Today we look at three common mean investment returns: arithmetic, geometric, dollar weighted.

When analyzing investment options, you will always see at least one of these returns.

It is important to understand them as each may arrive at different results. And different results usually lead to different conclusions.

**Mean versus Median Returns**

The mean is the average of all the individual data points.

The mean is not the median, another term you may encounter.

The median return is simply the physical mid-point for all the individual results.

For example, you have 5 years of results: 10, 5, 22, 12, 11.

The arithmetic mean is the average of all results. In this case, 12.

To calculate the median, you must rearrange all the data points from lowest to highest. Then simply find the exact mid-point.

In this example, we would re-list the data in ascending order as: 5, 10, 11, 12, 22. The number at the mid-point is 11. That is the median.

The median is useful in letting you know that half the results are above 11 and half are below.

Medians are less sensitive to extreme scores, so may be a better indicator for smaller sample sizes. Means can be impacted by a few extreme results, so provide better information in larger sample sizes.

In our example above, let us change the one data point from 22 to 220. The mean changes from 12 to 51.6. While it is indeed the arithmetic average, when compared to the other numbers in the sample, it appears unusual.

However, the median still remains at 11. The outlier had no effect on the median.

But other than its relevance for small sample sizes, I find the median of little use in investing.

The mean return though is quite useful in investing. Knowing the average results over a period of time or other criteria is important to the decision-making process.

However, there are a variety of possible mean calculations. And depending on the formula employed, the average return can differ significantly.

Further, certain mean returns are good for some calculations, but are less relevant for others.

**Arithmetic Mean Return**

In our example above, I used the arithmetic mean calculation.

An arithmetic mean return is simply the sum of all the returns divided by the number of returns.

For example, you are analyzing an investment whose returns over the last three years were -10%, 20%, and 5%.

The arithmetic average return is 5% = [(-10+20+5)/3].

Pretty simple. Something you often calculate in everyday life.

Arithmetic returns are useful when data in the series is independent from each other.

The arithmetic mean is relevant when calculating the average exam results for a class of students. The performance of each student is independent from the others (assuming no one is cheating by copying another student’s answers!). How you score is not affected by the students sitting around you.

Or when measuring the mean height of all the students in your class. The height of the student on your right should have no impact as to your own height.

However, in investing, performance between periods is inter-related.

For example, if you invest $1000 and earn 100% in the first year, you start year two with $2000 in capital. But if you lost 50% in year one, you would only have $500 in capital at the beginning of year two.

In year two, let us say that you had a 20% return. In one scenario, your $2000 would grow to $2400. However, under the second scenario, your $500 would only grow to $600.

Same percentage gain of 20%, but significantly different monetary change.

You can easily see how the cumulative investment performance is inter-related to past results.

A problem with using the arithmetic mean return for investment calculations is that negative returns skew average returns and sometimes make the results irrelevant.

For example, you invest $1000 on January 1, 2009. On December 31, the investment is worth $2000 and there were no cash flows during the year. Your holding period return for 2009 is 100%.

You hold the investment throughout 2010 and at December 31, the value has fallen back to $1000, with no cash flows. Your holding period return for 2010 is -50%.

Your arithmetic mean return for the two years is 25% = [(100-50)/2].

But at December 31, 2010 you have exactly what you initially invested on January 1, 2009. Your return is 0%. It did not increase, on average, by 25%.

Always remember that negative returns skew arithmetic mean return calculations.

Also remember that arithmetic mean returns bias the average upwards.

This is where the geometric mean return becomes important.

**Geometric Mean (Time Weighted) Return**

The geometric mean is also known as the time weighted rate of return.

It measures the compound growth rate of the portfolio’s beginning market value over the evaluation period. The geometric mean return assumes that all cash flows are reinvested in the portfolio.

To calculate the geometric mean you need to add 1 to each period’s return. Then, multiply the results together for each period. Next, take the root value using a root equal to the number of periods. Finally, subtract 1 and you get the result.

Not as easy a calculation as the arithmetic mean return, but not too complex.

In our arithmetic mean example, we had three year returns of 10%, 20%, 5%. Our arithmetic mean is 5%.

The geometric mean return though is = [(1-0.1)(1+.20)(1+0.05)]^{1/3}-1.0 = 4.3%

Note that the geometric mean is always less than the arithmetic mean. Good to know for exams and quick calculation checks.

In our second arithmetic example, we had two periods with results of 100% and -50%. Our arithmetic return is 25%. This made no sense though as we ended up with the same amount as we started with.

In looking at the geometric return, we see that this is addresses the illogical arithmetic result.

The geometric mean return = [(1+1.00)(1-0.50)]^{1/2}-1.0 = 0%

Unless you need to know the calculations for exams, I suggest you not worry too much about them.

The key is to know that arithmetic mean returns are useful for independent data, whereas geometric mean returns are best used for investment results where the data is interdependent to some degree.

Also, when comparing arithmetic to geometric returns, arithmetic results will always be higher for identical data.

**Dollar Weighted (Internal) Return**

You may also see comparisons between time weighted (i.e. geometric) and dollar weighted returns (i.e. internal rate of return).

Dollar weighted returns are computed by determining the interest rate that will equate the present value of the cash flows from all investment periods under consideration plus the end portfolio market value to the portfolio’s beginning market value.

In essence, it is the internal rate of return for the portfolio.

For example, on January 1, 2009 you invest $1000 in a 1 year term deposit earning 10% interest. On January 1, 2010 you reinvest the proceeds of $1000 into another 1 year term deposit earning 15%. You also invest an additional $2000 into the same term deposit. On December 31, 2010 you receive $3565 in cash.

Going through the manual calculations starts to get tricky here. I suggest you purchase a cheap, but effective, financial calculator. I use a Hewlett Packard (HP) 10BII. Not the sturdiest piece of machinery, but it does the job.

For something sturdier, and more expensive, I like the HP 12C. It has a few weaknesses – cost, a little slow on calculation time, different input format – but it is an excellent and durable tool.

If you do not like HP, Sharp and Texas Instrument also make decent models.

As for our example, by plugging the data into my HP we get a return of 13.66%.

Dollar weighted returns do provide useful information as to growth of a portfolio.

However, dollar weighted measures are not usually very useful in evaluating portfolio performance. That is because the return is affected by events outside the control of the portfolio manager.

Changes in funding, such as client contributions or withdrawals, will impact the dollar weighted return. This makes it difficult to compare the performance of two managers over time.

So when evaluating two separate funds in which you wish to invest, do not put much emphasis on the dollar weighted returns in your comparison.

That concludes our initial look at investment returns.

Next up, we will consider how risk and return relate to investor profiles.