Terms

# Modified Duration Formula, Calculation, and How to Use It

## What Is Modified Duration?

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

### Key Takeaways

• Modified duration measures the change in a bond’s value in response to a 100-basis-point (1%) change in interest rates.
• Modified duration is an extension of the Macaulay duration and requires calculating the Macaulay duration first.
• Macaulay duration calculates the weighted average time before a bondholder receives the bond’s cash flows.
• As a bond’s maturity increases, duration increases, while a bond’s coupon and interest rate increases result in a decrease in duration.

## Formula and Calculation of Modified Duration

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

where:

Macaulay Duration = Weighted average term to maturity of the cash flows from a bond

YTM = Yield to maturity

n = Number of coupon periods per year

Modified duration is an extension of the Macaulay duration, allowing investors to measure a bond’s sensitivity to changes in interest rates. Macaulay duration calculates the weighted average time before a bondholder receives the bond’s cash flows, and it must be calculated first to determine modified duration. The formula for the Macaulay duration is:

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Macaulay Duration = ∑ t = 1 n ( PV × CF ) × t / Market Price of Bond

where:

PV × CF = Present value of coupon at period t

t = Time to each cash flow in years

n = Number of coupon periods per year

Here, (PV) * (CF) represents the present value of a coupon at period t, and T equals the time to each cash flow in years. Perform this calculation and sum it 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 crucial metric for portfolio managers, financial advisors, and clients when selecting investments. Higher durations tend to indicate greater price volatility in bonds compared to those with lower durations. Duration calculations utilize various bond components, including price, coupon, maturity date, and interest rates.

Keep in mind the following duration principles. First, as maturity increases, duration increases, leading to higher bond volatility. Second, as a bond’s coupon increases, duration decreases, resulting in reduced bond volatility. Third, as interest rates increase, duration decreases, and the bond becomes less sensitive to further interest rate increases.

## Example of How to Use Modified Duration

Assume a \$1,000 bond with a three-year maturity, a 10% coupon, and 5% interest rates. Applying the basic bond pricing formula, the market price of this bond would be:

Market Price = \$100 / 1.05 + \$100 / 1.05^2 + \$1,100 / 1.05^3 = \$95.24 + \$90.70 + \$950.22 = \$1,136.16

Next, using the Macaulay duration formula, calculate the duration:

Macaulay Duration = (\$95.24 × 1 / \$1,136.16) + (\$90.70 × 2 / \$1,136.16) + (\$950.22 × 3 / \$1,136.16) = 2.753

This result indicates that it takes 2.753 years to recoup the bond’s true cost. With this number, you can now calculate the modified duration.

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To find the modified duration, divide the Macaulay duration by 1 + (yield-to-maturity / number of coupon periods per year). In this example, the calculation would be 2.753 / (1.05 / 1), resulting in 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%.

To find the modified duration, divide the Macaulay duration by 1 + (yield-to-maturity / number of coupon periods per year). In this example, the calculation would be 2.753 / (1.05 / 1), resulting in 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%.