decreases, the price is higher than otherwise (with more convexity, relative to linear tangent); i.e., "is more affected".
Thus we see that coupon bind on an average earned -8%+12%/2=2% whereas Z bond earned on average -9+15/2=3% on average thus Z Bond is more immune to losses that could arise by interest rate movement and always end on positive side of it anf thus overall interest rate affects negatively less the Z bond than coupon bonds.
Thus Z Bond is more sensitive to interest rate downshift as compared to when interest rate goes up. Also Z Bond seems to be more sensitive to interest rate movement as compared to coupon bearing bond because Z price changes much more in both the cases (9%>8% and 15%>12%) as compared to coupon bearing bond.
thanks

 

[flex]

hi, @David Harper CFA FRM.

Generally, WA-convexity calculation concept is a pretty clear. it's also approved basic (math) derivation of PV's statement.

But i'm confused by mismatching 'WA-approach' result (.5*[30^2 ]*bps^2, sorry for short) & 715.2 answers.
Any comment. thx, flex