### Key Concepts

 Hull Series : Duration and Convexity
Written by FemiByte
Tuesday, 29 September 2009 12:36

# Duration

Intuitively, the duration of a bond is how long on average the holder of the bond has to wait before receiving cash payments. Formally the duration is the percent change in a bond's price with respect to yield. It measures the sensitivity of the bond's price to interest rate movements.

The price of a bond B is given by

$B=\sum^n_{i=1}c_ie^{-yt_i}$

The duration is given by

$D=\frac{\sum^n_{i=1}t_ic_ie^{-yt_i}}{B}$

We can rewrite D as

$D=\frac{1}{B}\sum^n_{i=1}t_ic_ie^{-yt_i}$

=>

$BD=-\frac{dB}{dy}$

The negative of the 1st derivative of the bond price

$\frac{dB}{dy}=-BD$

is known as the Dollar Duration (DD) and is the absolute change in the bond's price with respect to the yield.

## Modified Duration

If y is expressed with a compounding frequency of m times per year, instead of the continuous assumption as above, then we have

$\frac{dB}{dy}=-\frac{BD}{1+y/m}$

We then define the modified duration as

$D^*=\frac{D}{1+y/m}$

## DV01

Risk is often measured as the dollar value of a basis point (DV01):

$DV01=DD * \Delta y=DD * 0.0001$

where 0.0001 repreesents 1 basis point or one-hundredth of 1 percent.

# Convexity

The convexity is a measure of the sensitivity of the duration of a bond with respect to a change in interest rates. It is a measure of the 2nd derivative of the bond price with respect to the interest rate.

The convexity is given by

$C=\frac{1}{B}\frac{d^2B}{dy^2}=\frac{\sum^n_{i=1}c_it^2_ie^{-yt_i}}{B}$

We can obtain the Taylor expansion for the change in the price of the bond, which is:

$\Delta B=-[D^** B](\Delta y)+1/2[C*B](\Delta y)^2 + \cdots$

Last Updated on Friday, 01 January 2010 22:36