## Macaulay Duration vs. Modified Duration: An Overview

The Macaulay duration and the modified duration are chiefly used to calculate the duration of bonds. The Macaulay duration calculates the weighted average time before a bondholder would receive the bond's cash flows. Conversely, the modified duration measures the price sensitivity of a bond when there is a change in the yield to maturity.

### Key Takeaways

- There are a few different ways to approach the concept of duration, or a fixed-income asset's price sensitivity to changes in interest rates.
- The Macaulay duration is the weighted average term to maturity of the cash flows from a bond, and is frequently used by portfolio managers who use an immunization strategy.
- The modified duration of a bond is an adjusted version of the Macaulay duration and is used to calculate the changes in a bond's duration and price for each percentage change in the yield to maturity.

## The Macaulay Duration

The Macaulay duration is calculated by multiplying the time period by the periodic coupon payment and dividing the resulting value by 1 plus the periodic yield raised to the time to maturity. Next, the value is calculated for each period and added together. Then, the resulting value is added to the total number of periods multiplied by the par value, divided by 1, plus the periodic yield raised to the total number of periods. Then the value is divided by the current bond price.

$\begin{aligned} &\text{Macaulay Duration}=\frac{\left( \sum_{t=1}^{n}{\frac{t*C}{\left(1+y\right)^t}} + \frac{n*M}{\left(1+y\right)^n } \right)}{\text{Current bond price}}\\ &\textbf{where:}\\ &C=\text{periodic coupon payment}\\ &y=\text{periodic yield}\\ &M=\text{the bond's maturity value}\\ &n=\text{duration of bond in periods}\\ \end{aligned}$

A bond's price is calculated by multiplying the cash flow by 1, minus 1, divided by 1, plus the yield to maturity, raised to the number of periods divided by the required yield. The resulting value is added to the par value, or maturity value, of the bond divided by 1, plus the yield to maturity raised to the total number of periods.

For example, consider a three-year bond with a maturity value of $1,000 and a coupon rate of 6% paid semi-annually. The bond pays the coupon twice a year and pays the principal on the final payment. Given this, the following cash flows are expected over the next three years:

$\begin{aligned} &\text{Period 1}: \$30 \\ &\text{Period 2}: \$30 \\ &\text{Period 3}: \$30 \\ &\text{Period 4}: \$30 \\ &\text{Period 5}: \$30 \\ &\text{Period 6}: \$1,030 \\ \end{aligned}$

With the periods and the cash flows known, a discount factor must be calculated for each period. This is calculated as 1 ÷ (1 + r)^{n}, where r is the interest rate and n is the period number in question. The interest rate, r, compounded semiannually is 6% ÷ 2 = 3%. Therefore, the discount factors would be:

$\begin{aligned} &\text{Period 1 Discount Factor}: 1 \div ( 1 + .03 ) ^ 1 = 0.9709 \\ &\text{Period 2 Discount Factor}: 1 \div ( 1 + .03 ) ^ 2 = 0.9426 \\ &\text{Period 3 Discount Factor}: 1 \div ( 1 + .03 ) ^ 3 = 0.9151 \\ &\text{Period 4 Discount Factor}: 1 \div ( 1 + .03 ) ^ 4 = 0.8885 \\ &\text{Period 5 Discount Factor}: 1 \div ( 1 + .03 ) ^ 5 = 0.8626 \\ &\text{Period 6 Discount Factor}: 1 \div ( 1 + .03 ) ^ 6 = 0.8375 \\ \end{aligned}$

Next, multiply the period’s cash flow by the period number and by its corresponding discount factor to find the present value of the cash flow:

$\begin{aligned} &\text{Period 1}: 1 \times \$30 \times 0.9709 = \$29.13 \\ &\text{Period 2}: 2 \times \$30 \times 0.9426 = \$56.56 \\ &\text{Period 3}: 3 \times \$30 \times 0.9151 = \$82.36 \\ &\text{Period 4}: 4 \times \$30 \times 0.8885 = \$106.62 \\ &\text{Period 5}: 5 \times \$30 \times 0.8626 = \$129.39 \\ &\text{Period 6}: 6 \times \$1,030 \times 0.8375 = \$5,175.65 \\ &\sum_{\text{ Period } = 1} ^ {6} = \$5,579.71 = \text{numerator} \\ \end{aligned}$

$\begin{aligned} &\text{Current Bond Price} = \sum_{\text{ PV Cash Flows } = 1} ^ {6} \\ &\phantom{ \text{Current Bond Price} } = 30 \div ( 1 + .03 ) ^ 1 + 30 \div ( 1 + .03 ) ^ 2 \\ &\phantom{ \text{Current Bond Price} = } + \cdots + 1030 \div ( 1 + .03 ) ^ 6 \\ &\phantom{ \text{Current Bond Price} } = \$1,000 \\ &\phantom{ \text{Current Bond Price} } = \text{denominator} \\ \end{aligned}$

(Note that since the coupon rate and the interest rate are the same, the bond will trade at par.)

