Module 05 Problem 5: Rate Of Return And Standard Deviation

Module 05problem 5 1rate Of Return And Standard Deviationporter Inc

Evaluate a security's expected return and standard deviation based on given data, compute holding period returns for specified periods, determine holding-period gains and returns for stock transactions, estimate required rates of return using the Capital Asset Pricing Model (CAPM), calculate portfolio expected return and beta, and analyze historical price data to find monthly returns and their standard deviations.

Paper For Above instruction

Financial analysis of securities and portfolio performance is fundamental to investment decision-making. This paper comprehensively addresses the evaluation of a security’s expected return and standard deviation, the computation of holding period returns, the assessment of holding-period gains, the application of the Capital Asset Pricing Model to estimate required returns, and the analysis of historical price data to derive insights into market behavior.

Expected Return and Standard Deviation of a Security

To evaluate a security, it is essential to determine its expected return and associated risk, which quantify its potential profitability and variability. Given the data with probabilities and corresponding returns, the expected return (E(R)) is calculated as the sum of the products of each outcome’s probability and its return:

E(R) = Σ [Probability(i) × Return(i)]

Using the provided data with probabilities of 0.20 for returns of 5.0%, and assuming other relevant probabilities for the missing data, the expected return can be computed accordingly. The variance, a measure of dispersion around the expected return, is calculated by summing the squared deviations weighted by probabilities:

Variance = Σ [Probability(i) × (Return(i) - E(R))²]

The standard deviation is the square root of the variance, offering a measure of risk in the same units as the return. These calculations provide insight into the expected profitability and stability of the security's return profile.

Holding Period Returns for Waters and Panner

The holding period return (HPR) measures the total return earned from holding an asset over a specific period, considering price changes and dividends. Given the prices for Waters and Panner across periods 2 through 4, the HPR is calculated as:

HPR = (Price at End of Period - Price at Start of Period) / Price at Start of Period

Applying this formula to each period for both stocks yields the respective returns, indicating their performance over the selected periods and aiding in comparative analysis and investment planning.

Holding-Period Gains and Returns for Apple Stock

Calculating the holding-period gain involves the difference between the selling and purchase prices, multiplied by the number of shares sold. The specific formula is:

Gain = (Selling Price - Purchase Price) × Number of Shares Sold

In the case with no dividends paid, the total gain directly reflects the profit. The holding-period rate of return (HPR) is then computed as:

HPR = Gain / (Purchase Price × Number of Shares Initially Held)

These calculations assess the profitability of the investment over the period, providing crucial information for decision-making.

Estimating Required Return Using CAPM

The Capital Asset Pricing Model (CAPM) estimates the expected return on a security based on its systematic risk, represented by beta, the risk-free rate, and the expected market return. The formula is:

Required Return = Risk-Free Rate + Beta × (Market Return - Risk-Free Rate)

Applying this to stocks with given betas, a risk-free rate of 5%, and a market return of 12%, yields the appropriate expected return for each stock. For instance, for stock A with beta 0.85:

Required Return = 5% + 0.85 × (12% - 5%) = 5% + 0.85 × 7% = 5% + 5.95% = 10.95%

This approach helps investors determine whether a security offers an acceptable expected return relative to its risk.

Portfolio Expected Return and Beta

Constructing a diversified portfolio involves calculating the weighted average expected return and beta. The expected return of the portfolio (E(Rp)) is:

E(Rp) = Σ (Weight of stock i × Expected return of stock i)

Using the provided weights and individual expected returns, the portfolio's overall return can be determined. The beta of the portfolio (βp) is similarly the weighted sum of individual betas:

βp = Σ (Weight of stock i × Beta of stock i)

These metrics enable evaluation of the portfolio's risk-return profile relative to market movements.

Analyzing Historical Price Data for Returns and Volatility

Historical monthly prices of ABC Corporation and the S&P 500 Index allow for calculating monthly holding-period returns, which reflect the percentage change from one month to the next:

Monthly Return = (Price at Month n - Price at Month n-1) / Price at Month n-1

Calculating these for each month provides a time series of returns, from which the average monthly returns and standard deviations are derived. These statistics inform about historical performance and volatility, aiding in risk management and forecasting future behavior.

Conclusion

In conclusion, the comprehensive financial analysis involving expected returns, standard deviation, holding period returns, CAPM-based risk assessments, portfolio analysis, and historical data evaluation offers vital insights for investors. Such evaluations support informed decision-making by quantifying profitability potential and associated risks, thereby optimizing investment strategies within diverse financial environments.

References

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