Foundation

What Is Duration?

Duration, in the context of fixed income securities, is a key measure of a bond's price sensitivity to changes in interest rates. It calculates the weighted average time, in years, an investor must wait to receive a bond's total cash flows, including both coupon payments and the final principal repayment. Understanding duration is fundamental to fixed income analysis and helps investors assess the inherent interest rate risk of their bond holdings. The longer a bond's duration, the more sensitive its bond prices will be to fluctuations in interest rates.

History and Origin

The concept of duration was introduced by Canadian economist Frederick Macaulay in 1938. [Macaulay's research aimed to provide a more structured measure of a bond's "effective maturity" and quantify the relationship between bond prices and interest rate fluctuations, laying the groundwork for modern bond valuation techniques.⁴⁴ Macaulay duration, also referred to as "pure" duration, became the most common measure.⁴², ⁴³ While Macaulay focused on the weighted average time to cash flows, economist John Hicks, in his 1939 work "Value and Capital," further developed duration as a measure of a bond's price elasticity with respect to changes in the discount rate.³⁹, ⁴⁰, ⁴¹ The significance of duration gained considerable attention in the 1970s when interest rates began to experience dramatic increases, leading investors to seek tools for assessing the price volatility of their fixed income investments.³⁸

Key Takeaways

• Duration measures a bond's price sensitivity to changes in interest rates.
• A higher duration indicates greater interest rate risk; bond prices will be more volatile.
• Macaulay duration calculates the weighted average time until a bond's cash flows are received.
• Duration is a critical tool for portfolio management and managing interest rate exposure.
• For zero-coupon bonds, duration is equal to its time to maturity.³⁷

Formula and Calculation

Macaulay duration is calculated as the weighted average of the time until each cash flow is received, where the weights are the present value of each cash flow as a percentage of the bond's full price.³⁶

The formula for Macaulay Duration is:

Where:

• ( D ) = Macaulay Duration
• ( CF_t ) = Cash flow (coupon payment or principal repayment) at time ( t )
• ( t ) = Time until cash flow ( CF_t ) is received (in years)
• ( y ) = Yield to maturity per period (or periodic discount rate)
• ( P ) = Current market price of the bond
• ( N ) = Total number of cash flows

To compute this, one typically discounts each future cash flow to its present value using the bond's yield to maturity, multiplies each present value by its respective time period, sums these results, and then divides by the bond's current market price.³⁴, ³⁵

Interpreting Duration

Duration is interpreted as the approximate percentage change in a bond's price for a 1% (or 100-basis point) change in interest rates.³², ³³ For example, a bond with a duration of 5 years implies that if interest rates increase by 1%, the bond's price is expected to decrease by approximately 5%. Conversely, if interest rates fall by 1%, the bond's price would increase by roughly 5%.³⁰, ³¹

This measure helps investors quantify the sensitivity of their bond holdings to interest rate movements. A longer duration signifies higher interest rate risk, meaning the bond's price will fluctuate more significantly with changes in prevailing rates. Conversely, bonds with shorter durations are less susceptible to interest rate changes.²⁹ Portfolio managers frequently use duration to assess risk and make informed investment decisions, including implementing immunization strategies.

Hypothetical Example

Consider a 3-year bond with a face value of $1,000, an annual coupon rate of 5%, and a yield to maturity (YTM) of 6%.

Step 1: Calculate Annual Cash Flows

• Year 1: $50 (coupon)
• Year 2: $50 (coupon)
• Year 3: $1,050 (coupon + principal repayment)

Step 2: Calculate the Present Value of Each Cash Flow (at 6% YTM)

• PV (Year 1) = ( \frac{50}{(1.06)^1} ) = $47.17
• PV (Year 2) = ( \frac{50}{(1.06)^2} ) = $44.50
• PV (Year 3) = ( \frac{1050}{(1.06)^3} ) = $881.56
• Current Bond Price (Sum of PVs) = $47.17 + $44.50 + $881.56 = $973.23

Step 3: Calculate Weighted Time for Each Cash Flow

• Year 1: ( \frac{47.17}{973.23} \times 1 ) = 0.0485
• Year 2: ( \frac{44.50}{973.23} \times 2 ) = 0.0915
• Year 3: ( \frac{881.56}{973.23} \times 3 ) = 2.7161

Step 4: Sum the Weighted Times to Find Macaulay Duration

• Macaulay Duration = 0.0485 + 0.0915 + 2.7161 = 2.8561 years

This means it would take approximately 2.86 years for the bondholder to receive the weighted average of the bond's cash flows. This figure helps in understanding the bond's interest rate sensitivity.

