The bivariate normal distribution extends the familiar normal distribution to two variables. The key additional ingredient is correlation: a measure of how strongly the two variables tend to move together. Move the correlation assumption below and watch both probability-density visualisations change in real time.
The correlation coefficient, commonly written as rho, measures the strength and direction of the linear relationship between the two variables.
Both charts below use exactly the same probability distribution. The 2D view shows the density from above, while the 3D view shows the probability-density surface.
For two standardised variables with correlation rho, the bivariate normal probability density is determined by both individual standard deviations and their correlation.
In this demonstration both means are zero. Changing the correlation does not simply move the distribution: it changes its shape.
Correlation is one of the most important concepts in portfolio, market and credit risk. Two assets can each be risky individually, but their combined risk depends heavily on how their returns move relative to one another.
When rho is negative, the variables tend to move in opposite directions. This can provide strong diversification because losses in one variable can be partly offset by gains in the other.
In the bivariate normal model, rho = 0 means independence. The two variables contain no linear relationship and their joint density is circular when their standard deviations are equal.
When rho is positive, the variables tend to move together. Diversification becomes weaker and portfolio risk increases. At very high positive correlation, the two risks behave increasingly like one risk.
The same mathematics appears directly in portfolio risk. Consider two assets with weights w1 and w2. Their portfolio variance is:
This is the key insight behind correlation in portfolio risk. Even if two assets have exactly the same individual volatility, combining them can substantially reduce overall volatility when their correlation is below one.
The slider stops at +/-0.99 rather than +/-1.00 because a correlation of exactly +1 or -1 makes the covariance matrix singular. The ordinary two-dimensional density then collapses onto a line rather than remaining a regular two-dimensional probability distribution.