La China Loca A M - Fuqua School of Business



BA 453

Global Asset Allocation

ASSIGNMENT 1

Fama-French Factors: Predictability

and Asset Allocation

TEAM

La China Loca Asset Management

John Bracchini

Dorris Chen

Tiago Eiro

James Krieger

Gabriel Michalup

March 5th, 2002

INDEX

1. Introduction and Objectives

2. Methodology

3. Introduction to the Fama-French Model

4. Choosing the Independent Variables

5. Factor Forecasting Model and Estimation of Significant Independent Variables

6. Out of Sample 3 Factors Forecast Testing

7. Industry Portfolios Betas

8. Estimating Returns based on Estimated Factors and Portfolios Betas

9. Allocation Strategy

10. Out of Sample Return Forecasting and Analysis

11. Conclusions

Appendices

References

1. Introduction and Objectives

The original Fama-French model suggested that four factors determine expected returns: a high minus low market-to-book portfolio, a small minus big capitalization portfolio, the excess return on corporate bonds, and the excess returns on long-term government bonds. We worked on the simplified Fama-French model with only three factors. Our strategy was to build forecasting models for these three factors and to test to see if, given the factor loadings and forecasted premiums, we could then implement a cross-sectional portfolio allocation strategy that would outperform our buy-and-hold benchmark.

2. Methodology

From the Fama-French explanatory model, we forecasted future returns by forecasting the Fama-French factors. The forecasts of these factors will allow us to forecast any asset class with a predefined Beta (β0, β1, β2, β3).

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This prediction will help in building a portfolio that should outperform a buy-and-hold strategy.

Trading Strategy

Based on our forecasted return of our assets, we selected the weight of our portfolio using a portfolio optimizer as depicted in the following diagram:

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3. Introduction to the Fama-French Model

The Fama-French model says that the expected return on a portfolio in excess of the risk free rate is explained by the sensitivity of the return to three factors: 1) the excess return of the market portfolio minus the risk free rate 2) the excess return of a small stock portfolio minus a large stock portfolio 3) the excess return of a high book-to-market ratio portfolio minus a low book-to-market ratio portfolio. The expected excess return on the portfolio i is mathematically expressed as:

E(Ri)-Rf = β0 + β1 [ E(Rm)-Rf ] + β2 E(SMB) + β3 E(HML) + error term

In which we have:

Rf as the risk free rate

Rm as the market portfolio return

SMB (Small Minus Big) as the average return on three small portfolios minus the average return on three big portfolios

HML (High Minus Low) as the average return on two value portfolios minus the average return on two growth portfolios

β1, β2, β3 The corresponding Beta of the asset (or portfolio)

In other words, the Fama-French model tries to explain the return of an asset not only based on a market risk premium (as a CAPM) but also on two additional factors, which depend on the asset capital market size and the growth.

A complete description of the factor returns is available in the article from Fama-French, 1993, “Common Risk Factors in the Returns on Stocks and Bonds,” Journal of Financial Economics.

Even though the Fama-French model is highly utilized, we do not believe it is perfect.

1. Explanation of the Additional Factors: There are differences between High and Low, or Small and Big firms, but it is unclear whether these variables are constructed optimally.

2. Controversial Theory: The way the factors affect the portfolios are backed up by strong statistical data but lacks a sound theory explaining how these factors really affect the returns[i].

3. Portfolio Construction Method: The model has been tested using different portfolios and the results are quite different. Some portfolios created by Fama and French are organized by high and low differences between HML and SMB, giving very good results to the tested model, but the same is not applicable to portfolios that do not follow the same construction mechanism. Our conclusion from our model is that the Fama-French explanatory factors explain well the industry portfolio performance.

4. Choosing the Independent Variables

In order to develop an accurate forecasting model for the HML, SMB, and Risk Premium factors in the Fama-French model, we selected a number of likely econometric variables that would impact the levels of these factors. The additional information gained from harvesting relationships between our variables and the Fama-French Factors should help us gain slightly better information than the average investor and allow our model to outperform the market.

