In 1986, Gary P. Brinson, L. Randolph Hood, and Gilbert L. Beebower published a study about the effects of asset allocation of 91 large corporate pension funds measured from 1974 to 1983. The result of the study showed that Asset Allocation (what they called Policy) explained 93.6% of Total Return Variation, Asset Allocation together with timing the market explained 95.3%, and Asset Allocation together with stock selection explained 97.8%.

In the study, Investment Policy identified the long-term asset class allocation plan of each fund and was determined by using the funds’ benchmark returns and weights, Policy with timing used actual weights but benchmark returns, and Policy with selection used benchmark weights with active returns (actual security returns in excess of benchmark returns).

**A Universal Myth**

Armed with the findings of this study, some financial advisers were quick to jump on the asset allocation bandwagon. If asset allocation was supposedly the key, then stock selection and timing would have little consequence.

But a survey by Nuttall & Nuttall (1998) demonstrates that out of 50 writers who quoted Brinson, only one quoted him correctly. Approximately 37 writers misinterpreted Brinson’s work as an answer to the question, “What percent of total return is explained by asset allocation policy?” and five writers misconstrued the Brinson conclusion as an answer to the question, “What is the impact of choosing one asset allocation over another?”

Ibbotson & Associates called it a universal myth:

“From the marketing materials of mutual fund companies and financial planning firms to the mouths of academics and financial representatives, there is a universal misunderstanding of the relationship between asset allocation and performance”.

What the Brinson study describes is the variability (not the return) of the portfolio’s performance over time due to asset allocation, i.e. Brinson found that more than 90 percent of the *movement* of one’s portfolio from quarter to quarter is due to market *movement* of the asset classes in which the portfolio is invested.

**Putting Things In Context**

Perhaps what contributed to the myth was the fact that Brinson made a connection between total returns and the variation of returns explained by asset allocation. In the study, Brinson mentioned that timing deducted 0.66% and stock selection deducted 0.36% from the benchmark annualized total return (not variability) over the 10-year period in his study. He then connected this to his statement on the determinants of portfolio performance.

In his study, Brinson used regression to view the relationship between:

- Asset allocation versus actual total return,
- asset allocation with timing versus actual total return, and
- asset allocation with stock selection versus actual total return.

Based on the presumably linear relationship between the independent and dependent variables, Brinson could get a *predicted* total return (the dependent variable) given any values of the independent variables using regression.

The variability of the *predicted* total returns around the actual average total return divided by the variability of the actual total returns around the same actual average total return turned out to be 93.6% for asset allocation, 95.3% for asset allocation with timing, and 97.8% for asset allocation with stock selection. This ratio is termed r-squared in statistical parlance.

Regression is a mathematical construct that relates the movements in portfolio return (relative to its mean) to any attributes or indeed any number of attributes that you choose.

It is a statistical relationship that does not necessarily imply cause-and-effect. If asset allocation explains the movement in portfolio returns, what does that actually tell you?

Ibbotson did their own study and instead of using “estimated policy weights and the same asset class benchmarks for all funds, the actual policy weights and asset class benchmarks of the pension funds were used. In each quarter, the policy weights were known in advance of the realized returns”.

The study found that, on average, the policy benchmarks match the actual portfolios, so the ratio is 1.0, or 100 percent. He described it succinctly when he said “Policy benchmarks match the actual portfolios because, if one averages the universe of funds, one gets the index...”

**Factor Models**

Factor models are another area of research that uses regression. Academics have attempted to explain return behavior attributed to factors such as value, momentum, size, market, quality, etc.

This began with the Capital Asset Pricing Model, which suggested all expected stock returns could be described via beta. Beta measured the extent to which a stock return moved with the market portfolio’s return (often approximated by an index such as the S&P 500 or Russell 3000). However, the evidence in support of this model was lacking.

Then in 1992, Eugene Fama and Ken French laid the empirical foundations for the Fama and French three-factor model which explained how size and value, in addition to the market, contributed to portfolio returns (or more accurately portfolio return variability).

Five years later in 1997, Mark Carhart created a new “four-factor model” which significantly enhanced the explanatory power of the original three-factor model.

Not to be outdone, and as if to head off competition from a "q-model" proposed by Zhang and company, Fama and French in 2014 extended their model to include five factors (operating profitability and YOY asset growth scaled by prior year's total assets being the two additional factors).

Indeed, one could continue adding a slew of factors to improve the explanatory power of the factor model. If factors explain the movement in portfolio returns, which factors are the true ones? Are these factors themselves related? Does portfolio return actually have a linear relationship with these factors?

**Explanatory Power**

R-squared is used by both factor models and asset allocation studies to denote the degree to which the chosen attributes explain movements in the dependent variable.

The dependent variable in the context of this article is portfolio return. It is aptly called a "dependent variable" because it is assumed to depend on attributes that have been chosen before-hand - attributes such as value, size, or asset allocation. These attributes are called independent variables.

In linear regression, the dependent variable is assumed to have a linear relationship with the independent variables. Assuming this relationship is correct, plugging values in the independent variables predicts values of the dependent variable.

Divide the variability of the predicted values by the variability of the actual values and you get the so-called explanatory power of the model i.e. r-squared (*variability* uses a simple formula that finds the sum of the squared differences between the predicted/actual values for each observation and the mean of the actual values).

Let me conclude by showing you a *linear relationship* which has an impressive r-squared of 97% in the chart below. The illustration should have *strong explanatory power* in letting readers know why regression and r-square, while great as analytical tools, can be difficult to interpret.

**Disclosure:** I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.

I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.