8.4 Correlation, Covariance & Regression

8.4 Correlation, Covariance & Regression

Modern Portfolio Theory (MPT) established that an investment's risk should not be evaluated in isolation, but rather by how it interacts with other holdings in a diversified portfolio. The mathematical framework developed by Harry Markowitz demonstrated that combining assets with imperfectly correlated returns can reduce overall portfolio volatility without sacrificing expected return. To construct optimal portfolios, evaluate asset comovements, and assess manager performance against market benchmarks, wealth managers must master covariance, Pearson correlation, and simple linear regression.

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Measuring Comovement: Covariance Mechanics

Covariance is a statistical measure that quantifies the joint variability and directional comovement of two random variables (such as the historical returns of two financial assets).

Covariance Formulations

• Population Covariance ($\text{Cov}(X,Y)$ or $\sigma_{XY}$):

$\text{Cov}(X,Y) = \frac{\sum_{i=1}^N (X_i - \mu_X)(Y_i - \mu_Y)}{N}$

• Sample Covariance ($s_{XY}$):

$s_{XY} = \frac{\sum_{i=1}^n (X_i - \bar{X})(Y_i - \bar{Y})}{n - 1}$

Where $X_i$ and $Y_i$ represent the simultaneous returns of assets $X$ and $Y$ in period $i$, and $\bar{X}$ and $\bar{Y}$ denote their respective sample mean returns.

Interpreting Covariance

• Positive Covariance ($\text{Cov} > 0$): Assets $X$ and $Y$ tend to deviate from their respective means in the same direction at the same time. When Asset $X$ achieves an above-average return, Asset $Y$ is also likely to generate an above-average return.
• Negative Covariance ($\text{Cov} < 0$): Assets $X$ and $Y$ tend to move in opposite directions relative to their means. An above-average return in Asset $X$ coincides with a below-average return in Asset $Y$.
• Zero Covariance ($\text{Cov} = 0$): There is no linear relationship between the deviations of the two assets.

Fundamental Limitation of Covariance

While covariance indicates the direction of comovement, its absolute numerical magnitude is unstandardized and depends entirely on the units of measurement (e.g., return percentages expressed as decimals vs. whole numbers). A covariance of $+0.045$ between two equities does not provide an intuitive sense of whether their relationship is moderately positive or nearly identical. To assess the strength of the linear association, covariance must be standardized.

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The Pearson Correlation Coefficient ($\rho$ or $r$)

The Pearson Correlation Coefficient (symbolized as $\rho_{XY}$ for population parameters or $r$ for sample statistics) standardizes covariance by dividing it by the product of the two assets' individual standard deviations:

$\rho_{XY} = \frac{\text{Cov}(X,Y)}{\sigma_X \times \sigma_Y}$

Mathematical Properties of Correlation

1. Strictly Bounded Range: The correlation coefficient is dimensionless and bounded strictly between $-1.0$ and $+1.0$:

$-1.0 \le \rho_{XY} \le +1.0$

1. Perfect Positive Linear Correlation ($\rho = +1.0$): The returns of Asset $X$ and Asset $Y$ move in identical proportions in the same direction. There is zero portfolio risk reduction from combining them.
2. Zero Linear Correlation ($\rho = 0.0$): There is no linear association between the asset returns. Combining them provides substantial diversification benefits.
3. Perfect Negative Linear Correlation ($\rho = -1.0$): The returns move in exactly opposite directions in fixed proportion. In theory, an investor can construct a risk-free portfolio with zero standard deviation.

Worked Example: Calculating Correlation from Covariance

A portfolio analyst evaluates two asset classes. Emerging Market Equities ($X$) exhibit a standard deviation of 20.0% ($\sigma_X = 0.20$), Global Real Estate ($Y$) exhibits a standard deviation of 15.0% ($\sigma_Y = 0.15$), and their sample covariance is $+0.018$:

$\rho_{XY} = \frac{0.018}{0.20 \times 0.15} = \frac{0.018}{0.030} = +0.60$

The assets display a moderately strong positive linear correlation of $+0.60$.