We decided to examine the following variables:

|Variables |Factors affected |Reason |

|Baa - Aaa |HML, SMB |Since corporate bond ratings are based on the stability of a firm’s cash |

| | |flows, they capture information about a firm’s product life cycle and its |

| | |business risk. In general, value stocks and stocks with large market |

| | |capitalization have higher bond ratings, this is opposite of growth firms |

| | |and firms with low book-to-market ratios. |

|US Risk Free Rate |HML, SMB, Rp |Key factor indicating risk free return that affects market risk premiums |

|(Annualized) | |and the cost of capital of all firms. |

|Aaa – T-bill |Rp, HML, SMB |Credit risk premium of investing in corporate bonds rather than government|

| | |securities. It affects firms’ cost of capital and equity risk premium. |

| | |In general, more established firms that have higher leverage would be more|

| | |affected by the change of this variable. |

|US Treasury, 10 yr to 3 mo|Rp, HML, SMB |Measures the spread of long-term risk free returns and short-term risk |

|spread | |free returns. It reflects general economic conditions that affect all |

| | |firms. |

|S & P 500 Price / Book |HML, SMB |Reflects the difference between a firm’s market value and its book value. |

|Ratio | |Usually, growth firms have higher price-to-book ratios versus the ratios |

| | |of value firms. |

|S & P Dividend Yield |Rp, HML, SMB |Reflects a firm’s dividend value as a portion of its stock price. Larger |

| | |and value firms should have higher dividend payout than smaller and growth|

| | |firms. |

|Oil Price / Change in Oil |Rp, HML, SMB |Oil price is imbedded into a firm’s manufacturing cost and into the |

|Price | |consumer price index. Change in oil price will have higher impact on |

| | |firms from traditional industries than firms from biotech or IT |

| | |industries, and the later usually have high growth rate and P/B ratio. |

|S&P 500 minus NASDAQ |Rp, HML, SMB |Measures return difference between more established firms and growth |

|Returns | |firms. |

|ROE of S&P 500 |Rp, HML, SMB |Reflects the value of net income of established firms compared to the |

| | |value of their shareholder’s equity. |

|Rp, HML, SMB |Rp, HML, SMB |We believe that there may be some auto-correlation/momentum effects. |

5. Factor Forecasting Model and Estimation of Significant Independent Variables

Our three-factor forecasting model was constructed by selecting a number of likely variables that could impact the values of SMB, HML, and Market Risk Premium. These independent variables were then used to forecast the likely values of these three factors in the coming month. We used our forecasted factors in the Fama-French three-factor model to predict the likely market returns for an asset we select. We then used this information to efficiently allocate our funds to optimize our portfolio returns without increasing our portfolio variance.

We used our selected variables and included them in our regression model to check if they would be able to predict our independent variables: Risk Premium factor, HML factor, and SMB factor. The selection of the independent variables was made by looking at both the p-value and the t-statistics of the various variables when used in the regression model.

After running regressions with our independent and the dependent variables, we arrived at optimal prediction models for Risk Premium, HML and SMB for the period of January 1983 – December 2000. We then checked our model with out-of-sample data for the full year of 2001.

After optimizing our portfolios for our initial January 2002 forecast, we arrived at an unlikely, extremely high return of about 5% monthly return. At this point, we adjusted the variables in our model, tested it again and used this improved model going forward.

Our prediction model seemed to predict more than we expected, having the following R-squares for the factors:

▪ Risk Premium – 7.0%

▪ HML – 6.9%

▪ SMB – 10.1%

Dependent Variables

The following are our selected independent variables for each of our dependent factors. Also included are the regression outputs from our model.

HML Factor Forecast

1. Annualized Baa minus Aaa Yields

2. S&P 500 Return on Equity Values

3. Lagged HML Factor Values (measures how much of this month’s HML is explained by HML factor of last month)

4. Oil Price

5. S&P 500 Return minus NASDAQ Return

6. Lagged equity premium return (measures how much of this month’s HML is explained by equity premium return of last month)

SMB Factor Forecast

1. Annualized Baa minus Aaa Yields

2. Dividend yield

3. Price to book ratio deviation from average

4. S&P 500 Return on Equity

5. Lagged Risk Premium Factor Values

6. Return on AAA bond on T-bill

Equity Premium Return Factor Forecast

1. Annualized Baa minus Aaa Yields

2. Dividend yield

3. S & P 500 minus NASDAQ

4. S&P 500 Return on Equity

5. Lagged Risk Premium Factor Values

6. Return on AAA bond on T-bill

7. Change in oil price

8. Oil price

9. Lagged SMB factors

6. Out of Sample 3 Factors Forecast Testing

To validate the results of our forecasting model, we tested our model with out-of sample data. We did this to check if our model would continue to yield usable data when used with data that wasn’t included in the sample data used to construct the model. This testing would help us identify the robustness of our model and would help us determine if it the model should continue to be used in constructing our trading portfolio and its optimized asset allocations. The following table shows our out-of-sample testing results:

