Correlation as probability: applications of Sheppard’s formula to financial assets

Abstract

In this paper, a result for bivariate normal distributions from statistics is transformed into a financial asset context in order to build a tool which can translate a correlation matrix into an equivalent probability matrix and vice versa. This way, the correlation coefficient parameter is more understandable in terms of joint probability of two stocks’ returns, and much more useful in terms of the information it provides. We validate, empirically, our result for a sample covering the three market capitalization categories in the S&P 500 index over a ten-year period. Finally, the accuracy of this new tool is measured theoretically and some applications from the practitioners’ point of view are offered. Such applications include, for instance, the calculation of the number of trading days in a year in which two stocks have same sign returns and how to split the average return of weighted stocks into four orthants.

Keywords:

• Correlation
• Stock returns
• Joint probability
• Bivariate normal distribution

JEL Classification:

• G10
• G110
• G170

Acknowledgements

The authors thank two anonymous referees and the editor for their insightful comments.