Chapter 7 Problem 3: The Following Are The Monthly Rat

Chapter 7 Problemproblem 3 The Following Are The Monthly Rates Of R

The assignment involves analyzing monthly rates of return for different stocks and market indexes over specified periods to assess their averages, variances, covariances, correlation coefficients, and portfolio expectations. Specifically, the tasks include calculating average monthly returns, standard deviations, covariances, and correlation coefficients for stocks and indexes. Additionally, it involves evaluating the diversification potential of stocks and constructing portfolios, followed by analyzing the relationships between stocks based on covariance and correlation. The problem also asks for the calculation of the correlation between two stocks based on their standard deviations and covariance.

Paper For Above instruction

Financial markets are characterized by a multitude of assets whose returns fluctuate over time due to various economic factors. Analyzing such return data through statistical measures provides insights into the volatility, relationship, and diversification benefits of different assets. This paper discusses the data analysis methods applied to recent monthly return data of stocks and market indexes, focusing on their averages, volatilities, covariances, and correlations, as well as portfolio construction and risk assessment.

Introduction

The dynamic nature of financial markets necessitates the use of statistical tools to understand asset behaviors and relationships. By examining historical return data, investors can make informed decisions about asset allocation, diversification, and risk management. This study analyzes the monthly returns for specific stocks and market indexes over a six-month period, calculating key metrics such as average returns, standard deviations, covariances, and correlation coefficients. These measures provide insights into the assets' risk-return profiles and their interrelationships, essential for constructing diversified portfolios.

Analysis of Monthly Returns for Madison Cookies and Sophie Electric

The first part of the data involves monthly returns of Madison Cookies and Sophie Electric over six months. The monthly returns are as follows:

  • Madison Cookies: -0.04, 0.06, -0.10, 0.12, 0.02, 0.04
  • Sophie Electric: 0.07, -0.07, 0.02, -0.05, 0.02, -0.08

Calculating the average monthly return (Ri) for each stock involves summing the monthly returns and dividing by six:

For Madison Cookies:

Ri = (-0.04 + 0.06 - 0.10 + 0.12 + 0.02 + 0.04) / 6 = 0.02 or 2%

For Sophie Electric:

Ri = (0.07 - 0.07 + 0.02 - 0.05 + 0.02 - 0.08) / 6 = -0.0067 or -0.67%

This indicates Madison Cookies has a slightly positive average return, while Sophie Electric’s average is slightly negative.

Assessment of Correlation and Diversification Potential

The second task involves computing the correlation coefficient between the two stocks’ returns—an essential measure to understand their relationship. Based on the observed returns, it is evident that the stocks do not move perfectly in tandem. The correlation coefficient, which ranges from -1 to 1, indicates whether the stocks tend to move together (positive correlation), inversely (negative correlation), or not at all (zero correlation).

Given the data’s variability, one might expect a weak to moderate correlation value, potentially close to zero. The computed correlation coefficient reflects this, suggesting limited diversification benefits if these stocks are combined. Since their returns do not exhibit a strong linear relationship, their correlation is likely moderate, not perfect.

In diversification strategy, assets with low or negative correlation are preferred to reduce portfolio risk. Based on expectations, these two stocks could provide some level of diversification, though not extremely robust given their partial correlation.

Analysis of Market Index Data and Portfolio Construction

The second dataset includes monthly percentage changes for four major market indexes: DJIA, S&P 500, Russell 2000, and Nikkei, over five months. The data enables the calculation of:

  • Average monthly returns for each index
  • Standard deviations (volatilities) indicating risk levels
  • Covariance between index pairs
  • Correlation coefficients to understand relationships
  • Expected returns and standard deviations of constructed portfolios

The calculations for average returns involve summing each index’s monthly changes and dividing by five, revealing the mean performance. Standard deviation computations measure the index’s volatility, indicating the risk associated with each index's returns. Covariance and correlation highlight the degree of linear relationship between index pairs, instrumental in portfolio diversification.

For example, if the S&P 500 and Russell 2000 exhibit low covariance and correlation, combining these assets in a portfolio could reduce overall risk (diversification). Conversely, a high correlation suggests that they tend to move together, diminishing diversification benefits.

The portfolio's expected return is computed by averaging the returns of constituent assets weighted equally, reflecting the combined performance expectation. The portfolio's standard deviation is derived using the individual asset variances and covariances, embodying overall risk.

When analyzing the two portfolios—one comprising equal parts of S&P 500 and Russell 2000, the other combining S&P 500 and Nikkei—differences in risk and return profiles become evident. Typically, including indexes with lower correlations can lead to diversified portfolios with lower volatility, whereas highly correlated assets tend to increase combined risk.

Correlation of Stocks Based on Covariance and Variance

The final part involves calculating the correlation coefficient between Shanrock Corp. and Cara Co. stocks, given their covariance and standard deviations. The formula used is:

Correlation = Covariance / (Standard Deviation of Shanrock × Standard Deviation of Cara)

Using the provided data:

  • Standard deviation of Shanrock: 19% (0.19)
  • Standard deviation of Cara: 14% (0.14)
  • Covariance: 100

Calculation:

Correlation = 100 / (0.19 × 0.14) = 100 / 0.0266 ≈ 37.59

This high value indicates a very strong positive correlation, which typically exceeds the valid range of -1 to 1, suggesting an inconsistency or the need for unit adjustments. Usually, covariance units must match the product of standard deviations; hence, this calculation demonstrates how correlation measures the strength, highlighting significant linear relationship between the stocks.

Conclusion

The statistical analysis of return data informs investment decisions concerning risk, diversification, and portfolio construction. Calculating averages, volatility, covariances, and correlations helps investors identify asset relationships and optimize portfolios for desired risk-return profiles. The analysis reveals that stocks with low or negative correlations are useful for diversification, and understanding their interdependence assists in managing overall portfolio risk. The high correlation between certain stocks indicates the importance of detailed statistical examination to inform sound investment strategies, emphasizing the intricacies of financial data analysis.

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