# Macaulay vs. Modified Duration: What’s the Difference?

Discover the Macaulay duration and the modified duration, how to compute them, and their dissimilarities.

If you are an investor or a finance professional, you may have come across the terms Macaulay duration and modified duration. These two concepts are crucial in the bond market as they help investors understand the sensitivity of a bond’s price to changes in interest rates. In this article, we will explore what these durations are, how to calculate them, and the difference between them.

What is Macaulay Duration?

Macaulay duration is a measure of a bond’s sensitivity to changes in interest rates. It is named after Frederick Macaulay, a Canadian economist who first introduced the concept in 1938. The Macaulay duration is the weighted average time to receive all cash flows from a bond, including both coupon payments and the principal repayment at maturity. The weights used in the calculation are the present values of each cash flow, divided by the bond’s current market price.

The formula for calculating Macaulay duration is as follows:

Macaulay duration = (C1 x t1 + C2 x t2 + … + Cn x tn + n x tn) / P

Where:

C1, C2, …, Cn = the cash flows received at times t1, t2, …, tn

tn = the time in years until each cash flow is received

n = the total number of cash flows

P = the current market price of the bond

For example, let’s say you own a bond with a face value of \$1,000 that pays a coupon rate of 5% annually and matures in five years. The bond is currently trading at \$950. The cash flows for this bond would be:

Year 1: \$50 (coupon payment)

Year 2: \$50 (coupon payment)

Year 3: \$50 (coupon payment)

Year 4: \$50 (coupon payment)

Year 5: \$1,050 (coupon payment + principal repayment)

Using the formula above, we can calculate the Macaulay duration as follows:

Macaulay duration = [(50 x 1) + (50 x 2) + (50 x 3) + (50 x 4) + (1,050 x 5)] / 950

Macaulay duration = 4.26 years

This means that the Macaulay duration for this bond is 4.26 years. It indicates that if interest rates were to increase by 1%, the bond’s price would decrease by approximately 4.26%.

What is Modified Duration?

Modified duration is another measure of a bond’s sensitivity to changes in interest rates. It is a modified version of the Macaulay duration that takes into account the bond’s yield to maturity (YTM). The modified duration measures the percentage change in a bond’s price for a given change in its YTM.

The formula for calculating modified duration is as follows:

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

Where:

Macaulay duration = the Macaulay duration of the bond

YTM = the yield to maturity of the bond

n = the number of coupon payments per year

For example, let’s say you own a bond with a face value of \$1,000 that pays a coupon rate of 5% annually and matures in five years. The bond has a YTM of 4%. The cash flows for this bond would be the same as in the previous example. Using the formula above, we can calculate the modified duration as follows:

Modified duration = 4.26 / (1 + 0.04/1)

Modified duration = 4.10 years

This means that if interest rates were to increase by 1%, the bond’s price would decrease by approximately 4.10%.

Difference between Macaulay Duration and Modified Duration

The main difference between Macaulay duration and modified duration is that the latter takes into account the bond’s yield to maturity. This means that modified duration is a more accurate measure of a bond’s sensitivity to changes in interest rates, as it reflects the impact of changes in both the timing and size of cash flows.

Another difference between the two durations is that Macaulay duration is expressed in years, while modified duration is expressed as a percentage. This makes modified duration more useful for comparing bonds with different maturities and coupon rates.

Conclusion

Macaulay duration and modified duration are essential concepts for investors and finance professionals who want to understand the sensitivity of a bond’s price to changes in interest rates. Macaulay duration measures the weighted average time to receive all cash flows from a bond, while modified duration takes into account the bond’s yield to maturity. By calculating these durations, investors can make informed decisions about their bond investments and manage their interest rate risk.