# The Calculation of Bond Duration

In the previous article, we have already learned about why the duration is considered to be a very important concept while investing in fixed income securities. However, it is important for the readers to understand how exactly duration is calculated and how it impacts the valuation of fixed income securities.

In this article, we will have a closer look at the methodology which is commonly used to calculate bond duration.

### Weighted Average of Cash Flows

Before we delve into the nitty-gritty of the calculation, it is important to understand what we are trying to achieve by calculating duration. As mentioned in the previous article, cash flows that are farther away in terms of time are impacted more by interest rate changes. Duration, therefore, tries to calculate a weighted average of the cash flows. The time factor is used as a weight to ensure that the negative impact related to cash flows that are farther into the future is captured.

1. Step #1: Calculate the Cash Flows:

The first step of the model is to calculate the nominal cash flows which will accrue to the bondholder. These nominal payments must then be converted to real payments by discounting them using the appropriate discount rate. This includes all the coupon payments as well as the principal repayment which generally occurs towards the maturity of the bond. It is important for investors to have complete visibility over the cash flow schedule of the bond before they begin calculating duration.

2. Step #2: Calculate the Proportion of Cash Flow in Every Coupon:

The next step is to calculate the proportion of cash flow that is being received in every coupon payment. For instance, the total value of the bond is \$100, but the first coupon payment accounts for \$7, then it can be said that 7% of the money will be received in that particular coupon payment. In order to calculate the duration, the proportion of funds being received as a result of every payment need to be calculated.

3. Step #3: Multiply by Time Factor:

The next step is to multiply each cash flow with the time factor. This ensures that the cash flows which occur later in the future will show greater sensitivity to interest rate changes as composed of cash flows that are closer.

4. Step #4: Summation:

The last and final step is to sum the weighted average of all cash flows. The product calculated above for each and every cash flow which is due to happen in the future is added in order to calculate a final sum. This sum represents the bond duration.

### Duration of a Portfolio

It is common for individual investors as well as mutual funds to calculate the duration of an entire portfolio instead of an individual bond. The method used for calculation is the same. However, given a large number of bonds in any portfolio, the calculation can turn out to be quite complex. This is why such investors do not generally use spreadsheets for their calculations. Instead, they have specialized tools to calculate the duration of the entire portfolio.

It is common for such investors to have a target bond duration. For example, if the investor believes that the interest rates will fall in the future, then they will increase the overall duration of their portfolio. This will help them lock in higher yields for a longer period of time.

The opposite of this is also true. If investors feel that interest rates will rise in the future, then they try to reduce the duration of the portfolio in order to reduce the impact of this interest rate fall.

### Factors Which Affect the Duration of a Bond

The duration of a bond is affected by the following factors.

• If a large portion of the coupon payments of the bond occurs in the early stages of the bond’s life, then such a bond has a lower duration. The manner in which coupons are structured has a huge impact on the duration of the bond.

• The time required for the bond to mature also has a huge impact on the duration of the bond. Every time, the bond makes a coupon payment, it reaches closer to maturity. Hence, the duration of a bond shortens as it reaches closer to maturity. On the other hand, if maturity is very far into the future, then the duration is higher.

• If a bond has a sinking fund, then such a bond pays extra cash flow in the early years of its existence. As a result of this extra cash flow, bonds that have sinking fund provisions have a lower duration as compared to other bonds.

• It is common for bonds that have call features to pay the principal at a faster rate. Since callable bonds are generally front-loaded i.e. they pay the principal faster, these bonds also tend to have a lower duration.

• The yield to maturity of a bond also has an inverse relationship with its duration. Hence, the higher the yield to maturity, the lower is the duration. This is because the cash flows which are farther into the future are overshadowed by the higher cash flows which are scheduled to happen in the near future.

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