Portfolio Variance

It is a risk metric that enables investors to understand the level of volatility of specific portfolios.

Portfolio variance is a risk metric that enables investors to understand the level of volatility of specific portfolios. It is computed using the standard deviation of the underlying portfolio's assets and the correlations (or covariance) of each asset pair in the portfolio. 

Portfolio Variance

It is the product of the squared weight of every equity by its variance, adding twice the weighted average weight multiplied by the covariance of every equity.

Correlation coefficients of equities in a portfolio are crucial to determining portfolio variance. A combination of high correlated securities in a portfolio will increase the variance. 

According to the Modern Portfolio Theory, it is possible to decrease the variance by picking assets that have a low or negative correlation. By doing so, the covariance will impact the variance negatively. 

A low portfolio variance is critical for investors looking to have a well-diversified portfolio, which can offset risks related to the price fluctuation of equities.

Formula and Interpretation

The following is the formula: 

Portfolio variance = w12σ12 + w22σ22 + 2w1w2Cov1,2

In which:

  • w1 is equal to the portfolio weight of the first stock;

  • w2 is equal to the portfolio weight of the second stock;

  • σis equal to the standard deviation of the first stock;

  • σ2 is equal to the standard deviation of the second stock;

  • Cov1,2 is equal to the covariance of the two stocks, which can also be written as p(1,2)σ1σ2, in which p(1,2) is the correlation coefficient between the two stocks.

Hence, the variance is given by adding the product of the squared weight of each asset by the squared standard deviation (w12σ12 + w22σ22). Lastly, it is necessary to add twice the product of both weighted averages and the covariance of the assets (2w1w2Cov1,2). 

Calculation

If the number of equity in the portfolio increases, the terms of the formula also grow exponentially. The variance formula is the weighted combination of the singular variances of every equity adjusted by their respective covariances. 

By doing so, the total portfolio variance is smaller than the weighted average of the singular variances of each equity in the portfolio. 

Example

Let's assume an investor has a portfolio of two stocks. 

  • Stock X is worth $100.000 with a portfolio weight (w1) of 57%;
  • Stock X has a standard deviation (σ1) of 35%;
  • Stock Y is worth $75.000 with portfolio weight (w2) being 43%;
  • Stock Y has a standard deviation (σ2) of 28%;
  • The correlation (Cov1,2) between Stock X and Stock Y is 0.48. 

Inputting the information, it is now possible to compute the variance of the portfolio. 

Variance= w12σ12 + w22σ22 + 2w1w2Cov1,2

Therefore,

Variance = (57%2 * 35%2) + (43%2 * 28%2) + (2 * 57% * 43% * 48%) = 28,96%

Computation

Assuming now a different scenario in which we have: 

  • Stock X is worth $100.000 with a portfolio weight (w1) of 57%;
  • Stock X with a standard deviation (σ1) of 35%;
  • Stock Y is worth $75.000 with portfolio weight (w2) being 43%;
  • Stock Y with a standard deviation (σ2) of 28%;
  • The correlation (Cov1,2) between Stock X and Stock Y is 0.15. 

In this situation, we have the same weights for both stocks and the same standard deviations; the only thing that has changed is the correlation between the stocks.  

Variance = (57%2 * 35%2) + (43%2 * 28%2) + (2 * 57% * 43% * 15%) = 12,78%

As we can see, the change in the correlation between assets has greatly impacted the variance. Bringing it down to 12,78%. 

Portfolio Variance and Modern Portfolio Theory 

Modern portfolio theory suggests that investors tend to avoid risk, meaning that they will choose a relatively less risky portfolio for a specific return. This investment theory helps investors build a portfolio with the best-expected return depending on the level of risk chosen. 

As a result, the theory implies that investors will be able to achieve higher rates of return if they choose to increase the level of risk. Variance and correlation are two statistical measures crucial in modern portfolio theory; they strongly influence the process that makes investors choose the appropriate assets. 

Stock

The modern portfolio theory (MPT) framework helps make it possible to build well-diversified investment portfolios. Portfolio variance and MPT are strictly connected. The theories behind the MPT of decreasing the portfolio risk are complementary to the financial application of the Portfolio Variance. 

By utilizing the Portfolio Variance, it is possible to build portfolios that follow the MPT. In addition, the risk is significantly decreased in MPT portfolios where investors allocate their funds to non-correlated assets. 

Key Takeaways 
  • Portfolio variance is an indicator that helps investors pick specific equity based on its volatility level;
  • It is calculated using the standard deviation of each portfolio's stock and the correlations (or covariance) of each stock pair in the portfolio. 
  • A low variance is achieved when the assets chosen have a low or negative correlation,where-in investors can offset volatility risks; 
  • Portfolio variance and the Modern Portfolio Theory are connected as computing the Portfolio Variance makes it possible to create portfolios following MPT rules. 
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Researched and authored by Alessandro Davì | LinkedIn

Reviewed and Edited by Abhijeet Avhale | LinkedIn

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