Duration

Updated on January 5, 2024
Article byWallstreetmojo Team
Edited byAnkush Jain
Reviewed byDheeraj Vaidya, CFA, FRM

Duration Meaning

Duration is a risk measure used by market participants to measure the interest rate sensitivity of a debt instrument, e.g., a Bond. It tells how sensitive a bond is with respect to the change in interest rates. This measure can be used for comparing the sensitivities of bonds with different maturities. There are three different ways to arrive at duration measures, viz. Macaulay DurationMacaulay DurationMacaulay Duration is the amount of time it takes for an investor to recover his invested money in a bond through coupons and principal repayment. This is the weighted average of the period the investor should stay invested in the security in order for the present value of the cash flows from the investment to be equal to the amount paid for the bond.read more, Modified DurationModified DurationModified Duration tells the investor how much the price of the bond will change given the change in its yield. To calculate it, the investor needs to calculate Macauley duration which is based on the timing of the cash flow.read more, and Effective DurationEffective DurationEffective Duration measures the duration of security with options embedded. It helps evaluate the price sensitivity and risk of hybrid securities (bonds and options) to a change in the benchmark yield curve. The modified duration can be called a yield duration.read more.

Duration

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Bond duration is vastly affected by factors such as coupon rate and time to maturity. However, the fundamental idea is that the bond’s value decreases with an increase in its duration. As the time frame rises, the bond’s price will experience a decrease and the interest rates might be higher.

Key Takeaways

  • Duration measures a bond’s price sensitivity to changes in interest rates.
  • Duration represents the weighted average time it takes to receive the bond’s cash flows, including coupon payments and the return of principal.
  • Investors and portfolio managers widely use duration to manage interest rate risk. By understanding the duration of their bond holdings, investors can make informed decisions about the impact of interest rate changes on their portfolio
  • Duration and interest rates have an inverse relationship. As interest rates rise, the duration of a bond decreases, indicating that its price will be more sensitive to interest rate changes. Conversely, as interest rates decline, the duration increases, indicating reduced price sensitivity.

Duration Explained

Duration is a measurement of the sensitivity of a debt instrument such as bonds and other debt instrument with relation to interest rates. The interest rates changes are majorly reliant on the time to maturity and coupon rate of the bond.

As the time frame of maturity rises, the base value of the bond experiences and dip and the interest rates are most likely to rise.

As bond price is inversely proportional to yield, it is highly sensitive to how yield changes. The duration measures defined above quantify the impact of this sensitivity on bond price.

A bond with a longer maturity will have a longer duration; hence, it is more sensitive to changes in interest rates.

A bond with a lower coupon rate will be more sensitive than a bond with a bigger coupon. However, the reinvestment riskThe Reinvestment RiskReinvestment risk refers to the possibility of failing to induce the profits earned or cash flows into the same scheme, financial product or investment. It even states the uncertainty of not getting the similar returns when such funds are invested in a new investment opportunity.read more will be higher in the case of a small coupon bond.

Effective duration calculator is an approximate measure of duration, and for an option-free bond, the modified and effective duration will be almost the same. Modified duration quantifies the sensitivity by specifying the percentage change in bond price for every 100-bps change in the interest rates.

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Formula

There are three different types to calculate durationCalculate DurationThe duration formula measures a bond’s sensitivity to changes in the interest rate. It is calculated by dividing the sum product of discounted future cash inflow of the bond and a corresponding number of years by a sum of the discounted future cash inflow.read more measures.

Each of these types of formula has its intricacies and can be used to calculate the bond duration and that of other debt securities. Let us understand each of the formulas through the discussion below.

