Introduction

In the world of fixed-income investments, understanding the concept of bond duration is crucial. Bond duration measures how sensitive a bond’s price is to changes in interest rates. This chapter delves into the three primary duration metrics: Macaulay duration, modified duration, and dollar duration. By comprehending these metrics, investors can make informed decisions about their fixed-income portfolios and manage interest rate risk more effectively.

Macaulay Duration

Macaulay duration is a fundamental concept that represents the weighted average time it takes to receive the bond’s cash flows, considering both coupon payments and the final principal repayment. Mathematically, it is calculated as the sum of the present value of each cash flow multiplied by the time until that cash flow is received, divided by the bond’s current price. This duration metric provides insight into the bond’s effective maturity and helps investors understand the timing of their cash flows. For example, consider a 5-year bond with annual coupon payments and a final principal repayment. If the bond’s cash flows are weighted more towards the distant future, the Macaulay duration will be greater.

Mathematically, it is calculated as follows:

$$\text { Macaulay Duration }=\frac{\sum_{t=1}^n \frac{C F_l}{(1+Y T M)^t \times P}}{P}$$ Where:

• $n$ is the number of periods until each cash flow is received.
• $C F_t$ is the cash flow at time $t$.
• $Y T M$ is the yield to maturity.
• $P$ is the current market price of the bond.

Example: Consider a 5-year bond with a coupon rate of 5% and a yield to maturity of 4%. The bond pays annual coupon payments and has a face value of USD 1,000. The cash flows are USD 50 each year for coupon payments and USD 1,000 for the principal repayment. If the bond’s current market price is USD 1,050, the Macaulay duration can be calculated as follows:

$$\text{Macaulay Duration} =\frac{50}{(1+0.04)^1 \times 1050}+\frac{50}{(1+0.04)^2 \times 1050}+ \ldots \times + \frac{1050}{(1+0.04)^5 \times 1050} ≈ \text{4.42 years} $$

Modified Duration

Modified duration is a modified version of Macaulay duration that quantifies the bond’s price sensitivity to changes in yield. It is a percentage change in bond price for a 1% change in yield. Modified duration is a valuable tool for assessing interest rate risk, as it provides an estimate of the approximate percentage change in bond price for a given change in yield. Mathematically, modified duration is calculated by dividing the Macaulay duration by the sum of one plus the yield to maturity. For instance, a bond with a modified duration of 5 years will see its price decrease by approximately 5% for a 1% increase in yield.

Modified duration is calculated as follows:

$$\text{Modified Duration} = \dfrac{\text{Macaulay Duration}}{\text{1+YTM}}$$

Example: If the Macaulay duration of a bond is calculated to be 5 years and the yield to maturity is 6%, the modified duration would be:

$$\text{Modified Duration} = \dfrac{\text{5}}{\text{1+0.06}}≈ \text{4.72 years}$$

Dollar Duration

Dollar duration extends the concept of modified duration by incorporating the bond’s current market value. It represents the expected change in the bond’s value for a one-unit change in yield and is measured in dollars. Dollar duration is calculated by multiplying the modified duration by the bond’s current price. This metric helps investors estimate the potential impact of interest rate changes in terms of actual monetary value. For example, a bond with a dollar duration of USD 10,000 implies that a 1% increase in yield would lead to an approximate USD 10,000 decrease in the bond’s price.

Dollar duration is calculated as follows:

$$\text{Dollar Duration} = \text{Modified Duration } \times P$$ Where $P$ is the current market price of the bond.

Example: If a bond has a modified duration of 4.5 years and a current market price of USD 1,200, the dollar duration would be:

$$\text{Dollar Duration} = 4.5 \times 1200 = \text{ USD 5,400}$$