The Following Information Is Available On The Percentage Rat

The Following Information Is Available On The Percentage Rates Of R

The assignment involves analyzing asset returns, calculating optimal portfolio weights, examining the Capital Market Line (CML), evaluating security plotting on the Security Market Line (SML), assessing statements regarding the CAPM and diversification, determining the impact of covariance changes on stock prices, and evaluating market efficiency based on historical fund performance.

Paper For Above instruction

The analysis of asset returns and portfolio optimization is fundamental to portfolio management and asset allocation strategies in finance. The provided data on percentage returns over three years offers a basis for applying various financial models, including the computation of minimum-variance portfolios, the derivation of the Capital Market Line (CML), and the assessment of securities within the broader market context.

Part A: Portfolio Weights and Risk-Return Calculations for Shares A and B

The initial step involves determining the portfolio of Shares A and B that minimizes the total risk, as measured by standard deviation. This classic optimization problem relies on the mean-variance framework, where the goal is to find the weights \(w_A\) and \(w_B\) such that the overall portfolio risk attains its lowest point. The relevant formulas include the calculation of the portfolio variance based on the individual variances, covariances, and weights associated with the securities.

The expected returns of the securities are computed as the average of their historical returns:

  • Share A: \((-6 - 18 + 60)/3 = 78/3 = 26\%\)
  • Share B: \((32 - 22 + 35)/3 = 45/3 = 15\%\)

The variance and covariance calculations require computing the deviations of each year's return from the mean, then squaring and averaging these deviations for the variances, and multiplying deviations for covariance. Using these, the optimal weights are found by solving the formula for the minimum-variance portfolio:

\[

w_A^{*} = \frac{\sigma_B^2 - \text{Cov}(A,B)}{\sigma_A^2 + \sigma_B^2 - 2 \text{Cov}(A,B)}

\]

The exact calculation involves numerical substitution of the variances, covariance, and solving for \(w_A^{}\) and \(w_B^{}\). Once the weights are determined, the expected return of the portfolio becomes:

\[

E(R_p) = w_A E(R_A) + w_B E(R_B)

\]

and the standard deviation (risk) is derived from the square root of the portfolio variance.

Part B: The Numerical Equation of the Capital Market Line (CML)

The CML represents combinations of the market portfolio and the risk-free asset, describing the highest attainable return for a given level of risk. Its equation, based on the Capital Asset Pricing Model (CAPM), is: 

 E(R_p) = R_f + \frac{E(R_M) - R_f}{\sigma_M} \times \sigma_p

Where:

- \(R_f\) is the risk-free rate (10% in this context),

- \(E(R_M)\) is the expected market return (average of the market returns),

- \(\sigma_M\) is the standard deviation of the market portfolio,

- \(\sigma_p\) is the standard deviation of the portfolio.

Plugging in the known values: \(R_f = 10\%\), \(E(R_M) = 50\%\), \(\sigma_M\) calculated from the data, the equation becomes a linear function of \(\sigma_p\) that defines the CML.

Part C: Plotting Shares on the Security Market Line (SML)

To determine whether Shares A and B plot on the SML, their Betas must be calculated as:

\[

\beta_i = \frac{\text{Cov}(R_i, R_M)}{\sigma_M^2}

\]

Once Betas are obtained, the expected return of each security can be compared to the return predicted by the SML:

\[

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

\]

If the actual expected return equals the SML return for the calculated Beta, the security plots on the SML. Deviations indicate over- or under-valuation, with implications for market efficiency.

Part 2: Evaluative Statements on CAPM and Diversification

Statement (a): Under the CAPM, securities with the same Beta should offer similar expected returns, regardless of their idiosyncratic risk. Higher idiosyncratic risk is diversifiable and should not command a higher return. Therefore, the statement is false; higher expected return for stock A due to idiosyncratic risk contradicts CAPM assumptions.

Statement (b): When two risky securities are perfectly positively correlated, diversification offers no risk reduction. Within a traditional portfolio, diversification benefits arise from imperfect correlations; perfect correlation means they move in lockstep, so combining them doesn't reduce overall risk, making this statement true.

Part 3: Effect of Covariance Change on Cella, Inc.'s Price

Based on CAPM, the security's expected return relates to its Beta:

\[

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

\]

Given the initial price (\$50), dividend (15%), and market conditions, the Beta is determined before and after the covariance doubles. The doubling of covariance impacts Beta proportionally, increasing it, which raises the security's systematic risk. Since the expected return remains constant at 15%, an increase in Beta shifts the equilibrium price according to the CAPM relation:

\[

P_{new} = \frac{D}{E(R_i) - R_f} = \frac{0.15 \times 50}{0.15} = \$50

\]

However, as covariance doubles, Beta increases, implying higher required return or a different equilibrium price if the expected return were to adjust accordingly. Because everything else remains constant, the key is that increased systematic risk potentially inflates the stock's required return, thus lowering its equilibrium price.

Part 4: Market Efficiency and Fund Performance

The fact that Lynch’s funds outperformed the S&P 500 in 11 out of 13 years, with an average outperformance of 10%, is compelling but not definitive evidence against market efficiency. Market efficiency posits that all available information is reflected in security prices, making consistent outperformance improbable without access to superior information or skill. Lynch's success could reflect skill, timing, or informational advantages, suggesting that markets are not perfectly efficient but does not conclusively prove inefficiency. Therefore, these facts are suggestive but not conclusive evidence against market efficiency.

References

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  • Sharpe, W. F. (1964). Capital Asset Prices: A theory of market equilibrium under conditions of risk. Journal of Finance, 19(3), 425-442.
  • Lintner, J. (1965). The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets. The Review of Economics and Statistics, 47(1), 13-37.
  • Markowitz, H. (1952). Portfolio Selection. The Journal of Finance, 7(1), 77-91.
  • Ross, S. A. (1976). The Arbitrage Theory of Capital Asset Pricing. Journal of Economic Theory, 13(3), 341-360.
  • Siegel, J. J. (2014). Stocks for the Long Run. McGraw-Hill Education.
  • Brealey, R. A., Myers, S. C., & Allen, F. (2017). Principles of Corporate Finance. McGraw-Hill Education.
  • Elton, E. J., Gruber, M. J., Brown, S. J., & Goetzmann, W. N. (2014). Modern Portfolio Theory and Investment Analysis. Wiley.
  • Fama, E., & French, K. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics, 33(1), 3-56.
  • Malkiel, B. G. (2003). The Efficient Market Hypothesis and Its Critics. Journal of Economic Perspectives, 17(1), 59-82.

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