Bond duration is measured in years and reflects how sensitive a fixed income portfolio is to changes in interest rates. It represents the average time to receive the bond’s principal and coupon payments. In more technical terms, duration represents the weighted average time to receive the bond's coupon and principal payments, with the weights being the present value of those cash flows.

Here’s how to calculate the two main types of bond duration:

Macauley Duration

Macaulay duration measures the weighted average time until all cash flows are paid. It accounts for the present value of future bond payments, helping evaluate bonds independently of their maturity.

To calculate the Macaulay duration:

1. Collect the bond's details, such as the annual coupon payment, the years to maturity, the face value (or par value), and the yield to maturity (YTM).

2. Calculate the present value of each cash flow, including both the coupon payments and the final face value payment.

3. Determine the weight of each cash flow by dividing the present value of the cash flow by the price of the bond.

4. Multiply the weight of each cash flow by the time period in which the cash flow will be received to get the weighted time periods.

5. Sum the weighted time periods to obtain the Macaulay Duration.

Example:

Let's say we have a bond with the following characteristics:

Annual coupon rate: 5%

Face value: $1,000

Years to maturity: 3 years

Yield to maturity (YTM): 4%

1. Calculate the present value of each cash flow:

For the annual coupon: Coupon payment is 5% of face value = 0.05 * $1,000 = $50.

To find the present value (PV) of these payments, you discount them back to their present value using the YTM.

PV of a coupon = Coupon payment / (1 + YTM)^(Year number)

Year 1 Coupon PV = $50 / (1 + 0.04)^1 = $48.08

Year 2 Coupon PV = $50 / (1 + 0.04)^2 = $46.23

Year 3 Coupon PV, including face value = ($50 + $1,000) / (1 + 0.04)^3 = 1,050/1.124864 = $933.45[MP1] [MB2]

2. Calculate the bond price (sum of the present values):

Bond price = $48.08 (Year 1) + $46.23 (Year 2) + $933.45 (Year 3) = $1,027.76 [MP3] [MB4]

3. Calculate the weights of each cash flow:

Year 1 Weight = $48.08 / $1,027.76

Year 2 Weight = $46.23 / $1,027.76

Year 3 Weight = $933.45 / $1,027.76

4. Multiply by the time period to get the weighted time periods:

Weighted Time Year 1 = Year 1 Weight * 1

Weighted Time Year 2 = Year 2 Weight * 2

Weighted Time Year 3 = Year 3 Weight * 3

5. Sum the weighted time periods to find the Macaulay Duration:

Macaulay Duration = Weighted Time Year 1 + Weighted Time Year 2 + Weighted Time Year 3

Now, let's calculate the weights and the weighted times:

Year 1 Weight = $48.08 / $1,027.76 ≈ 0.0468

Weighted Time Year 1 = 0.0468 * 1 = 0.0468

Year 2 Weight = $46.23 / $1,027.76 ≈ 0.0450

Weighted Time Year 2 = 0.0450 * 2 = 0.09

Year 3 Weight = $933.45 / $1,027.76 ≈ 0.9082

Weighted Time Year 3 = 0.9082 * 3 = 2.7246

Now, sum the weighted time periods:

Macaulay Duration ≈ 0.0468 + 0.09 + 2.7246 = 2.8614 years

In this example, the Macaulay Duration of the bond is approximately 2.8614 years. This means, on average, it would take about 2.8614 years to receive the present value equivalent of the future cash flows from the bond. [MP5] [MB6]

Modified Duration

Modified duration measures the expected change in a bond’s price if there is a 1% rise or fall in interest rates. It is an adjustment of the Macauley duration that accounts for changes in interest rates, so you must calculate the Macauley duration before determining the modified duration. This calculation is useful if an investor is concerned about an upcoming rate change.

Example:

Let's use the same bond, as well as the Macauley duration calculated above:

Annual coupon rate: 5%

# of Coupon payments per year: 1

Face value: $1,000

Years to maturity: 3 years

Yield to maturity (YTM): 4%

Macauley Duration (MacD): 2.8614 years

The steps to calculate the modified duration of a bond are:

1. Gather Bond Information: Collect all the necessary details about the bond, including the yield to maturity (YTM), Macaulay duration, face value, coupon rate, and frequency of coupon payments.

2. Determine Coupon Payment Frequency: Identify the number of coupon payments per year (denoted as C). For example, annual payments would be 1, semi-annual would be 2, quarterly would be 4, and so on.

3. Calculate Periodic Yield to Maturity: If the bond's YTM is given on an annual basis (which is usually the case), you'll need to convert it to the periodic rate based on the frequency of coupon payments. This is done by dividing the YTM by the number of coupon payments per year (YTM/C).

4. Apply the Modified Duration Formula: ModD = (MacD)/ 1+(YTM/C)

5. Calculate the Modified Duration: Using the formula and the information gathered, calculate the modified duration of the bond by substituting the values into the formula.

6. Interpret the Result: The modified duration will give you the approximate percentage change in the bond's price for a 1% change in yields. A higher modified duration indicates that the bond price is more sensitive to changes in interest rates.

See calculation below based on a calculated Macauley duration of 2.8614 years:

ModD= (2.8614)/1+(0.04/1)

ModD= (2.8614)/(1.04)

ModD ≈ 2.751 years

Based on the calculation above, the modified duration of the bond, given a 4% yield to maturity, is about 2.751 years. This is an approximate measure as it assumes a linear relationship between changes in yields and changes in bond prices.