What Is Modified Duration?

Modified d🍨uration follows the concept that interest rates and bond prices move in opposite ꦦdirections. 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.

Formula to Calculate Modified Duration

澳洲幸运5开奖号码历史查询: T✤he formula to calculate modified 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 th♊e cash flows fro🧸m a bondYTM=Yield to maturityn=Number&💟nbsp;of coupon periods per year​

Modified duration is an extension of the 澳洲幸运5开奖号码历史查询: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开奖号码历史查询: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 ꧋;at period tt=Time ♕;to each c💎ash flow in yearsn=Number of coup🌌on periods per year​

Here, (PV) * (CF) is the 澳洲幸运5开奖号码历史查询:present value of a coupon at per🌼iod 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开奖号码历史查询:term to maturity of a bond. It is a very important number for portfolio managers, 澳洲幸运5开奖号码历史查询: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开奖号码历史查询:volatility than bonds with lower durations.

There are many types of duration, and all compo🃏nents 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:

$\begin{aligned} &\text{Market Price} = \frac{ \$100 }{ 1.05 } + \frac{ \$100 }{ 1.05 ^ 2 } + \frac{ \$1,100 }{ 1.05 ^ 3 } \\ &\phantom{\text{Market Price} } = \$95.24 + \$90.70 + \$950.22\\ &\phantom{\text{Market Price} } = \$1,136.16 \\ \end{aligned}$​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 duration formula, the duration is calcuꦚlated 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 possꦜible 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 e🐓very 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.