# Duration

February 25, 2019

## Basics

$D=-\frac{d P}{d i}\cdot\frac{1}{P}=-\frac{d\ln(P)}{d i}$

Modified duration (sometimes abbreviated MD) is a price sensitivity measure, defined as the percentage derivative of price with respect to yield (the logarithmic derivative of bond price with respect to yield).

Wikipedia ~ Bond duration

As this is a measure of slope we can obviously take any linear combination of slopes from various lines (weighted average) to compute a Modified Duration for that sum of assets (or liabilities)

We can also apply the above derivative to the simple price calculation used in this exam: $\frac{w_t}{(1+i)^{t+0.5}}$ [t = 0, 1, 2,…] payments made during mid-point of year. $\sum w_t = 1$

\begin{aligned} P &=\sum \frac{w_t}{(1+i)^{t+0.5}}\\ \frac{dP}{di}&=-(t+0.5)\sum \frac{w_t}{(1+i)^{t+0.5+1}}\\ -\frac{dP}{di}&=(t+0.5)\frac{1}{1+i}\sum \frac{w_t}{(1+i)^{t+0.5}}\end{aligned}

The above formula represents the Modified duration, which is derived from the Macauly aka useless duration (it doesn’t measure duration at all):

$\mbox{Modified} = \frac{\mbox{Macauly}}{1+i}$

We can also calculate a effective duration (direct):

\begin{aligned} \mbox{effective} &= \frac{\Delta P}{\Delta i} \times \frac{1}{P}\\&=\frac{\Delta \ln\left(P\right)}{\Delta i} \end{aligned}

## Duration Purpose

To compute the interest rate risk margin, a duration and an interest rate shock factor are applied to the fair value of interest rate sensitive assets and liabilities.

MCT Guidelines – 2018

## Duration – Claim Liabilities

It’s important to realize that claim liabilities are generally (not always) assumed to be paid at the mid-point of the year as premiums are earned uniformly throughout the year. As such the average payment date will be the mid-point of any year or: t+0.5; this is consistent with present value calculations, but is more important here.

$\frac{1}{1+i}\cdot\sum \frac{(t+0.5)\cdot w_t}{(1+i)^{t+0.5}}$

Where

$w_t = \frac{\mbox{paid}_{t} - \mbox{paid}_{t-1}}{\mbox{paid}_{t=t_{0}}}$

## Duration – Unearned Premium Liabilities

It’s important to realize that premium liabilities are generally (not always) assumed to be paid at a decreasing rate throughout the year as unearned premiums (from the previous year) are more heavily weighted towards the beginning of the year. As such the average payment date will be the third-point of any year or: t+0.333; this is consistent with present value of premium liabilities calculations, but is even more important here.

$\frac{1}{1+i}\cdot\sum \frac{(t+0.3333)\cdot w_t}{(1+i)^{t+0.3333}}$

Where

$w_t = \frac{\mbox{paid}_{t} - \mbox{paid}_{t-1}}{\mbox{paid}_{t=t_{0}}}$

## Duration – Transformation

The exam includes materials related to transforming a mid-point payment calculation (1/2) to a unearned premium calculation based on third-point (1/3) for unearned premiums. One can easily adjust the Macaulay Duration by exactly -1/6 (1/3 – 1/2). This is because, the weighted average is all multiplied by the same factor: $\frac{1}{(1+i)^{-1/6}}$ so the only thing that matters is the fact that the duration weights are smaller by 1/6.

Macaulay’ = Macaulay – 1/6

Since Macaulay isn’t actually a measure of duration we can transfer this to the useful modified duration:

Macaulay’/(1+i) = Macaulay/(1+i) – (1/6)/(1+i)
Modified’ = Modified – (1/6)/(1+i)