Why duration is calculated

In general, the higher the duration, the more a bond's price will drop as interest rates rise and the greater the interest rate risk. The duration of a bond in practice can refer to two different things. The Macaulay duration is the weighted average time until all the bond's cash flows are paid. By accounting for the present value of future bond payments, the Macaulay duration helps an investor evaluate and compare bonds independent of their term or time to maturity.

The second type of duration is called modified duration. Unlike Macaulay's duration, modified duration is not measured in years. In order to understand modified duration, keep in mind that bond prices are said to have an inverse relationship with interest rates. Therefore, rising interest rates indicate that bond prices are likely to fall, while declining interest rates indicate that bond prices are likely to rise.

Macaulay duration finds the present value of a bond's future coupon payments and maturity value. Fortunately for investors, this measure is a standard data point in most bond searching and analysis software tools. Because Macaulay duration is a partial function of the time to maturity, the greater the duration, the greater the interest-rate risk or reward for bond prices. Macaulay duration can be calculated manually as follows:.

The previous formula is divided into two sections. The first part is used to find the present value of all future bond cash flows. The second part finds the weighted average time until those cash flows are paid. When these sections are put together, they tell an investor the weighted average amount of time to receive the bond's cash flows. In order to find the Macaulay duration, the first step will be to use this information to find the present value of all the future cash flows as shown in the following table:.

This part of the calculation is important to understand. However, it is not necessary if you already know the YTM for the bond and its current price.

This is true because, by definition, the current price of a bond is the present value of all its cash flows. To complete the calculation, an investor needs to take the present value of each cash flow, divide it by the total present value of all the bond's cash flows and then multiply the result by the time to maturity in years. This calculation is easier to understand in the following table. The "Total" row of the table tells an investor that this three-year bond has a Macaulay duration of 2.

Traders know that, the longer the duration is, the more sensitive the bond will be to changes in interest rates. If the YTM rises, the value of a bond with 20 years to maturity will fall further than the value of a bond with five years to maturity.

This is an important number if an investor is worried that interest rates will be changing in the short term. The modified duration of a bond with semi-annual coupon payments can be found with the following formula:.

Unfortunately, as the YTM changes, the rate of change in the price will also increase or decrease. The acceleration of a bond's price change as interest rates rise and fall is called " convexity. Investors need to be aware of two main risks that can affect a bond's investment value: credit risk default and interest rate risk interest rate fluctuations.

Duration is used to quantify the potential impact these factors will have on a bond's price because both factors will affect a bond's expected YTM. For example, if a company begins to struggle and its credit quality declines, investors will require a greater reward or YTM to own the bonds. In order to raise the YTM of an existing bond, its price must fall.

The same factors apply if interest rates are rising and competitive bonds are issued with a higher YTM. The duration of a zero-coupon bond equals its time to maturity since it pays no coupon. A bond is essentially a loan between two counterparties.

The traditional bond structure includes a series of cash flows, such as coupon payments that occur before the bond matures, culminating with a maturity where the principal is fully repaid.

The time to maturity is certainly useful in assessing interest rate risk, as the farther into the future a bond matures, the more likely its value could be impacted by changing interest rates. However, maturity should not be viewed in isolation because it does not take into account either the timing of intermittent cash flows before the maturity date, or the potential changes to the ultimate principal repayment date.

In , Canadian economist Frederick R. Macaulay Duration, as it became known, is the average number of years it will take to receive payments on a bond; importantly, this average is weighted by the capital recovered in each payment. As such, the purpose of Macaulay Duration is to calculate the average time horizon for an investment, rather than to measure price volatility resulting from interest rate fluctuations.

Modified Duration adjusted the formula 2 for Macaulay Duration to create a new, important calculation. For example, the cash flows of bonds with optionality 4 can change with the rise or fall of interest rates. One example of bonds with optionality is callable bonds.

This typically occurs when interest rates are falling and issuers are able to call bonds with higher coupons and reissue debt at the new, lower prevailing market interest rates. The difference between the Modified and Effective Duration for option-free i. However, for some bonds with optionality, the difference can be substantial. Effective Duration has become an essential tool for assessing the interest rate risks of bonds with optionality, such as callable municipal bonds and mortgage-backed securities MBS , where the timing of principal repayment is highly dependent on the level of interest rates.

For most bonds, as yields change, bond prices will become more, or less sensitive to yield changes. Therefore, Effective Duration becomes a less accurate estimation of price sensitivity to interest rates for larger changes in rates. When evaluating fixed income investments, understanding the type of duration used in portfolio reporting and the associated risks of duration is critical.

They are current as of the date s indicated but are subject to change without notice. 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, 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, 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. 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. Milken Institute. New York University. Fixed Income Essentials. Corporate Bonds. Your Privacy Rights. To change or withdraw your consent choices for Investopedia. At any time, you can update your settings through the "EU Privacy" link at the bottom of any page.

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