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Portfolio Diversification and the Markowitz Effect

The mathematical essence of Modern Portfolio Theory is captured in the formula for the variance of a two-asset portfolio ($\sigma_p^2$):

$\sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \text{Cov}(A,B)$

Substituting $\text{Cov}(A,B) = \rho_{AB} \sigma_A \sigma_B$, the formulation becomes:

$\sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \rho_{AB} \sigma_A \sigma_B$

$\sigma_p = \sqrt{\sigma_p^2}$

Where $w_A$ and $w_B$ represent the portfolio capital weights ($w_A + w_B = 1.0$), and $\sigma_A$ and $\sigma_B$ represent their respective individual standard deviations.

The Condition for Diversification Benefits

A critical mathematical deduction emerges from this equation: as long as the correlation coefficient is strictly less than $+1.0$ ($\rho < 1.0$), the portfolio standard deviation will be strictly less than the weighted average of the individual asset standard deviations:

$\sigma_p < w_A \sigma_A + w_B \sigma_B \quad \text{for all } \rho_{AB} < 1.0$

Numerical Demonstration of the Diversification Effect

Consider an equal-weighted portfolio ($w_A = 0.50, w_B = 0.50$) consisting of Asset A ($\sigma_A = 14.0%$) and Asset B ($\sigma_B = 22.0%$). The weighted average standard deviation is $0.50(14.0%) + 0.50(22.0%) = 18.0%$. Let us observe how varying correlation shifts actual portfolio volatility:

1. Perfect Positive Correlation ($\rho = +1.0$):

$\sigma_p = \sqrt{(0.5)^2(14)^2 + (0.5)^2(22)^2 + 2(0.5)(0.5)(1.0)(14)(22)} = \sqrt{49 + 121 + 154} = \sqrt{324} = 18.0\%$

1. Moderate Positive Correlation ($\rho = +0.40$):

$\sigma_p = \sqrt{49 + 121 + 2(0.25)(0.40)(308)} = \sqrt{170 + 61.6} = \sqrt{231.6} \approx 15.22\%$

1. Uncorrelated ($\rho = 0.0$):

$\sigma_p = \sqrt{49 + 121 + 0} = \sqrt{170} \approx 13.04\%$

Notice that when $\rho = 0.0$, the portfolio volatility (13.04%) is lower than the volatility of BOTH individual assets (even Asset A at 14.0%)!

1. Moderate Negative Correlation ($\rho = -0.40$):

$\sigma_p = \sqrt{170 - 61.6} = \sqrt{108.4} \approx 10.41\%$

1. Perfect Negative Correlation ($\rho = -1.0$):

$\sigma_p = \sqrt{49 + 121 - 154} = \sqrt{16} = 4.0\%$

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Scatter Plots and Linear Relationship Patterns

A scatter plot provides a visual representation of bivariate financial data by plotting paired observations $(X_i, Y_i)$ on a two-dimensional Cartesian plane.

• Upward-Sloping Ellipse: Indicates positive correlation. Data points cluster along a line rising from bottom-left to top-right.
• Downward-Sloping Ellipse: Indicates negative correlation. Data points cluster along a line descending from top-left to bottom-right.
• Circular / Spherical Cloud: Indicates zero correlation. Data points are dispersed randomly with no directional alignment.
• Non-Linear Relationships: It is vital to note that the Pearson correlation coefficient measures strictly linear relationships. If two assets share a deterministic non-linear relationship (such as a quadratic U-shape, common in option volatility smiles), the computed Pearson correlation may be approximately 0.0, despite the variables being strongly linked.

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Simple Linear Regression and Ordinary Least Squares (OLS)

While correlation measures the degree of linear association, simple linear regression models the functional relationship between an independent (explanatory) variable ($X$) and a dependent (response) variable ($Y$).

In portfolio management, linear regression is frequently used to evaluate how a fund manager's portfolio returns ($Y$) react to movements in a benchmark market index ($X$).

The Linear Regression Equation

$Y_i = \alpha + \beta X_i + \epsilon_i$

Where:

• $Y_i$ = Return of the dependent variable (e.g., investment fund) in period $i$
• $X_i$ = Return of the independent variable (e.g., benchmark index) in period $i$
• $\alpha$ (Alpha / Intercept): The expected value of $Y$ when $X = 0$. In wealth management, Jensen's Alpha measures a manager's risk-adjusted excess performance attributable to security selection skill.
• $\beta$ (Beta / Slope): The sensitivity of the dependent variable to changes in the independent variable. It measures systematic, non-diversifiable market risk.
• $\epsilon_i$ (Epsilon / Residual Error): The random disturbance term reflecting specific, unsystematic factors unique to the fund in period $i$.

