In the previous article, we have already learned about why the duration is considered to be a very important concept while investing in fixed income securities. However, it is important for the readers to understand how exactly duration is calculated and how it impacts the valuation of fixed income securities. In this article, we will have a closer look at the methodology which is commonly used to calculate bond duration.

### Weighted Average of Cash Flows

Before we delve into the nitty-gritty of the calculation, it is important to understand what we are trying to achieve by calculating duration. As mentioned in the previous article, cash flows that are farther away in terms of time are impacted more by interest rate changes. Duration, therefore, tries to calculate a weighted average of the cash flows. The time factor is used as a weight to ensure that the negative impact related to cash flows that are farther into the future is captured.

1. Step #1: Calculate the Cash Flows:The first step of the model is to calculate the nominal cash flows which will accrue to the bondholder. These nominal payments must then be converted to real payments by discounting them using the appropriate discount rate. This includes all the coupon payments as well as the principal repayment which generally occurs towards the maturity of the bond. It is important for investors to have complete visibility over the cash flow schedule of the bond before they begin calculating duration.
2. Step #2: Calculate the Proportion of Cash Flow in Every Coupon:The next step is to calculate the proportion of cash flow that is being received in every coupon payment. For instance, the total value of the bond is \$100, but the first coupon payment accounts for \$7, then it can be said that 7% of the money will be received in that particular coupon payment. In order to calculate the duration, the proportion of funds being received as a result of every payment need to be calculated.
3. Step #3: Multiply by Time Factor:The next step is to multiply each cash flow with the time factor. This ensures that the cash flows which occur later in the future will show greater sensitivity to interest rate changes as composed of cash flows that are closer.
4. Step #4: Summation:The last and final step is to sum the weighted average of all cash flows. The product calculated above for each and every cash flow which is due to happen in the future is added in order to calculate a final sum. This sum represents the bond duration.

### Duration of a Portfolio

It is common for individual investors as well as mutual funds to calculate the duration of an entire portfolio instead of an individual bond. The method used for calculation is the same. However, given a large number of bonds in any portfolio, the calculation can turn out to be quite complex. This is why such investors do not generally use spreadsheets for their calculations. Instead, they have specialized tools to calculate the duration of the entire portfolio. It is common for such investors to have a target bond duration. For example, if the investor believes that the interest rates will fall in the future, then they will increase the overall duration of their portfolio. This will help them lock in higher yields for a longer period of time. The opposite of this is also true. If investors feel that interest rates will rise in the future, then they try to reduce the duration of the portfolio in order to reduce the impact of this interest rate fall.

### Factors Which Affect the Duration of a Bond

The duration of a bond is affected by the following factors. The weighted average of the time to receive the cash flows from a bond

## What is Macaulay Duration?

Macaulay duration is the weighted average of the time to receive the cash flows from a bond. It is measured in units of years. Macaulay duration tells the weighted average time that a bond needs to be held so that the total present value of the cash flows received is equal to the current market price paid for the bond. It is often used in bond immunization strategies. ### Summary

• Macaulay duration measures the weighted average of the time to receive the cash flows from a bond so that the present value of cash flows equals the bond price.
• A bond’s Macaulay duration is positively related to the time to maturity and inversely related to the bond’s coupon rate and interest rate.
• Modified duration measures the sensitivity of a bond’s price to the change in interest rates.

### How to Calculate Macaulay Duration

In Macaulay duration, the time is weighted by the percentage of the present value of each cash flow to the market price of a bond. Therefore, it is calculated by summing up all the multiples of the present values of cash flows and corresponding time periods and then dividing the sum by the market bond price. Where:

• PV(CFt) – Present value of cash flow (coupon) at period t
• t – Time period for each cash flow
• C – Periodic coupon payment
• n – Total number of periods to maturity
• M – Value at maturity
• Y – Periodic yield

For example, a 2-year bond with a \$1,000 par pays a 6% coupon semi-annually, and the annual interest rate is 5%. Thus, the bond’s market price is \$1,018.81, summing the present values of all cash flows. The time to receive each cash flow is then weighted by the present value of that cash flow to the market price. The Macaulay duration is the sum of these weighted-average time periods, which is 1.915 years. An investor must hold the bond for 1.915 years for the present value of cash flows received to exactly offset the price paid. ### Factors that Affect Macaulay Duration

The Macaulay duration of a bond can be impacted by the bond’s coupon rate, term to maturity, and yield to maturity. With all the other factors constant, a bond with a longer term to maturity assumes a greater Macaulay duration, as it takes a longer period to receive the principal payment at the maturity. It also means that Macaulay duration decreases as time passes (term to maturity shrinks). Macaulay duration takes on an inverse relationship with the coupon rate. The greater the coupon payments, the lower the duration is, with larger cash amounts paid in the early periods. A zero-coupon bond assumes the highest Macaulay duration compared with coupon bonds, assuming other features are the same. It is equal to the maturity for a zero-coupon bond and is less than the maturity for coupon bonds. Macaulay duration also demonstrates an inverse relationship with yield to maturity. A bond with a higher yield to maturity shows a lower Macaulay duration.

### Macaulay Duration vs. Modified Duration

Modified duration is another frequently used type of duration for bonds. Different from Macaulay duration, which measures the average time to receive the present value of cash flows equivalent to the current bond price, Modified duration identifies the sensitivity of the bond price to the change in interest rate. It is thus measured in percentage change in price. Modified duration can be calculated by dividing the Macaulay duration of the bond by 1 plus the periodic interest rate, which means a bond’s Modified duration is generally lower than its Macaulay duration. If a bond is continuously compounded, the Modified duration of the bond equals the Macaulay duration. In the example above, the bond shows a Macaulay duration of 1.915, and the semi-annual interest is 2.5%. Therefore, the Modified duration of the bond is 1.868 (1.915 / 1.025). It means for each percentage increase (decrease) in the interest rate, the price of the bond will fall (raise) by 1.868%. Another difference between Macaulay duration and Modified duration is that the former can only be applied to the fixed income instruments that will generate fixed cash flows. For bonds with non-fixed cash flows or timing of cash flows, such as bonds with a call or put option, the time period itself and also the weight of it are uncertain. Therefore, looking for Macaulay duration, in this case, does not make sense. However, Modified duration can still be calculated since it only takes into account the effect of changing yield, regardless of the structure of cash flows, whether they are fixed or not.

### Macaulay Duration and Bond Immunization

In asset-liability portfolio management, duration-matching is a method of interest rate immunization. A change in the interest rate affects the present value of cash flows, and thus affects the value of a fixed-income portfolio. By matching the durations between the assets and liabilities in a company’s portfolio, the change in interest rate will move the value of assets and the value of liabilities by exactly the same amount, but in opposite directions. Therefore, the total value of this portfolio remains unchanged. The limitation of duration-matching is that the method only immunizes the portfolio from small changes in interest rate. It is less effective for large interest rate changes.

Thank you for reading CFI’s guide on Macaulay Duration. To keep learning and developing your knowledge of financial analysis, we highly recommend the additional resources below:

• Discount Rate
• Effective Duration
• Yield Curve
• Modified Duration

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