Return Distribution

1. Definition

• In finance, a return is the gain or loss of an investment over a period, expressed as a fraction of the initial value.

$R_t = \frac{P_t – P_{t-1}}{P_{t-1}}$

where $P_t$ = asset price at time $t$.

• The return distribution describes the probability distribution of returns over time.
• Instead of saying “average return = 5%,” we capture the entire range of possible returns and their probabilities.

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2. Key Characteristics of Return Distributions

• Mean (Expected Return): average return investors expect.
• Variance / Volatility: spread of returns → risk measure.
• Skewness: asymmetry (positive skew = large gains possible; negative skew = large losses possible).
• Kurtosis (fat tails): likelihood of extreme events compared to a Normal distribution.

In finance, returns often have fat tails and negative skewness, meaning:

• Extreme losses are more likely than a Normal distribution predicts.

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3. Common Models for Return Distributions

Normal Distribution (simplest assumption)

• Early finance theory (e.g., Black-Scholes) assumed returns ~ Normal($\mu,\sigma^2$).
• Easy to work with, but underestimates extreme events (crashes).

Student’s t-Distribution

• Heavy-tailed alternative.
• Better for capturing extreme losses.

Empirical Distributions

• Using historical returns directly to build a distribution (non-parametric).

Mixture Distributions

• Combine several normals to model volatility clustering.

GARCH Models

• Model time-varying volatility of returns → conditional return distributions change over time.

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4. Examples

Stock Market

• Average daily return ≈ 0.05% (small).
• Volatility ≈ 1% per day.
• Distribution: centered near 0, but fat-tailed (crashes more likely than Normal).

Portfolio Returns

• Built from weighted sum of individual asset return distributions.
• Diversification shapes the overall distribution.

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5. Why Return Distributions Matter

• Risk Management:
  • Need distribution to compute VaR and CVaR (Expected Shortfall).
• Option Pricing:
  • Payoffs depend on full return distribution, not just average.
• Portfolio Optimization:
  • Balancing expected return vs risk (variance, skewness, kurtosis).
• Stress Testing:
  • Assess likelihood of extreme losses.

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6. Visualization (Intuition)

• Normal curve: symmetric, thin tails.
• Actual stock returns: fatter tails, slightly negative skew (losses more extreme than gains).
• This mismatch explains why “crash risk” is higher than classical models predict.

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Summary:
A return distribution describes the probability distribution of investment returns. Key features are mean, variance, skewness, and kurtosis. While the Normal distribution is the simplest assumption, real-world returns often show fat tails and asymmetry, requiring heavy-tailed or time-varying models (t-distribution, GARCH, empirical distributions). They are essential for risk management, option pricing, and portfolio optimization.