1.1 Descriptive Statistics, Probability Distributions & Return Metrics

1.1 Descriptive Statistics, Probability Distributions & Return Metrics

Quick Summary: Quantitative portfolio management relies on descriptive statistics to evaluate historical returns, model probability distributions, and quantify multi-asset co-movements. Understanding the spread between arithmetic and geometric returns—driven by volatility drag—alongside higher-order moments (skewness and kurtosis), is essential for institutional portfolio construction and risk mitigation.

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Measures of Central Tendency: Arithmetic vs. Geometric Mean

Evaluating asset returns requires selecting the appropriate averaging metric based on the analytical objective:

1. Arithmetic Mean Return

The arithmetic mean (R_A) is the simple unweighted average of periodic returns over n periods:

$R_A = \frac{1}{n} \sum_{t=1}^{n} R_t$

• Primary Role: Serves as the unbiased estimator of expected return for a single upcoming period.
• Limitation: Ignores compounding and the sequence of returns over multiple horizons.

2. Geometric Mean Return (CAGR)

The geometric mean (R_G, or Compound Annual Growth Rate) measures the constant rate per period that compounds an initial investment to its final value over n periods:

$R_G = \left[ \prod_{t=1}^{n} (1 + R_t) \right]^{1/n} - 1$

• Primary Role: Accurately reflects historical compound wealth accumulation.
• Mathematical Boundary: Under Jensen's Inequality, R_G ≤ R_A, with equality holding only when all periodic returns are identical (zero variance).

Volatility Drag (Variance Drain)

The divergence between arithmetic and geometric returns is caused by return dispersion, known as volatility drag. For returns with variance σ², this is approximated by:

$R_G \approx R_A - \frac{\sigma^2}{2}$

Worked Example: Volatility Drag in Action

Consider a $1,000,000 portfolio with two consecutive annual returns: +50% in Year 1 and -50% in Year 2.

• Arithmetic Mean: R_A = (+50% - 50%) / 2 = 0.00%
• Portfolio Value:
  • Year 1 End: $1,000,000 × 1.50 = $1,500,000
  • Year 2 End: $1,500,000 × 0.50 = $750,000
• Total Loss: -25.00% (a net loss of $250,000).
• Exact Geometric Mean: $R_G = \sqrt{(1 + 0.50)(1 - 0.50)} - 1 = \sqrt{0.75} - 1 = -13.40\%$
• Volatility Drag Approximation: With variance σ² = 0.25, R_G ≈ 0.00 - (0.25 / 2) = -12.50%.

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Dispersion and Co-Movement: Variance, Covariance, and Correlation

Measuring multi-asset risk requires evaluating both individual asset volatility and pairwise co-movements.

Variance and Standard Deviation

Sample variance (s²) and sample standard deviation (s) quantify dispersion around the arithmetic mean:

$s^2 = \frac{1}{n - 1} \sum_{t=1}^{n} (R_t - R_A)^2, \quad s = \sqrt{s^2}$

To annualize standard deviation for i.i.d. returns:

• Monthly to Annual: σ_annual = σ_monthly × √12
• Daily to Annual: σ_annual = σ_daily × √252

Covariance and Correlation

Sample covariance (Cov(A,B) or σ_AB) measures the co-movement of two return series:

$\text{Cov}(A,B) = \frac{1}{n - 1} \sum_{t=1}^{n} (R_{A,t} - R_A)(R_{B,t} - R_B)$

The correlation coefficient (ρ_AB) standardizes covariance between -1.0 and +1.0:

$\rho_{AB} = \frac{\text{Cov}(A,B)}{\sigma_A \sigma_B}$

Portfolio Diversification Implications

For a two-asset portfolio with weights w_A and w_B (where w_A + w_B = 1):

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

• ρ = +1.0 (Perfect Positive): σ_P = w_A σ_A + w_B σ_B. No risk reduction occurs.
• ρ < +1.0 (Imperfect Correlation): σ_P < w_A σ_A + w_B σ_B. Diversification reduces total portfolio risk without reducing expected return.
• ρ = -1.0 (Perfect Negative): Complete risk elimination is mathematically possible at weights w_A = σ_B / (σ_A + σ_B).

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Higher-Order Statistical Moments: Skewness and Kurtosis

Financial returns routinely diverge from Gaussian assumptions, requiring higher-order moments.

Third Moment: Skewness (Asymmetry)

Skewness measures directional asymmetry around the mean:

$\text{Skewness} = \frac{E[(R - \mu)^3]}{\sigma^3}$

• Symmetric (Skew = 0): Mean = Median = Mode (normal distribution).
• Positive Skew (Skew > 0): Long right tail with Mean > Median > Mode. Frequent small losses and rare large gains (e.g., venture capital, long call options).
• Negative Skew (Skew < 0): Long left tail with Mean < Median < Mode. Frequent small gains and rare, severe drawdowns (e.g., high-yield credit, short volatility strategies).

Fourth Moment: Kurtosis (Tail Fatness)

Kurtosis measures the concentration of values in tails versus shoulders:

$\text{Kurtosis} = \frac{E[(R - \mu)^4]}{\sigma^4}, \quad \text{Excess Kurtosis} = \text{Kurtosis} - 3$

• Mesokurtic: Kurtosis = 3 (Excess = 0). Normal benchmark.
• Leptokurtic: Kurtosis > 3 (Excess > 0). Fat tails and high central peak; extreme market moves occur far more frequently than predicted by normal models.
• Platykurtic: Kurtosis < 3 (Excess < 0). Lighter tails and flatter peak.

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Probability Distributions in Financial Modeling

Normal vs. Lognormal Distributions

• Normal Distribution: Symmetrical bell curve defined by mean (μ) and standard deviation (σ). Empirical rule: 68.27% within ±1σ, 95.45% within ±2σ, and 99.73% within ±3σ. Flaw: Spans (-∞, +∞), allowing negative prices and violating limited liability.
• Lognormal Distribution: If continuously compounded return r = ln(S_t / S_0) is normal, price S_t follows a lognormal distribution bounded at zero (S_t ≥ 0), with positive skewness. Widely used for asset prices and option valuation.

Non-Normal Realities in Portfolio Consulting

Because empirical market returns exhibit negative skewness, fat tails, and volatility clustering, CIMA professionals supplement standard deviation with downside metrics such as semivariance, Sortino ratios, Value at Risk (VaR), and Conditional VaR (CVaR).