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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

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    $\begingroup$ Typically we say that X is a convex function of Y (or simply X is convex in Y). Here X is duration, I am not sure what Y is. $\endgroup$
    – nbbo2
    Aug 11 at 9:44

2 Answers 2

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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.

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  • $\begingroup$ Note that currently the proof only goes through assuming that the $c_i$ show no dependence on $r$; this assumption does not hold for a large class of bonds such as MBS. In fact, you can get bonds with negative duration in the MBS market but $D$ is bounded below by 0 in your formulation. $\endgroup$
    – Sharad
    Aug 11 at 16:07
  • $\begingroup$ Yes, it requires a fixed rate bond. $\endgroup$ Aug 11 at 19:20
  • $\begingroup$ MBS bonds are fixed-rate but the cash flows depend on prepayment rates which are a function of $r$. Another example of a fixed-rate rate dependent bond would be GSE callable debt. $\endgroup$
    – Sharad
    Aug 11 at 19:39
  • $\begingroup$ Put differently: We are assuming a vanilla bond here, of course. $\endgroup$ Aug 13 at 10:30
  • $\begingroup$ Thanks, but the duration I know is not exactly the one you mentioned here. Should it not be the percentage change per interest rate change, instead of the absolute change of NPV? So to calculate the duration,I also need to divide your D by NPV, which also is convex and decreasing. I am not sure how to determine which of the effect is stronger. $\endgroup$ Aug 14 at 7:56
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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.

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