Analyze Time Series Of Monthly Returns For Three Securities ✓ Solved

Analyze time series of monthly returns for three securities

Analyze time series of monthly returns for three securities from 10/01/2017 to 09/30/2020. Compute the monthly return data for the period for the selected three securities using the formula Return for Month t = (Adj Close_t / Adj Close_{t-1}) – 1.

Calculate the average arithmetic monthly return, variance, and standard deviation of returns for the period for the three securities.

Calculate the covariance and correlation of returns between Company A and Company B for the 10/01/2017 – 09/30/2020 period and interpret the correlation.

Compute the stock betas for Company A and Company B for the same period. Hint: identify a market proxy and specify what goes into known-xs and known-ys arrays.

Suppose you invest 50% in Company A and 50% in Company B. Determine the portfolio’s expected return and standard deviation.

Compare the average returns and standard deviations of Company A and Company B from earlier calculations with the portfolio’s performance and discuss insights.

Paper For Above Instructions

Introduction and objective. This paper demonstrates a structured approach to analyzing time-series monthly returns for three securities over the window from October 2017 to September 2020. The focus is on calculating monthly returns, summarizing risk and return with descriptive statistics, assessing inter-security relationships via covariance and correlation, estimating market betas, and evaluating a simple 50/50 portfolio comprised of two securities. The analysis follows standard financial theory, drawing on the foundational ideas of portfolio theory, CAPM, and empirical asset pricing (Markowitz, 1952; Sharpe, 1964; Fama & French, 1993). Throughout, the goal is to translate formulas into reproducible calculations and interpret the results in terms of diversification and risk/return trade-offs (Campbell, Lo, & MacKinlay, 1997).

Data and methodology. The core data are monthly adjusted closing prices for the three securities over the specified period. The monthly return for month t is computed as r_t = AdjClose_t / AdjClose_{t-1} – 1. This simple arithmetic approach aligns with conventional practice in time-series return analysis (Elton et al., 2014). For clarity, the first month in the series has no prior month to differ against and is therefore excluded from the monthly return series.

Descriptive statistics. For each security, the monthly return series is summarized with:

- Mean (average arithmetic monthly return): the simple average of the monthly returns.

- Variance and standard deviation (volatility): variance = sum[(r_t – mean)^2] / (n – 1), and sd = sqrt(variance).

These measures provide a direct sense of typical monthly performance and risk relative to the mean (Welch & Anderson, 1999). The calculations mirror the standard practices described in portfolio theory texts (Markowitz, 1952) and are consistent with empirical finance literature (Campbell et al., 1997). The results are interpreted in terms of which security exhibits higher expected monthly gains and which shows greater dispersion of returns.

Covariance and correlation. The interdependence between Company A and Company B is captured by covariance cov(r_A, r_B) = sum[(r_A,t – mean_A)(r_B,t – mean_B)] / (n – 1) and the corresponding correlation corr(r_A, r_B) = cov(r_A, r_B) / (sd_A * sd_B). A positive correlation indicates that the two securities tend to move together, which has implications for diversification benefits in portfolio construction (Fama & French, 1993; Bodie et al., 2014). The interpretation focuses on whether the correlation is strong, weak, or moderate, and how this affects the expected risk reduction from combining the two assets.

Betas and market proxy. The beta of each security with respect to a market proxy (e.g., a broad market index like the S&P 500) is estimated as beta_i = cov(r_i, r_m) / var(r_m), where r_m is the market proxy return. Equivalently, beta can be interpreted as the slope in the regression r_i = alpha_i + beta_i r_m + error. This framing mirrors the Capital Asset Pricing Model (CAPM) and the intuition that beta measures systematic risk relative to the market (Sharpe, 1964; Ross, 1976). The known-xs and known-ys arrays correspond to (r_m) and (r_i) in the regression setup (i.e., market returns as the independent variable and security returns as the dependent variable).

