The following is a simple (if I did a good job explaining it) math lesson on how to calculate important multiyear metrics used to measure asset manager or portfolio performance.  These are compound annual growth rate (also known as CAGR) and compound return (also known as cumulative return). 

Say you have a fund that reports total returns for the past three years (and you trust the firm and its numbers):

Year1 = 16%                                        
Year2 = 25%
Year3 = -4%

Convert percentages to decimal form (remove % and move decimal point two places to left and fill empty space with zero when percentage is less than 10%):

For year 1 let TR1 = .16
For year 2 let TR2 = .25
For year 3 let TR3 = -.04

The compound return will be:

compoundReturn = (1+TR1) (1+TR2) (1+TR3) – 1

                                 = (1+.16) (1+.25) (1-.04) – 1

                                 = (1.16) (1.25) (0.96) – 1

                                 =1.392– 1

                                 = 0.392

Converting back into percentage (move decimal point two places to the right and add % symbol) we get compound return for years 1 to 3 = 39.2%

Two things worth mentioning: we would get the same result no matter what order the individual year results occurred and we could accommodate more years simply by tacking on additional “(1+TR)” factors in the compound return equation. In this case there will be a factor for each year of return data for a total of N years.

compoundReturn = (1+TR1) (1+TR2) (1+TR3)… (1+TRN) – 1

If you have compound return from a prior period (say years 1 to 3 calculated above), you can update this with a year 4 return (say 25%) as follows.

For existing compound return let ECR = 0.392

For year 4 let TR4 = 0.25

compoundReturn = (1+ECR) (1+TR4) – 1

                                 = (1.392) (1.25) – 1

                                 = 1.74 – 1

                                 = 0.74

And the updated compound return is 74%

Next, we can calculate the compound annual growth rate by asking what single rate results in the same compound return (using the same 3 years of data from above):

compoundReturn = (1+CAGR) (1+CAGR) (1+CAGR) – 1

Adding 1 to both sides,

compoundReturn + 1 = (1+CAGR) (1+CAGR) (1+CAGR)

compoundReturn + 1 = (1+CAGR) ^3          ^3 means raised to the third power

if we take the 3^rd root of both sides and rearrange,

1+CAGR = 3^rd root (compoundReturn +1)

Subtracting 1 from both sides

CAGR = [3^rd root (compoundReturn +1)] -1

CAGR = [3^rd root(.392+1)] -1

CAGR = [3^rd root(1.392)] -1

CAGR = 1.116554 – 1

CAGR = 0.116554

CAGR as a percentage is approximately 11.66%

As before, we could accommodate more years (say N) simply by changing 3^rd root to Nth root in the CAGR equation.

CAGR = [N^th root (compoundReturn +1)] -1

If you would like to learn how to calculate Total Return, you can use the following method from Kenneth Marshall (see page 22 of his book Good Stocks Cheap):

totalReturn = (capitalAppreciation + realizedGains + dividends + interest) /assetsUnderManagement