Ordinary Least Squares (OLS) Mechanics

The regression line of best fit is calculated using the Ordinary Least Squares (OLS) method, which minimizes the sum of the squared vertical differences (residuals) between actual data points and the fitted regression line:

$\min \sum_{i=1}^n \epsilon_i^2 = \min \sum_{i=1}^n (Y_i - (\alpha + \beta X_i))^2$

Mathematical Formulation for Beta and Alpha

Using sample statistics, the slope coefficient ($\beta$) and intercept ($\alpha$) are calculated as:

$\beta = \frac{\text{Cov}(X,Y)}{\sigma_X^2} = \frac{\rho_{XY} \times \sigma_Y}{\sigma_X}$

$\alpha = \bar{Y} - \beta \bar{X}$

Interpreting Beta in Financial Markets

• $\beta = 1.0$: The asset possesses average market sensitivity. If the market index rises by 10%, the asset is expected to rise by 10%.
• $\beta > 1.0$ (Aggressive Asset): The asset is more volatile than the benchmark market (e.g., $\beta = 1.3$ implies a 10% market gain generates an expected 13% asset gain, but a 10% market drop produces a 13% decline).
• $0 < \beta < 1.0$ (Defensive Asset): The asset exhibits lower sensitivity to market swings (e.g., utilities or consumer staples with $\beta = 0.65$).
• $\beta < 0$ (Inverse Asset): The asset moves counter to the market index (e.g., dedicated short funds or gold during certain market crises).

Worked Example: Calculating Beta and Alpha

A wealth manager evaluates a UK equity fund ($Y$) against the FTSE 100 Index ($X$). Statistical analysis reveals the following parameters:

• Average fund return ($\bar{Y}$) = 11.0%
• Average benchmark return ($\bar{X}$) = 8.0%
• Fund standard deviation ($\sigma_Y$) = 16.0%
• Benchmark standard deviation ($\sigma_X$) = 10.0%
• Correlation coefficient ($\rho_{XY}$) = +0.75

1. Calculate Beta ($\beta$):

$\beta = \frac{\rho_{XY} \times \sigma_Y}{\sigma_X} = \frac{0.75 \times 16.0\%}{10.0\%} = \frac{12.0\%}{10.0\%} = 1.20$

1. Calculate Alpha ($\alpha$):

$\alpha = \bar{Y} - \beta \bar{X} = 11.0\% - (1.20 \times 8.0\%) = 11.0\% - 9.6\% = +1.40\%$

The fund exhibits a beta of 1.20 (20% more sensitive than the FTSE 100) and generated +1.40% of positive alpha, indicating value added through active portfolio management above the return dictated by systematic market exposure.

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Coefficient of Determination ($R^2$)

The Coefficient of Determination, symbolized as $R^2$ (or R-squared), measures the proportion of the total variance in the dependent variable ($Y$) that is explained by the independent variable ($X$) in the regression model.

Mathematical Formulation

In simple univariate linear regression, $R^2$ is mathematically equal to the square of the Pearson correlation coefficient:

$R^2 = (r_{XY})^2$

$R^2$ takes values strictly between $0.0$ and $1.0$ (or 0% to 100%):

• $R^2 = 1.0$ (100%): All return variability in the fund is completely explained by movements in the benchmark index; residuals are zero.
• $R^2 = 0.0$ (0%): The benchmark index provides zero explanatory power regarding fund returns.

Application to Risk Decomposition: Systematic vs. Specific Risk

In wealth management, $R^2$ provides a direct decomposition of portfolio risk into systematic and unsystematic components:

$\text{Total Portfolio Variance} = \text{Systematic Variance (Market)} + \text{Unsystematic Variance (Specific)}$

$\text{Percentage Systematic Risk} = R^2$

$\text{Percentage Specific (Idiosyncratic) Risk} = 1 - R^2$

In our UK equity fund example above, the correlation was $r = 0.75$. Therefore:

$R^2 = (0.75)^2 = 0.5625 = 56.25\%$

$\text{Specific Risk} = 1 - 0.5625 = 43.75\%$

This reveals that 56.25% of the fund's return volatility is driven by broader macroeconomic movements in the FTSE 100 (systematic risk), while 43.75% is attributable to fund-specific factors (unsystematic risk), such as individual stock selection and sector overweights.