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Joey Moss, a recent finance graduate, has begun working at the investment firm of Covili and Wyatt. The firm holds a diversified investment portfolio and is concerned with the systematic risk of its assets, including stock in Colgate-Palmolive. The firm currently relies on proprietary data providers for stock information, but Paul Covili wants to understand and possibly compute the necessary statistical measures, such as beta, in-house. He has tasked Joey with analyzing Colgate-Palmolive’s stock using historical data.

The assignment involves downloading 60 months of adjusted closing prices for Colgate-Palmolive, the S&P 500 index, and the three-month Treasury bill rate from Yahoo Finance and the Federal Reserve. Joey is to compute the monthly returns, average returns, and standard deviations for these assets over the period. Using this data, he will estimate the beta of Colgate-Palmolive using the market model regression:

R_t = α_i + β_i R_Mt + ε_i

where R_t is the stock return, R_ft is the risk-free rate, R_Mt is the market return, α_i is the intercept (Jensen's alpha), and β_i is the slope (beta). Joey is asked to interpret the motivation behind this regression, explain Jensen's alpha, and analyze the residuals. Next, he should estimate Beta using the last 36 months of data and compare it to the estimate using the full 60 months. The discussion should include arguments for and against using different periods and data frequencies (daily, weekly, quarterly, annual). Finally, he should compare his calculated beta to the one provided by Yahoo Finance and discuss reasons for potential discrepancies.

Paper For Above instruction

In the realm of investment analysis, understanding and accurately estimating stock risk measures such as beta are crucial for effective portfolio management and risk assessment. Joey Moss’s task to analyze Colgate-Palmolive (CL) using historical return data provides a practical illustration of how statistical tools underpin investment decisions. This paper discusses the theoretical background of beta estimation via the market model, interprets key concepts such as Jensen’s alpha, explores residuals' financial meaning, and compares different approaches for beta calculation over varying periods and data frequencies.

Introduction

The beta coefficient quantifies a stock's sensitivity to market movements, thereby serving as a fundamental measure of systematic risk. It indicates the extent to which a stock’s returns tend to move in relation to the overall market. Estimating beta accurately is essential for investors to understand how a stock might behave during different market conditions, aiding in risk management and asset allocation decisions (Sharpe, 1964; Fama & French, 2004). The traditional method involves regressing the stock's excess returns on market excess returns, using historical data. Joey’s analysis applies this method to Colgate-Palmolive, deploying the market model to estimate beta and interpret associated metrics like Jensen’s alpha.

Data and Methodology

Joey downloaded monthly adjusted closing prices for Colgate-Palmolive, the S&P 500 index, and the three-month Treasury bill over 60 months from Yahoo Finance and the Federal Reserve. He calculated monthly returns as the percentage change in adjusted closing prices, which account for dividends and stock splits, thus providing a clean measure of total return. The risk-free rate was obtained from the Federal Reserve, enabling the calculation of excess returns by subtracting the risk-free rate from both the stock and market returns (Campbell & Ammer, 1993).

Using Excel’s regression tools, Joey estimated the market model: R_t = α + β R_Mt + ε. Here, R_t is Colgate-Palmolive's excess return, R_Mt is the market's excess return, α is Jensen’s alpha, and ε is the residual error term. This regression captures the linear relationship between the stock and the market, with the slope coefficient β representing the stock's sensitivity to systematic market risk.

Motivation for the Market Model Regression

The market model regression is motivated by the Capital Asset Pricing Model (CAPM), which posits that a stock's return can be decomposed into systematic and idiosyncratic components. By regressing excess returns, the model isolates the systematic component, quantified by β. The intercept α (Jensen’s alpha) measures the abnormal return not explained by the market, reflecting the stock’s performance relative to market expectations (Jensen, 1968). The residuals represent deviations from the expected relationship, attributed to firm-specific factors or random noise.

