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Duration

A tool used in assessing the price volatility of a fixed-income security

What is Duration?

Duration is one of the fundamental characteristics of a fixed-income security (e.g., bond) alongside maturity, yield, coupon, and call features. It is a tool used in the assessment of the price volatility of a fixed-income security.

Since the interest rate is one of the most significant drivers of a bond’s value, duration measures the sensitivity of the value fluctuations to the changes in the interest rates. The general rule states that a longer duration indicates a greater likelihood that the value of a bond will fall as the interest rate increases.

 

Duration

 

Duration is commonly used in portfolio and risk management of fixed-income instruments. Using interest rate forecasts, a portfolio manager can change a portfolio’s composition to align its duration with the expected level of interest rate.

However, duration only reveals one side of a fixed-income security. A full analysis of the fixed-income asset must be done using all available characteristics.

 

CFI’s Fixed Income Fundamentals Course covers the essential topics for fixed-income valuation.

 

Types of Duration

The duration characteristic comes in several modifications. The most common are Macaulay duration, modified duration, and effective duration.

 

1. Macaulay Duration

Macaulay duration is a weighted average of the times until the cash flows of a fixed-income instrument are received. The concept was introduced by Canadian economist Frederick Macaulay. It is a measure of a time required for an investor to be repaid the bond’s price by the bond’s total cash flows. The Macaulay duration is measured in units of time (e.g., years).

The Macaulay duration for coupon-paying bonds is always higher than the bond’s time to maturity. For zero-coupon bonds, the duration equals the time to maturity.

The formula for the calculation of Macaulay duration is expressed in the following way:

 

Macaulay Duration - Formula

 

Where:

  • ti – the time until the ith cash flow from the asset will be received
  • PV– the present value of the ith cash flow from the asset
  • V – the present value of all cash flows from the asset

 

2. Modified Duration

Modified duration is a measure of the sensitivity of the price of a bond to the fluctuations in interest rates. Relative to the Macaulay duration, modified duration is a more precise measure of price sensitivity. It is primarily applied to bonds, but it can also be used with other types of securities that can be considered as a function of yield.

The modified duration figure indicates the percentage change in the bond’s value given the X% interest rate change. Unlike Macaulay duration, modified duration is measured in percentages.

The modified duration is considered as an extension of Macaulay duration. It is supported by the following mathematical formula:

 

Modified Duration - Formula

 

Where:

  • YTM – the yield to maturity of a bond
  • n – the frequency of compounding

 

3. Effective Duration

Effective duration is a measure of the duration for bonds with embedded options (e.g., callable bonds). Unlike modified duration and Macaulay duration, effective duration considers fluctuations in the bond’s price movements relative to the changes in the bond’s yield to maturity (YTM). In other words, the measure takes into account possible fluctuations in the expected cash flows of a bond.

The effective duration is calculated using the following formula:

 

Effective Duration - Formula

 

Where:

  • VΔy  – the bond’s value if yield falls by y%
  • V+Δy – the bond’s value if yield rises by y%
  • V0 – the present value of all cash flows of a bond
  • Δy – the yield changes

 

Related Readings

CFI is the official provider of the global Financial Modeling & Valuation Analyst (FMVA)™ certification program, designed to help anyone become a world-class financial analyst. To keep advancing your career, the additional resources below will be useful:

  • Bond Pricing
  • Debt Capital Markets
  • Equity Risk Premium
  • Fixed Income Trading

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