# Macaulay Duration Overview, How To Calculate, Factors

15 julio, 2016
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18 diciembre, 2022 The higher the Macaulay duration of a bond, the higher the resulting modified duration and volatility to interest rate changes. Modified duration illustrates the concept that bond prices and interest rates move in opposite directions – higher interest rates lower bond prices, and lower interest rates raise bond prices. Once familiar with the basic concepts, duration should be a welcome find when encountered in the practical actuarial problems we attempt to understand and solve. One macaulay duration meaning of the problems actuaries face is the disappearance of DB plans as a result of decreasing interest rates, longevity risk, and increasingly higher market volatility. Implementing a liability-driven investment (LDI) strategy is one way actuaries have taken on this problem, attempting to create an environment sensitive to risks such as interest rate and market volatility. The opposite is true of low convexity bonds, whose prices don’t fluctuate as much when interest rates change.

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What adds to the confusion is that these negative returns happen in some schemes, while some Debt Funds in the portfolio might be doing well. A full analysis of the fixed-income asset must be done using all available characteristics. There are many ways to calculate duration, and the Macaulay duration is the most common due to its simplicity. If you are unfamiliar with any of the terms, you can refer to the Fixed Income Glossary. Amanda Bellucco-Chatham is an editor, writer, and fact-checker with years of experience researching personal finance topics.

## Understanding Macaulay Duration, Modified Duration and Convexity

To understand how duration is calculated, knowing when and how much cash flows into the hands of retirees is key. Investors can also consider ‘average maturity’, a tool that measures the weighted average maturity of the all bonds constituting a portfolio. 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. 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. Effective duration is a measure of the duration for bonds with embedded options (e.g., callable bonds).

The shaded area reveals the difference between the duration estimate and the actual price movement. As indicated, the larger the change in interest rates, the larger the error in estimating the price change of the bond. Key rate durations require that we value an instrument off a yield curve and requires building a yield curve. Formally, modified duration is a semi-elasticity, the percent change in price for a unit change in yield, rather than an elasticity, which is a percentage change in output for a percentage change in input. Modified duration is a rate of change, the percent change in price per change in yield.

## Modified Duration Formula

This bond duration tool can calculate the Macaulay duration and modified duration based on either the market price of the bond or the yield to maturity (or the market interest rate) of the bond. However, investors may find the Modified Duration an even better measure of a fund’s sensitivity to interest rates. As the name suggests, this is a modified version of the Macaulay model that accounts for changing interest rates. Modified duration is calculated simply dividing the Macaulay Duration by the portfolio yield. It measures the change in the value of a fixed income security that will result from a one per cent change in the interest rate. A bond with a longer maturity period is more sensitive to changes in interest rates than a bond with a short duration. The formula can also be used to calculate the DV01 of the portfolio (cf. below) and it can be generalized to include risk factors beyond interest rates. From the series, you can see that a zero coupon bond has a duration equal to it’s time to maturity – it only pays out at maturity. The modified duration provides a good measurement of a bond’s sensitivity to changes in interest rates.

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Now that we understand and know how to calculate the Macaulay duration, we can determine the modified duration. In order to arrive at the modified duration of a bond, it is important to understand the numerator component – the Macaulay duration – in the modified duration formula. Modified duration, a formula commonly used in bond valuations, expresses the change in the value of a security due to a change in interest rates. 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%.

## Using Average Maturity while Shortlisting Debt Funds

When using duration, cash flows are assumed to remain unaffected by changing interest rates and they are discounted at all future times using the same interest 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.

• The most common are the Macaulay duration, modified duration, and effective duration.
• If you have all of the details of the bond and know the market price, click the blue «You Know Market Price» button.
• In layman’s term, the Macaulay duration measures, in years, the amount of time required for an investor to be repaid his initial investment in a bond.

It also means that Macaulay duration decreases as time passes (term to maturity shrinks). As an example, a \$1,000 bond that can be redeemed by the holder at par at any time before the bond’s maturity (i.e. an American put option). No matter how high interest rates become, the price of the bond will never go below \$1,000 (ignoring counterparty risk). This bond’s price sensitivity to interest rate changes is different from a non-puttable bond with otherwise identical cash flows. This represents the bond discussed in the example below – two year maturity with a coupon of 20% and continuously compounded yield of 3.9605%. If these circles were put on a balance beam, the fulcrum (balanced center) of the beam would represent the weighted average distance (time to payment), which is 1.78 years in this case.

## Macaulay Duration and Bond Immunization

The article concludes by discussing how duration is present in the understanding and applying of more advanced actuarial topics. 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. 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.

• In the simplest terms, if the Modified Duration of a Bond is 5 years and the market Interest Rate decreases by 1%, then the Bond’s price will increase by 5%.
• Portfolio duration plays a key role in helping debt fund investors measure the risk of the fund they are buying into.
• So, one way in which you can minimize the impact of rising Interest Rates on Your Debt Portfolio is to increase your investments in Debt Funds with low Average Maturity.
• The difference between the two modified durations is the modified duration of the interest rate swap.

(You can also compute the Macaulay and modified duration of an entire portfolio by summing cash flow). The modified duration of a bond is a measure of the sensitivity of a bond’s market price to a change in interest rates. It’s the percentage change of a bond’s price based on a one percentage point move in market interest rates. For a fixed coupon bond, there are two risks that is caused by the change of interest rate, one is the bond price risk and the other is the coupon reinvestment risk. The modified Duration of a bond is a measure of how much the price of a Bond changes because of a change in its Yield To Maturity (YTM) or interest rate. In the simplest terms, if the Modified Duration of a Bond is 5 years and the market Interest Rate decreases by 1%, then the Bond’s price will increase by 5%. 