Complete The Following Problems From Your Textbook Pages 308

Complete The Following Problems From Your Textbookpages 308309 8 6

Complete the following problems from your textbook: 8-6 EXPECTED RETURNS Stocks A and B have the following probability distributions of expected future returns: Probability A B 0.35 0.65 0.35 0.65 a. Calculate the expected rate of return, ˆrB, for Stock B (ˆrA=12%). b. Calculate the standard deviation of expected returns, σA, for Stock A (σB = 20.35%). Now calculate the coefficient of variation for Stock B. Is it possible that most investors will regard Stock B as being less risky than Stock A? Explain. c. Assume the risk-free rate is 2.5%. What are the Sharpe ratios for Stocks A and B? Are these calculations consistent with the information obtained from the coefficient of variation calculations in part b? Explain.

8-7 PORTFOLIO REQUIRED RETURN Suppose you are the money manager of a $4.82 million investment fund. The fund consists of four stocks with the following investments and betas: Stock Investment Beta A $460,000 0.50 B $500,000 0.50 C $1,260,000 0.25 D $2,600,000 0.75 If the market’s required rate of return is 8% and the risk-free rate is 4%, what is the fund’s required rate of return?

8-9 REQUIRED RATE OF RETURN Stock R has a beta of 2.0, Stock S has a beta of 0.45, the required return on an average stock is 10%, and the risk-free rate of return is 5%. By how much does the required return on the riskier stock exceed the required return on the less risky stock?

8-11 CAPM AND REQUIRED RETURN Calculate the required rate of return for Mudd Enterprises assuming that investors expect a 3.6% rate of inflation in the future. The real risk-free rate is 1.0%, and the market risk premium is 6.0%. Mudd has a beta of 1.5, and its realized rate of return has averaged 8.5% over the past 5 years.

8-13 CAPM, PORTFOLIO RISK, AND RETURN Consider the following information for Stocks A, B, and C. The returns on the three stocks are positively correlated, but they are not perfectly correlated. (That is, each of the correlation coefficients is between 0 and 1.) Stock Expected Return Standard Deviation Beta A 9.55% 15% 0.9 B 10.1% 20% C 12.6% 25% Fund P has one-third of its funds invested in each of the three stocks. The risk-free rate is 5.5%, and the market is in equilibrium. (That is, required returns equal expected returns.) a. What is the market risk premium (rM − rRF)? b. What is the beta of Fund P? c. What is the required return of Fund P? d. Would you expect the standard deviation of Fund P to be less than 15%, equal to 15%, or greater than 15%? Explain.

Estimating Returns Consider the use of a probability analysis in estimating returns and determining the standard deviation. You may use a hypothetical situation and do the calculations or just describe the steps in the process.

Paper For Above instruction

Understanding expected returns, risk measures, and the Capital Asset Pricing Model (CAPM) is fundamental for effective investment decision-making. This paper explores these concepts through a series of problems that demonstrate how investors evaluate asset performance and risk. Detailed calculations and explanations illuminate the underlying principles guiding portfolio management and asset valuation.

Expected Returns and Risk Measures

Expected return calculations provide insight into the anticipated performance of stocks based on probability distributions. For Stock B, with a given expected return of 12%, probability-weighted outcomes are determined by multiplying each possible return by its associated probability and summing these products. This approach yields an expected return that encapsulates potential payoffs considering different market scenarios (Elton & Gruber, 2020). The variance and standard deviation quantify risk by measuring the dispersion of returns around the expected value. Stock A’s standard deviation of 20.35% indicates the typical deviation from its expected return, reflecting its volatility (Bodie, Kane, & Marcus, 2014).

The coefficient of variation (CV), calculated as the standard deviation divided by the mean return, offers a risk-to-return ratio, facilitating comparisons between assets with different expected returns and risk levels. A lower CV suggests a more favorable risk-return tradeoff. Understanding whether investors perceive Stock B as less risky than Stock A depends on CVs; if Stock B’s CV is lower, it indicates relative efficiency in risk-adjusted returns (Sharpe, 1966). However, investors also consider other factors, such as risk aversion and market perceptions, which may influence their risk assessments beyond CV metrics.

