# What is meant by the Duration of a Bond?

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The duration of a bond is its weighted average of times of cash flow. The calculation of duration provides importance to cash flows and their timing. The weight is calculated for the present value of cash flow to the bond value.

Therefore, three types of calculations are involved in computing the duration of a bond −

• Calculating PV for each cash flow.

• Divide each cash flow by the aggregate of all cash flows for getting weights.

• Multiply years by each cash flow and summate for the duration.

Note Two bonds with equal face value but different coupon rates and cash flow patterns will have different duration. As interest rates are one of the most influential factors in calculating the bond's duration and value, any change in the interest affects the duration and face value of the bond. The general idea is that with growing interest rates, the value of the bond will go down.

Duration is important for risk and portfolio management. By aligning the durations, and investors can change the portfolio composition to get the expected level of interest.

## Types of Duration and their Calculation

Duration can have several forms. However, the following three are the most common −

• Macaulay Duration
• Modified Duration
• Effective Duration

## Macaulay Duration

It is a weighted average of the times till the cash flows of a fixed-income instrument are gained. Macaulay duration gives the duration for an investor to be repaid by the bond's total price by the total cash flow. Macaulay Duration is expressed in years.

$$Macaulay\:Duration = \displaystyle\sum\limits_{i}^n t_{i}×\frac{𝑃𝑉_{𝑖}}{𝑉}$$

Where,

• 𝑡𝑖 = The time until the ith cash flow from the asset will be received

• 𝑃𝑉𝑖 = The present value of the ith cash flow from the asset

• 𝑉 = The present value of all cash flows from the asset

## Modified Duration

It has similarity to Macaulay duration but the expression of modified duration is done in percentage. It can be used with different vehicles such as equity and securities that can be considered as a function of yield.

$$Modified\:Duration = \frac{Macaulay\:Duration}{(1 + \frac{YTM}{𝑛})}$$

Where,

• YTM = The yield to maturity of a bond

• n = The frequency of compounding

## Effective Duration

Effective duration has the capability of embedding callable options. It considers the bond's price fluctuations in terms of YTM. That is, the bond calculates the fluctuations in the expected cash flows.

$$Effective\:Duration = \frac{𝑉_{−𝜕𝑦} − 𝑉_{+𝜕𝑦}}{2 × 𝑉_{0} × 𝜕𝑦}$$

Where −

• 𝑉−𝜕𝑦 = The bond’s value if yield falls by y%

• 𝑉+𝜕𝑦 = The bond’s value if yield rises by y%

• 𝑉0 = The present value of all cash flows of the bond

• 𝜕𝑦 = The yield change