Introduction

Effective duration is a fundamental concept in fixed income analysis. It measures the sensitivity of a fixed income security’s price to changes in yield or interest rates. This chapter explores the definition, calculation, and interpretation of effective duration and its significance in risk management for fixed income investments.

Effective Duration

Effective duration quantifies how a bond’s price is expected to change in response to a 1% change in its yield. It takes into account the bond’s cash flows, maturity, and the impact of changes in interest rates. Unlike Macaulay duration, which measures the weighted average time until a bond’s cash flows are received, effective duration considers the non-linear relationship between bond prices and yields.

Effective duration is calculated as follows:

$$\text{Effective Duration}=-\frac{1}{P}\cdot \frac{\partial P}{\partial y}$$

Where:

• $P$= Current Price of the Bond
• $\frac{\partial P}{\partial y}$= The change in the bond’s price for a $1\mathrm{\%}$ change in yield $(\mathrm{\Delta }y)$

Interpreting Effective Duration

1) Magnitude of Effective Duration: The higher the effective duration, the greater the bond’s price sensitivity to changes in yield. Longer-term bonds typically have higher effective durations.

2) Effective Duration and Interest Rates: If a bond has an effective duration of 5 years, a 1% increase in interest rates would lead to an approximate 5% decrease in its price.

3) Comparing Bonds: Effective duration allows for easy comparison of the price sensitivity of different bonds. Bonds with higher effective durations are riskier in terms of interest rate changes.

Example: Let’s consider a 10-year bond with a current price of $$ 1,000$andanannualyieldof$5 \%$.Iftheyieldincreasesby$1 \%$, we can calculate the effective duration as follows:

$$\begin{array}{rl}& \text{ Effective Duration }=-\frac{1}{\$1,000}\cdot \frac{\partial (\$1,000)}{\partial 0.01}\\ & \text{ Effective Duration }=-\frac{1}{\$1,000}\cdot (\$1,000-\$950)=0.05\end{array}$$

In this example, the effective duration is 0.05 years, indicating that for a 1% increase in yield, the bond’s price is expected to decrease by approximately 5%.