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Geometric Vs. Arithmetic Mean of Investment Returns

Project Description

Having a basic understanding of descriptive statistics can go a long way when managing your own money. Even if you have no direct contact with statistics in your professional life, you probably still have a vague recollection of the “mode, median and mean” from your school days. These are referred to as measures of central tendency and allow us to infer a commonality among a set of numbers and the the degree to which they revolve around a common, or middle value. Alongside these measures of central tendency, we have measures of central variance; the most common of which being Standard Deviation, explaining how much the results tend to deviate from the middle value. These simple tools allow us to describe a set of numbers and draw certain conclusions. These measures are used universally when dealing with large populations of numbers, for example: test scores, votes, scientific experiments, accounting, product sales, medical trials and insurance. When assessing the performance of an investment it would be practical to be able to make accurate inferences, with a view to assessing suitability, and also allowing meaningful comparisons among other similar investments.

Most commonly you may encounter the arithmetic mean. This is arguably the simpler of the two. It is also the least valuable of the two for assessing investment performance because unlike the geometric mean, it does not account for compounding. The annualized return, computed via the geometric mean, is less than the average annual return, computed by the arithmetic mean. It is calculated by adding together all of the numbers in the series and dividing the result by the number of entries in the series. For example:

Series 1
2, 3, 4, 5, 6, 7

Calculation
2 + 3+ 4+ 5+ 6+ 7 = 27 27 / 6 = 4.5

Arithmetic mean =  4.5

If we are to look at the same series as investment returns:

Series 2
+2%, +3%, +4%, +5%, +6%,+7%

Calculation

[0.02 + 0.03 + 0.04 + 0.05 + 0.06 + 0.07 ] / 6 = 0.045

Arithmetic mean =  0.045 or 4.5%

The geometric for this same series of numbers will be quite different. It is the geometric mean, not the arithmetic mean, that tells you what the average financial rate of return would have had to have been over the entire investment period to achieve the end result. This term is also so called the Compound Annual Growth Rate or CAGR. To calculate the geometric mean you multiply all of the numbers in the series by eachother (e.g result X result X result) and then work out the root of the numbers of results in the series. This may sound more difficult than it is. Using our previous data set:

Series 1
2, 3, 4, 5, 6, 7

Calculation

(2 x 3 x 4 x 5 x 6 x 7) ^ (1/6 or .166)

Geometric mean = 4.2

If we are to look at the same series as investment returns we must make an adjustment.

1. First add 1 to each number in the sequence. This is to avoid problems with negative numbers.
2. Multiple each number in the sequence.
3. Raise the answer to the power of one divided by the number of data points in the series. 4. Subtract one from the result.

Series 2
+2%, +3%, +4%, +5%, +6%,+7%

Calculation

(1.02*1.03*1.04*1.05*1.06*1.07) ^ (1/6 or .166) -1 = 0.044

Geometric mean =  4.4% CAGR

The difference between the results is relevant, and goes to show the importance of the different methods of calculating the geometric mean when dealing with financial returns. Using our Series 1 data example the two methods produce a percentage which differs by 0.28%.  The inequality of the arithmetic and geometric mean, and the affect that volatility has on growth rates forms the basis of performance drain theory. The affect of compound interest is substantial over time and when the principal is large, even the smallest improvements in performance can potentially result in tens, if not hundreds of thousands of dollars in the future. Another common area in which it pays to be aware of the workings of compound interest is bank accounts which often quote the nominal rate of interest and not the annual equivalent rate (commonly referred to as AER).

For example

Bank account nominal rate of interest: 2% p.a
presumed growth on 100,000 USD after 12 months: 102,000 USD

Bank account nominal rate of interest: 2% p.a
Frequency of payment:
quarterly
AER calculation:

[2% / 4 periods = 0.5% per period]

decimalized = 0.02/4 = 0.005
raised to the power of 1 = 1.005

1.005^4 = 1.0201

(1.0201 – 1)

0.02015

0.02015 * 100 = 2.01

AER = 2.01% p.a The difference may seem trivial, but that extra .01% too will go on to compound indefinitely alongside the principal. The frequency with which interest payments are made is important and in with certain environments can drastically improve the return-profile. In closing consider a US Property REIT (real estate investment trust) which pays its dividend monthly, and assumes dividends are reinvested.

Annual dividend: 7.5% p.a
Payment frequency: monthly
Calculation:

.075/12 = .00625

(.00625 + 1)

1.00625^12 = 1.0776

(1.0776 – 1)

0.0776*100 = 7.76

AER = 7.76%

On a 1,000,000 investment that .21% is an extra 2,100 USD in the first year, just for having dividends paid monthly instead of annually.  Literally nothing else has changed. Compound interest is your friend. Why not download our free interest calculator.

[Sources]

– The Handbook of  Traditional  and Al ternative Investment  Vehicles: Investment  Characteristics and Strategies by  Mark J. P. Anson, Frank  J. Fabozzi  and Frank J. Jones
– Handbook of Means and Their Inequalities, Kluwer Acad. Publ., Dordrecht, 2003
– Fundamentals of Descriptive Statistics 1st Edition by Zealure C Holcomb
– A Handbook of Statistics An Overview of Statistical Methods, Darius Singpurwalla

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Summary Article Name
Whats The Difference: Geometric Vs. Arithmetic Mean
Description
Learn how to use the statistical Mean to calculate investment returns.
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