Harry Markowitz and the Covariance of Asset Returns
21st March 2015
It is interesting to note that Harry Markowitz introduced the statistical concept of "Covariance" in the early 1950s while working on the portfolio selection problem and the Mean-Variance optimization technique.
Markowitz was the first to think of the statistical concept of variance of a financial asset's returns as the measure of a financial security's "risk", or what is today commonly known as "market risk". If the return of the asset prices follows a Gaussian (Normal) distribution then the variance of these returns, or the standard deviation (volatility) which is the square of the variance, measured over a historical time period is the measure of that asset's risk. However, if there are two assets, each of whose returns follow a Gaussian distribution, then how do we measure the portfolio risk? This is the question that Markowitz asked.
The problem is compounded because besides the two variances (or standard deviations), one for each asset, there is also the correlation coefficient which measures the co-movement between them. However, the correlation coefficient is an imperfect measure of co-movement as it only captures the direction of co-movement, but not the quantum of variation. Markowtiz's insight was that if we are to simply take the average of the variances of the two assets then we would not be able to capture the co-movement between them, as the correlation would be left out. By itself correlation is not sufficient to capture both the direction (co-movement) and the quantum of movement of each asset. Therefore, the best way to capture portfolio risk would be to multiply the correlation with the product of each asset's variance. This was the only way in which portfolio risk of two assets can reduce to the risk of a single asset if one of them were to disappear. The covariance of a financial asset with itself will simply become the variance of that asset because the correlation coefficient of an asset with itself is +1. If then , which is the variance of the asset.
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