# Modified Duration

Modified duration is a measure of a bond price sensitivity to changes in its yield to maturity. It is calculated by dividing the Macaulay’s duration of the bond by a factor of (1 + y/m) where y is the annual yield to maturity and m is the total number of coupon payments per period.

Duration is a measure of interest rate risk of a bond, the risk of decrease in bond price due to increase in market interest rates. In general, the degree to which bond price moves due to a change in yield i.e. interest rate is directly proportional to the time to maturity. It means the longer the bond cash flows are stretched the more pronounced the price movement is. Macaulay’s duration measures the weighted average time till the bond cash flows. Modified duration adjusts Macaulay’s duration so that it can be used to estimate the price movement given a change in yield.

Modified duration measures the change in price assuming that the change in price for an increase or decrease in yield is the same which is not the case in reality. The change in bond price with reference to change in yield is convex in nature. A convexity adjustment is needed to improve the estimate for change in price.

## Formula

If the Macaulay duration value is available, modified duration can be easily calculated using the following formula:

$$ Modified\ Duration=\frac{Macaulay\ Duration}{(1+\frac{y}{m})} $$

Where y is the annual yield to maturity and m is the number of compounding periods per year.

Macaulay duration can be calculated using the following formula:

$$ Macaualy\ Duration=\frac{{PV}_1}{PV}\times T_1+\frac{{PV}_2}{PV}\times T_2+...+\frac{{PV}_n}{PV}\times T_n $$

Where PV_{1}, PV_{2} and PV_{n} refer to the present value of cash flows that occur T_{1}, T_{2} and T_{n} years in future and PV is the price of the bond i.e. the sum of present value of all the bond cash flows at time 0.

The unit of Macaulay’s duration and the modified duration is the same as the units in which maturities are entered. For example if we enter the time period in months, we get the monthly duration, which can be annualized by simple multiplication with 12.

### Approximate modified duration

Approximate modified duration can be estimated using by taking the difference between the price when interest rates increase and the price when interest rates decrease divided by 2 times the product of change in yield and the base price:

$$ Approximate\ Modified\ Duration=\frac{P_d-P_i}{2\times P_0\times deltaY} $$

Where P_{d} is the price after a decrease in yield, P_{i} is the price after an increase in yield, P_{0} is the base price i.e. before any increase or decrease in yield and deltaY is the change in yield.

Given a modified duration value, an approximate change in bond price given a change in yield can be worked out using the following formula:

$$ \%\ Change\ in\ Bond\ Price=-D\times deltaY $$

Where deltaY is the change in yield.

## Example

You have a $1,000 par value 6%-annual coupon bond matures in 2 years yielding 6.2%. Calculate the bond’s modified duration and expected percentage change in bond price given a 0.5% decrease in yield.

We first need to calculate the Macaulay’s duration, which is the average maturity of the bond cash flows weighted based on their relevant contribution to the present value of the bond.

$$ Macaualy\ Duration=\frac{\frac{$60}{{(1+6.2\%)}^1}}{\frac{$60}{{(1+6.2\%)}^1}+\frac{$1,000+$60}{{(1+6.2\%)}^2}}\times1+\frac{\frac{$1,000+$60}{{(1+6.2\%)}^2}}{\frac{$60}{{(1+6.2\%)}^1}+\frac{$1,000+$60}{{(1+6.2\%)}^2}}\times2=1.94 $$

Modified duration equals Macaulay’s duration divided by (1 + y/m):

$$ Modified\ Duration=\frac{1.94}{(1+\frac{6.2\%}{1})}=1.83 $$

Modified duration works out to 1.83 which means the bond prices increases (decreases) by 1.83% given a 1% decrease (increase) in bond price.

The percentage bond price change given a 0.5% decrease in yield equals 0.915% (i.e. -1.83×(-0.5%)).

### Weakness of Modified Duration

While modified duration is better than Macaulay's duration, it has a few serious weaknesses:

- It assumed that the bond price changes equally for any increase or decrease in yield, which is not the case. Convexity adjustment is needed.
- t measures sensitivity with reference to the yield to maturity and not the overall structure of yield curve. Effective duration is a better measure than modified duration.