Chapter 8 Problem 6: The Following Are The Historic

Chapter 8 Problem 6chapter 86the Following Are The Historic Retur

The assignment requires analyzing historic financial data for Chelle Computer Company and the general market index to compute the correlation coefficient and beta of Chelle Computer. Additionally, it involves evaluating two mutual funds (Fund T and Fund U) using CAPM principles by calculating their expected returns based on given forecasts and market data, and determining if these funds are correctly priced according to the security market line (SML). The task also includes drawing security market lines under specified conditions, calculating and comparing beta estimates for Rader Tire using different indices, and assessing the performance of these indices based on market data. Further, the assignment asks for estimating expected returns for three stocks (QRS, TUV, WXY) using single and multi-factor models incorporating market and macroeconomic risks, analyzing exposure to macroeconomic variables, and evaluating the usefulness of the models. Lastly, it involves calculating expected stock prices based on factor models, identifying mispriced securities for arbitrage opportunities, and analyzing factor betas via regression for further insights into risk exposures, including the relevance of Fama-French factors and classifying funds as growth or value based on factor loadings. This comprehensive analysis combines statistical computation, financial modeling, valuation, and risk assessment.

Sample Paper For Above instruction

Financial analysis and risk modeling are critical tools in contemporary investment management, enabling firms and investors to identify, quantify, and manage potential risks and returns. This paper explores various aspects of financial modeling, including the calculation of historical correlations, beta coefficients, expected returns using the Capital Asset Pricing Model (CAPM), security market line (SML) assessments, multi-factor models, and arbitrage opportunities based on mispricings.

Historical Return Analysis and Correlation Measurement

Understanding the relationship between a firm's stock and the overall market is essential for portfolio diversification and risk management. The correlation coefficient, a statistical measure of linear association between two variables, is computed to quantify this relationship. Using the historic returns of Chelle Computer and the general market index, we calculate the correlation coefficient, which indicates the strength and direction of their relationship.

The correlation coefficient formula is given by:

\( \rho = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} \)

where Cov(X,Y) is the covariance between Chelle Computer's returns and the market index, and \(\sigma_X\), \(\sigma_Y\) are the standard deviations of their returns. A high positive correlation suggests the stock moves closely with the market, while lower or negative correlations indicate diversification benefits or different risk factors.

Similarly, the beta coefficient, a measure of systematic risk, is derived from regression analysis of Chelle Computer's returns against the market index returns. Beta quantifies how sensitive Chelle Computer's stock returns are relative to market movements. Calculating beta involves performing a linear regression where Chelle's returns are the dependent variable and the index returns are the independent variable. The slope of the regression line is the beta coefficient, which helps determine if the stock is more volatile than the market.

Expected Returns and the CAPM Framework

Applying the CAPM allows for the estimation of expected returns based on the risk-free rate, the market's excess return, and the stock's systematic risk (beta). The CAPM formula is:

\( E(R_i) = R_f + \beta_i (E(R_M) - R_f) \)

where \( R_f \) is the risk-free rate, \( E(R_M) \) is the expected market return, and \( \beta_i \) is the beta for stock i.

For Fund T with a forecasted return of 9.0% and a beta of 1.20, using a risk-free rate of 3.9% and an expected market risk premium of 6.1%, the calculated expected return via CAPM is:

\( E(R_T) = 3.9\% + 1.20 \times 6.1\% = 3.9\% + 7.32\% = 11.22\%\)

Similarly, for Fund U with a forecasted return of 10.0% and beta of 0.80, the CAPM expected return is:

\( E(R_U) = 3.9\% + 0.80 \times 6.1\% = 3.9\% + 4.88\% = 8.78\%\)

The comparison of these CAPM estimates with respective forecasts reveals whether the funds are over- or undervalued. If the market-based expected return exceeds the forecasted return, the fund might be undervalued and vice versa.

Security Market Line (SML) Analysis

Plotting the SML involves graphing the relationship between expected return and beta, illustrating the trade-off between risk and return. Under the specified conditions, the SML equations are derived using the given risk-free rates and market premiums, facilitating the assessment of whether funds are correctly priced relative to their risk profiles.

For example, if the risk-free rate is 8%, and the market premium is 0.15, the SML line can be plotted accordingly, and funds can be evaluated for pricing efficiency based on their positioning relative to this line.

Beta Estimation Using Different Indices

Calculating and comparing beta coefficients for Rader Tire using proxy and true indices help analyze how different market measures influence perceived security risk. The beta estimates are derived via regression analyses, and their significance can be tested statistically. If the current period market return is 12% and Rader Tire's return is 11%, analyzing whether the beta estimates indicate superior or inferior relative risk-adjusted performance is essential for investment decisions.

Multi-Factor Models and Stock Return Estimation

Expanding beyond single-factor models, the use of multi-factor models, incorporating market, and macroeconomic variables, provides a more comprehensive view of expected stock returns. The models estimate how different risk exposures influence returns, using factor loadings and risk premiums.

For stocks QRS, TUV, and WXY, expected returns calculated from the single-factor model depend solely on the market risk premium, while the multi-factor model incorporates macroeconomic risks through factors MACRO1 and MACRO2. The differences between these models often reflect improved explanatory power with multiple variables, endorsing multifactor approaches as more practical for actual investment analysis.

Pricing, Arbitrage, and Riskless Profit Opportunities

By analyzing the current and expected future prices of stocks A, B, and C against their model-based fair values, arbitrage opportunities can be identified. Mispricings offer scope for riskless profit through strategies like constructing portfolios that exploit these deviations. For instance, buying undervalued stocks and short-selling overvalued ones, followed by balancing the portfolio, can generate arbitrage profits, assuming transaction costs are minimal.

Regression Analysis and Risk Factor Significance

Regression analysis of stock returns against identified risk factors helps estimate factor loadings or betas, which quantify each factor's influence. Statistical significance tests determine which loadings are meaningfully different from zero, indicating a genuine association. Additionally, evaluating how well models explain return variations involves assessing metrics like R-squared values, with higher values indicating better explanatory power.

In the context of the Fama-French three-factor model, distinguishing the market factor from size (SMB) and value (HML) factors provides insights into the characteristics driving stock performance. Classifying portfolios as growth or value stocks based on factor loadings supports strategic investment decisions.

Conclusion

Overall, integrating statistical analysis, financial theory, and empirical data enhances decision-making in asset valuation and risk management. Determining correlations, betas, expected returns, and arbitrage scenarios allows investors to optimize portfolios aligned with their risk appetite and return objectives. Advanced models incorporating multiple risk factors provide a nuanced understanding of stock behavior, aiding in more accurate valuation, risk assessment, and strategic allocation.

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