By

Konstantin Vasilev Member of the Board of Directors of Cbonds, Ph.D. in Economics

Updated July 1, 2023

**What Is Modified Duration?**

Modified duration is a mathematical expression that quantifies the change in a security’s value when there is a variation in interest rates. It is based on the inverse relationship between interest rates and bond prices. This formula is utilized to assess the impact of a 100-basis-point (1%) fluctuation in interest rates on the price of a bond.

Modified duration calculates how much a bond’s price will change in response to changes in interest rates. It is derived from the weighted average time until the bond’s cash flows are received. By considering the number of coupon periods and the present value of each cash flow, the modified duration formula determines the bond’s price sensitivity to interest rate movements.

## Formula to Calculate the Modified Duration

To determine the modified duration, investors can divide the Macaulay duration by the sum of 1 and the ratio of the bond’s yield-to-maturity to the number of coupon periods per year.

*Modified Duration = Macaulay Duration / (1 + YTM/n)*

Modified duration builds upon the concept of Macaulay duration, enabling investors to assess how sensitive a bond is to changes in interest rates. Macaulay duration computes the average time it takes for a bondholder to receive the bond’s cash flows, considering their respective weights. To calculate modified duration, one must first calculate the Macaulay duration, which involves multiplying the time period by the periodic coupon payment and dividing the result by 1 plus the periodic yield raised to the time to maturity.

## Example of Modified Duration

Let’s say you have a bond with a modified duration of 4.5 years. If interest rates were to increase by 1%, you can estimate that the bond’s price would decrease by approximately 4.5%.

For instance, if the bond’s current price is $1,000, a 1% increase in interest rates would lead to a potential price decrease of $45 (4.5% of $1,000). Therefore, the new estimated price of the bond would be $955 ($1,000 - $45).

Similarly, if interest rates were to decrease by 1%, you can estimate that the bond’s price would increase by approximately 4.5%. In this case, the bond’s price would potentially rise to $1,045 ($1,000 + $45).

## Example of How to Use Modified Duration

Suppose you are considering an investment in a bond with a modified duration of 5 years. The bond’s current price is $1,000, and its yield-to-maturity is 4% (0.04 as a decimal). You anticipate that interest rates will decrease by 0.5%.

To estimate the potential price change of the bond due to the expected interest rate decrease, you can use modified duration. The formula to calculate the percentage change in bond price is:

*Percentage Change in Bond Price = - Modified Duration × Change in Yield*

In our example, the change in yield is -0.005 (0.5% decrease). Let’s calculate the percentage change in bond price:

*Percentage Change in Bond Price = -5 × (-0.005) = 0.025 or 2.5%*

According to the calculation, you can expect the bond’s price to increase by approximately 2.5% if the interest rates decrease by 0.5%.

To estimate the new price of the bond, you can apply the percentage change to the current price:

*New Price of Bond = Current Price × (1 + Percentage Change in Bond Price)*

*New Price of Bond = $1,000 × (1 + 0.025) = $1,025*

Based on the estimated price change, the bond’s new price would be $1,025.

Using modified duration allows you to estimate the potential impact of interest rate changes on bond prices, helping you make informed investment decisions.

## What Can Modified Duration Tell You?

Modified duration is a crucial metric portfolio manager, financial advisors, and clients utilize to assess the average cash-weighted time until a bond reaches maturity. It plays a vital role in investment selection as it indicates the potential price volatility of bonds, assuming all other risk factors remain constant. The calculation of duration takes into account various bond attributes such as price, coupon, maturity date, and interest rates.

There are several key principles associated with duration that are important to consider. Firstly, as the maturity period of a bond lengthens, its duration increases, resulting in higher volatility. Secondly, when a bond’s coupon rate rises, its duration decreases, leading to reduced volatility. Lastly, as interest rates climb, the duration of a bond decreases, diminishing its sensitivity to further interest rate increases.

## The Modified Duration and Interest Rate Swaps

The modified duration, which is a measure of a bond’s interest rate sensitivity and risk, is calculated by dividing the dollar value of a one basis point change in an interest rate swap leg or series of cash flows by the present value of the cash flows. The resulting value is then multiplied by 10,000. This calculation helps assess the impact of changes in interest rates on bond prices and investment strategies.