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Understanding Different Types of Averages and Average Returns as an Investor

Understanding Different Types of Averages and Average Returns as an Investor

Introduction

As an investor whether a passive or aggressive, understanding the different types of how returns are averaged over time is essential. You may be thinking isn’t an average the same no matter what, well its a little complicated, in ordinary language an average is simply the ‘sum of numbers divided by the number of number’. However, in statistics, the average(also called the mean) is the number that measures the central tendency of a set of numbers. In other words its a number that best represents a set of numbers. Don’t be confused by average or mean, they both mean the same thing. The following are the most essential types of a averages an investor must be familiar with:

Arithmetic Mean Return

This is the simple average that everybody is familiar with. $$ \text{Arithmetic Mean Return} = \frac{(R_1 + R_2 + \cdots + R_N)}{N} $$ Here ‘n’ refers to the number of periods, which is usually in years. \ If you made the returns: 10%, -5%, and 7% in a time frame of 3 years, the arithmetic mean would be about 4% $$ \frac{10 + (-5) + 7}{3} = 4 $$ The things to keep in mind are:

This does not reflect compounding, the returns are independent.
Only good for a quick estimate of the expected return over the course of a single time period(e.g. one year).

Geometric Mean Return

This is a compound rate. This is the mean return to keep in mind when our returns are compounding the investment principal. $$ \text{Geometric Mean Return} = \sqrt[n]{(1 + R_1)(1 + R_2)\cdots(1 + R_N)} - 1 $$ Here ‘n’ is the number of periods(usually in years).

Let us intuitively understand how this rate of return actually works. Say, you have $1 as your principal. If you made the returns: 10%, -5% and 7% over a time frame of 3 years. The value of $1 of principal after every year: $$ \text{Year 1}: 1 \times (1 + \frac{10}{100}) = 1.1 $$ $$ \text{Year 2}: 1.1 \times (1 + \frac{-5}{100}) = 1.045 $$ $$ \text{Year 3}: 1.045 \times (1 + \frac{7}{100}) = 1.118 $$

So after 3 years $1 –> $1.118

So , What is the compound return over 3 years that would turn $1 into $1.118?

NOTE: we know that future value formula : $$FV = PV(1+r)^n$$ $$ 1(1+r)^3 = 1.118 => 1 + r = \sqrt[3]{1.118} = 1.0378 $$ $$ r \approx 0.0378 $$ Therefore the Geometric Mean Return = 3.78%

The things to keep in mind are:

This is best used for finding the compound rate of returns over multiple time periods.
$$\text{Geometric Mean Return} \leq \text{Arthimetic Mean Return}$$ NOTE: They can only be equal when all the returns are identical.
The difference between Geometric Mean Return and Arithmetic Mean Returns increases as the spread(dispersion) between the returns increases.

Harmonic Mean

This is the average we would use mainly to do, averaging of the rates of quantities or multiples, where one part of the fraction (numerator or denominator) is in fixed quantity and the other part varies. The example of such rates are Cost per share, P/E ratios and so on. Usually averaging anything which is a rate or a multiple. $$ \text{Harmonic Mean} = \frac{N}{\displaystyle\sum_{i=1}^NX_i} $$ Where there N values of Xi.

Here Xi are the rates or ratios in question. In the ratios one part must be fixed(for all of them) and the other part can vary.(This statement is a little confusing, but the upcoming example will dispel your doubts my friend)

To truly build the intuition surrounding Harmonic mean we need to go though a few examples.

Example 1: Speed

Speed is also a ratio or rate. It is distance over time. Say you travel from A to B which is 60 miles at 60 mph. Then you travel back from B to A at 30 mph. What would the average speed in this case be? If you take arithmetic mean it would be: $$ \frac{60+30}{2} = 45mph $$ But this answer would be wrong. If we did the whole process of finding the speed manually, by finding total time and total distance we would get a different answer.

Total time = 60 miles at 60 mph + 60 miles at 30 mph = 1 hour + 2 hours = 3 hours Total distance = 60 miles + 60 miles = 120 miles $$ \text{Speed} = \frac{120}{3} = 40mph $$

Therefore as speed is a rate. In the problem above the distance is fixed same value whereas time for each of the speed is varying producing different speed. As slower speed will produce more time. So by just calculating harmonic mean: $$ \text{Harmonic Mean} = \frac{2}{\frac{1}{60} + \frac{1}{30}} = \frac{2}{\frac{3}{60}} = \frac{2 \times 60}{3} = 40mph $$

Cost per share is also a ratio. It is the cost or price over number of shares. Suppose an investor purchases $1,000 of mutual fund shares each month, and over the last three months, the prices paid per share were $8, $9, and $10. What is the average cost per share?

Here we can clearly see that cost or price is varying whereas the total number of shares remains fixed(the total shares is fixed as a specific amount has been bought and added to the whole).

$$ \text{Harmonic Mean} = \frac{3}{\frac{1}{8} + \frac{1}{9} + \frac{1}{10}} = 8.926 \text{ per share} $$

If you want to check this result, let us just manually calculate the total shares bought over 3 months: $$ \frac{1000}{8} + \frac{1000}{9} + \frac{1000}{10} = 336.11 \text{ shares} $$ Thus the average price of each share is
$$ \frac{3000}{336.11}= 8.926 \text{ per share} $$