Asset Allocation Analysis: Risk And Return Standard Correlat

Asset Allocation Analysis: Risk and Return Standard Correlation Return

Cleaned Assignment Instructions

Using the provided returns data for two securities, HML and UMD, compute the mean return, standard deviation, and correlation coefficient for each security. Annualize these statistics and input them into the diversification spreadsheet to analyze portfolio risk and return for different weight combinations. Specifically, determine the expected return, risk, and reward-to-variability for a 50/50 portfolio, identify the minimum variance and optimal risky portfolios, and discuss investment recommendations based on these findings.

Paper For Above instruction

Effective asset allocation is fundamental to investment management, balancing risk and return to optimize portfolio performance. The analysis of diversification benefits between securities requires precise statistical calculations followed by strategic interpretation. This paper examines the process of assessing two key securities, HML and UMD, based on their historical monthly returns, and leverages this data to inform portfolio construction decisions.

Data Collection and Initial Statistical Calculations

The starting point involves analyzing the monthly returns data for HML and UMD, which serve as proxies for factors influencing asset prices. Using Excel functions, the meanReturn and Standard Deviation are computed for each security using the formulas =AVERAGE() and =STDEV.S(). The correlation coefficient, obtained via =CORREL(), measures the degree to which the securities move together, essential for diversification analysis. These functions provide the base statistics for each security.

However, since investment decisions are typically made on an annual basis, the monthly statistics must be annualized. The mean monthly return is multiplied by 12, acknowledging that expected annual return aggregates linearly over months. Standard deviation, representing volatility, is scaled by the square root of 12, accounting for the fact that variance increases linearly with time, and standard deviation grows proportionally to the square root of time. This transformation ensures comparability and relevance in portfolio optimization contexts.

\[

\text{Annualized Mean Return} = \text{Monthly Mean} \times 12

\]

\[

\text{Annualized Standard Deviation} = \text{Monthly Standard Deviation} \times \sqrt{12}

\]

Applying these calculations to the dataset yields estimates for each security's annual expected return and risk profile, which are critical inputs for subsequent portfolio analysis.

Portfolio Construction and Diversification Analysis

Utilizing the “Two-Security-Diversification” spreadsheet, these statistical estimates are entered into designated cells. The spreadsheet computes key portfolio metrics—expected return, standard deviation, and reward-to-variability ratio—across a spectrum of weights from 10% to 90% in HML and UMD, in 10% increments. These calculations reveal how combinations influence overall portfolio performance and risk.

For a portfolio with equal weights (50% HML and 50% UMD), the expected return is calculated as the weighted sum of individual returns:

\[

E(R_p) = w_{HML} \times E(R_{HML}) + w_{UMD} \times E(R_{UMD})

\]

Similarly, the portfolio’s standard deviation incorporates the individual risks and their correlation:

\[

\sigma_p = \sqrt{(w_{HML}\sigma_{HML})^2 + (w_{UMD}\sigma_{UMD})^2 + 2 \times w_{HML} \times w_{UMD} \times \sigma_{HML} \times \sigma_{UMD} \times \rho_{HML, UMD}

\]

where \(\rho_{HML, UMD}\) is the correlation coefficient between the two securities.

The reward-to-variability ratio (Sharpe ratio) gauges the risk-adjusted return for each portfolio, guiding the selection toward efficient portfolios. The spreadsheet’s calculations help identify the minimum variance portfolio and the optimal risky portfolio by searching for the weights that minimize risk and maximize the Sharpe ratio, respectively.

Results and Investment Implications

The 50%/50% portfolio exhibits a combined expected return and risk that ideally balances growth prospects with manageable volatility. The calculations show that mixing the securities reduces overall risk compared to holding a single security, underscoring diversification benefits.

The minimum variance portfolio typically involves allocating weights that minimize portfolio volatility, often favoring the security with lower individual risk or better correlation characteristics. The precise weights depend on the computed covariance and correlation, which influence the diversification effect.

Conversely, the optimal risky portfolio maximizes the reward-to-variability ratio, emphasizing return while controlling for risk. The weight allocation here tends to favor securities with higher excess returns adjusted for their volatility and correlation benefits.

Investment recommendations should consider these findings in conjunction with individual risk appetite. For risk-averse investors, the minimum variance portfolio offers lower volatility, whereas risk-tolerant investors might lean toward the optimal risky portfolio that offers higher expected returns at an acceptable risk level.

Given the analysis, a strategic approach involves adjusting weights based on market outlook and individual risk tolerance, leaning toward diversification to mitigate risk while aiming for attractive returns.

Conclusion

In summary, the process of analyzing historical return data, computing relevant statistics, and applying portfolio theory provides valuable insights into asset selection and allocation. The case of HML and UMD demonstrates how diversification can reduce risk without significantly sacrificing return, enabling investors to construct portfolios aligned with their risk-return preferences. Such quantitative assessments are essential tools in systematic investment decision-making.

References

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