Portfolio Variance and the Bound on Portfolio Volatility

Summary

The discussion examines a claim that portfolio variance is always below the weighted average of the component variances. For two assets, it writes portfolio variance in terms of each asset’s variance and their correlation, then uses the fact that correlation cannot exceed one to bound portfolio variance by the square of the weighted sum of the assets’ standard deviations. This gives a bound on portfolio standard deviation, rather than the stated bound on variance.

The answer therefore identifies a wording error in the original claim: the general result is that portfolio standard deviation is no greater than the weighted average of component standard deviations, assuming the usual nonnegative portfolio weights. The discussion is brief and focuses on two assets; it does not establish a corresponding bound on variance by the weighted average of variances. The distinction matters when describing diversification mathematically.

Key ideas

• Portfolio variance includes a covariance term that depends on the correlation between asset returns.
• Correlation is bounded above by one, which bounds portfolio standard deviation by the weighted sum of component standard deviations for nonnegative weights.
• The bound applies to standard deviation, not to variance compared with the weighted average of individual variances.
• The two-asset derivation illustrates the point but does not discuss extensions or cases involving negative weights.

Tags

# Portfolio variance $<=$ weighted average of individual variances


# Portfolio variance $<=$ weighted average of individual variances












In portfolio theory, I often (with some justifications but the message is the same) come across the following statement:

"The most important quality of portfolio variance is that its value is a weighted combination of the individual variances of each of the assets adjusted by their covariances. This means that the overall portfolio variance is lower than a simple weighted average of the individual variances of the stocks in the portfolio."

Link of quote: https://www.investopedia.com/terms/p/portfolio-variance.asp

So, the text in bold is my problem. "The overall portfolio variance is lower than a simple weighted average of the individual variances of the stocks in the portfolio."

Can anyone prove this or refer to an link where it is proven?

## Answer by D Stanley (score 2)

https://quant.stackexchange.com/a/60521

The key phrase is "adjusted by their covariances". The formula for the variance of a portfolio of two assets is

$\sigma _{p}^{2}=w_{A}^{2}\sigma _{A}^{2}+w_{B}^{2}\sigma _{B}^{2}+2w_{A}w_{B}\sigma _{A}\sigma _{B}\rho _{AB}$

, which, since $\rho _{AB} <= 1$, is always less than

$ w_{A}^{2}\sigma _{A}^{2}+w_{B}^{2}\sigma _{B}^{2}+2w_{A}w_{B}\sigma _{A}\sigma _{B} $

$= (w_{A}\sigma _{A}+w_{B}\sigma _{B})^{2}$

That said, the wording in the article is inaccurate; it's more accurate to say that the standard deviation of a portfolio is lower (or at worst, equal to) a weighted average of the standard deviations of returns of the constituents.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.