## The Relationship Between Arithmetic and Compounded Average Returns

I first wrote about this topic on 6/2/08 and Dr. Kris herself covered another aspect of the subject on 4/13/09. My article was entitled “MPT Part VI: Arithmetic vs Compounded Returns.” Since that time monthly data has been assembled for each of the nine asset classes back to 1928 and the development of the SMC Analyzer software now permits graphic analysis. So let’s have another look at this important topic.

When optimizing the allocation of assets in an investment portfolio the goal is that they will produce a certain average annual return in addition to doing so with a minimum variation about this average. There are two ways in which returns can be evaluated, as an Arithmetic Average or as a Compound Average. These two can be significantly different from each other and the consequences of this difference to the value of a portfolio can grow significantly with time.

Examples:

An example of an Arithmetic Average return is as follows: An asset class produces annual returns over a three year period of +8%, -5% , and +12%, the Arithmetic Average return is:

(8 – 5 + 12) / 3 = 5.00%

The Compounded Average annual return is a bit more complicated to calculate. For this same example it is:

(((1 + 8/100) x (1 + 5/100) x (1 + 12/100) )1/3 – 1 ) x 100 = 4.74%

The Arithmetic Average is higher because it unfairly 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. 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 in this case would be -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 Arithmetic Average standard deviation of the data. In the following approximation formula the variance is the standard deviation squared.

Compound Average = Arithmetic Average-variance/(2*(1+Arithmetic Average))

An ideal compounding rate would be unbiased in the weighting of gains and losses. For short investment horizons the Arithmetic Average is closer to this ideal and is a better measure for predicting what is more likely to occur in a single time period. A Compounded Average on the other hand is more representative for what is likely to occur over many time periods in terms of the long term accumulation of wealth.

Portfolio Return Data:

Let’s look now at some actual portfolio return data using data for nine asset classes provided with the SMC Analyzer to compute the approximate relationship between Arithmetic Average and Compound Average for two strategies. The first graph is the application of classic Modern Portfolio Theory optimum asset allocation where all assets classes are held in the recommended allocation percentages with no market timing strategy applied. These graphs were not computed using the approximation formula above. You can look up the Compound Average return along the bottom of the graph and see what equivalent Arithmetic Average return would be required to produce about the same level of portfolio growth. In this scenario a 13% Arithmetic Average rate of return would be required to equal a 10% Compounded Average annual return.

The two return values will have the same standard deviation about their respective averages. What makes this approximate is that the portfolio allocations are not the same and the actual distributions of returns are not exactly the same. The equality of the measures of risk (standard deviations) implies equality in level of return.

The graph below shows the relationship between the two averages using the Modern Portfolio Theory market timing enhancement of placing a particular allocation for an asset class in the safety of T-Bills when a significant and durable downward market trend in that asset class is identified. As can be seen, this graph expands the range of returns achievable to 17% Compounded and 19% Arithmetic Average. You will notice that the two averages are closer to each other than in the graph further above for the classic Modern Portfolio Theory application. This is due to the lower variance (lower risk) in the distribution of returns with the market timing strategy applied.