# How to Calculate the Bond Duration (example included)

In this short post, you’ll see how to calculate the bond duration. More specifically, you’ll see how to calculate the:

• Macaulay duration; and
• Modified duration

To start, here is the formula that you can use to calculate the Macaulay duration (MacD):

```          (t1*FV)(C)                  (tn*FV)(C)           (tn*FV)
MacD = (m*PV)(1+YTM/m)mt1  + ... + (m*PV)(1+YTM/m)mtn  + (PV)(1+YTM/m)mtn
```

Where:

• m = Number of payments per period
• YTM = Yield to Maturity
• PV = Bond price
• FV = Bond face value
• C = Coupon rate
• ti = Time in years associated with each coupon payment

Once you calculated the Macaulay duration, you can then apply the following formula to get the Modified Duration (ModD):

```          MacD
ModD =  (1+YTM/m)
```

## Example of calculating the bond duration

Imagine that you have a bond, where the:

• Coupon rate is 6% with semiannually payments
• Yield to maturity (YTM) is 8%
• Bond’s price is 963.7
• Bond’s face value is 1000
• Bond matures in 2 years

(1) What is the bond’s Macaulay duration?
(2) What is the bond’s Modified duration?

### The Solution – Calculating the Macaulay duration

Since we are dealing with semiannually payments each year, then the number of payments per period (i.e., per year) is 2. Therefore, for our example, m = 2.

Here is a summary of all the components that can be used to calculate Macaulay duration:

• m = Number of payments per period = 2
• YTM = Yield to Maturity = 8% or 0.08
• PV = Bond price = 963.7
• FV = Bond face value = 1000
• C = Coupon rate = 6% or 0.06

Additionally, since the bond matures in 2 years, then for semiannual bond you’ll have a total of 4 coupon payments (one payment every 6 months), such that:

• t1 = 0.5 years
• t2 = 1 years
• t3 = 1.5 years
• t4=tn= 2 years

Pay special attention for the last period (t4=tn= 2 years) which requires both coupon payment as well as final principal repayment.

Recall that the formula to calculate Macaulay duration (MacD) is:

```          (t1*FV)(C)                  (tn*FV)(C)           (tn*FV)
MacD = (m*PV)(1+YTM/m)mt1  + ... + (m*PV)(1+YTM/m)mtn  + (PV)(1+YTM/m)mtn
```

Let’s now plug all those components within the MacD formula:

```          (0.5*1000)(0.06)           (1*1000)(0.06)           (1.5*1000)(0.06)          (2*1000)(0.06)            (2*1000)
MacD = (2*963.7)(1+0.08/2)2*0.5  + (2*963.7)(1+0.08/2)2*1 + (2*963.7)(1+0.08/2)2*1.5 + (2*963.7)(1+0.08/2)2*2 + (963.7)(1+0.08/2)2*2
```

MacD = 0.01496 + 0.02878 + 0.04151 + 0.05322 + 1.7740 = 1.9124

Now let’s see how to calculate the Modified duration.

### Calculating the Modified duration

To calculate the Modified duration (ModD), you’ll need to use this formula:

```          MacD
ModD =  (1+YTM/m)
```

In the context of our example:

```          1.9124
ModD =  (1+0.08/2)
```

The Modified duration is therefore =  1.838