MPT Part VI: Arithmetic vs Compounded Returns

I'm in Wisconsin this week chowing down on BBQed brats and watching the Brewers. My guest columnist, Dr. Pat, has graciously agreed to write this week's blogs. He'll be sharing his profound knowledge on Modern Portfolio Theory, revising his previous asset allocation charts to incorporate monthly daily instead of yearly data. He also compares portfolio allocations based on arithmetic averaging to compounded averaging of returns. Which approach an investor should use depends on her portfolio goals and investment horizon.The week will be closed out with a discussion on Post-Modern Portfolio Theory. Professor Pat casts a jaundiced eye on the claims of the theory, noting its good points, bad points, and points of sheer absurdity.

Professor Pat's discourse is mathematically rooted and although it may take an academician to grasp all the nuances, even the average investor can easily employ his charts to determine the optimum asset allocation for his portfolio. It is with tremendous gratitude and deep respect for Prof. Pat's knowledge and time that I give my blog over to him for the next five days.

MPT: Arithmetic vs Compounded Returns
Previous installments of this series on Modern Portfolio Theory have presented optimum portfolio allocations for various Required Returns. These returns have been based on the arithmetic average of the history of returns for each asset class. This installment will present the optimum portfolio allocation results based on compounded annual returns. Remember that a return is said to be compounded if the investment amount is adjusted based on the previous return. For example, if you have $10,000 invested in the Small Stocks asset class and it gains 10% in a year, then you'll be adding that gain to next year's investment amount for a total of $11,000. (And the reverse process goes for losses.)

The arithmetic average annual return for an asset class is simply the sum of the percent returns divided by the number of years. For example, if the annual returns over a three year period were +8%, -5% , and +12%, the arithmetic average return would be (8 - 5 + 12) / 3 = 5.00%.

The compounded average annual return is more complicated. For this same example it is:

To further illustrate the differences between arithmetic average return and compound average return let's consider another example. The Small Stock asset class is the highest returning asset class over the 81 years from 1927-2007, but the difference between the arithmetic return is significantly different from the compounded return. For this class the arithmetic average annual return is about 17.3% while the compound average annual return is approximately 12.6%. The arithmetic average is higher because it weights percentage gains equally with percentage losses.

If one asset class experiences a 50% loss in any one year then a 50% gain the next year will not entirely recoup the earlier loss but will in fact recoup only one half that loss (resulting in 75% of the value of the original investment). The arithmetic average return of the two years is zero but there is still an overall loss over the two years as shown by the compound average return which is -13.4%. It actually takes a 100% gain to recover from a 50% loss. In this way, a compounded average return actually numerically weights losses more heavily than gains. In general, the arithmetic average will be greater than or equal to the compounded average with the difference between the two averages positively related to the standard deviation of the data.

An arithmetic average analysis is better at predicting what is more likely to occur in a single time period such as a month or a year. A retiree who needs to withdraw portfolio gains on a periodic basis would use the arithmetic return tables. In contrast, the compounded average analysis is backward looking but is more representative for what is likely to occur over multiple time periods. Investors who don't need to make portfolio withdrawals and are interested in the long-term accumulation of wealth would elect to use the compounded return tables.

Comparison of Asset Classes in a Basic Portfolio
For the six asset classes considered by Ibbotson & Associates in their Stocks, Bonds, Bills, and Inflation (SBBI) yearbook, the results of configuring the asset allocation optimization algorithm, also known as the mean-variance optimizer, to obtain the best asset class allocations for the range of possible compounded annual Required Returns is shown in the table below. This table ranges to 12.6% because that is the highest possible compound average annual return that can be obtained. That is accomplished with a 100% allocation to the highest returning, but riskiest, asset class--Small Stocks.

As before, all of my calculations below are based on annual return data from 1927-2007 for all of the asset classes described earlier. (See April 21-24 plus April 27 blogs.)

This can be compared to the results using an arithmetic average presented in Part II of this article and reproduced below for convenience.

Comparison of Asset Classes in an Extended Portfolio
For the expanded list of asset classes discussed in Part III where REIT's, International Stocks, and International Bonds are added to the mix the optimum allocations based on compound Required Returns are shown in the following table.

This can be compared to the results using an arithmetic average presented in MPT Part III and reproduced below for convenience.

In general the differences in the tables can be summarized by noting that a greater allocation toward Large Stocks relative to Small Stocks is preferred when compounded returns are the objective rather than arithmetic average returns. Note that the allocations for the same standard deviation (measure of portfolo risk) are only the same at the low extreme of Required Return. They are not the same elsewhere.

Conclusion
Usage of these tables can be best explained by recommending that the allocations derived using the arithmetic average Required Returns are preferred in the short-term. This is applicable to situations where a particular return is needed for income and any portfolio gains are withdrawn is they are generated. The allocations based on compounded returns are best for long-term financial planning and wealth accumulation.

Tomorrow I'll be comparing the asset allocations based on annual data with monthly data.

Posted by Prof. Pat