Duration Finance Formula

Duration is a crucial concept in finance, particularly for fixed-income securities like bonds. It measures the sensitivity of a bond's price to changes in interest rates. In simpler terms, it tells you how much a bond's value is likely to fluctuate for every 1% change in interest rates. A higher duration implies greater price sensitivity.

What is Duration?

While often confused with maturity, duration is *not* the same. Maturity is the time until a bond's face value is repaid. Duration, on the other hand, considers the timing and size of all cash flows (coupon payments and the face value repayment) to provide a weighted average time until those cash flows are received. This weighted average reflects the bond's price sensitivity.

The Macaulay Duration Formula

The most common type of duration is Macaulay Duration, named after its creator, Frederick Macaulay. The formula is as follows:

D = ∑ [t * (CFt / (1 + y)^t)] / P

Where:

• D = Macaulay Duration
• t = Time period until the cash flow is received (e.g., year 1, year 2, etc.)
• CFt = Cash flow received at time t (coupon payment or face value)
• y = Yield to maturity (YTM) – the expected return if the bond is held until maturity
• P = Current market price of the bond
• ∑ = Summation (adding up all the terms for each time period)

Let's break down the formula:

1. Calculate the Present Value of Each Cash Flow: For each coupon payment and the face value repayment, calculate the present value by discounting it back to today using the yield to maturity (y). This is represented by `CFt / (1 + y)<sup>t</sup>`.
2. Multiply Present Value by Time: Multiply the present value of each cash flow by the time period (t) when it will be received. This gives you the time-weighted present value of each cash flow. `t * (CFt / (1 + y)<sup>t</sup>)`
3. Sum the Time-Weighted Present Values: Add up all the time-weighted present values calculated in step 2. This is the numerator of the formula. `∑ [t * (CFt / (1 + y)<sup>t</sup>)]`
4. Divide by the Bond's Price: Divide the sum from step 3 by the current market price of the bond (P). This normalizes the duration, giving you the weighted average time until the bond's cash flows are received.

Modified Duration

While Macaulay Duration provides a measure of time, Modified Duration is more directly used to estimate the percentage change in a bond's price for a given change in yield. The formula is:

Modified Duration = Macaulay Duration / (1 + (YTM / n))

Where:

• YTM = Yield to maturity
• n = Number of compounding periods per year (e.g., 2 for semi-annual bonds)

The percentage price change can then be approximated as:

Percentage Price Change ≈ - Modified Duration * Change in Yield

For example, a bond with a modified duration of 5 would be expected to decrease in value by approximately 5% if interest rates rise by 1% and increase by approximately 5% if interest rates fall by 1%.

Importance of Duration

Understanding duration is critical for several reasons:

• Risk Management: It helps investors assess and manage interest rate risk in their fixed-income portfolios.
• Portfolio Immunization: Duration can be used to construct a portfolio that is immunized against interest rate changes, meaning the portfolio's value will be less sensitive to rate fluctuations.
• Bond Pricing: Duration helps traders and analysts understand the relationship between bond prices and yields, and to identify potentially mispriced bonds.