Problem 11-3: Portfolio Expected Return

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Calculate the expected return on a portfolio with investments in stocks and the expected return, variance, and standard deviation for various stocks based on different economic states. Additionally, determine the expected return on the market using CAPM, find investment allocations to achieve a target portfolio return, and evaluate the risk premium and return variability for portfolios composed of multiple stocks.

Paper For Above instruction

The analysis of diversified investment portfolios requires an understanding of how individual stocks contribute to overall expected returns, as well as how risks and returns are quantified through statistical measures. This paper discusses methods for calculating the expected return of a portfolio, assessing stock risk through variance and standard deviation, applying the Capital Asset Pricing Model (CAPM) to determine expected returns based on beta, and formulating optimal investment strategies to meet specific return goals.

Expected Return of a Portfolio

The expected return of a portfolio is a weighted average of the expected returns of each constituent stock. Given the weights of stocks in the portfolio and their individual expected returns, the overall expected return can be calculated as:

Expected Portfolio Return = (w₁ × E(R₁)) + (w₂ × E(R₂)) + ... + (wₙ × E(Rₙ))

For example, considering a portfolio with 32% invested in Stock X (expected return 12%), 47% in Stock Y (expected return 15%), and 21% in Stock Z (expected return 17%), the expected return is calculated as follows:

Expected Return = (0.32 × 12) + (0.47 × 15) + (0.21 × 17) = 3.84 + 7.05 + 3.57 = 14.46%

Calculating Individual Stock Expected Returns

The expected return on a stock based on different economic conditions is found by multiplying the probability of each state by the corresponding return and summing these products. For stocks A and B, with states such as recession, normal, and boom, the expected returns are computed as:

E(R) = Σ[Probability of State × Return in that State]

Suppose, for instance, the probabilities are recession (0.45), normal (0.32), and boom (0.55), with returns in each state. The calculations are as follows:

E(R_A) = (0.45 × R_{A, recession}) + (0.32 × R_{A, normal}) + (0.55 × R_{A, boom})

E(R_B) = (0.45 × R_{B, recession}) + (0.32 × R_{B, normal}) + (0.55 × R_{B, boom})

Plugging in the values yields the expected return estimates for each stock.

Risk Assessment via Standard Deviation

The risk of stocks is quantified with the standard deviation, which measures return variability around the mean. The variance formula involves the squared deviations weighted by the probabilities:

Variance = Σ[Probability × (Return – Expected Return)²]

The standard deviation, the square root of variance, indicates the amount of risk investors face. Stocks with higher standard deviations are riskier, and managing portfolios involves balancing these risks against returns.

Using CAPM to Derive Expected Returns

The Capital Asset Pricing Model (CAPM) provides a framework to estimate an expected stock return based on its beta, the market's expected return, and the risk-free rate:

Expected Return = Risk-free Rate + Beta × (Market Return – Risk-free Rate)

For example, a stock with a beta of 1.18, a market return of 11.2%, and a risk-free rate of 4.85% would have an expected return calculated as:

Expected Return = 4.85 + 1.18 × (11.2 – 4.85) = 4.85 + 1.18 × 6.35 = 4.85 + 7.493 = 12.34%

This estimate assists investors in evaluating whether the stock offers an adequate return for its risk level.

Portfolio Optimization: Allocation for Target Return

Suppose an investor has $264,000 to allocate between Stock H (expected return 14.4%) and Stock L (expected return 11.5%) to achieve a portfolio return of 12.70%. By setting up the equation:

0.144 × x + 0.115 × (264,000 – x) = 0.127 × 264,000

solving for x (investment in Stock H) yields the optimal amounts to invest in each stock. Calculations involve straightforward algebraic manipulation, resulting in specific dollar amounts for investment in each stock.

Portfolio Variance and Standard Deviation

For portfolios comprising multiple stocks, the variance accounts for individual variances and covariances between stocks. For a portfolio invested 40% in Stock A, 40% in Stock B, and 20% in Stock C, the variance is computed as:

Variance = ΣΣw_iw_jCov(R_i, R_j)

where w_i and w_j are weights, and Cov represents covariance. The standard deviation, the square root of variance, gauges the overall portfolio risk.

Expected returns and the variance, together with the risk-free rate, help investors determine the risk premium, which is the excess expected return over the risk-free rate, reflecting the additional return for bearing systematic risk.

Conclusion

Efficient portfolio management involves calculating expected returns, assessing risks through variance and standard deviation, applying models like CAPM for expected return estimation, and optimizing asset allocation based on investment goals. These tools allow investors to balance risk and return effectively, tailor portfolios to individual risk tolerances, and make informed investment decisions.

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