$\begin{aligned} &\text{Macaulay Duration} = \$5,579.71 \div \$1,000 = 5.58 \\ \end{aligned}$

Note that this duration calculation is for 5.58 half-years, since the bond pays out semi-annually. The annual duration is thus 5.58/2 =2.79 years, which is less than the three years in which the bond matures.

## The Modified Duration

$\begin{aligned} &\text{Modified Duration}=\frac{\text{Macauley Duration}}{\left( 1 + \frac{YTM}{n}\right)} \\ &\textbf{where:}\\ &YTM=\text{yield to maturity}\\ &n=\text{number of coupon periods per year} \end{aligned}$

The modified duration is an adjusted version of the Macaulay duration, which accounts for changing yield to maturities. The formula for the modified duration is the value of the Macaulay duration divided by 1, plus the yield to maturity, divided by the number of coupon periods per year. The modified duration determines the changes in a bond's duration and price for each percentage change in the yield to maturity.

For example, let's look at our bond from the example above, which was calculated to have a Macaulay duration of 5.58 years. The modified duration for this bond would be:

(2.79)/((1+0.06)/2) = 2.71%

The formula to calculate the percentage change in the price of the bond is the change in yield multiplied by the negative value of the modified duration multiplied by 100%. This resulting percentage change in the bond, for an interest rate increase from 8% to 9%, is calculated to be -2.71%. Therefore, if interest rates rise 1% overnight, the price of the bond is expected to drop 2.71%..

### The Modified Duration and Interest Rate Swaps

Modified duration could be extended to calculate the number of years it would take an interest rate swap to repay the price paid for the swap. An interest rate swap is the exchange of one set of cash flows for another and is based on interest rate specifications between the parties.

The modified duration is calculated by dividing the dollar value of a one basis point change of an interest rate swap leg, or series of cash flows, by the present value of the series of cash flows. The value is then multiplied by 10,000. The modified duration for each series of cash flows can also be calculated by dividing the dollar value of a basis point change of the series of cash flows by the notional value plus the market value. The fraction is then multiplied by 10,000.

The modified duration of both legs must be calculated to compute the modified duration of the interest rate swap. The difference between the two modified durations is the modified duration of the interest rate swap. The formula for the modified duration of the interest rate swap is the modified duration of the receiving leg minus the modified duration of the paying leg.

For example, assume bank A and bank B enter into an interest rate swap. The modified duration of the receiving leg of a swap is calculated as nine years and the modified duration of the paying leg is calculated as five years. The resulting modified duration of the interest rate swap is four years (9 years – 5 years).

## Key Differences

Since the Macaulay duration measures the weighted average time an investor must hold a bond until the present value of the bond’s cash flows is equal to the amount paid for the bond, it is often used by bond managers looking to manage bond portfolio risk with immunization strategies.

In contrast, the modified duration identifies how much the duration changes for each percentage change in the yield while measuring how much a change in the interest rates impacts the price of a bond. Thus, the modified duration can provide a risk measure to bond investors by approximating how much the price of a bond could decline with an increase in interest rates. It's important to note that bond prices and interest rates have an inverse relationship with each other.

## What's the Difference Between Macaulay and Modified Duration?

Macaulay duration is the is the weighted average term to maturity of the cash flows from a bond.

Modified duration is a bond's price sensitivity to changes in interest rates, which takes the Macaulay duration and adjusts it for the bond's yield to maturity (YTM).

## Is the Modified Duration Always Less than Macaulay Duration?

Because modified duration divides the Macaulay duration by one plus the modified yield to maturity, it will always be less than the Macaulay duration—except in the rare case if the modified YTM is equal to zero, in which case they will both be the same since the denominator will be 1 + 0% = 1.

## What Is Dollar Duration?

Dollar duration measures the dollar change in a bond’s value to a change in the market interest rate, providing a straightforward dollar-amount computation given a 1% change in rates.