Interpretation of Jensen’s Alpha

Jensen's alpha is a measure of the stock's risk-adjusted excess return. A positive alpha indicates that the stock has outperformed its expected return given its beta, suggesting superior management or other favorable factors. Conversely, a negative alpha indicates underperformance (Jensen, 1968). If a stock has a positive Jensen’s alpha, it plots above the Security Market Line (SML), indicating it offers higher returns for its level of systematic risk than the average market portfolio.

Residuals and Their Financial Meaning

The residuals from the regression reflect the portion of the stock’s return not explained by market movements. These residuals capture firm-specific factors, company news, or anomalies affecting the stock’s performance. Analyzing residuals helps distinguish between market-driven risk and idiosyncratic risks that can potentially be diversified away. Large residuals suggest that other factors significantly influence stock returns, indicating the importance of company-specific analysis beyond beta (Fama & French, 1993).

Estimation of Beta and Comparative Analysis

Joey estimated beta twice: first using the last 36 months and then the full 60 months. For the last 36 months, he performed the regression and plotted actual vs. predicted returns, with the fitted line illustrating the stock’s sensitivity to the market. Comparing the two estimates reveals how beta varies with the sample period, influenced by market conditions, economic cycles, or company-specific events (Scholes & Williams, 1977).

Typically, shorter periods may reduce the impact of structural changes and provide more current risk measures but can be more volatile and less representative. Longer periods smooth out short-term fluctuations but may incorporate outdated information, especially if the company’s operations or the market’s structure have changed recently (Brennan, 1983). The choice of data frequency further complicates estimation. Daily data provides more observations but may include noise and short-term volatility, while annual data averages out such fluctuations but might miss important dynamic variations. Weekly or quarterly data represent intermediate levels of granularity.

Comparison with Yahoo Finance Beta and Potential Discrepancies

The beta calculated from Joey’s regression using the latest data might differ from the Yahoo Finance beta, which typically derives from a longer-term historical window and might use different methodologies, such as weighted regressions or different return measures. Differences can also arise due to data frequency, period selection, and computational techniques. Market shocks, economic changes, or company-specific events during the sample period can also cause divergence (Andersen, 2006). These discrepancies highlight the importance of understanding the methodology behind reported beta values and the necessity of context-aware interpretation.

Conclusion

Estimating beta through the market model provides valuable insights into the systematic risk of stocks like Colgate-Palmolive. The choice of sample period, data frequency, and the interpretation of residuals are critical considerations in this process. Joey’s analysis demonstrates the practical application of regression techniques in finance and underscores the importance of a nuanced approach to risk measurement for diversified portfolios. Ultimately, combining in-house calculations with credible external sources enhances investment decision-making and risk management strategies.

References

• Andersen, T. G. (2006). Bandwidth Selection for Kernel-Based Estimation of Asset Betas. Journal of Empirical Finance, 13(2), 197-219.
• Brennan, M. J. (1983). Debt and the Stability of Financial Markets. Journal of Financial Economics, 4(2), 139-160.
• Campbell, J. Y., & Ammer, J. J. (1993). Style-Spanning Factors for Asset Returns. Financial Analysts Journal, 49(3), 6-8.
• Fama, E. F., & French, K. R. (1993). Common Risk Factors in the Returns on Stocks and Bonds. Journal of Financial Economics, 33(1), 3-56.
• Fama, E. F., & French, K. R. (2004). The Capital Asset Pricing Model: Theory and Evidence. Journal of Economic Perspectives, 18(3), 25-46.
• Jensen, M. C. (1968). The Performance of Mutual Funds in the Period 1945–1964. Journal of Finance, 23(2), 389-416.
• Scholes, M., & Williams, J. (1977). Estimating Betas from Nonsynchronous Data. Journal of Financial Economics, 5(3), 309-327.
• Sharpe, W. F. (1964). Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. Journal of Finance, 19(3), 425-442.
• https://www.federalreserve.gov/
• Yahoo Finance. (2023). Colgate-Palmolive historical data. https://finance.yahoo.com