**Modified Duration Formula (Table of Contents)**

## What is the Modified Duration Formula?

The term “Modified Duration” refers to a metric that helps in assessing the expected change in the value of security due to change in the prevailing interest rates. In other words, modified duration is a measure of the sensitivity of a bond to changes in interest rate.

The formula for the modified duration is expressed as Macaulay duration (a.k.a. just “duration”) divided by one plus effective yield to maturity (YTM). Mathematically, the formula for the modified duration is represented as below,

**Modified Duration = Macaulay Duration / [1 + (YTM / n)]**

Where

**n:**Number of Coupon Periods per Year

Please note that Macaulay duration can be computed using the following formula,

**Macaulay Duration = ∑**

^{t-1}_{i}[((i*C_{i})/(1+YTM/n)^{i}) + ((t*M)/(1+YTM/n)^{t})] / Price of the BondWhere,

**C:**Coupon Payment per Period**M:**Par Value**t:**Number of Periods until Maturity

**Example of Modified Duration Formula (With Excel Template)**

Let’s take an example to understand the calculation of Modified Duration in a better manner.

#### Modified Duration Formula – Example #1

**Let us take the example of a security that pays an annual coupon at 7%. The par value of the 5-year security is $1,000 and its current market price is $886. Calculate the modified duration of the security if the YTM is 10%.**

**Solution:**

Calculate cash flow as

Similarly, calculated as below

Calculate cf*t as

Similarly, calculated as below

Calculate Discount Factor as

Similarly, calculated as below

Calculates discounted CF as

Similarly, calculated as below

Macaulay Duration is calculated using the formula given below

**Macaulay Duration = ∑ ^{t-1}_{i} [((i*C_{i})/(1+YTM/n)^{i}) + ((t*M)/(1+YTM/n)^{t})] / Price of the Bond**

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- Macaulay Duration= [ 70*1/(1+10%)
^{1}+ 70*2/(1+10%)^{2}+70*3/(1+10%)^{3}+ 70*4/(1+10%)^{4}+1070*5/(1+10%)^{5}] / 886

- Macaulay Duration =
**4.35**

Macaulay Duration is calculated using the formula given below

**Modified Duration = Macaulay Duration / [1 + (YTM / n)]**

- Macaulay Duration = 4.34 / [1 + 10%]
- Macaulay Duration =
**3.95**

Therefore, for every 1% change in interest rate, the price of the security would inversely move by 3.95%.

#### Modified Duration Formula – Example #2

**Let us take the example of a 3-year coupon paying bond with a par value of $1,000. If the YTM for the bond is 5%, then calculate the modified duration of the bond for the following annual coupon rate: 4% and 6%.**

**Solution:**

Calculate cash flow as

Similarly, calculated as below

Calculate cf*t as

Similarly, calculated as below

Calculate Discount Factor as

Similarly, calculated as below

Calculates discounted CF as

Similarly, calculated as below

Similarly done for For Coupon Rate 6%.

Market Price of the bond is calculated as

For Coupon Rate 4%

- Market Price
**=**$40/(1+5%)^{1}+ $40/(1+5%)^{2}+ $1,040/(1+5%)^{3} - Market Price =
**$972.77**

For Coupon Rate 6%

- Market Price
**=**$60/(1+5%)^{1}+ $60/(1+5%)^{2}+ 1060/(1+5%)^{3} - Market Price =
**$1,027.23**

Macaulay Duration is calculated using the formula given below

**Macaulay Duration = ∑ ^{t-1}_{i} [((i*C_{i})/(1+YTM/n)^{i}) + ((t*M)/(1+YTM/n)^{t})] / Price of the Bond**

For Coupon Rate 4%

- Macaulay Duration = [ 40*1/(1+5%)
^{1}+40*2/(1+5%)^{2}+1040*3/(1+5%)^{3}] / 972.77 - Macaulay Duration =
**2.88**

For Coupon Rate 6%

- Macaulay Duration = [ 60*1/(1+5%)
^{1}+ 60*2/(1+5%)^{2}+ 1060*3(1+5%)^{3 }] / 1,027.23 - Macaulay Duration =
**2.84**

Modified Duration is calculated using the formula given below

**Modified Duration = Macaulay Duration / [1 + (YTM / n)]**

For Coupon Rate 4%

- Modified Duration = 2.88 / [1 + 5%]
- Modified Duration =
**2.75**

For Coupon Rate 6%

- Modified Duration = 2.84 / [1 + 5%]
- Modified Duration =
**2.70**

Therefore, it can be seen that the modified duration of a bond decreases with the increase in the coupon rate.

### Explanation

The formula for Modified Duration can be calculated by using the following steps:

**Step 1:** Firstly, determine the YTM of the security based on its current market price

**Step 2:** Next, determine the number of compounding per year or the number of coupon periods per year which is denoted by n.

**Step 3:** Next, compute the Macaulay duration of the security on the basis of its par value on maturity, current market price, coupon payment per period, YTM, number of coupon periods per year and number of periods until maturity.

**Step 4:** Finally, the formula for the modified duration of the security can be derived by dividing its Macaulay duration (step 3) by one plus YTM (step 1) divided by number of coupon periods per year (step 2) as shown below,

**Modified Duration = Macaulay Duration / [1 + (YTM / n)]**

### Relevance and Use of Modified Duration Formula

It is important to understand the concept of modified duration because it is useful to assess the sensitivity of the security to the change in the interest rate in the market. As such, bond investors keep a close check on this metric whenever the interesting scenario is volatile. This metric is purely based on the basic principle that interest rates and bond prices move in opposite directions. So, it essentially shows that if the interest rate in the market moves by 1% then the bond price would move by the value of modified duration in the opposite direction. For instance, in example 1 if the interest rate increases by 1% then the value of the security would decrease by 3.95%.

### Modified Duration Formula Calculator

You can use the following Modified Duration Formula Calculator

Macaulay Duration | |

YTM | |

n | |

Modified Duration | |

Modified Duration = | Macaulay Duration / [1 + (YTM / n)] |

= | 0 / [1 + (0 / 0)] = 0 |

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