## Monday, 11 January 2010

### Methods to Calculate Historical Market Returns – Arithmetic Means versus Compound Annual Growth Rates

When I described previously how I was building my low charge investment portfolio and forecasting potential future retirement dates it turns out that in mathematics terms I have generally been using arithmetic means to calculate percentage returns from various data sets. That means I've taken the yearly percentage change for each entry of the dataset, summed these values and then divided by the number of items in the dataset.

It looks like this might be too bullish a method and instead I potentially should be using the compound annual growth rate (CAGR) which is the smoothed annualised gain. The formula is CAGR = (End Value/Start Value)^(1/number of years)-1.

Let me demonstrate with an example. At the end of year 0 your index is worth 100, by the end of year 1 your index has increased to 200 and then by the end of year 2 your index has decreased to 150. Using the arithmetic mean the mean annual return is (100% [year 1 gain] + -25% [year 2 loss])/2 = 37.5%. This can’t be correct as you don’t have 100 x 137.5% x 137.5% = 189 at the end of year 2. Now using the CAGR the return is (150/100)^(1/2)-1 = 22.5%. Checking this 100 x 122.5% x 122.5% = 150.

Let me now calculate a stock market example, the real (ie inflation adjusted) S&P 500 for the period January 1871 to January 2009. Using the arithmetic mean we get an average annual real price increase of 3.7% and an average annual real dividend of 4.5% for an average annual real return of 8.2%. Now using the CAGR (more complicated to calculate as I had to first calculate a real total return for the S&P 500) for an average real return of 6.7% which is significantly less than the 8.2% arithmetic mean calculation.

I'm now considering calculating real CAGR returns for my (S&P 500, ASX 200 and Gold) datasets and using these when projecting my retirement dates and amounts.

As always I would be very interested to hear others experience here.

You might also be interested in calculating portfolio year to date returns, annualised returns or multiple year returns.

#### 1 comment:

1. Hi

FWIW, I have done a fair amount of financial analysis in my work. And n o question - you are correct. CAGR is correct; adding %s to get an arithmetic average is wrong.

Great site by the way. (I've only disocvered it now.) Thanks for your work on it.

Cheers
T