The following charts display a comparison of our out-of-sample results with historical results:

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As shown by our comparisons above, we can note that our model is better at predicting the SMB and HML factors. The Market Risk Premium is not forecasted as accurately in our out-of-sample test. This is likely due to larger number of independent variables used to forecast this factor. Generally, forecasting models do not perform well when tested with out-of-sample data if they are over parameterized. We clearly see this effect on our model. We could refine our model by reconstructing the Market Risk Premium model such that it would make use of fewer independent variables. It is also interesting to note the number of times our forecasted factors accurately predict the sign of our historical out-of-sample factors. This is summarized in the following table:

|Factor |Count when Sign is Correct |% Correct |

|SMB |7 |58.3% |

|HML |9 |75.0% |

|Mkt Rp |5 |41.7% |

7. Industry Portfolios Betas

We are picking the 10 industry portfolios calculated by Fama-French[ii]. We decided to use these portfolios because of their high correlation with the Risk Factors. To calculate the Beta Coefficients of each industry portfolio we run the following regressions:

E(Ri)-R(f) = β0 + β1 [ E(Rm)-R(f) ] + β2 E(SMB) + β3 E(HML) + error term

Where:

Ri: Return of industry portfolio i (i = 1…10)

βj: Betas for each of the Fama-French variables (j = 1…10)

Rf: Risk free rate from US 3 month T-Bill

Risk Premiumt: Fama-French Market Risk Premium

SMBt: Small Minus Big Fama-French risk

HMLt: High Minus Low Fama-French risk

εt: Residual coefficient.

The regression outputs of the Betas and Intercept, including the statistical results, can be found in Appendices D to M, but in the next table please find the summary results of the Beta coefficients.

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8. Estimating Returns based on Estimated Factors and Portfolios Betas

After constructing the forecast equations for the Risk Factors and the Explanatory equations for the Returns of the industry portfolios, we combined all of these to forecast the returns of the portfolios for the following month.

The first step is to estimate coming month’s Risk Factors (Risk Premium, HML and SMB) from the forecast equations by introducing the values of the most recent data, which in this case are from December 2001. After getting these factors, we plug them into each portfolio explanatory equation so we can calculate the forecasted returns for the next month. The returns for each industry portfolio can be found in Appendix C.

9. Allocation Strategy

To determine the weights to put in each of the 10 portfolios, we ran an optimizer model where we set the variance of the portfolio equal to the historical variance of the S&P 500 Index. Then we ran Excel Solver to calculate the allocation weights to maximize the Return. The idea is to set a monthly portfolio in order to maximize the returns. The constraints used in the optimizer to build the portfolio were: maximum short sell weight of 50%, and maximum long position weight of 100%. Appendix C shows the weights, standard deviation, variance, and return for each portfolio.

After the optimizer was constructed and the weights found, we compared the return of the portfolio with the historical return of the S&P 500, so we could contrast if our portfolio strategy is better than a Buy-and-Hold the Index Strategy. The table below compares some statistical results of both, the portfolio and the S&P 500, concluding that constructing the portfolio gives better returns than the S&P 500 with the same standard deviation.

| |Portfolio |S&P 500 |

|Historical Mean Return (monthly) | |0.94% |

|Estimated Return (monthly) |1.24% | |

|Standard Deviation |4.16% |4.16% |

|Variance |17.26% |17.26% |

Please see Appendix C for the recommended weights and variance of the portfolio.

10. Out of Sample Return Forecasting And Analysis

In order to test the soundness of our model, we introduced out of sample data for the three factors from the Fama-French model as well as risk-free rate and market return data from January 2001 to October 2001. We calculated expected return on each of the 10 industries by using Beta values that we obtained in earlier linear regression analysis. Matching S&P 500 return volatility, we optimized the portfolio weight for each of the 10 industries across these 10 months and obtained weighted portfolio excess return. Then, we calculated the real portfolio excess return at the end of each month by timing real return with the same weight we used earlier. These results are compared with our benchmark, the S&P 500. The following table and chart summarize the results of our analysis:

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We can see that the return of our model outperformed the S&P 500 in 6 months out of the 10 and that our total real return for the 10-month out-of-sample data set is substantially above the S&P 500. However, there are some differences between our model’s projected return and real portfolio return. This means that the accuracy of our model is still under question, and there should be some uncertainty whether we can consistently beat our benchmark.