Practical Applications

Duration is a vital concept in various financial applications:

• Risk Management: Investors and institutions use duration to manage interest rate risk within their bond portfolios. By matching the duration of assets and liabilities, financial entities like pension funds can employ immunization strategies to protect against adverse interest rate movements.²⁷, ²⁸
• Bond Selection: Duration allows investors to compare the interest rate sensitivity of different bonds, even those with varying maturities and coupon payments. This enables them to select bonds that align with their interest rate outlook and risk tolerance.²⁵, ²⁶
• Portfolio Construction: Portfolio managers use duration to construct portfolios with a desired level of interest rate exposure. They can adjust the aggregate duration of a bond portfolio by investing in bonds with different characteristics, such as short-term or long-term Treasury bonds.²³, ²⁴
• Central Bank Operations: Central banks, such as the Federal Reserve Board, analyze duration when assessing the impact of their monetary policy decisions on financial markets. Their bond purchasing programs, for instance, can significantly alter the duration profile of assets held in the market.²²

Limitations and Criticisms

While duration is a powerful tool, it has several important limitations:

• Linear Approximation: Duration assumes a linear relationship between bond prices and interest rates, which is only accurate for small changes in yields.²⁰, ²¹ In reality, the relationship between bond prices and yield to maturity is convex, meaning prices fall at a decreasing rate as rates rise and increase at an increasing rate as rates fall.¹⁹ This can lead duration to overestimate price declines and underestimate price increases for larger interest rate changes. The concept of convexity was developed to address this non-linear relationship.¹⁷, ¹⁸
• Yield Curve Shifts: Duration typically assumes a parallel shift in the yield curve, meaning all maturities change by the same amount.¹⁵, ¹⁶ However, in practice, the yield curve can twist or steepen, with short-term and long-term interest rates moving differently.¹³, ¹⁴
• Bonds with Embedded Options: Macaulay duration and even modified duration (a derivative of Macaulay duration that measures price sensitivity more directly) may not accurately reflect the interest rate sensitivity of bonds with embedded options, such as callable bonds.¹¹, ¹² These bonds have cash flows that can change based on interest rate movements, making their duration more complex. For such securities, "effective duration" is often used.⁹, ¹⁰ A paper published by Atlantis Press highlights these limitations and proposes modified duration and effective duration as solutions.⁸

Duration vs. Modified Duration

While closely related, Macaulay duration and Modified Duration serve distinct purposes. Macaulay duration measures the weighted average time, in years, an investor waits to receive the bond's cash flows, essentially indicating the bond's "effective maturity" or repayment period.⁷ It's a measure of time. Modified Duration, on the other hand, directly quantifies the percentage change in a bond's price for a 1% change in its yield to maturity.⁵, ⁶ It is derived from Macaulay duration and the bond's yield.⁴ Modified duration is therefore a more direct measure of a bond's price sensitivity to interest rate changes. If a bond has a Macaulay duration of D and a yield to maturity of y (per period), its Modified Duration can be approximated as ( \frac{D}{1 + y} ).³

FAQs

What does a higher duration mean?

A higher duration means that a bond's price is more sensitive to changes in interest rates. This implies greater interest rate risk; if rates rise, the bond's price will fall more significantly, and vice versa.

Is duration the same as maturity?

No, duration is not the same as maturity. While maturity is the fixed date when a bond's principal is repaid, duration is a weighted average time to all of a bond's cash flows, including coupon payments and principal. For a zero-coupon bond, duration equals its maturity because all cash flows occur at maturity. For coupon-paying bonds, duration is always less than its maturity.¹, ²

How does duration help in managing bond portfolios?

Portfolio management relies on duration to gauge and manage interest rate risk. By calculating the weighted average duration of a portfolio, managers can adjust their holdings to align with their interest rate outlook, for example, by shortening duration when expecting rates to rise. This can also involve immunization strategies to match asset and liability durations.