What Is Modified Duration?

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 t𒅌he price of a bond.

Formula to Calculate Modified Duration

澳洲幸运5官方开奖结果体彩网: The formula t﷽o calculate modꦇified duration is:

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

Modified duration is an extension of the 澳洲幸运5官方开奖结果体彩网: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. To calculate modified duration, the Macaulay duration 澳洲幸运5官方开奖结果体彩网:must first be calculated. The formula for the Macaulay duration is:

$\begin{aligned}&\text{Macaulay Duration}=\frac{\sum^n_{t=1}(\text{PV}\times \text{CF})\times \text{t}}{\text{Market Price of Bond}}\\&\textbf{where:}\\&\text{PV}\times \text{CF}=\text{Present value of coupon at period }t\\&\text{t}=\text{Time to each cash flow in years}\\&n=\text{Number of coupon periods per year}\end{aligned}$​Macaulay Duration=Market Price of Bond∑t=1n​(PV×CF)×t​where:PV×CF=Present value of coupon atꦇ period tt=Time to each ca💧sh flow in yearsn=Nu💛mber of coupon periods per&⛎nbsp;year​

Here, (PV) * (CF) is the 澳洲幸运5官方开奖结果体彩网: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 澳洲幸运5官方开奖结果体彩网:term to maturity of a bond. It is a very important number for portfolio managers, 澳洲幸运5官方开奖结果体彩网:financial advisors, and clients to consider when selecting investments because—all other risk factors equal—bonds with higher durations have greater price 澳洲幸运5官方开奖结果体彩网: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 🎃inte⭕rest 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%. Thi🐎s bond, following the basic bond pricing formula, would have a market price of:

$\begin{aligned} &\text{Market Price} = \frac{ \$100 }{ 1.05 } + \frac{ \$100 }{ 1.05 ^ 2 } + \frac{ \$1,100 }{ 1.05 ^ 3 } \\ &\phantom{\text{Market Price} } = \$95.24 + \$90.70 + \$950.22\\ &\phantom{\text{Market Price} } = \$1,136.16 \\ \end{aligned}$​Market Price=1.05$100​+1.052$100​+1.053$1,100​Market Price=$95.24+$90.70+$950.22Market Price=$1,136.16​

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

$\begin{aligned}\text{Macaulay Duration}&=\bigg(\$95.24\times\frac{1}{\$1,136.16}\bigg)\\&\quad+\bigg(\$90.70\times\frac{2}{\$1,136.16}\bigg)\\&\quad+\bigg(\$950.22\times\frac{3}{\$1,136.16}\bigg)\\&=2.753\end{aligned}$Macaulay Duration​=($95.24×$1,136.161​)+($90.70×$1,136.162​)+($950.22×$1,136.163​)=2.753​

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 b💟e 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%.

The Bottom Line

Bond prices and interest rates have an inverse relationship. Modified duration helps investors understand this relationship by measuring the sensitivity of a bond's price to changes in interest rates. This helps investors make smart investment decisions and manage investment risk.