Talking about eigenvalues in a client meeting is asking for trouble because the mathematical explanation can quickly derail a meeting and cause confusion. Instead, I find it preferable, if asked, to explain eigenvalues from a financial perspective, rather than to worry too much about communicating the theory.
So, if you find yourself in a risk management meeting with the conversation turning to correlation matrices and the client asks something like, “I heard that the correlations have to be ‘positive definite’ or something? What is that?” instead of giving a formal definition of positive definiteness or getting into an eigenvalues discussion, it’s better to apply the answer to the real-life situation you’re discussing in the meeting.
First, some background. To get a little bit mathy, a positive definite matrix is, in some important senses, a multidimensional version of a positive number. It’s positive in all “directions”. It also has analogous properties to positive numbers, such as square roots1: and multiplicative inverses2:.
Consider a single asset with normally distributed returns
The variance of this distribution must be a positive number3:. If there is zero variance, it’s risk-free. If it has negative variance...well that’s not a thing, so that is just broken.
The variance can also be characterized in terms of the width of an interval set at some probability level.
A positive variance means that the interval has some width, and the bigger the variance, the wider the interval.
Consider two assets. Here you can “look down” on the distribution and see a level set.
This shows the variance of Home Depot on the x-axis and Lowe’s on the y-axis. By drawing a line at some angle, the variance of a weighted portfolio of Home Depot and Lowe’s is given.
Quite reasonably, we want these possible portfolios to have a positive variance; if we can invest some capital in Home Depot, some capital in Lowe’s, and wind up with a “less than riskless” portfolio, then the model is broken. A positive definite covariance matrix is precisely the formalization of that requirement.
So in short, a good response to the client’s question, instead of mumbling, “Don’t worry, these are just a technical requirement that we have taken care of” and shutting down your client’s interest in your model, you could respond to her question, “Yes. Positive definite matrices are a thing. They encode the conditions necessary to ensure all portfolios of risky assets, are themselves risky.”
What other topics do you encounter in client meetings that threaten to derail everything? Let me know at email@example.com.
The analogy for the square root is the Cholesky factorization ↩
The analogy for 1/x is matrix inverse. Note that the inverse of the covariance matrix is required in the probability density function of the multivariate normal distribution ↩
A jump from a positive correlation matrix to a positive covariance matrix is necessary here. I find it is not usually an issue. Saying, “You’re multiplying a positive vector of variances by a positive matrix correlations” is a little vague, but usually gets the job done. ↩
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