Monday, January 10, 2011

Calculating Average Returns - Arithmetic vs. Geometric Average

The most basic of all calculations a hedge fund will want to report to their investors is their Average returns.

Despite the simple nature of this calculation, the difference between the Arithmetic and Geometric averages as well as the difference between the terminology used to refer to these terms, for example Compounded Annual Rate of Return vs. Compound Annual Growth Rate, have lead to confusion as to what number is being displayed.

The Average, or more accurately Arithmetic Mean is, what most people think of when they refer to the word "average".

Taking the 2010 returns for the S&P 500 Total Returns Index, we'll first generate the Arithmetic Mean

1 January, 2010 -3.597%
2 February, 2010 3.098%
3 March, 2010 6.034%
4 April, 2010 1.579%
5 May, 2010 -7.985%
6 June, 2010 -5.235%
7 July, 2010 7.010%
8 August, 2010 -4.514%
9 September, 2010 8.924%
10 October, 2010 3.810%
11 November, 2010 0.013%
12 December, 2010 6.683%
13 Arithmetic Mean 1.318% =AVERAGE(B1:B12)

The Geometric Mean, is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root (where n is the count of numbers in the set) of the resulting product is taken.

Here we see the same dataset using the Geometric Average

16 1000
17 January, 2010 964.03 =(B1*B16)+B16
18 February, 2010 993.90
19 March, 2010 1053.87
20 April, 2010 1070.51
21 May, 2010 985.03
22 June, 2010 933.46
23 July, 2010 998.90
24 August, 2010 953.81
25 September, 2010 1038.92
26 October, 2010 1078.51
27 November, 2010 1078.65
28 December, 2010 1150.73
29 Geometric Average 1.177% =(B28/B16)^(1/12)-1

Looking at the dataset used above, we see that instead of working off of the percentage gained we're using the decimal multiplier equivalents. In hedge funds, these numbers would be referred to as your VAMI (Value-Added Monthly Index) and very likely seen in a "Growth of $1000" chart.

So when do you use the Geometric Average over a Arithmetic Mean? When it comes to Hedge Funds, it's pretty safe to say always.

In closing, when you hear Compounded Monthly Rate of Return or Monthly CAGR, they are also referring to your Geometric Average.

Additional Reading:


  1. Great point Aaron because this is an important distinction. This is a critical consideration in portfolio management because a drawdown has a disproportionate impact on returns. For instance, if a $100M fund goes up 50% in one year and down 50% the next, the fund only has $75M left. This dynamic highlights the importance of downside estimation in portfolio construction:

  2. Thanks for the comment Cameron. I've added your blog to my blogroll. I hope to be able to provide more statistically focused blog entries in the upcoming weeks.

  3. Aaron, thanks for the concise explanation of the difference between the two calculations. I have a follow up question (or questions) for you. I recently calculated my investment returns versus the S&P 500. I used two methodologies for measuring the compounded annual return of the S&P 500 – first, I simply took the dividend/split adjusted S&P 500 level from the first day I invested (August 2005) and the most recent closing date (March 24, 2011) and compared that return to my split/dividend adjusted returns (using an XIRR formula). Second, I attempted to recreate a “time-adjusted” S&P 500 return. I began investing in August 2005 and made subsequent investments in different stocks across that period. It struck me as a little misleading to compare my returns to the S&P 500 returns by taking the first and last day without considering that I had invested at many more periods along the way (basically, if I were reviewing my own returns, I would want to know if I did better than the S&P simply because many of my investments were made during low points across the measured period). So, I created a mechanism by which at any point I invested in stocks (actual investments used in my own return calculation), a corresponding “investment” in a share of the S&P 500 was made (I did this by multiplying the dollar amount invested by the S&P 500 close on that day, making the assumption that I bought “shares” in the S&P 500 equal to the amount theoretically invested). Then, I took the implied number of S&P 500 “shares” purchased since August 2005 and multiplied that amount by the March 25, 2011 S&P 500 adjusted close and used an XIRR formula to calculate the return.

    Long story to get to my questions but hopefully that gave you some context. First, is either of these ways the “correct” or generally accepted way to compare returns to the S&P 500. Second, if I try to use the geometric average (which it sounds like a hedge fund would use in this situation) to compare returns, do I need to account for timing differences – if so, what is the best way to do that, if not, could you elaborate on the theoretical reasons why it is either already factored into the calculation or why this isn’t a theoretical concern when analyzing return comparisons? Thanks in advance!

  4. Dear Mr.Aaron Wormus,

    The basic Calculation is expalined in very simple & Understandable.

    Thank you very much for your inoformation on this website.