# Effective Duration Calculator

Created by Wei Bin Loo
Reviewed by Dominik Czernia, PhD and Rijk de Wet
Last updated: Jan 18, 2024

With this effective duration calculator, we aim to help you calculate the effective duration of a bond. This metric allows you to assess the interest rate risk of your bond investments. To understand more on this topic, you can check out our bond price calculator and interest rate calculator.

We have written this article to help you understand what effective duration is and how to apply the bond effective duration formula. We will also demonstrate some practical examples to help you understand the concept.

## What is effective duration?

Effective duration is one of the best metrics used to analyze the interest rate sensitivity of bonds with embedded options. Bonds can be embedded with call or put options. Callable bonds allow the issuers to buy back the bonds at a specified price, whereas putable bonds enable the bondholders to sell the bonds back to the issuers at a specified price.

This flexibility has made the cash flows unpredictable. For instance, when the interest rate drops, the bond price tends to increase. The issuer will then trigger the options to buy back the callable bond, hence lowering the cash flows to the investors, and vice versa.

Thus, before investing in bonds embedded with options, it is critical to understand its sensitivity to interest rate changes. The bond effective duration is a metric used just for that.

After understanding what the effective duration is, we can now discuss how to calculate it.

## How to calculate effective duration? The effective duration calculator

To understand more about the effective duration definition, let's take Bond Alpha with the following properties as an example:

• Face value: $1,000 • Annual coupon rate: 5% • Coupon Frequency: Annual • Years to maturity: 10 years • Yield to maturity (YTM): 8% • Yield differential: 1% The bond effective duration calculation requires four steps: 1. Calculate the coupon per period. The first step is to calculate the coupon per period using the following formula: coupon per period = face value × coupon rate / frequency As this is an annual bond, the frequency = 1, and the coupon for Bond A is: $1,000 × 5% / 1 = $50 2. Calculate the bond price We can calculate the bond price using this formula: $\footnotesize\qquad\!\!\text{bond price} = \sum_{k=1}^n\left[\rm{cf} / (1 + r)^k\right]$ with $\rm{cf}$ representing the cash flows, $r$ the yield to maturity ($\rm{YTM}$) and $n$ the years to maturity. The bond price calculation looks like this: $\footnotesize \qquad \!\! \begin{split} \text{bond price}& = \\ & \50 / (1 + 8\%)^1 \\ & + \50 / (1 + 8\%)^2 \\ & + \50 / (1 + 8\%)^3 \\ & + ...\ \\ & + \50 / (1 + 8\%)^9 \\ & + \1050 / (1 + 8\%)^{10} \\ &= \798.70 \end{split}$ 1. Calculate the bond price after shifting the bond yield We need to calculate the upwards bond price and downwards bond price of the bond to analyze the bond price's sensitivity to a change in interest rates. The upwards bond price is the bond price when the bond yield shift downwards by the amount of yield differential, whereas the downwards bond price is the bond price when the yield shift upwards. You can check out our bond yield calculator to understand more. For the calculation of upwards bond price, we use the bond yield of 8% - 1% = 7%, and for the calculation of downwards bond price, we will use 8% + 1% = 9% as the bond yield. Hence, the calculation of upwards bond price is: $\footnotesize \begin{split} \qquad \!\! \text{upwards } & \text{bond price} = \\ &\50 / (1 + 7\%)^1 \\ & +\50 / (1 + 7\%)^2 \\ & +\50 / (1 + 7\%)^3 \\ & + ... \ \\ & + \50 / (1 + 7\%)^9 \\ & + \1050 / (1 + 7\%)^{10} \\ &= \859.53 \end{split}$ On the other hand, the calculation of downwards bond price is: $\footnotesize \begin{split} \qquad \!\! \text{downwards } & \text{bond price} = \\ &\50 / (1 + 9\%)^1 \\ &+ \50 / (1 + 9\%)^2 \\ & +\50 / (1 + 9\%)^3 \\ & + ...\ \\ & +\50 / (1 + 9\%)^9 \\ & +\1050 / (1 + 9\%)^{10} \\ & =\743.29 \end{split}$ 1. Calculate the effective duration The last step is to calculate the effective duration using the bond effective duration definition. Using the formula below, the Bond Alpha's effective duration is ($859.53 - $743.29) / (2 ×$798.70 × 1%) = 7.277.

$\footnotesize\quad \text{effective duration} = \\ \qquad (\text{upwards bond price} \\ \qquad\quad -\ \text{downwards bond price}) \\ \qquad / (2 \times \text{bond price} \\ \qquad\quad \times\ \text{yield differential})$

## How to interpret the effective duration?

Now that we understand how to use the effective duration calculator, let's talk about the ways to interpret the results:

1. From the example above, the bond effective duration is 7.277. But what does this mean? As explained above, the effective duration measures the sensitivity of the bond price to the change in interest rates. Hence, when the effective duration is said to be 7.277, it means that when there is a change in the interest rate of 100 basis points (equivalent to 1%), the bond price will change by 7.277%.

2. Essentially, if the interest rate goes up by 1%, the bond price will decrease by 7.277%. And if the interest rate goes down by 1%, the bond price will increase by 7.277%.

3. Finally, it is essential to note that this is not 100% accurate as the relationship between the interest rates and bond price is not linear. To calculate a more precise version, we need to calculate the bond's effective convexity.

## FAQ

### How do I calculate the effective duration?

You can calculate the effective duration using the following steps:

1. Find the coupon per period.

2. Determine the bond price.

3. Calculate the bond price after shifting the bond yield by yield differential.

4. Apply the effective duration formula:

effective duration = (upwards bond price - downwards bond price) / (2 × bond price × yield differential)

### What is an embedded option?

The embedded options in a bond allows either the bondholder to sell the bond back to the issuer at a specified price or the issuer to buy the bond back at a specified price.

### Does effective duration measure interest rate risk?

Yes, the effective duration is the metric used to measure the interest rate risk of bonds with embedded options.

### What is the effective convexity?

Similar to the effective duration, the effective convexity measures the sensitivity of the bond price to the changes in interest rates. But, unlike the effective duration, the effective duration takes into account the non-linear relationship between the bond prices and the interest rates.

Wei Bin Loo
Coupon per period
Face value
$Annual coupon rate % Coupon frequency Annually Coupon per period$
Annual coupon
$Bond price Years to maturity year(s) Yield to maturity (YTM) % Bond price$
Effective duration
Yield differential
%
Effective duration
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