Portfolio construction and risk. With a 50%/50% portfolio allocation between Company A and Company B, the monthly portfolio return is r_p,t = 0.5 r_A,t + 0.5 r_B,t. The portfolio’s expected monthly return is the weighted average of the individual means: E[r_p] = 0.5 E[r_A] + 0.5 E[r_B]. The portfolio variance is Var(r_p) = w_A^2 Var_A + w_B^2 Var_B + 2 w_A w_B Cov(A,B), which for equal weights simplifies to Var_p = 0.25 Var_A + 0.25 Var_B + 0.5 Cov(A,B). The portfolio standard deviation is the square root of Var_p. These formulas are standard results from Markowitz portfolio theory and are widely employed to assess diversification effects (Markowitz, 1952; Elton et al., 2014).

Illustrative results (hypothetical). To ground the discussion, consider illustrative results derived from the computations (these numbers are for demonstration and would be replaced by actual data in practice). Suppose the average monthly returns are:

- Company A: 0.0082 (0.82%)

- Company B: 0.0065 (0.65%)

- Company C: 0.0059 (0.59%)

Corresponding monthly volatilities (standard deviations) are:

- sd_A = 0.042

- sd_B = 0.038

- sd_C = 0.035

Assume a positive cross-correlation between A and B with corr(A,B) ≈ 0.65, so Cov(A,B) ≈ 0.00104. The implied betas relative to a market proxy r_m (monthly) would be Beta_A ≈ 1.05 and Beta_B ≈ 0.92, reflecting A's somewhat higher sensitivity to market movements (consistent with CAPM intuition). These numbers are for illustration; actual results depend on the data.

Portfolio performance and interpretation. With equal weights w_A = w_B = 0.5, the portfolio’s expected monthly return is E[r_p] ≈ 0.00735 (0.735%). The portfolio variance is Var_p ≈ 0.00132, giving a portfolio monthly standard deviation of sd_p ≈ 0.0364 (3.64%). The corresponding annualized figures are obtained by scaling, typically by multiplying monthly figures by 12 for return and by sqrt(12) for risk, yielding an annualized expected return of roughly 8.8% and annualized volatility around 12.2%, under the simplifying assumption of stable returns and independence across months (Campbell et al., 1997). The key takeaway is that diversification via a 50/50 mix reduces risk relative to holding the riskier individual asset (A with higher sd) but does not eliminate risk entirely, especially when the assets are positively correlated (Fama & French, 1993). The beta estimates suggest how much systematic risk remains in the portfolio, with A contributing more to market-risk exposure than B (Sharpe, 1964; Bodie et al., 2014).

Discussion and implications. The comparison across securities shows that an individual security with the higher average return may also possess higher idiosyncratic risk (volatility). The portfolio, while offering a moderate return, benefits from diversification by combining assets with positive but not perfectly correlated returns. The observed correlation informs the magnitude of diversification benefits: a correlation well below 1 implies meaningful risk reduction from combination, while a correlation near 1 implies limited diversification gains (Markowitz, 1952; Elton et al., 2014). The beta analysis highlights how market risk contributes to each security’s performance, with higher beta assets expected to move more in line with market fluctuations, affecting both expected return (via CAPM) and risk management considerations (Co­chrane, 2005). In practice, practitioners would replace the illustrative numbers with actual data to produce decision-relevant estimates and would test robustness across different market regimes (Campbell et al., 1997; Fama & French, 1993).

References

  • Sharpe, W. F. (1964). Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. Journal of Finance, 19(3), 425-442.
  • Treynor, J. L. (1965). How to Rate Management of Investment Funds. Harvard Business Review, 43(1), 63-75.
  • Markowitz, H. (1952). Portfolio Selection. Journal of Finance, 7(1), 77-91.
  • 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. (1992). The cross-section of expected stock returns. Journal of Finance, 47(2), 427-465.
  • Ross, S. A. (1976). The Arbitrage Theory of Capital Asset Pricing. Journal of Economic Theory, 13(3), 341-360.
  • Campbell, J. Y., Lo, A. W., & MacKinlay, A. C. (1997). The Econometrics of Financial Markets. Princeton University Press.
  • Bodie, Z., Kane, A., & Marcus, A. J. (2014). Investments. McGraw-Hill Education.
  • Elton, E. J., Gruber, M. J., Brown, S. J., & Goetzmann, W. N. (2014). Modern Portfolio Theory and Investment Analysis. Wiley.
  • Cochrane, J. H. (2005). Asset Pricing. Princeton University Press.

Related Assignments