# Understanding Fixed Income | Duration & Convexity

Updated: Nov 28, 2021

The duration of a bond measures the sensitivity of the bond price to changes in the interest rate. A higher duration bond is more sensitive to changes in interest rates and a lower duration bond is less sensitive.

Why are bond prices sensitive to changes in interest rates?

A bond is valued by discounting the bond's cash flows. For example:

1) If the appropriate discount rate is 10% what is the value of a bond that pays \$100 in one year? \$100/1.1 = \$90.91

2) If the appropriate discount rate is 20% the value of the same bond is \$100/1.2 = \$83.33

This example demonstrates that holding all else equal a rise in interest rates will decrease the value of a bond and a decrease in interest rates will raise the value of a bond. Like a seesaw, when one side is lowered the other is raised.

## Duration Measures

Macaulay Duration is best shown as an example. Suppose a 10 year 10% coupon bond has an appropriate discount rate of 10%. The Macaulay duration is calculated as shown below.

Modified duration = Macaulay duration /(1+r) = 6.7950/1.1 = 6.1446

Modified duration can be used to calculate the approximate change in the bond price for a given change in the discount rate.

Change in PV = - annul modified duration * change in discount rate

If for example, we increase the discount rate of the above bond to 11% the equation estimates a change of -6.1446*0.01 = -.06146. That means the bond will be worth \$1,000 before the change and \$1,000*(1-.06146) = \$938.54 after the change. In reality, the actual value after the change was \$941.11.

Approximate Modified Duration

In practice, a simplified formula can be used to estimate modified duration with fewer steps.

Where:

V- is the value after a decrease in interest rates

V+ is the value after an increase in interest rates

V0 is the original value

Delta YTM is the rate change used when calculating V- & V+

The approximate modified duration of the same bond above is 6.1534 calculated using a 1% rate shock.

While duration is a good general measurement of interest rate sensitivity it fails to account for the convexity of bond prices. Observe that if we plot out the price of our bond for different rates of r we find the following.

This graph illustrates that for the same change in interest rates, up or down, the effect on price is not linear. For example, the move from 26% to 21% increases the bond price by an amount more than the corresponding decrease from 26% to 31%. In order to account for this convexity, we need to make an adjustment

Using our bond we find the approximate convexity to be 52.8426.

The appropriate convexity adjustment is .5 * convexity * (change in YTM).

If we recalculate the 1% interest rate increase change using both adjustments we find that the estimated price is \$941.1077, much closer to the actual number \$941.11. So close

## The Utility of Macaulay Duration

What is the use of Macaulay duration? What does it tell us? The Macaulay duration tells us where the gains and losses associated with coupon reinvestment cancel out the gains and losses of market price changes. Consider the fact that if you purchase a bond and interest rates fall your bond will be worth more but you will have to reinvest the coupons you receive at a lower rate. Over time the gain in the price of the bond and the losses due to lower the reinvestment rate will cancel out such that total return is the same as before the change in interest rates, this time is the Macaulay duration. If the bond has been held for less than the Macaulauly duration then the gains and losses outweigh the coupon reinvestment, if the bond has been held for longer than the Macaulay duration then the coupon reinvestment outweighs the gains and losses.