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.

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.

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.

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.

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.

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.

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.