Portfolio Variance

What Is Portfolio Variance?

Portfolio variance is a measurement of risk, of how the aggregate actual returns of a set of securities making up a portfolio change over the long haul. This portfolio variance statistic is calculated utilizing the standard deviations of every security in the portfolio as well as the correlations of every security pair in the portfolio.

Grasping Portfolio Variance

Portfolio variance takes a gander at the covariance or correlation coefficients for the securities in a portfolio. Generally, a lower correlation between securities in a portfolio brings about a lower portfolio variance.

Portfolio variance is calculated by duplicating the squared weight of every security by its comparing variance and adding two times the weighted average weight increased by the covariance of all individual security pairs.

Modern portfolio theory says that portfolio variance can be reduced by picking asset classes with a low or negative correlation, like stocks and bonds, where the variance (or standard deviation) of the portfolio is the x-pivot of the efficient frontier.

Formula and Calculation of Portfolio Variance

The main quality of portfolio variance is that its value is a weighted combination of the individual variances of every one 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.

The formula for portfolio variance in a two-asset portfolio is as follows:

Portfolio variance = w₁²\u03c3₁² + w₂²\u03c3₂² + 2w₁w₂Cov_1,2

Where:

w₁ = the portfolio weight of the main asset
w₂ = the portfolio weight of the subsequent asset
\u03c3₁= the standard deviation of the main asset
\u03c3₂ = the standard deviation of the subsequent asset
Cov_1,2 = the covariance of the two assets, which can in this manner be communicated as p_(1,2)\u03c3₁\u03c3₂, where p_(1,2) is the correlation coefficient between the two assets

The portfolio variance is equivalent to the portfolio standard deviation squared.

As the number of assets in the portfolio develops, the terms in the formula for variance increase dramatically. For instance, a three-asset portfolio has six terms in the variance calculation, while a five-asset portfolio has 15.

Portfolio Variance and Modern Portfolio Theory

Modern portfolio theory (MPT) is a system for developing an investment portfolio. MPT takes as its central reason the possibility that rational investors need to augment returns while likewise limiting risk, some of the time measured utilizing volatility. Investors look for what is called an efficient frontier, or the lowest level of risk and volatility at which a target return can be accomplished.

Risk is lowered in MPT portfolios by investing in non-associated assets. Assets that may be risky all alone can really lower the overall risk of a portfolio by introducing an investment that will rise when different investments fall. This reduced correlation can reduce the variance of a hypothetical portfolio.

In this sense, an individual investment's return is less important than its overall contribution to the portfolio, in terms of risk, return, and diversification.

The level of risk in a portfolio is many times measured utilizing standard deviation, which is calculated as the square root of the variance. On the off chance that data points are far away from the mean, the variance is high, and the overall level of risk in the portfolio is high also. Standard deviation is a key measure of risk utilized by portfolio managers, financial advisors, and institutional investors. Asset managers regularly remember standard deviation for their performance reports.

Illustration of Portfolio Variance

For instance, expect there is a portfolio that comprises of two stocks. Stock An is worth $50,000 and has a standard deviation of 20%. Stock B is worth $100,000 and has a standard deviation of 10%. The correlation between the two stocks is 0.85. Given this, the portfolio weight of Stock An is 33.3% and 66.7% for Stock B. Connecting this data into the formula, the variance is calculated to be:

Variance = (33.3%^2 x 20%^2) + (66.7%^2 x 10%^2) + (2 x 33.3% x 20% x 66.7% x 10% x 0.85) = 1.64%

Variance is certainly not an especially simple statistic to decipher all alone, so most analysts work out the standard deviation, which is essentially the square root of variance. In this model, the square root of 1.64% is 12.81%.

Highlights

Portfolio variance is a measure of a portfolio's overall risk and is the portfolio's standard deviation squared.
A lower correlation between securities in a portfolio brings about a lower portfolio variance.
Portfolio variance (and standard deviation) characterize the risk-pivot of the efficient frontier in modern portfolio theory (MPT).
Portfolio variance considers the weights and variances of every asset in a portfolio as well as their covariances.