Models of Stock Returns–A Comparison

Kon, Stanley J., “Models of Stock Returns–A Comparison,” The Journal of Finance, Vol 39, No 1 (1984), 147-165.

Purpose:  To explain the observed kurtosis (fat tails) and positive skewness in the distribution of stock returns.

Findings:  The discrete mixture of normal distributions proposed in this paper explains these moments better than existing models, including the Student-t.

Motivation:  Mean-variance portfolio theory, option pricing models, and empirical tests of capital asset pricing models and efficient markets make assumptions about the distribution of stock returns.  It has been shown that stock returns cannot be approximated using a single normal distribution, but the normal is just what the theoretical models assume.

Methods:  Returns may be driven by multiple normal distributions–one for random shocks, one for firm-specific information, one for macroeconomic information, etc.  In other words, it is not necessary that each observation of stock return be drawn from the same distribution as all others.  This paper uses a mix of up to five normal distributions and likelihood ratio tests to model the returns of the 30 Dow Jones stocks.

  • Let  r_t = \alpha_i + u_{it}, where r_t is the observed return for period t, and u_{it} is normally distributed with mean zero and variance \sigma_i^2.
  • Let \underline{\gamma}_i be a normal distribution with mean \alpha_i and mean \sigma_i^2.
  • Let \lambda_i = T_i/T be the observations associated with information set I_i over total observations, or the proportion of total observations that are associated with information set I_i.
  • Let \underline{\theta} = \{\alpha_i, ..., \alpha_N, \sigma_i^2, ..., \sigma_N^2, \lambda_i, ..., \lambda_{N-1}\} be a vector of parameters.
  • Let \underline{r} be the vector of returns.
  • The vector of parameters, given a vector of returns, can be found by solving

max \ell(\underline{\theta}/\underline{r}) = \Pi_{t=1}^T \left[\sum_{i=1}^N \lambda_i p(r_t|\underline{\gamma_i})\right].

Results:

  • All 30 Dow Jones stocks can be explained by a mixture of 2, 3, or 4 normals.
  • Returns of the S&P500, value-weighted, and equal-weighted market indices were also explained by mixtures of normals.
  • The mixture of normals describes 27 of the 30 stocks, and all three indices, better than does the student-t distribution, which also has fat tails and has been used to model stock returns.
  • Differences in the means of the normal distributions can explain the skewness of observed stock returns.
  • Differences in the variance of the normals can explain the kurtosis of stock returns.
  • Stock returns do appear to be normally distributed, but the distribution parameters are time-varying, and the timing of parameter shifts also varies across stocks.