Duration Formula

What is the Duration Formula?

The formula for the duration is a measure of a bond’s sensitivity to changes in the interest rate, and 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 inflowDiscounted Future Cash InflowDiscounted cash flow analysis is a method of analyzing the present value of a company, investment, or cash flow by adjusting future cash flows to the time value of money. This analysis assesses the present fair value of assets, projects, or companies by taking into account many factors such as inflation, risk, and cost of capital, as well as analyzing the company's future performance.read more. The cash inflow basically comprises of coupon payment and the maturity at the end. It is also known as 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.

Mathematically, the equation for the duration is represented as below,

Duration Formula = [ ∑in-1 i*Ci/(1+r)i + n*M/(1+r)n] / [∑in-1  Ci/(1+r)i + M/(1+r)n]

where,

  • C = Coupon payment per period
  • M= Face or Par value
  • r =Effective periodic rate of interest
  • n = Number of periods to maturity

Further, the denominator, which is the summation of the discounted cash inflow of the bond, is equivalent to the present value or price of the bond. Therefore, the formula for the duration can be further simplified as below,

duration formula 2

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For eg:
Source: Duration Formula (wallstreetmojo.com)

Explanation of the Duration Formula

The equation for the duration can be computed by using the following steps:

  1. Firstly, the face or par value of the bond issuance is figured out, and it is denoted by M.

  2. Now, the coupon payment of the bond is calculated based on the effective periodic rate of the interest. Then the frequency of the coupon payment is also determined. The coupon payment is denoted by C, and the effective periodic rate of interest is denoted by r.

  3. Now, the total number of periods till maturity is computed by multiplying the number of years till maturity and the frequency of the coupon payments in a year. The number of periods till maturity is denoted by n. Also, the time of the periodic payment is noted, which is denoted by i.

  4. Finally, based on the available information, the equation for the duration can be derived as below,


    duration formula 1

Examples of Duration Formula (with Excel Template)

Let’s see some simple to advanced types of durationTypes Of DurationDuration 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 is a bond 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 duration measures, viz. Macaulay Duration, Modified Duration, and Effective Duration.read more formula to understand it better.

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

Duration Formula Formula – Example #1

Let us take an example of a bond with annual coupon payments. Let us assume that company XYZ Ltd has issued a bond having a face value of $100,000, carrying an annual coupon rate of 7% and maturing in 5 years. The prevailing market rate of interest is 10%.

Given, M = $100,000

  • C = 7% * $100,000 = $7,000
  • n = 5
  • r = 10%

The denominator or the price of the bond is calculated using the formula as,

duration formula examplee 1.1
  • Bond price = 84,281.19

Calculation of the numerator of Duration formula is as follows –

duration formula examplee 1.2

= (6,363.64 + 11,570.25 + 15,777.61 + 19,124.38 + 310,460.70)

= 363,296.50

Therefore, the calculation of duration of the bond will be as below,

duration formula example 1.3

Duration = 363,296.50 / 84,281.19

  • Duration =4.31 years

Duration Formula Formula – Example #2

Let us take an example of a bond with annual coupon payments. Let us assume that company XYZ Ltd has issued a bond having face value of $100,000 and maturing in 4 years. The prevailing market rate of interest is 10%. Calculate the bond duration for the following annual coupon rate: (a) 8% (b) 6% (c) 4%

Given, M = $100,000

  • n = 4
  • r = 10%

Calculation for Coupon Rate of 8%

Coupon payment (C)= 8% * $100,000 = $8,000

The denominator or the price of the bond is calculated using the formula as,

duration formula examplee 2.1
  • Bond price = 88,196.16

Calculation of the numerator of the Duration formula will be as follows –

duration formula examplee 2.2

= 311,732.81

Therefore, the calculation of duration of the bond will be as below,

duration formula example 2.3

Duration = 311,732.81/ 88,196.16

  • Duration = 3.53 years

Calculation for Coupon Rate of 6%

Coupon payment (C) = 6% * $100,000 = $6,000

The denominator or the price of the bond is calculated using the formula as,

duration formula example 2.4
  • Bond price = 83,222.46

Calculation of the numerator of the Duration formula will be as follows –

duration formula example 2.5

= 302,100.95

Therefore, the calculation of the duration of the bond will be as below,

duration formula examplee 2.6

Duration = 302,100.95 / 83,222.46

  • Duration = 63 years

The calculation for Coupon Rate of 4%

Coupon payment = 4% * $100,000 = $4,000

The denominator or the price of the bondPrice Of The 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 is calculated using the formula as,

duration formula examplee 2.7
  • Bond price = 78,248.75

Calculation of the numerator of the Duration formula will be as follows –

duration formula example 2.8

= 292,469.09

Therefore, the calculation of the duration of the bond will be as below,

duration formula example 2.9

Duration Formula = 292,469.09 / 78,248.75

  • Duration = 3.74 years

From the example, it can be seen that the duration of a bond increases with the decrease in coupon rate.

Relevance and Use of Duration Formula

It is important to understand the concept of duration as it is used by bond investors to check a bond’s sensitivity to changes in interest rates. The duration of a bond basically indicates how much the market priceMarket PriceMarket price refers to the current price prevailing in the market at which goods, services, or assets are purchased or sold. The price point at which the supply of a commodity matches its demand in the market becomes its market price.read more of a bond will change owing to the change in the rate of interest. It is noteworthy to remember that rate of interest and bond price move in opposite directions, and as such, bond price rise when the rate of interest falls and vice versa.

In case investors are seeking benefits from a fall in interest rate, the investors will intend to buy bonds with a longer duration, which is possible in the case of bonds with lower coupon payment and long maturity. On the other hand, investors who want to avoid the volatility in interest rate, the investors will be required to invest in bonds that have a lower duration or short maturity and higher coupon payment.

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This has been a guide to Duration Formula. Here we discuss how to calculate the Duration of Bond using practical examples and a downloadable excel template. You can learn more about financial analysis from the following articles –

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