### What Is Compound Annual Growth Rate?

Compound annual growth rate (CAGR) is the rate of return that would be required for an investment to grow from its beginning balance to its ending balance, assuming the profits were reinvested at the end of each year of the investment’s lifespan.

The formula for CAGR is as follows:

﻿\begin{aligned} &CAGR=\frac{EB}{BB}^{\frac{1}{n}}-1\\ &\textbf{where:}\\ &EB = \text{Ending balance}\\ &BB = \text{Beginning balance}\\ &n = \text{Number of years} \end{aligned}﻿

### How to Calculate CAGR

To calculate the CAGR of an investment:

1. Divide the value of an investment at the end of the period by its value at the beginning of that period.
2. Raise the result to an exponent of one divided by the number of years.
3. Subtract one from the subsequent result.

### What CAGR Tells You

The compound annual growth rate isn't a true return rate, but rather a representational figure. It is essentially a number that describes the rate at which an investment would have grown if it had grown the same rate every year and the profits were reinvested at the end of each year. In reality, this sort of performance is unlikely. However, CAGR can be used to smooth returns so that they may be more easily understood when compared to alternative investments.

### Key Takeaways

• CAGR is one of the most accurate ways to calculate and determine returns for anything that can rise or fall in value over time.
• Investors can compare the CAGR of two alternatives in order to evaluate how well one stock performed against other stocks in a peer group or against a market index.
• CAGR does not reflect investment risk.

### Examples of CAGR

Imagine you invested $10,000 in a portfolio with the returns outlined below: • From Jan 1, 2014, to Jan 1, 2015, your portfolio grew to$13,000 (or 30% in year one).
• On Jan 1, 2016, the portfolio was $14,000 (or 7.69% from Jan 2015 to Jan 2016). • On Jan 1, 2017, the portfolio ended with$19,000 (or 35.71% from Jan 2016 to Jan 2017).

We can see that on an annual basis, the year-to-year growth rates of the investment portfolio were quite different as shown in the parenthesis.

On the other hand, the compound annual growth rate smooths the investment’s performance and ignores the fact that 2014 and 2016 were so different from 2015. The CAGR over that period was 23.86% and can be calculated as follows:

﻿$CAGR=\left(\frac{\19,000}{\10,000}\right )^{\frac{1}{3}}-1=23.86\%$﻿

The compound annual growth rate of 23.86% over the three-year investment period can help an investor compare alternatives for their capital or make forecasts of future values. For example, imagine an investor is comparing the performance of two investments that are uncorrelated. In any given year during the period, one investment may be rising while the other falls. This could be the case when comparing high-yield bonds to stocks, or a real estate investment to emerging markets. Using CAGR would smooth the annual return over the period so the two alternatives would be easier to compare.

The compound annual growth rate can be used to calculate the average growth of a single investment. As we saw in our example above, due to market volatility, the year-to-year growth of an investment will likely appear erratic and uneven. For example, an investment may increase in value by 8% in one year, decrease in value by -2% the following year and increase in value by 5% in the next. CAGR helps smooth returns when growth rates are expected to be volatile and inconsistent.

CAGR can be used to compare investments of different types with one another. For example, suppose in 2013 an investor placed $10,000 into an account for 5 years with a fixed annual interest rate of 1% and another$10,000 into a stock mutual fund. The rate of return in the stock fund will be uneven over the next few years so a comparison between the two investments would be difficult.

Assume that at the end of the five-year period, the savings account’s balance is $10,510.10 and, although the other investment has grown unevenly, the ending balance in the stock fund was$15,348.52. Using CAGR to compare the two investments can help an investor understand the difference in returns:

﻿$\text{Savings Account CAGR} =\, \left ( \frac{\ 10,510.10}{\ 10,000} \right )^{\frac{1}{5}}-1 = 1.00\%$﻿

And:

﻿$\text{Stock fund CAGR} =\, \left ( \frac{\ 15,348.52}{\ 10,000} \right )^{\frac{1}{5}}-1 = 8.95\%$﻿

On the surface, the stock fund may look like a better investment with nearly nine times the return of the savings account. On the other hand, one of the drawbacks to CAGR is that by smoothing the returns, CAGR cannot tell an investor how volatile or risky the stock fund was.

CAGR can also be used to track the performance of various business measures of one or multiple companies alongside one another. For example, over a five-year period, Big-Sale Stores’ market share CAGR was 1.82%, but its customer satisfaction CAGR over the same period was -0.58%. In this way, comparing the CAGRs of measures within a company reveals strengths and weaknesses.

Comparing CAGRs of business activities across similar companies will help evaluate competitive weaknesses and strengths. For example, Big-Sale’s customer satisfaction CAGR might not seem so low when compared with SuperFast Cable’s customer satisfaction CAGR of -6.31% during the same period.

### Modifying CAGR in the Real World

An investment is rarely made on the first day of the year and then sold on the last day of the year. Imagine an investor who wants to evaluate the CAGR of a $10,000 investment that was entered on June 1st, 2013 and sold for$16,897.14 on September 9th, 2018.

Before the CAGR calculation can be performed, the investor will need to know the fractional remainder of the holding period. They held the position for 213 days in 2013, a full year in 2014, 2015, 2016, and 2017, and 251 days in 2018. This investment was held for 5.271 years, which calculated by the following:

• 2013 = 213 days
• 2014 = 365
• 2015 = 365
• 2016 = 365
• 2017 = 365
• 2018 = 251

The total number of days the investment was held was 1,924 days. To calculate the number of years, divide the total number of days by 365 (1,924/365), which equals 5.271 years.

The total number of years the investment was held can be placed in the denominator of the exponent inside CAGR’s formula as follows:

﻿$\text{Investment CAGR} =\, \left ( \frac{\ 16,897.14}{\ 10,000} \right )^{\frac{1}{5.271}}-1 = 10.46\%$﻿

For example, imagine that an investor knows that they need $50,000 for a child’s college education in 18 years and they have$15,000 to invest today. How much does the average rate of return need to be in order to reach that objective? The CAGR calculation can be used to find the answer to this question as follows:
﻿$\text{Required Return} =\, \left ( \frac{\ 50,000}{\ 15,000} \right )^{\frac{1}{18}}-1 = 6.90\%$﻿