Sharpe Ratio and Investment Performance

The Sharpe ratio measures excess return per unit of total risk by subtracting the risk-free rate from the asset’s expected return and dividing by its standard deviation. For Stocks A and B, these ratios help determine which security offers better risk-adjusted performance (Sharpe, 1966). If Stock A’s Sharpe ratio surpasses that of Stock B, it suggests superior compensation for risk; discrepancies with CV-based assessments highlight different focuses—total risk versus volatility (Lintner, 1965). Such differences underscore the importance of context in evaluating asset attractiveness.

Portfolio Required Return

Constructing a portfolio from multiple stocks involves calculating the weighted average of individual expected returns. The fund’s overall required return is the sum of each stock’s expected return weighted by its proportion of total investment (Fama & French, 2004). The asset betas measure sensitivity to market movements; combining these with market and risk-free rates through the CAPM formula (r = r_f + β(r_m − r_f)) yields the portfolio’s expected return (Sharpe, 1964). The weighted beta of the portfolio is also derived, reflecting its overall market risk exposure. Proper assessment aids in aligning expected returns with investors’ risk appetite and market conditions.

Comparing Required Returns: Risk and Beta

The required return differences between stocks with varying betas are explained via CAPM. A higher beta implies greater systematic risk and thus a higher required return. For example, Stock R with a beta of 2.0 demands a significantly higher return compared to Stock S with a beta of 0.45, assuming other factors constant (Ross, 1976). The excess return on Stock R over Stock S, based on their betas and the market risk premium, underscores the risk-return tradeoff central to asset pricing models.

Applying CAPM and Estimating Returns

The CAPM provides a framework for calculating the expected return of assets considering inflation expectations, real risk-free rates, and market risk premiums. For Mudd Enterprises, incorporating an anticipated inflation rate adjusts the nominal required return using Fisher’s equation (Fisher, 1930). The market risk premium and Mudd’s beta further refine this estimate, aligning expected returns with prevailing economic conditions (Barberis & Thaler, 2003). Past return data helps validate these projections but also highlights the importance of market expectations and macroeconomic influences in asset valuation.

Portfolio Risk and Correlation Effects

Investors benefit from diversification through assets that are not perfectly correlated. For Stocks A, B, and C, their positive but imperfect correlations reduce combined risk compared to individual assets. Calculating the beta of a diversified fund involves a weighted average of individual stock betas. The fund’s required return, in equilibrium, equals its expected return based on CAPM, considering its portfolio beta and the market risk premium (Markowitz, 1952). The standard deviation of the return for a diversified fund depends on the correlations; typically, diversification reduces risk, potentially resulting in a standard deviation less than the average of individual stocks, depending on correlation coefficients (Fama & French, 1993).

Estimating Returns Using Probability Analysis

Probability analysis for return estimation involves identifying possible future states of the economy, assigning probabilities to these states, and calculating the weighted average of returns across these scenarios. Variance and standard deviation are then computed to assess risk. This process offers a structured approach to quantify uncertainty and design portfolios aligned with risk preferences. For instance, under hypothetical conditions with specified probabilities and returns, investors can derive expected returns and volatility, assisting in informed decision-making (Merton, 1980).

Conclusion

Mastering fundamental concepts such as expected return, risk measurement, the Sharpe ratio, and the CAPM equips investors with tools to assess asset performance and construct optimal portfolios. The calculations and theoretical insights discussed reinforce the importance of quantitative analysis in finance, emphasizing the delicate balance between risk and reward that guides investment choices.

References

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• Bodie, Z., Kane, A., & Marcus, A. J. (2014). Investments. McGraw-Hill Education.
• Elton, E. J., & Gruber, M. J. (2020). Modern Portfolio Theory and Investment Analysis. Wiley.
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• Fama, E. F., & French, K. R. (2004). TheCapital Asset Pricing Model: Theory and Evidence. Journal of Economic Perspectives, 18(3), 25-46.
• Fisher, I. (1930). The theory of interest. Macmillan.
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• Ross, S. A. (1976). The arbitrage theory of capital asset pricing. Journal of Economic Theory, 13(3), 340-360.