Portfolio Variance
A risk metric that enables investors to understand the level of volatility of specific portfolios.
What Is Portfolio Variance?
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.
It is the sum of the products of the squared weight of each asset, its variance, and the covariance with every other asset in the portfolio.
Correlation coefficients of equities in a portfolio are crucial to determining portfolio variance. A combination of highly correlated securities in a portfolio will increase the variance.
According to the Modern Portfolio Theory, it is possible to decrease the portfolio variance by selecting assets that have a low or negative correlation. By doing so, the negative covariance will lower the overall portfolio variance.
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.
Key Takeaways
- Portfolio variance assesses portfolio volatility using asset standard deviations and correlations. It plays a vital role in Modern Portfolio Theory, emphasizing diversified portfolios with minimized variance for optimal risk-return trade-offs.
- The formula, Portfolio variance = w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov₁,₂, encapsulates weights, variances, and covariances. Lower variance indicates effective diversification, aligning with MPT's goal of maximizing returns for a given risk level.
- Portfolio variance is a specific risk measure, while Modern Portfolio Theory offers a comprehensive framework, integrating weights, returns, and correlations to construct well-diversified portfolios.
Portfolio Variance Formula and Calculation
The following is the formula:
Portfolio variance = w12σ12 + w22σ22 + 2w1w2Cov1,2
In which:
-
w1is equal to the portfolio weight of the first stock
-
w2is equal to the portfolio weight of the second stock
-
σ1 is equal to the standard deviation of the first stock
-
σ2is equal to the standard deviation of the second stock
-
Cov1,2is 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).
If the number of equities in the portfolio increases, the terms of the formula also grow quadratically. By doing so, the total portfolio variance is smaller than the weighted average of the singular variances of each equity in the portfolio.
Note
The variance formula is the weighted combination of the singular variances of every equity adjusted by their respective covariances.
Example of Portfolio Variance
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%
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 change 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 Vs. Modern Portfolio Theory
Modern portfolio theory suggests that investors avoid risk, meaning 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 suggests that investors may have the potential to achieve higher rates of return by accepting a higher level of risk.
Variance and correlation are two statistical measures crucial in modern portfolio theory; they strongly influence investors' decision-making process to choose the appropriate assets.
The modern portfolio theory (MPT) framework helps make it possible to build well-diversified investment portfolios. Portfolio variance plays a crucial role in the context of Modern Portfolio Theory, especially in the optimization process of building well-diversified investment portfolios.
The theories behind MPT, which involve optimizing portfolios for a given level of risk, align with the financial application of Portfolio Variance to build well-diversified portfolios.
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.
To understand better, take a look at the table below:
| Aspect | Portfolio Variance | Modern Portfolio Theory |
|---|---|---|
| Definition | Measure of the dispersion of returns of a portfolio | Investment theory that constructs portfolios to maximize return for a given level of risk |
| Objective | Quantify risk in a portfolio | Optimize portfolio to achieve the highest return for a given level of risk |
| Focus | Specific measure of risk | Comprehensive approach to portfolio construction |
| Components | Weights, individual asset variances, and covariances | Expected returns, standard deviations, and correlations |
| Optimization | Limited to risk management at the portfolio level | Utilizes diversification to achieve the optimal risk-return trade-off |
| Single vs. Comprehensive Approach | Single metric for risk | Comprehensive framework considering multiple assets and their interactions |
| Example | If you have two assets, A and B, the portfolio variance would consider weights, individual variances, and the covariance between A and B. | MPT might involve constructing a portfolio with a mix of assets to achieve a desired level of return with minimum risk. |
or Want to Sign up with your social account?