# What is the difference between Holding Period Return, Arithmetic Return and Geometric Return?

We want to take this opportunity to discuss the main differences between these various return calculations and the main reasons why and when you should use one formula instead of another while analyzing financial data. Although this might sound intuitive for some candidates preparing for the upcoming December 7th, 2013 CFA Level 1 exam, we find that this will be a great refresher for many.

To make all the required calculations, we will use the following table with monthly returns for Pfizer (PFE). Source: Yahoo.com

## Holding Period Return (HPR)

HPR is probably the most intuitive and widely used financial return calculation. It is very straightforward, simple and does not leave much room for calculation errors. All you need are three variables: the beginning and end market values of a portfolio (or a share) and the total cash flow received from the portfolio (if any) during the time period analyzed. If we analyze the annual HPR for PFE using the twelve monthly returns and dividends received in the table above, we would use three values: \$24.85 (beginning) \$28.73 (end) and \$0.94 (total cash-flow). Specifically, we write the HPR formula in this way: ### What does the result tell us?

The most straight forward way to interpret the formula and the result is to say that an investor made 19.4% HPR on his investment in PFE held for one year.

### When to use the Holding Period Return?

HPR is widely used when you want to have a quick and simple overview of your investment in a particular product (stock, portfolio, etc). HPR is simple and quick to calculate so its definitely a great tool to use when you are simply interested in a quick and general return number.

### What to look out for?

If the stock did not pay any dividends during the period in question, then the Po in the formula would simply result in a \$0. One of the main disadvantages of HPR is the fact the the formula ignores any movement in the share price between the beginning and end value of the investment. Sometimes, the investment might have performed very well during a given period, but on the date when the HPR was calculated the ending value was negatively affected by the market forces and the overall return appeared weaker.

## Arithmetic Return

Arithmetic return is simply the average of shorter returns. In our example, since we want to calculate the Arithmetic Return for PFE over the 12 months we first need to calculate 12 monthly returns and then calculate the overall average of these monthly returns to arrive at annual return. In a way, we calculate a series of shorter HPR and then calculate the mean of these returns. For example, we would first calculate the HPR for September 2012 as (\$24.87 – \$24.85 + \$0)/\$24.85 = 0.001 or 0.1%. Then we would calculate the HPR for October 2012 (\$25.02 – \$24.87 + \$0.22)/\$24.87 = 0.0149 or 1.49%. We would proceed calculating monthly HPR until the twelfth month’s return: (\$28.73 – \$28.21 + \$0)/ \$28.21 = 0.0184 or 1.84%. Then once we have all the monthly returns, we would proceed to calculate the Arithmetic Return using the following formula: ### What does the result tell us?

The results tells us that the investor made an average monthly return of 18.9% over the previous 12 months investing in PFE.

### When to use the Arithmetic Return?

You can use the Arithmetic Return in you would like to know an average return you made by investing in a particular product. Its interesting to note how Arithmetic Return solves one of the main shortcomings of Holding Period Return. Since the Arithmetic Return uses more data (twelve monthly returns) instead of only one as in the case with HPR, you have a better understanding of the investment’s performance throughout the year.

### What to look out for?

The primary disadvantage of the Arithmetic Return is the fact that it ignores the effect of compounding returns. Similar to the notion of compounded interest, compounded return results when the return in one month is influenced by the return obtained in the previous month. If you would look at your investment as a series of independent monthly returns, then the effect of compounded returns would not be a concern. However, since your financial returns are all linked together and since your current return depends on your past return, the effect of compounding interest should not be ignored. Thankfully, the Geometric Return (discussed next) solves the main issue faced with Arithmetic Return.

## Geometric Return

Our final return analysis deals with the concept of Geometric Return. This return concept uses the monthly returns calculated in the previously discussed Arithmetic Return, but then added the effect of compounding between each period (in our case, it’s monthly returns). Instead of adding monthly returns and dividing them by the number of months as we did with Arithmetic Return, we multiply the monthly returns and then we take the twelfths root of the answer to arrive at a monthly return. See the following formula on Geometric Return: ### What does the result tell us?

The result indicate that the investor made a compounded monthly return of 19.7% on his investment in the previous 12 months.

### When to use the Geometric Return?

Geometric Return provides the most accurate return result among the three returns covered in this article. In a way, Geometric Return solves the issue found with the Holding Period Return since the Geometric Return analyzes returns throughout the year instead of simply the beginning and the end values like with HPR. In addition, Geometric Return also solves the issue found with Arithmetic Return since it considers the very important power of compounding from one period over to the next period. The result obtained with the Geometric Return is more representative of the return the investor actually received from his investment.

### What to look out for?

One of the main drawbacks of the Geometric Return is the fact that it required quite a bit of calculations and this increases the possibility of calculation errors on the exam day.

We know that these concepts might appear a bit complex and abstract for some of you, but these different formulas come back on all three levels of the CFA exam. Holding Period Return, Arithmetic Return and Geometric Return are widely used in the financial community and we highly recommend that you take the time to understand them now while preparing for the level 1 exam. Understanding the basics now, will definitely make your overall CFA journey much more pleasant and hopefully more successful as well.

Our goal here at the Financial Analyst Warrior is to help you master the study materials and pass your CFA exams. Use the comments sections below or send us an email and let us know what areas of the required studies you have the most difficulty with and we will gladly help you out.

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