# 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).

• I should add that I know convexity is additive in a portfolio – Permian Jul 23 '16 at 15:18
• To keep it simple consider only ZCB of 2, 3 and 4 years. For ZCB the Convexity is proportional to $T^2$. It is obvious that $\frac{1}{2}2^2+\frac{1}{2}4^2 = 10$ is bigger than $3^2 = 9$. QED ;) – Alex C Jul 23 '16 at 15:40
• In other words I have just proved to you that a barbell portfolio of 2 and 4 year zero coupon bonds has the same duration (3 years) but a higher Convexity than a bullet portfolio of 3 year bonds. The general case is left to the reader. – Alex C Jul 24 '16 at 23:13
• @AlexC Put this as answer and I will accept – Permian May 9 '18 at 17:55

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 ($N$),