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Fixed Income Volatility: Macaulay Duration

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Macaulay Duration of a Bond

Macaulay Duration is a widely used measure in bond analysis that helps assess the sensitivity of a bond's price to changes in interest rates. It provides a valuable tool for estimating the average time it takes to receive the bond's cash flows, including both coupon payments and the return of principal. Understanding the Macaulay duration of a bond, its relationship with modified 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.

Introduction

Macaulay Duration is a widely used measure in bond analysis that helps assess the sensitivity of a bond's price to changes in interest rates. It provides a valuable tool for estimating the average time it takes to receive the bond's cash flows, including both coupon payments and the return of principal. Understanding the Macaulay duration of a bond, its relationship with modified 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.

Macaulay Duration Calculation

The Macaulay duration is calculated as the weighted average time to receive cash flows, where each cash flow is weighted by its present value. The general formula for Macaulay duration is as follows:

Macaulay Duration = (CF1 * t1 + CF2 * t2 + ... + CFn * tn) / (Bond Price)

Where:

  • CFi = Cash flow in period i
  • ti = Time until receipt of cash flow i
  • n = Total number of cash flows
  • Bond Price = Present value of all cash flows

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%

To calculate the Macaulay duration of this bond, we first determine the present value of each cash flow and its weighted contribution:

Period 1:

  • Cash flow = $50 (coupon payment)
  • Time (t) = 1 year
  • Present value = $50 / (1 + 0.04) = $48.08
  • Weighted contribution = $48.08 * 1 = $48.08

Period 2:

  • Cash flow = $50 (coupon payment)
  • Time (t) = 2 years
  • Present value = $50 / (1 + 0.04)^2 = $46.32
  • Weighted contribution = $46.32 * 2 = $92.64

Period 3:

  • Cash flow = $50 (coupon payment)
  • Time (t) = 3 years
  • Present value = $50 / (1 + 0.04)^3 = $44.73
  • Weighted contribution = $44.73 * 3 = $134.19

Period 4:

  • Cash flow = $50 (coupon payment)
  • Time (t) = 4 years
  • Present value = $50 / (1 + 0.04)^4 = $43.27
  • Weighted contribution = $43.27 * 4 = $173.08

Period 5:

  • Cash flow = $1,050 ($1,000 face value + $50 coupon payment)
  • Time (t) = 5 years
  • Present value = $1,050 / (1 + 0.04)^5 = $916.17
  • Weighted contribution = $916.17 * 5 = $4,580.83

Bond Price = $48.08 + $92.64 + $134.19 + $173.08 + $4,580.83 = $5,028.82

Macaulay Duration = ($48.08 + $92.64 + $134.19 + $173.08 + $4,580.83) / $5,028.82 ≈ 4.93 years

The Macaulay Duration of this bond is approximately 4.93 years.

Comparison to Modified Duration

Modified duration is a modified version of Macaulay duration that measures the percentage change in bond price in response to a 1% change in yield. It provides a more direct measure of bond price sensitivity to interest rate changes. Modified duration is calculated as:

Modified Duration = Macaulay Duration / (1 + Yield)

The key difference is that modified duration accounts for the impact of compounding and adjusts the duration for changes in yield.

Benefits of Macaulay Duration

  1. Time Measurement: Macaulay Duration provides a measure of the average time it takes to receive the bond's cash flows, helping investors assess the bond's repayment timeline.
  2. Price Sensitivity: Macaulay Duration provides an estimate of the bond's price sensitivity to changes in interest rates. Bonds with longer durations are more sensitive to interest rate changes.

Drawbacks of Macaulay Duration

  1. Limited Accuracy: Macaulay Duration assumes a linear relationship between bond price and yield changes, which may not hold true for large interest rate fluctuations.
  2. Macaulay Duration vs. Investment Horizon: Macaulay Duration may not align with an investor's specific investment horizon. It measures the bond's repayment timeline, which may differ from an investor's holding period.

Common Misuses

  1. Ignoring Convexity: Macaulay Duration does not account for convexity, which measures the curvature of the bond's price-yield relationship. Ignoring convexity can lead to inaccuracies in predicting bond price changes for larger interest rate movements.
  2. Absolute Comparison: Comparing Macaulay Durations across different bonds is misleading. Macaulay Duration is a relative measure of price sensitivity within a specific bond, but it cannot be directly compared between bonds with different cash flow patterns or maturities.

Conclusion

Macaulay Duration is a valuable tool for bond investors, providing an estimate of the average time to receive cash flows and measuring price sensitivity to interest rate changes. While it offers useful insights, it should be complemented with modified duration and considerations of convexity for accurate risk assessment. Understanding the benefits, drawbacks, and potential misuses of Macaulay 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.