## MERp MER t apb

inferences.

5. A priori beta estimate: Developing forecasts of betas (ex-ante) for assessing ex-post performance.

6. Value added: Using a different approach to examine returns behavior.

7. Controlling for public information: Explaining variations in realized excess returns on a portfolio after adjusting for dividend yield, interest rates, and benchmark portfolio returns.

8. Style analysis: Estimating skill based on selection returns after adjusting for style returns.

9. Controlling for size and value: Estimating variations in realized excess returns after adjusting for size and value factors.

### Statistical Refinements

Return-based performance assessment techniques, like alpha, the information ratio, and the Sharpe ratio, involve the use of regression techniques and/or tests of significance of estimates. Advanced analysis refines and broadens the accuracy and efficiency of statistical results.

### 1. Hetroskedasticity

One of the underlying assumptions of the regression model is that die variance of error terms is the same across all observations in the sample. Violation of this assumption generates hetroskedastic error terms. That is, error terms have unequal variances. Unequal error variances produce biased standard deviation of the error terms (called standard error of estimate, SEE), which in turn produces biased estimated standard error for regression coefficients as well as the intercept. Consequently, all'tests of significance for coefficients and models will be biased due to hetroskedastic error terms. Various refinements allow researchers to detect the presence of hetroskedasticity (such as a residual plot) and correct the problem using various adjustments in order to achieve unbiased standard errors and valid tests of significance.

### 2. Serial Correlation

Another underlying assumption of the regression model is that the error terms are independent of one another. If they are not, they are said to be serially correlated. A positive (negative) serial correlation underestimates (overestimates) the variance of error terms which in turn generates underestimated (overestimated) variance for regression coefficients leading to inflated (deflated) i-statistics, producing incorrect inferences. Various refinements allow researchers to detect seriaLâ[email protected]<i|ii|W.ùi a time series and fix the problem in order to achieve valid i-tests for testing the hypotheses and drawing correct inferences.

### 3. Bayesian Correction

In a return-based performance assessment technique, one observes an ex-post distribution of alphas (superior performance) and betas across managers. Bayesian correction allows one to match this ex-post distribution to an ex-ante distribution, based on prior information. Thus, one can quickly detect the large and unexpected discrepancies between the two distributions and make necessary judgments and required adjustments about the true value of alphas and betas based on prior information.

Financial Theory Based Refinements

### 1. Benchmark Timing

Benchmark timing is an extension of basic return regression models offering a tool to assess superior market timing skills of a portfolio manager. A market timer will include high (low) beta stocks in her portfolio if she expects an up (down) market. If her forecasts are accurate, her portfolio will outperform the benchmark portfolio. Using a benchmark timing model, we can empirically test whether there is evidence of superior market timing skills exhibited by the portfolio manager. The regression equation is:

Where Dt is a dummy variable that is assigned a value of zero for down-market and one for up-market. Mp is the difference between the up-market and down-market betas and will be positive for a successful market timer.

### 2. A Priori Beta Estimate

Return-based models use standard regression techniques in which realized returns are regressed against market returns and estimates for alpha and beta (slope) are generated. An improvement over this method is to calculate ex-ante or forecasted beta as opposed to realized beta. Use of forecasted beta may alter the magnitude of alpha (superior performance) as obtained by the return-based model. Also, forecasted beta will be useful in overcoming the appearance of correlation, when there is, in fact, no correlation between portfolio returns and benchmark returns.

The value added concept examines the behavior of portfolio returns from a different perspective. It raises the question of how much an investor will be willing to pay for an investment. An investor will be willing to pay a higher price for a portfolio (higher than the fairly priced portfolio) if she believes the portfolio will produce higher returns. Assuming risk-free assets and market returns are fairly priced equal to one unit, an investor may be willing to pay 1.039 to receive portfolio returns indicating that the value added is 3.9% for the investor.

### 4. Controlling for Public Information

According to CAPM, portfolio returns are solely explained by benchmark market portfolio returns (i.e., the CAPM is a single factor model). Returns over and above risk-adjusted market returns captured by alpha are attributed to the superior skills of the portfolio manager. However, some believe that there are factors other than the market, that account for variation in realized portfolio returns. Two such factors are dividend yield and interest rates, which are believed to have the ability to predict market conditions. Information is publicly available on both of these variables. Once these two additional market predictive variables are included into the standard CAPM-based regression model, the bar to achieve superior performance becomes higher. In addition to market returns, now interest rates and dividend yield returns are also subtracted from the realized returns to estimate alpha. The regression takes the form of:

RPt = ap + (3 x RB(t) + (3dy [RB(t) x DYt_j] + (3IN [RB(t) x INJ + sp(c)

Where the second and third terms are additions to the standard return generating regression equation. DY j is the lagged dividend yield and IN ^ is the lagged interest rate variable assumed to have predictive ability of explaining variations in portfolio returns at time t.

### 5. Style Analysis

Returns-based models, including many advanced refinements, estimate alpha by subtracting benchmark portfolio returns (usually a market portfolio like the S&P 500) from the actively managed portfolio returns. A positive and significant alpha is taken as evidence of superior performance. However, the superior performance may be the result of a particular style, such as value investing, and not skill. Investors pay higher fees to the portfolio managers that use their skill and not style to produce superior returns. Investors can invest in a value index and earn the style returns themselves. Thus, the returns achieved by an active portfolio manager should be compared against the relevant style index and not the general market index.

Advanced models refine the basic models by decomposing the realized returns into style and selection components. An active manager is credited for achieving selection returns-over and above the benchmarked style returns. In order to assess the achievement level of the fund manager, the information ratio (IR) can be constructed based on the mean and standard deviation of selection returns. As discussed earlier, a higher IR offers a higher evidence of skill compared to a lower IR.

### 6. Controlling for Size and Value

Standard one factor (market returns) regression models can be extended to include two additional variables capturing size and value effects. So, variation in realized returns is explained by size and value factors in addition to a benchmark return.

Fama and French1 (1993) proposed performance analysis techniques based on the following regression.

RP(t) = ap + (3 x RB(t) + ps x VLt + (3v x MVt + ep(t) where:

VL = the return to a long portfolio (on small capitalization stocks) and short portfolio (on large capitalization stocks) MV = the return to long portfolio (on high book to price stocks) and short (on low book to price stocks)

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