Fixed Income Volatility: Modified Duration

Introduction

Modified duration is a key measure used in bond analysis to assess the price sensitivity of a bond to changes in interest rates. It provides investors with an estimate of the percentage change in bond price for a 1% change in yield. Understanding the modified duration of a bond, its relationship with Macaulay duration, as well as its benefits, drawbacks, and common misuses, is essential for bond valuation, risk management, and investment decision-making in debt and money markets.

Modified Duration Calculation

Modified duration is calculated by dividing the Macaulay duration of a bond by a factor of (1 + Yield). The formula for modified duration is as follows:

Modified Duration = Macaulay Duration / (1 + Yield) Where: Macaulay Duration = Weighted average time to receive cash flows Yield = Yield to maturity or yield to call/put

Example

Consider a bond with the following characteristics:

Face value (F) = $1,000

Coupon rate = 5% (annual coupon payment rate)

Coupon payment (C) = $50 (5% of $1,000)

Time to maturity (n) = 5 years Yield to maturity (YTM) = 4%

The Macaulay duration of this bond was calculated in the previous example as approximately 4.93 years.

To calculate the modified duration: Modified Duration = 4.93 years / (1 + 0.04) ≈ 4.73 years

The modified duration of this bond is approximately 4.73 years.

Comparison to Macaulay Duration

The key difference between modified duration and Macaulay duration lies in their interpretation and usage. Macaulay duration measures the weighted average time to receive cash flows, providing a time-based measure. On the other hand, modified duration measures the percentage change in bond price in response to a 1% change in yield, providing a price sensitivity measure.

Benefits of Modified Duration

1. Price Sensitivity: Modified duration provides a direct measure of how bond prices are affected by changes in interest rates. It helps investors estimate the potential impact on bond prices and manage interest rate risk.

2. Comparative Analysis: Modified duration allows for the comparison of different bonds, even if they have varying coupon rates, maturities, or yields. It standardizes the measure of price sensitivity across different bonds.

Drawbacks of Modified Duration

1. Convexity Consideration: Modified duration assumes a linear relationship between bond price and yield changes, which may not hold true for larger interest rate fluctuations. Ignoring the impact of convexity can lead to inaccuracies in predicting bond price changes.

2. Sensitivity to Yield Changes: Modified duration is more accurate for small changes in yields but may be less precise for larger changes. It becomes less reliable when interest rates deviate significantly from the initial yield.

When to Use Modified Duration

Modified duration is particularly useful for bond investors when:

- Assessing interest rate risk: It helps measure the potential impact of interest rate changes on bond prices.

- Comparing bonds: It enables the comparison of bonds with different characteristics, such as coupon rates, maturities, or yields.

Common Misuses

1. Absolute Comparison: Modified duration should not be compared directly between bonds with different durations. It is a relative measure within a specific bond and does not indicate the price sensitivity of a bond compared to others.

2. Ignoring Convexity: Modified duration does not consider the curvature of the bond's price-yield relationship, known as convexity. Neglecting convexity can lead to inaccuracies in estimating bond price changes, particularly for larger interest rate movements.

Conclusion

Modified duration is a valuable tool for bond investors, providing an estimate of price sensitivity to changes in interest rates. It allows for the comparison of bonds with different characteristics and assists in managing interest rate risk. While it offers useful insights, modified duration should be complemented with an understanding of convexity for more accurate predictions of bond price changes. Understanding the benefits, drawbacks, and potential misuses of modified duration enables investors to make informed decisions in debt and money markets, considering the specific characteristics and objectives of their investment portfolios.

This article takes inspiration from a lesson found in FIN 4243 at the University of Florida.