Duration Definition and Its Use in Fixed Income Investing

What Is Duration?

Duration measures how long it takes, in years, for an investor to be repaid a bond’s price through its total cash flows. Duration can also be used to measure how sensitive the price of a bond or fixed-income portfolio is to changes in interest rates.

A bond’s duration is easily confused with its term or time to maturity because some duration measurements are also calculated in years.

However, a bond’s term is a linear measure of the years until the repayment of its principal is due. It does not change with the interest rate environment. Duration is nonlinear, accelerating as the time to maturity lessens.

How Duration Works in Investing

Duration is a measure of the sensitivity of the price of a bond or other debt instrument to a change in interest rates.

In general, the higher the duration, the more a bond’s price will drop as interest rates rise. This also indicates a higher level of interest rate risk. For example, if rates were to rise 1%, a bond or bond fund with a five-year average duration would likely lose about 5% of its value.

Different factors can affect a bond’s duration, including the time to maturity and the coupon rate.

Time to Maturity

The longer the maturity, the higher the duration, and the greater the interest rate risk. Consider two bonds that each yield 5% and cost $1,000, but have different maturities. A bond that matures in one year would repay its true cost faster than a bond that matures in 10 years. Therefore, the shorter-maturity bond would have a lower duration and less risk.

Coupon Rate

A bond’s coupon rate, or yield that it pays, is a key factor in the calculation of duration. If two bonds are identical except for their coupon rates, the bond with the higher coupon rate will pay back its original costs faster than the bond with a lower yield. The higher the coupon rate, the lower the duration, and the lower the interest rate risk.

Types of Duration

In practice, the duration of a bond 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.
• Modified duration is not measured in years. Modified duration measures the expected change in a bond’s price given a 1% change in interest rates.

To understand modified duration, keep in mind that bond prices generally 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

Macaulay duration finds the present value of a bond’s future coupon payments and maturity value. This measure is a standard data point in most bond searches and analysis software tools, which makes it easy for investors to find and use.

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:

$\begin{aligned}&MacD=\sum^n_{f=1}\frac{CF_f}{\left(1+\frac{y}{k}\right)^f}\times\frac{t_f}{PV}\\&\textbf{where:}\\&f = \text{cash flow number}\\&CF = \text{cash flow amount}\\&y = \text{yield to maturity}\\&k = \text{compounding periods per year}\\&t_f = \text{time in years until cash flow is received}\\&PV = \text{present value of all cash flows}\end{aligned}$​MacD=f=1∑n​(1+ky​)fCFf​​×PVtf​​where:f=cash flow numberCF=cash flow amounty=yield to maturityk=compounding periods per yeartf​=time in years until cash flow is receivedPV=present value of all cash flows​

The 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.

Macaulay Duration Calculation Example

Imagine a three-year bond with a face value of $100 that pays a 10% coupon semiannually ($5 every six months) and has a yield to maturity (YTM) of 6%. 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 yield to maturity (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 shown in the following table.

The “Total” row of the table tells an investor that this three-year bond has a Macaulay duration of 2.684 years.

The longer the duration of a bond is, the more sensitive it 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.

How much the bond’s price will change for each 1% the YTM rises or falls is called modified duration.

Modified Duration

The modified duration of a bond helps investors understand how much a bond’s price will rise or fall if the YTM rises or falls by 1%. This is an important number if an investor is concerned that interest rates will change in the short term.

The modified duration of a bond with semiannual coupon payments can be found with the following formula:

$ModD=\frac{\text{Macaulay Duration}}{1+\left(\frac{YTM}{2}\right)}$ModD=1+(2YTM​)Macaulay Duration​

Using the numbers from the previous example, you can use the modified duration formula to find how much the bond’s value will change for a 1% shift in interest rates, as shown below:

$\underbrace{\$2.61}_{ModD}=\frac{2.684}{1+\left(\frac{YTM}{2}\right)}$ModD$2.61​​=1+(2YTM​)2.684​

In this case, if the YTM increases from 6% to 7% because interest rates are rising, the bond’s value should fall by $2.61. Similarly, the bond’s price should rise by $2.61 if the YTM falls from 6% to 5%. 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.

Strategies for Using Duration

In general, the term “long” in investing is used to describe a position in which the investor owns the underlying asset or an interest in the asset that will appreciate in value if the price rises. The term “short” means that the investor has borrowed an asset or has an interest in the asset (through derivatives for example) that will rise in value when the price falls in value.

However, long and short mean something different when used to describe trading strategies based on duration.

A long-duration strategy describes an investing approach in which an investor focuses on bonds with a high duration value. The investor is likely buying bonds with a long time before maturity and greater exposure to interest rate risks. A long-duration strategy works well when interest rates are falling, which usually happens during recessions.

A short-duration strategy is one in which a fixed-income or bond investor is focused on buying bonds that mature soon. A strategy like this would be used by an investor who thinks interest rates will rise and wants to reduce the risk of the investment.

The Bottom Line

Fixed-income investors need to be aware of two main risks that can affect a bond’s value: credit risk (the risk that the issuer will default on the payments) and interest rate risk (interest rate fluctuations). 

Duration is used to quantify the potential impact that both of these factors will have on a bond’s value. For example, if a company begins to struggle and its credit quality declines, investors will require a greater reward or yield to maturity to own the bonds.

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 yield to maturity.