The Johnson model is a flexible family of distributions that can capture various shapes of return distributions, going beyond the limitations of the normal distribution. Crucially, it can accommodate skewness and kurtosis, which are often observed in real-world asset returns.
- Normal Distribution as a Special Case: The normal distribution, a cornerstone of many financial models, is a restricted case of the Johnson model where skewness and excess kurtosis are both zero. This implies symmetry and specific tail behavior.
- Lognormal Distribution as a Special Case: The lognormal distribution, often used to model asset prices, is also a special case of the Johnson model. By setting appropriate skewness and kurtosis parameters, the Johnson model can replicate the lognormal's characteristic positive skewness.
- Skewness: A positive skew indicates a longer tail on the right (more frequent large positive returns), while a negative skew signifies a longer tail on the left (more frequent large negative returns). Investors generally prefer positive skewness.
- Kurtosis: Excess kurtosis measures the "fatness" of the tails compared to the normal distribution. Positive excess kurtosis (leptokurtic) implies more frequent extreme events (both positive and negative), while negative excess kurtosis (platykurtic) suggests fewer extreme events. Investors generally prefer lower excess kurtosis.
Portfolio Optimization:
The provided formulas are fundamental for constructing and managing portfolios:
- Expected Return: The portfolio's expected return is the weighted average of the expected returns of its constituent assets.
- Portfolio Variance: Portfolio variance measures the overall risk. It's not simply the weighted average of individual asset variances because it also considers the correlation between assets. The formula shows how diversification (combining assets with low or negative correlations) can reduce portfolio variance.
- Portfolio Standard Deviation (Volatility): The square root of the portfolio variance gives the portfolio's standard deviation, a more interpretable measure of risk.
Key Insights for Portfolio Construction:
- Diversification: The core principle is to combine assets that are not perfectly positively correlated. This reduces overall portfolio risk without necessarily sacrificing return. The correlation coefficient (ρ) plays a critical role; the lower the correlation (ideally closer to -1), the greater the diversification benefit.
- Skewness and Kurtosis: Beyond mean and variance, investors should consider skewness and kurtosis. Positive skewness is desirable, as it indicates a higher probability of large positive returns. Lower excess kurtosis is also preferred, as it reduces the likelihood of extreme losses. The Johnson model allows for the explicit incorporation of these higher moments.
- Practical Application: To apply these concepts, one needs to estimate the expected returns, standard deviations, and correlations of the assets under consideration. Historical data is often used for this purpose, but it's important to recognize that past performance is not indicative of future results. Furthermore, the Johnson model's parameters (including skewness and kurtosis) would also need to be estimated, adding another layer of complexity.
Connecting the Concepts:
The Johnson model provides a more realistic framework for modeling asset returns than the traditional normal distribution, especially when skewness and kurtosis are significant. By using the Johnson model, investors can better estimate the true distribution of potential portfolio returns and make more informed decisions about asset allocation. This, combined with the principles of portfolio optimization, allows for the construction of portfolios that are better tailored to an investor's risk preferences and expectations.
A three-asset portfolio example
https://gist.github.com/viadean/c35f97933622477dd88d3678e04593d9
For the synthetic three-asset portfolio with equal weights: