Is duration of a bond a convex function?

Question

I understand that in general, the NAV of a bond is a convex function.

However, I am not too sure if the same can be said for its duration.

Are there references on this? Thanks

Answer 1

The generic bond pricing function is

$$ PV = \sum_i^n c_iD(t_i)+D(t_n) $$

Convexity of PV01

Let's identify its duration with the negative of its first derivate, and let's set $D(t_i)=e^{-rt_i}$

$$ D\equiv-\frac{\partial PV}{\partial r}=\sum_i^nt_ic_ie^{-rt_i}+t_ne^{-rt_n} $$

A function is (locally) convex, if its second derivative is (locally) positive. The second derivative of the duration equals the third derivative of the bond pricing function (w.r.t. $r$):

$$ \frac{\partial ^2D}{\partial r^2}\equiv-\frac{\partial ^3 PV}{\partial r^3}=\sum_i^nt_i^3c_ie^{-rt_i}+t_n^3e^{-rt_n} $$

As $D(r)\geq 0 \forall r$, and $t_i\geq 0$ as well, this function is strictly positive on the whole domain. This result holds irrespective of the used rate definition, and it holds strictly for any $c\geq0$.

Convexity of the Duration

Let us now identify duration as

$$ D\equiv -\frac{\frac{\partial PV}{\partial r}}{PV} $$

i.e. (negative of) first derivative over present value. Then

$$ \frac{\partial ^2D}{\partial r^2}=\frac{\frac{\partial^3PV}{\partial r^3}}{PV}-3\frac{\frac{\partial PV}{\partial r}\frac{\partial^2 PV}{\partial r^2}}{PV^2}+2\frac{\left(\frac{\partial PV}{\partial r}\right)^3}{PV^3} $$

We know that

$$ \begin{align} O(PV)&=1,\\ O(D)=O\left(\frac{\partial PV}{\partial r}\right)&=T,\\ O\left(\frac{\partial^2 PV}{\partial r^2}\right)&=T^2,\\ O\left(\frac{\partial^3 PV}{\partial r^3}\right)&=T^3 \end{align} $$ But with some trial-and-error, we find that for positive coupons $c$, the $k$ derivative increase "slower" than $T^k$. At the moment, I cannot find a mathematical proof, but trial-and-error shows that:

$$ \frac{\partial ^2 D}{\partial r^2}\leq 0 $$ valid for all $c\geq 0$, all rates $r$ and any $n\geq 1$.

Hence, under this definition, duration is concave.

Answer 2

Recall that duration is defined as the average time to receive the cashflow, with the weights being the present values of the cashflows. So when interest rates rise very high, the long dated cashflows have very low weights, and the duration goes monotonically down and asymptotically tends towards the time of the first coupon. When rates go to zero, the duration is much higher. Hence , thinking of duration as a function of rates, we have a function which must be positively convex in the region of positive interest rates.

In the region of negative rates, I believe it is negatively convex because it is bounded above by the time of the last cashflow.