## Friday, November 27, 2009

### Rates of return ... come again?

I stumbled upon a website today that provided the following brief explanation about returns:

"To evaluate the performance of a portfolio manager, you measure average portfolio returns. A rate of return (ROR) is a percentage that reflects the appreciation or depreciation in the value of a portfolio or asset"

We measure "average" returns? I don't think so. Average returns have been shown to have zero value. A classic example: Year 1 +100 return, Year 2 - 50% return, average= (100 - 50) / 2 = 25 percent. Now, let's use some dollars: start with \$100; at the end of year 1 you're at \$200, then at the end of year 2 you're at \$100, meaning zero percent return.

In addition, while there are times when a return will reflect the appreciation or depreciation, once we introduce cash flows, forget about it! Recall that time-weighting can yield funny situations, like having a positive return but losing money.

I guess the lesson is: be careful about what you read on the Internet ... may not always be correct.

1. It's important to remember that arithmetic averages are simply expected returns. That means that their correct use is forward looking. They answer the question: "What is the most reasonable return to expect in any single period going forward?" Geometric average returns (AKA Time Weighted Returns) answer the question: "What was the average rate of growth per year over the performance measurement period ASSUMING NO ADDITIONS OR SUBTRACTIONS OF THE INVESTED AMOUNT." Different question. Different return needed.

2. Its not just the internet. Look at some mutual fund presentation, the description may state the following: "Average" annualized return. That's not forward looking. It just some professionals in need to study the CIPM exams.

3. Part of the problem here is that the technical terms we use can have more than a single meaning, depending on the context. The term "average return" can have several meanings, but which one is "correct?" That depends on the context - it depends on what question you are trying to answer. Now, whether you are trying to look forward (arithmetic average) or look backward (geometric average) we find that BOTH of these terms are AVERAGES.

There is nothing incorrect or misleading about an average. In all cases, an average is simply a single number that represents a group of individual values. In the case of arithmetic returns, the average is the "middle" of the distribution of returns - it's a measure of central tendency. Assuming you have a representative sample of returns, then you can derive a reasonable estimate of a likely single period return going forward. With geometric returns, the average is the single rate of growth that reconciles the beginning and ending values of a fixed invested amount over the specified time period. Again, one makes a forward looking inference, while the other measures what actually happened. Now that we have a context, we have clarity.

Regarding mutual fund returns: these are NEVER forward looking. We have all heard the disclaimer that "past performance does not guarantee future performance." The geometric average returns that are posted have only one purpose: to measure the return that actually occurred on a single investment over the performance measurement period. These are not meant to be forward looking. Remember, any forward looking number is simply an OPINION - it's not a fact. We may use fancy words such as "forecast" or "inference" and we may even state these with credible sounding comments about "statistical significance" but these are still nothing more than opinions. And opinions are often wrong. So, when we stick with the facts (things that have already occurred) then we use geometric returns.

4. Of course there is no such thing as an "average" annualized return. Annualized returns are averages that take into consideration the impact of compounding. Thus, to say you have an average average is a bit redundant and superfluous and misleading.

5. In our intro course I always say "avoid simple averages" when it comes to returns. While we see their use in certain statistics (e.g., Sharpe ratio, Jensen's alpha) as a reporting statistic their meaning, to me, is overshadowed by their potential misuse. Since returns compound, our average should take this into consideration: thus, the use of annualized returns.

6. I challenge this concept in the past. I provided the annualized and average calculation and the concepts but was defeated because: 1) I was told the wording was an industrial standard (in my opinion, this is just BS. The presenters had no clue or didn't know the differences), 2) Wikipedia has more reliable information: http://en.wikipedia.org/wiki/Average_Annualized_Rate_of_Return (I'm not sure if I can post a link - my apology if I couldn't). Yes, this website is reliable but to a certain degree.

To many readers, average annualized or annualized probably makes no differences. Maybe as more people take the CIPM, more people will relize this little issues and make the necessary adjustment.