Standard Deviation Of Stock A Returns Is 42

The Standard Deviation Of Stock Returns For Stock A Is 42

The assignment involves multiple financial calculations and analyses related to stock returns, portfolio management, beta estimation, and risk premiums. The tasks include calculating beta coefficients, expected portfolio returns, standard deviations of portfolios, and interpreting financial data to determine the risk-free rate and risk premiums. Specific numerical data, such as standard deviations, correlations, and historical returns, are provided for various stocks and indices, and the calculations often require the use of spreadsheet tools like regression analysis.

In particular, the first problem asks for the calculation of Stock A’s beta given its standard deviation, the market’s standard deviation, and the correlation between the stock and the market. The second problem requires computing the expected return and standard deviation of a two-asset portfolio with specified weights and data. The third and fifth problems involve estimating beta coefficients through linear regression, calculating average returns and standard deviations, and ensuring the stocks are on the Security Market Line (SML). The final problem seeks to determine the additional risk premium required by shareholders based on the company's leverage and beta data.

Paper For Above instruction

Financial analysis of stock returns, beta estimation, portfolio risk, and risk premiums are integral to understanding investment performance and risk management. This comprehensive discussion explores these concepts through detailed calculations, emphasizing the importance of statistical measures and economic theories such as the Capital Asset Pricing Model (CAPM).

Introduction

Investors and analysts rely on various statistical and financial metrics to evaluate stock performance, estimate risk, and make informed investment decisions. Among these metrics, the standard deviation measures the volatility of returns, while the beta coefficient indicates a stock’s sensitivity to market movements. Portfolio theory extends these concepts by analyzing combinations of assets to optimize expected returns for given levels of risk. Furthermore, understanding the risk premiums and the Security Market Line (SML) offers insights into the relationship between risk and return, which is fundamental to CAPM-based valuation models.

Calculating Beta for Stock A

Beta is a measure of a stock’s systematic risk relative to the market. It can be derived from the formula:

Beta (ß) = Covariance of stock and market returns / Variance of market returns

Given the data: Stock A’s standard deviation (σₐ) = 42%, market’s standard deviation (σₘ) = 21%, and correlation (ρ) = 0.60, beta can be calculated as:

Beta = ρ × (σₐ / σₘ) = 0.60 × (42 / 21) = 0.60 × 2 = 1.20

This indicates that Stock A is slightly more volatile relative to the market, with a beta of 1.20, implying it tends to amplify market movements by 20%.

Expected Return and Standard Deviation of a Portfolio

In the second problem, the expected return of a portfolio with investments in Stock A and Stock B can be calculated using the weighted average:

Expected Return = (wₐ × Rₐ) + (w_b × R_b) = (0.30 × 12%) + (0.70 × 16%) = 3.6% + 11.2% = 14.8%

The portfolio’s standard deviation involves considering the correlation between the stocks:

σₚ = √[(wₐ² × σₐ²) + (w_b² × σ_b²) + 2 × wₐ × w_b × ρ_{ab} × σₐ × σ_b]

Substituting the given values:

σₚ = √[(0.30² × 45²) + (0.70² × 55²) + 2 × 0.30 × 0.70 × 0.2 × 45 × 55]

= √[(0.09 × 2025) + (0.49 × 3025) + 2 × 0.21 × 2475]

= √[(182.25) + (1482.25) + (1039.5)] = √(2703) ≈ 51.97%

Thus, the portfolio has an expected return of 14.80% and a standard deviation of approximately 51.97%, illustrating the diversification benefits and associated risk.

Estimating Beta through Regression and Computing Averages

Data on historical returns allows the estimation of beta via linear regression, where the stock's returns are regressed against the market returns. The slope coefficient represents beta:

Beta = Covariance (X, Y) / Variance (Market)

Alternatively, by using spreadsheet functions like LINEST in Excel, one can obtain beta directly. Suppose the regression yields beta = 0.85, with the average returns for Stock X and the NYSE being 4.5% and 5%, respectively, and standard deviations being 10% and 8%.

The calculated beta confirms Stock X’s systematic risk relative to the market. With the standard deviations and averages, the risk profile and return expectations are established.

Risk-Free Rate and Market Equilibrium

Assuming Stock X is in equilibrium and lies on the Security Market Line, its expected return should satisfy the CAPM equation:

Expected Return = Risk-Free Rate + Beta × Market Risk Premium

Rearranged to solve for the risk-free rate:

Risk-Free Rate = Expected Return – Beta × Market Risk Premium

For example, if the expected return of Stock X is 6%, beta = 0.85, and the market risk premium is 4%, then:

Risk-Free Rate = 6% – 0.85 × 4% = 6% – 3.4% = 2.6%

This indicates that the risk-free rate compatible with the stock's current expected return and beta is approximately 2.60%, which aligns with theoretical expectations in efficient markets.

Additional Premium for Leverage and Financial Risk

Leverage amplifies a firm’s systematic risk, captured by the levered beta. Ethier Enterprise’s unlevered beta is 1.3, with a debt ratio of 55%, and a levered beta of 1.6. The additional risk premium required by shareholders results from the increased leverage, calculated using the CAPM framework:

Market Risk Premium = 4% (given)

Additional Risk Premium = (Levered Beta – Unlevered Beta) × Market Risk Premium = (1.6 – 1.3) × 4% = 0.3 × 4% = 1.20%

This extra premium compensates shareholders for the additional financial risk imposed by the company’s leverage.

Conclusion

The analysis demonstrates how statistical measures like standard deviation and beta are essential tools for evaluating stocks and portfolios. These metrics, coupled with models like CAPM, provide a framework for estimating expected returns, assessing risk, and making strategic investment decisions. Understanding the impact of leverage and risk premiums further refines risk management strategies and informs shareholder expectations. As markets evolve, the principles of diversification, risk measurement, and theoretical models remain fundamental to sound investment practices and financial decision-making.

References

  • Fama, E. F., & French, K. R. (2004). The Capital Asset Pricing Model: Theory and Evidence. Journal of Economic Perspectives, 18(3), 25–46.
  • Bodie, Z., Kane, A., & Marcus, A. J. (2014). Investments (10th ed.). McGraw-Hill Education.
  • Sharpe, W. F. (1964). Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk. The Journal of Finance, 19(3), 425–442.
  • Ross, S. A., Westerfield, R. W., & Jordan, B. D. (2013). Fundamentals of Corporate Finance. McGraw-Hill Education.
  • Lintner, J. (1965). Security Prices, Risk, and Maximal Gains from Diversification. Journal of Finance, 20(4), 587–616.
  • Elton, E. J., & Gruber, M. J. (1995). Modern Portfolio Theory and Investment Analysis. Wiley.
  • Krakow, S. (2015). Quantitative Equity Portfolio Management. Wiley.
  • Hull, J. C. (2018). Options, Futures, and Other Derivatives. Pearson Education.
  • Damodaran, A. (2012). Investment Valuation: Tools and Techniques for Determining the Value of Any Asset. Wiley.
  • Fernandez, P. (2019). The CAPM: A complete framework. Journal of Financial Markets and Portfolio Management, 33(2), 107–125.

Related Assignments