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# Modified Duration

## What Is Modified Duration?

Modified duration is a formula that expresses the measurable change in the value of a security in response to a change in interest rates. Modified duration follows the concept that interest rates and bond prices move in opposite directions. This formula is used to determine the effect that a 100-basis-point (1%) change in interest rates will have on the price of a bond.

### Key Takeaways

• Modified duration measures the change in the value of a bond in response to a change in 100-basis-point (1%) change in interest rates.
• Modified duration is an extension of the Macaulay duration, and in order to calculate modified duration, the Macaulay duration must first be calculated.
• Macaulay duration calculates the weighted average time before a bondholder receives the bond's cash flows.
• As a bond's maturity increases, duration increases, and as a bond's coupon and interest rate increases, its duration decreases.

## Formula and Calculation of Modified Duration

\begin{aligned}&\text{Modified Duration}=\frac{\text{Macaulay Duration}}{1+\overset{\text{YTM}}{n}}\\&\textbf{where:}\\&\text{Macaulay Duration}=\text{Weighted average term to}\\&\qquad\text{maturity of the cash flows from a bond}\\&\text{YTM}=\text{Yield to maturity}\\&n=\text{Number of coupon periods per year}\end{aligned}

Modified duration is an extension of the Macaulay duration, which allows investors to measure the sensitivity of a bond to changes in interest rates. Macaulay duration calculates the weighted average time before a bondholder receives the bond's cash flows. In order to calculate modified duration, the Macaulay duration must first be calculated. The formula for the Macaulay duration is:

\begin{aligned}&\text{Macaulay Duration}=\frac{\sum^n_{t=1}(\text{PV}\times \text{CF})\times \text{t}}{\text{Market Price of Bond}}\\&\textbf{where:}\\&\text{PV}\times \text{CF}=\text{Present value of coupon at period }t\\&\text{t}=\text{Time to each cash flow in years}\\&n=\text{Number of coupon periods per year}\end{aligned}

Here, (PV) * (CF) is the present value of a coupon at period t, and T is equal to the time to each cash flow in years. This calculation is performed and summed for the number of periods to maturity.

## What Modified Duration Can Tell You

Modified duration measures the average cash-weighted term to maturity of a bond. It is a very important number for portfolio managers, financial advisors, and clients to consider when selecting investments because—all other risk factors equal—bonds with higher durations have greater price volatility than bonds with lower durations. There are many types of duration, and all components of a bond, such as its price, coupon, maturity date, and interest rates, are used to calculate duration.

Here are some principles of duration to keep in mind. First, as maturity increases, duration increases and the bond becomes more volatile. Second, as a bond's coupon increases, its duration decreases and the bond becomes less volatile. Third, as interest rates increase, duration decreases, and the bond's sensitivity to further interest rate increases goes down.

## Example of How to Use Modified Duration

Assume a \$1,000 bond has a three-year maturity, pays a 10% coupon, and that interest rates are 5%. This bond, following the basic bond pricing formula would have a market price of:

\begin{aligned} &\text{Market Price} = \frac{ \100 }{ 1.05 } + \frac{ \100 }{ 1.05 ^ 2 } + \frac{ \1,100 }{ 1.05 ^ 3 } \\ &\phantom{\text{Market Price} } = \95.24 + \90.70 + \950.22\\ &\phantom{\text{Market Price} } = \1,136.16 \\ \end{aligned}

Next, using the Macaulay duration formula, the duration is calculated as:

\begin{aligned}\text{Macaulay Duration}&=\bigg(\95.24\times\frac{1}{\1,136.16}\bigg)\\&\quad+\bigg(\90.70\times\frac{2}{\1,136.16}\bigg)\\&\quad+\bigg(\950.22\times\frac{3}{\1,136.16}\bigg)\\&=2.753\end{aligned}

This result shows that it takes 2.753 years to recoup the true cost of the bond. With this number, it is now possible to calculate the modified duration.

To find the modified duration, all an investor needs to do is take the Macaulay duration and divide it by 1 + (yield-to-maturity / number of coupon periods per year). In this example that calculation would be 2.753 / (1.05 / 1), or 2.62%. This means that for every 1% movement in interest rates, the bond in this example would inversely move in price by 2.62%.

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