3 Ways to Calculate Duration

#1 – Macaulay Duration

The Mathematical Definition: “Macaulay Duration of a coupon-bearing bond is the weighted average time period over which the cash flows associated with the bond are received.”  In simple terms, it tells how long it will take to realize the money spent to buy the bond in the form of periodic coupon payments and the final principal repayment.

formula #1

where:

  • Ct: Cashflow at time t
  • r: Interest rates/ Yield to maturity
  • N: Residual Tenure in Years
  • t: Time/ Period in Years
  • D: Macaulay Duration

#2 – Modified Duration

The Mathematical Definition: “Modified Duration is the percentage change in Price of a BondPrice Of A BondThe bond pricing formula calculates the present value of the probable future cash flows, which include coupon payments and the par value, which is the redemption amount at maturity. The yield to maturity (YTM) refers to the rate of interest used to discount future cash flows.read more for a unit change in yield.” It measures the price sensitivityPrice SensitivityPrice Sensitivity, also known and calculated by Price Elasticity of Demand, is a measure of change (in percentage term) in the demand of the product or service compared to the changes in the price. It is used widely in the business world to decide the pricing of a product or study consumer behavior.read more of a bond to changing interest rates. The interest rates are picked from the market yield curve, adjusted for the riskiness of the bondBondBonds refer to the debt instruments issued by governments or corporations to acquire investors’ funds for a certain period.read more and the appropriate tenure.

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

Where:

#3 – Effective Duration

If a bond has some options attached to it, i.e., the bond is puttable or callable before maturity. Effective duration takes into consideration the fact that as interest rate changes, the embedded options may be exercised by the bond issuer or the investor, thereby changing the cash flows and hence the duration.

Deffective = – [Pup – Pdown / 2 * Δi * P]

Where:

  • Pup: Bond price with yield up by Δi
  • Pdown: Bond price with yield down by Δi
  • P: Bond price at current yield
  • Δi: Change in yield (usually taken as 100 bps)

Example

Let us understand the duration calculator with the help of an examples. This detailed calculation will help us understand the intricate details about the concept and how investors can calculate the duration of their debt instruments.

Consider a bond with the face value of 100, paying a semi-annual coupon of 7% PA compounded annually, issued on 1 Jan 19 and with a tenure of 5 years and trading at par, i.e., the price is 100 and yield is 7%.

You can download this Duration Excel Template here – Duration Excel Template

Calculation of three types of duration is as follows –

duration example

Please download the above Excel template for detailed calculation.

Limitations

Although highly used as a duration calculator and one of the prominent risk measures for fixed-income securities, the duration is restricted for wider use because of underlying assumptions of interest rates movement. It assumes:

  • Market yield will be the same for the entire tenure of the bond
  • There will be a parallel shift in market yield, i.e., Interest rate changes by the same amount for all the maturities.

Both limitations are handled by considering regime-switching models, which provide for the fact that there can be different yields and volatility for different periods, thereby ruling out the first assumption. And by dividing the tenure of bonds into certain key periods basis, the availability of rates or basis the majority of cash flows lying around certain periods. This helps in accommodating non-parallel yield changes, hence taking care of the second assumption.

Advantages

Let us understand the advantages of using bond duration calculators and for other debt instruments. These pointers will give us a deeper understanding of the concept.

Disadvantages

Despite various advantages, there are a few factors that prove to be hassle or pain points for investors. Let us understand the disadvantages of using a duration calculator through the explanation below.

ΔB/B = -D Δy + 1/2 C(Δy)2

Where

The Convexity in the above formula can be calculated using the below formula:

CE = P + P+ – 2P0 / 2(Δy)2 P0

Where

  • CE : Convexity of the bond
  • P_: Bond Price with yield down by Δy
  • P+: Bond Price with yield up by Δy
  • Po: Original bond price
  • Δy: Change in yield (usually taken as 100 bps)

Frequently Asked Questions (FAQs)

Is duration the same as maturity?

No, duration and maturity are not the same. Duration measures the bond’s price sensitivity to interest rate changes, while maturity refers to the time it takes to repay its principal. Course considers both coupon payments and the timing of the principal repayment, providing a more comprehensive measure of price sensitivity.

How is duration calculated?

Duration can be calculated using various formulas, such as Macaulay, modified, or effective duration. The specific calculation depends on coupon payments, time to maturity, and the bond’s yield-to-maturity. Financial software or tools can provide accurate duration calculations.

Can duration be negative?

No, the duration cannot be negative. Duration is a positive value representing the weighted average time until the bond’s cash flows are received. A negative duration would not have a meaningful interpretation in the context of bond pricing and interest rate sensitivity.

Recommended Articles

This has been a guide to what is Duration and its definition. Here we discuss the 3 different ways to arrive at duration measures along with an example, advantages & disadvantages. You can learn more about fixed income from the following articles –

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