Why Barbell Bond Portfolios Can Have Greater Convexity
Summary
The discussion explains convexity for a portfolio containing two zero-coupon bonds with different maturities. It derives portfolio convexity by taking the second derivative of total price with respect to yield and normalizing by portfolio price. The resulting expression is a price-weighted combination of the component bonds’ convexities, with each bond’s contribution weighted by its share of total value.
Because convexity increases nonlinearly with maturity in the setup, a portfolio split across shorter and longer maturities can have greater convexity than a bullet portfolio concentrated around an intermediate maturity, even when the portfolios are compared at a similar duration. The answer proposes plotting convexity against maturity to visualize this relationship. Its derivation uses zero-coupon bonds and a common yield, so actual portfolios with coupons, different curve exposures, or changing market conditions may require a more detailed analysis.
Key ideas
- Portfolio convexity is the price-weighted average of component bond convexities under the stated setup.
- The derivation uses two zero-coupon bonds with distinct maturities and a shared yield.
- Convexity grows nonlinearly with maturity in the example, supporting higher convexity for a barbell than a bullet.
- A plot of convexity against maturity can help visualize the portfolio comparison.
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# Why does a barbell portfolio have higher convexity than a bullet porfolio
# Why does a barbell portfolio have higher convexity than a bullet porfolio
I cannot quite understood absolutely why a barbell portfolio has higher convexity than a bullet porfolio.
I can easily understand how the parallel line represents duration but I cannot see what the curve of the convexity would look like. My guess is that this curve would start very high and then decrease to zero. If so this would explain why a barbell portfolio has a higher convexity than a bullet portfolio purely due to buying the very high yielding bond. I am also not sure whether this solution would be stable as the original bond price-yield curve may move causing changes in the mathematical derivatives.
Moreover I cannot see how convexity could be pictured using the yield curve, ie on the graph below, which I believe may be key to fully understanding convexity.
Please assume normal market conditions and "normal" bond curves, ie positive convexity (at least primarily).
## Answer by hmmmmm (score 1, accepted)
https://quant.stackexchange.com/a/40668
Assume that there are two zero coupon bond with maturities $N_1$ and $N_2$ with prices $P_1 = \frac{CF_1}{(1+y)^{N_1}}$ and $P_2 = \frac{CF_2}{(1+y)^{N_2}}$ respectively. If we construct a bond portfolio by purchaing one each of the two ZCB, the price of the portfolio is $P=P_1+P_2$. Now, the convexity of the portfolio is
$\begin{align} {Convexity}_p &= -\frac{d^2P}{dy^2}\cdot\frac{1}{P} \\ &=\frac{1}{(1+y)^2}\left[\frac{CF_1}{(1+y)^{N_1}}N_1(N_1+1)+\frac{CF_2}{(1+y)^{N_2}}N_2(N_2+1)\right]\cdot\frac{1}{P} \\ &=\frac{1}{(1+y)^2}\left[\frac{P_1}{P}N_1(N_1+1)+\frac{P_2}{P}N_2(N_2+1)\right] \end{align}$.
Notice that $\frac{P_1}{P}+\frac{P_2}{P}=1$
Now, plot convexity against maturity ($N$),
We may see that the convexity of the barbell portfolio (in blue line) is above the convexity of the bullet portfolio (in black line).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.