11. Conclusions

In the early 1990’s, Eugene Fama and Kenneth French questioned the CAPM Model’s ability of adequately explaining the relationship between equity risk and return. They proposed a three-factor model to explain the expected excess return of a stock portfolio. This project intended to forecast these three factors and use them to forecast the expected return of our stock portfolio by using our return optimizer spreadsheet. Our optimized return was then compared to benchmark returns (S&P 500) to test the empirical value of the Fama-French three-factor model.

Our total return on the out of sample data outperformed the S&P 500 and 6 out of our 10 portfolios tested were actually above the S&P 500. Our model also seems to add some degree of predictability to asset returns; therefore, we should be able to outperform the market over time by exploiting this predictability.

Future work could involve examining our model to check for over parameterization. If our models are truly over parameterized then we could potentially be missing out on greater returns. Additionally, we could also consider using dynamic beta since beta values were calculated as an average number over almost 30 years, while in real life, beta should be a dynamic output a of certain variable, such as P/B ratio. With these further checks, we could further refine our model to increase its accuracy and predictability.

Appendices

- Appendix A

We chose to use industry classes as our assets in constructing our traded “basket of goods”. The industry classes selected for our “basket” include the following:

1. Chemicals and Allied Products (Chemicals)

2. Consumer Durables (Durables)

3. Consumer Non-Durables (No Durables)

4. Finance (Money)

5. Manufacturing (Manufacturing)

6. Oil, Gas, and Coal Extraction and Products (Oil)

7. Telephones and Television (Telecommunications)

8. Utilities (Utilities)

9. Wholesale, Retail, and Some Services (Shop)

10. Everything Else (Other)

Each of the assets is a composition of NYSE, AMEX, and NASDAQ stocks that are assigned to each industry portfolio according to there SIC codes as selected by Fama and French. A regression is run on each of our assets to characterize their returns and volatility. We then use our Optimizer and our Forecast Model to find the most efficient asset weighting for our trading portfolio.

Industry Portfolios with Associated SIC Codes[iii]

1 Consumer Non-Durables

0100-0999

2000-2399

2700-2749

2770-2799

3100-3199

3940-3989

2 Consumer Durables

2400-2439

2500-2519

2590-2599

3000-3099

3630-3659

3710-3711

3714-3714

3716-3716

3750-3751

3792-3792

3910-3939

3990-3999

3 Oil, Gas, and Coal Extraction and Products

1200-1399

2900-2999

4 Chemicals and Allied Products

2800-2899

5 Manufacturing

2440-2499

2520-2589

2600-2699

2750-2769

3200-3629

3660-3709

3712-3713

3715-3715

3717-3749

3752-3791

3793-3909

6 Telephones and Television

4800-4899

7 Utilities

4900-4949

8 Wholesale, Retail, and Some Services

5000-5999

7000-7999

9 Finance

6000-6999

10 Everything Else

- Appendix B

Optimize Portfolio Weights

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- Appendix C

Chemical Portfolio

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- Appendix D

Durable Portfolio

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- Appendix E

Manufacturing Portfolio

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- Appendix F

Money Portfolio

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- Appendix G

No Durable Portfolio

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- Appendix H

Oil Portfolio

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- Appendix I

Others Portfolio

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- Appendix J

Shop Portfolio

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- Appendix K

Telecommunications Portfolio

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- Appendix L

Utilities Portfolio

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References

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[i] Taken from Global Asset Allocation and Stock Selection class discussion, taught by Professor Campbell Harvey, Fuqua School of Business.

[ii] Please see:

[iii]Kenneth R. French’s website; ; February 26, 2002

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Forecasted return of Asset N for next period

Portfolio weight for the next period

Portfolio optimizer

Asset N

Covariance between assets on historical data

Beta of each industry to each factor

Return variance

Lagged economic variables

Forecasted 3 factors

Historical 3 factors

Historical return on 10 industries

Forecasted 10 industries return

Out of sample 3 factors

Optimized weight

Real 10 industries return on out of sample period

Out of sample forecasted return

Bench mark return

Return comparison

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