# Compound Annual Growth Rate

Compound annual growth rate (CAGR) is a single annual rate that captures the compounded growth of an investment or loan over multiple years. Given an investment’s value at time 0 called the present value, its value at certain future date called the future value and the time duration between the two values, we can calculate CAGR.

Investments, revenues, expenses, etc. grow at different rates in different periods, which makes comparison between them difficult. CAGR, being a standardized measure of annual compound growth regardless of the time duration, reflects the cumulative effect of multiple periods and enables us to make comparisons.

CAGR can be calculated only when we have present value, future value and time duration of a single sum. It can’t be calculated for a stream of cash flows, revenues, etc. CAGR is different from the holding period return, the cumulative total growth rate on an investment between two dates.

## Formula

CAGR can be calculated using the following formula:

CAGR = (FV/PV)^{(1/n)} - 1

PV stands for present value, the value at t=0 FV stands for future value, the ending value at t=n N is the total number of years between PV and FV.

Alternatively, we can use Excel RATE and/or RRI functions.

## Understanding the Math

You must be wondering how we came up with this fancy formula. It is not that difficult.

CAGR formula can be derived by simple mathematical manipulation of the formula for present value or future value of a single sum of money.

The relationship between present value (PV) and future value (FV) of a single sum of money invested for n number of periods at annual percentage interest rate i is represented by the following expression:

FV = PV × (1 + i)^{n}

Rate stands for the annual compound growth rate and *n* is total number of years. Let us substitute i with CAGR:

FV = PV × (1 + CAGR)^{n}

Let’s divide both sides by PV and raise both sides to (1/n):

(FV/PV)^{(1/n)} = ((1 + CAGR)^{n})^{(1/n)}

n and 1/n cancel each other, and we get:

(FV/PV)^{(1/n)} = 1 + CAGR

Subtracting 1 from both sides we get:

(FV/PV)^{(1/n)} − 1 = CAGR

## Example

You work in an accounting firm that has three divisions: audit, tax and advisory. Audit division revenues were $12 million 5 years back and $15 million now. Tax department revenues were $20 million two years back and they grew by 5% and 7% in the past two years respectively. Advisory revenues in the last three years were $5 million, $5.5 million and $6.2 million respectively. Your partner-in-charge has asked you to find out how much each division grew on average per year.

### Solution

You need to find out compound annual growth rate for each division.

In case of Audit, CAGR is:

CAGR of Audit Division

= ($15/$12)^{(1/5)} − 1

= 4.56%

You can also calculate it by entering the following in any Excel cell “=RATE(5,0,-12,15)”

In case of the Tax Division, we need to first find the revenues today given the revenue two years back and two-year growth rates. Revenue today equals $22.47 million (=$20 million × (1 + 5%) × (1 + 7%)). This can also be calculated using FVSCHEDULE function.

Now, we can calculate CAGR for Tax Division:

CAGR of Tax Division

= ($22.47/$20)^{(1/2)} − 1

= 6%

Alternatively, you can work out CAGR for tax by using the following Excel formula: “=RRI(2,20,-22.47)”

In case of Advisory Division, we have actual revenue figures but since we are making comparison between the start date and end date, we base CAGR on $5 million, the earliest revenue value and $7 million, the latest revenue.

CAGR of Advisory Division

= ($6.2/$5)^{(1/2)} − 1

= 11.36%

The comparison shows that the Advisory Division is the biggest growth area.

by Obaidullah Jan, ACA, CFA and last modified on

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