## Formula to Calculate Macaulay Duration

Macaulay Duration can be defined as the weighted average maturity of the cash flows and can be stated as the length of the period taken to receive the invested money through principal repayment and coupons, by the investor. The formula for Macaulay duration can be calculated as below

Where,

- t = time period
- C = Coupon payments
- y = yield
- n = frequency of periods
- M = Maturity Value
- CBP = Current market price of the bond which is nothing but the present value of cash flows of the bond.

### Calculation of Macaulay Duration (Step by Step)

The below steps need to be followed to calculate the Macaulay duration.

**Step 1:**Find out the cash flows of the bond over its lifetime which are coupon payments and principal repayment till its maturity.**Step 2:**Figure out the current yield of the market to be used as a discount factor.**Step 3:**Multiply the cash flows with the discount factor for those periods.**Step 4:**Now again multiply the values calculated in step 3 by the time period in which the cash flows are occurring for period 1 it will be 1 and period 5 it will be period 5, this will give weights to the cash flows.**Step 5:**Estimate the present value of cash flows using current yield that shall give your present value of bond which shall be its market price.**Step 6:**Divide value arrived in step 4 by value arrived in step 5 to arrive at the duration of the bond.

### Examples

#### Example #1

Consider a bond with following cash flows from year 1 to year 5 which is the year of maturity. The bond pays a 2.5% coupon every year and its par value is 100.

Assume interest rates to be 2.5% as well. Based on the above information you are required to calculate the Macaulay duration.

**Solution:**

We are given coupon cash flows along with maturity value, further the yield is also 2.5%.

The above formula can be used to calculating the Macaulay duration.

First, the cash flows will be multiplied with the time factor and then discounted using 2.5% as a discount factor

= ((1*10) / (1+2.5%)^1 + (2*10) / (1+2.5%)^2 + (3*10) / (1+2.5%)^3 + (4*10) / (1+2.5%)^4 + (5*10)/(1+2.5%)^5 + (5*100) / (1+2.5%)^5))/100

=579.008

Then the above result shall be divided by CBP which is 100

**So, Macaulay Duration will be –**

**Macaulay Duration = 579.008 / 100 = 5.79**

Therefore, the Duration is 5.79

#### Example #2

Mr. Pam works as a fixed income analyst at JP Morgan and Chase. The fund manager wants to perform duration matching to execute its immunization strategy and therefore the manager has asked Mr. Pam to calculate the duration for one of the bonds it has in its portfolio.

- CUSIP: 313WRZ889
- Coupon frequency: semi-annually
- Face Value of the bond: 1,000
- Maturity of the bond: 4 years
- The strike rate of coupon: 4.5%
- Yield: 5%
- Current Market Price of the bond: 982.07

Based on the above details you are required to calculate the Macaulay duration.

**Solution:**

We shall calculate the coupon cash flows along with maturity value, further the yield is 5%.

Since this coupon pays semi-annually hence the coupon rate would be 4.5%/2 which is 2.25% and the discount factor will be 5%/2 which is 2.5%

The above formula can be used to calculating the Macaulay duration.

First, the cash flows will be multiplied with the time factor and then discounted using 2.5% as a discount factor

=( (1*22.50) / (1+2.5%)^1 + (2*22.50) / (1+2.5%)^2 + (3*22.50) / (1+2.5%)^3 + (4*22.50) / (1+2.5%)^4 + (5*22.50) / (1+2.5%)^5 + (6*22.50) / (1+2.5%)^6 + (7*22.50) / (1+2.5%)^7 + (8*22.50) / (1+2.5%)^8 + (8*100) / (1+2.5%)^8 ) / 982.07

=7,271.05

Then the above result shall be divided by CBP (current market price of bond) which is 982.07.

**So, Macaulay Duration will be –**

**Macaulay Duration = 7,271.05 / 982.07****=7.4038**

Therefore, Duration is 7.4038 and when divided by 2 it will be 3.7019.

#### Example #3

Consider a zero-coupon bond with a face value of $1,000 which is maturating in 3 years. Suppose the market rate is 5%, you are required to calculate Macaulay Duration of this zero-coupon bond assuming that is was traded at $751.31

**Solution:**

Since this is a zero-coupon bond there are no cash flows and hence year 1 and year 2 have zero cash flows and only payment it shall make 1,000 at end of year 3 and when you take a weighted average of same and divide same by current bond price you will get exact 3 years per below calculation.

You can download and refer to the above-given template for the detail calculation of Macaulay duration.

### Relevance and Uses

Macaulay’s duration can be stated as a measurement of the price of the bond’s sensitivity to changes in the rate of interest prevailing in the market. This acts as a beta for the stock and is the most basic measure. It estimates that the investment in the bond will be recovered in what average span of time. Hence, the zero-coupon bonds have the maturity as their duration as it shall only receive the cash flow at the end of the period.

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