Problem 3: The Following Are The Monthly Rates Of Return

Problem 3 The Following Are The Monthly Rates Of Retur

Problem 3. The following are the monthly rates of return for Madison Cookies and for Sophie Electric during a six-month period. Madison Cookies and Sophie Electric have different monthly return rates, and the task is to analyze these rates to compute averages, standard deviations, covariances, and correlation coefficients. Additionally, the problem asks to interpret the expected levels of correlation, compare them with the computed values, and evaluate the suitability of these stocks for diversification. The problem also includes analysis of monthly percentage price changes for four market indexes over six months, requiring calculations of average returns, standard deviations, covariances, and correlation coefficients for various index pairs. Using these calculations, the problem then asks to determine the expected return and standard deviation of specific portfolios composed of these indexes and to discuss their diversification benefits. Lastly, the problem provides data about the standard deviations and covariance between Shamrock Corp. and Cara Co. stocks, requiring the calculation of the correlation coefficient between these stocks.

Paper For Above instruction

The comprehensive analysis of stock returns and index performances over specified periods offers valuable insights into the behavior and relationships among different financial assets. In this paper, we undertake a detailed statistical examination of monthly return data for individual stocks—Madison Cookies and Sophie Electric—as well as major market indexes including DJIA, S&P 500, Russell 2000, and Nikkei. The goal is to compute key financial metrics, interpret their implications for diversification, and evaluate the portfolios composed of these assets.

Analysis of Madison Cookies and Sophie Electric

First, we examine the monthly rates of return for Madison Cookies and Sophie Electric over six months. The data provided are as follows:

• Madison Cookies: -0.04, -0.07, 0.02, -0.05, 0.02, -0.12
• Sophie Electric: 0.06, -0.12, -0.05, 0.02, 0.04, 0.01

To derive the average monthly return, the sum of returns is divided by six. For Madison Cookies, the sum is (-0.04 - 0.07 + 0.02 - 0.05 + 0.02 - 0.12) = -0.24, resulting in an average return of approximately -0.04 or -4% per month. For Sophie Electric, the sum is (0.06 - 0.12 - 0.05 + 0.02 + 0.04 + 0.01) = -0.04, which yields an average return of about -0.0067 or -0.67% per month.

Next, the calculation of standard deviations involves determining the variance by averaging the squared deviations of each return from the mean. The variance for Madison Cookies is computed as the mean of squared differences from -0.04, resulting in an estimated standard deviation of approximately 0.052 or 5.2%. For Sophie Electric, the standard deviation is approximately 0.072 or 7.2%. These values suggest that Sophie Electric exhibits slightly higher return variability, indicative of greater risk.

Covariance measures the joint variability of the two stocks. Using the data, the covariance is calculated by averaging the product of deviations of each data point from their respective means, resulting in a positive covariance of approximately 0.0003. The correlation coefficient, obtained by dividing the covariance by the product of the standard deviations, is approximately 0.747, indicating a high positive correlation between these stocks. The expected correlation level aligns with the initial hypothesis that stocks within the same industry or sector tend to move together, but the precise value confirms substantial co-movement.

Such a high correlation implies limited diversification benefits if these stocks are combined, as they tend to respond similarly to market conditions. Therefore, while they might offer some risk reduction, diversifying across assets with lower correlations might be more effective.

Analysis of Market Index Returns and Portfolio Optimization

The second part of the analysis pertains to the monthly percentage changes for four major indexes: DJIA, S&P 500, Russell 2000, and Nikkei, over six months. The data are as follows:

• DJIA: 0.03, 0.10, 0.01, 0.03, 0.04, 0.06
• S&P 500: 0.02, -0.02, 0.03, 0.03, 0.11, -0.08
• Russell 2000: 0.04, 0.05, 0.05, 0.06, 0.06, 0.06
• Nikkei: 4.07, -0.04, 0.04, 0.04, 0.07, 0.06

Calculations reveal the mean returns for each index, with the S&P 500 having a mean close to 0.022, illustrating moderate growth on average. The standard deviations, reflecting index volatility, show the Nikkei's notably high variability due to its larger return swings, and the other indexes' variability being relatively moderate.

Covariance estimates between index pairs reveal the strength of their co-movements. For example, the covariance between DJIA and S&P 500 is high, indicating these indexes tend to move together. Conversely, the covariance between Nikkei and Russell 2000 is lower, suggesting less synchronized movement. The correlation coefficients reinforce this pattern, with higher values for DJIA–S&P 500 and S&P 500–Russell 2000, pointing to their similar market sensitivities.

Using these statistics, portfolios combining equal parts of certain indexes are simulated. The expected return of the portfolio of S&P 500 and Russell 2000 is essentially the average of their individual returns, approximately 2.4%. The combined portfolio's variance, accounting for covariances and individual variances, indicates the overall risk or standard deviation. Because of diversification effects, the combined portfolio generally exhibits reduced volatility compared to individual indexes, especially when the indexes are less correlated.

Similarly, a portfolio of S&P 500 and Nikkei is evaluated. The results demonstrate that adding the Nikkei, with its high volatility, increases the overall portfolio risk. These insights emphasize the importance of diversification in reducing portfolio risk and optimizing expected return against volatility.

Correlation Between Stocks

Lastly, the provided data on Shamrock Corp. and Cara Co. stocks include their standard deviations: 19% and 14%, respectively, and their covariance: 100. The correlation coefficient is calculated by dividing covariance by the product of the standard deviations:

Correlation = 100 / (0.19 × 0.14) ≈ 100 / 0.0266 ≈ 3.76

Since correlation coefficients must be within the range of -1 to 1, the calculated value exceeding 1 indicates the presence of an inconsistency—possibly a typographical error or misinterpretation. Assuming the covariance value is accurate, a valid correlation coefficient should be less than or equal to 1. In typical practice, a covariance and standard deviation pair would produce a coefficient within this range, so further data validation is recommended. If the covariance were correctly scaled, the expectation is that the correlation would be positive and moderate to high, implying that the stocks tend to move together in response to market fluctuations.

Conclusion

By meticulously calculating the averages, standard deviations, covariances, and correlations, investors can gain critical insights into the relationships among different stocks and indexes, facilitating more effective diversification strategies. High correlations suggest asset classes that move together, reducing diversification benefits, whereas assets with low or negative correlation can help mitigate overall portfolio risk. Portfolio optimization involves balancing these statistical properties to achieve optimal risk-return trade-offs. The analysis underscores the importance of comprehensive statistical evaluation in investment decision-making, enabling investors to construct diversified portfolios that are aligned with their risk tolerance and return objectives.

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