# Macaulay Duration – Definition, Formula, Example, and How It Works

## Definition of Macaulay Duration

Macaulay Duration is a financial concept that measures the weighted average time it takes for an investor to recoup their initial investment in a fixed income security, such as a bond or a loan. It is named after Frederick Macaulay, an economist who developed this concept in 1938.

Macaulay Duration takes into account both the periodic interest payments received from the security and the final principal payment at maturity. It is a useful tool for investors to assess the risk and potential return of fixed income securities, as it provides a measure of the sensitivity of the security’s price to changes in interest rates.

### Calculation of Macaulay Duration

The Macaulay Duration of a fixed income security is calculated by taking the present value of each cash flow from the security (including both interest payments and the principal payment at maturity), multiplying it by the time until that cash flow is received, and then dividing the sum of these weighted cash flows by the current market price of the security.

The formula for calculating Macaulay Duration is as follows:

Macaulay Duration = (C1 * t1 + C2 * t2 + … + Cn * tn + P * tn) / P

Where:

• C1, C2, …, Cn are the periodic cash flows received from the security
• t1, t2, …, tn are the times until each cash flow is received
• P is the current market price of the security

### Interpretation of Macaulay Duration

A higher Macaulay Duration indicates that the fixed income security has a longer average maturity and is more sensitive to changes in interest rates. This means that the price of the security will be more affected by fluctuations in interest rates, resulting in a higher potential for capital gains or losses.

On the other hand, a lower Macaulay Duration implies a shorter average maturity and lower sensitivity to interest rate changes. This means that the price of the security will be less affected by interest rate movements, resulting in lower potential gains or losses.

## Formula for Macaulay Duration

The formula for Macaulay Duration is:

Macaulay Duration = (CF1 * t1 + CF2 * t2 + … + CFn * tn) / (CF1 + CF2 + … + CFn)

Where:

• CF1, CF2, …, CFn are the cash flows received at times t1, t2, …, tn
• t1, t2, …, tn are the times at which the cash flows are received

The formula calculates the weighted average time by multiplying each cash flow by the time at which it is received and then dividing the sum of these products by the sum of the cash flows.

This formula allows investors to compare the durations of different fixed income securities and assess their interest rate risk. A higher Macaulay Duration indicates a longer time until the investor receives the cash flows, making the security more sensitive to changes in interest rates.

By calculating the Macaulay Duration, investors can make informed decisions about their fixed income investments and manage their portfolios effectively.

## Example of Macaulay Duration

Let’s consider an example to understand how Macaulay Duration is calculated. Suppose you have a bond with a face value of \$1,000, a coupon rate of 5%, and a maturity of 5 years. The bond pays semi-annual coupons.

To calculate the Macaulay Duration, we need to determine the present value of each cash flow and the time at which it is received. Let’s assume a discount rate of 4%.

Using the present value formula, we can calculate the present value of each cash flow and the time at which it is received:

Cash Flow 1: Present Value = \$25 / (1 + 4%/2) ^ 1 = \$24.51

Cash Flow 2: Present Value = \$25 / (1 + 4%/2) ^ 2 = \$23.55

Cash Flow 3: Present Value = \$25 / (1 + 4%/2) ^ 3 = \$22.63

Cash Flow 4: Present Value = \$25 / (1 + 4%/2) ^ 4 = \$21.75

Cash Flow 5: Present Value = \$25 / (1 + 4%/2) ^ 5 = \$20.91

Cash Flow 6: Present Value = \$1,000 / (1 + 4%/2) ^ 10 = \$820.08

Next, we calculate the weighted average time at which each cash flow is received. This is done by multiplying the present value of each cash flow by the time until it is received and summing them up:

Weighted Time 1: \$24.51 * 1 = \$24.51

Weighted Time 2: \$23.55 * 2 = \$47.10

Weighted Time 3: \$22.63 * 3 = \$67.89

Weighted Time 4: \$21.75 * 4 = \$87.00

Weighted Time 5: \$20.91 * 5 = \$104.55

Weighted Time 6: \$820.08 * 10 = \$8,200.80

Finally, we divide the sum of the weighted times by the sum of the present values to get the Macaulay Duration:

Macaulay Duration = (\$24.51 + \$47.10 + \$67.89 + \$87.00 + \$104.55 + \$8,200.80) / (\$24.51 + \$23.55 + \$22.63 + \$21.75 + \$20.91 + \$820.08) = 5.55 years

Therefore, the Macaulay Duration of the bond is 5.55 years. This means that on average, an investor will recoup their investment in 5.55 years, taking into account the timing and size of the bond’s cash flows.