Return, Risk, Correlation, and Diversification in 2 Equations
Two equations that summarize the four essential components of a portfolio
Disclaimer: This article is meant to provide some basic mathematical analysis of key concepts used in portfolio management, and in no way should be considered as investment advice.
In portfolio management, there are 4 important concepts or terminologies that can be described using 2 equations. There parameters are: Return, Risk, Correlation and Diversification.
Return is the expected portfolio return and is given by
where r_i is the expected return from the ith asset class, and omega_i are the asset allocation parameters, and obey
Risk is a measure of portfolio variance or volatility and is given by
Here, Sigma is the covariance matrix, which measures the degree of comovement or correlation between assets:
In this equation, the rho’s are the correlation coefficients. The diagonal terms represent the variance for each of the asset classes.
Diversification is measured by the parameter n in the summation. For example, n = 2 for a portfolio consisting of 2 asset classes, for instance stocks and bonds. Diversification helps to reduce portfolio variance or risk because in a diversified portfolio, the covariance matrix Sigma will be sparse, and this would lead to a smaller portfolio variance or risk.
Case Study: Two asset class portfolio
For a portfolio consisting of stocks and bonds, if we assume that the correlation between stocks and bonds is negligible, then the portfolio return is given by
and the portfolio risk is given by
In summary, we’ve discussed 4 important concepts in portfolio management using two main equations: the return equation, and the risk equation. These two equations contain important information about asset allocation, portfolio diversification, and correlation between assets in a portfolio. A diversified portfolio can generally lead to lower risk, if the asset classes are uncorrelated or have negligible correlations. Another important factor that could reduce portfolio risk is time. Generally, a long time horizon is less riskier compared to a short time horizon. For example, when the time horizon is a day (day trading), the risk is very high, as day trading captures the day to day volatility of the market. This can generally lead to a zero sum game (gains equal losses over time). However, when the time horizon is long (long term investment), the risk is low, as time is needed to separate the signal from the noise.