What is mean variance portfolio?
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Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. It is a formalization and extension of diversification in investing, the idea that owning different kinds of financial assets is less risky than owning only one type. Its key insight is that an asset's risk and return should not be assessed by itself, but by how it contributes to a portfolio's overall risk and return. It uses the variance of asset prices as a proxy for risk.
Mean-variance analysis is the process of weighing risk, expressed as variance, against expected return. Investors use mean-variance analysis to make decisions about which financial instruments to invest in, based on how much risk they are willing to take on in exchange for different levels of reward. Mean-variance analysis allows investors to find the biggest reward at a given level of risk or the least risk at a given level of return.
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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 fluctuate over time. This portfolio variance statistic is calculated using the standard deviations of each security in the portfolio as well as the correlations of each security pair in the portfolio.
The portfolio variance is equivalent to the portfolio standard deviation square .
Understanding Portfolio Variance
Portfolio variance looks at the covariance or correlation coefficients for the securities in the portfolio. Generally, a lower correlation between securities in a portfolio results in a lower portfolio variance.
Portfolio variance is calculated by multiplying the squared weight of each security by its corresponding variance and adding twice the weighted average weight multiplied by the covariance of all individual security pairs.
Modern portfolio theory says that portfolio variance can be reduced by choosing asset classes with a low or negative correlation, such as stocks and bonds, where the variance (or standard deviation) of the portfolio is the x-axis of the efficient frontier.
KEY TAKEAWAYS
Portfolio variance is a measure of a portfolio's overall risk, and is the portfolio's standard deviation squared.
Portfolio variance takes into account the weights and variances of each asset in a portfolio as well their covariances.
Portfolio variance (and standard deviation) define the risk-axis of the efficient frontier in Modern Portfolio Theory.
Equation for Portfolio Variance
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.
The equation for the portfolio variance of a two-asset portfolio, the simplest portfolio variance calculation, takes into account five variables:
w1 = the portfolio weight of the first asset
w2 = the portfolio weight of the second asset
σ1= the standard deviation of the first asset
σ2 = the standard deviation of the second asset
cov(1,2) = the covariance of the two assets, which can thus be expressed as:
p(1,2)σ1σ2, where p(1,2) is the correlation coefficient between the two assets.
As the number of assets in the portfolio grows, the terms in the formula for variance increase exponentially. For example, a three-asset portfolio has six terms in the variance calculation, while a five-asset portfolio has 15.
Two-Asset Portfolio Variance Example
For example, assume there is a portfolio that consists of two stocks. Stock A 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 A is 33.3% and 66.7% for Stock B. Plugging in this information 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 not a particularly easy statistic to interpret on its own, so most analysts calculate the standard deviation, which is simply the square root of variance. In this example, the square root of 1.64% is 12.82%.
Portfolio Variance and Modern Portfolio Theory
Modern Portfolio Theory is a framework for constructing an investment portfolio. MPT takes as its central premise the idea that rational investors want to maximize returns while also minimizing risk, sometimes measured using volatility. Investors seek what is called an efficient frontier, or the lowest level or risk and volatility at which a target return can be achieved.
Risk is lowered in MPT portfolios by investing in non-correlated assets. Assets that might be risky on their own can actually lower the overall risk of a portfolio by introducing an investment that will rise when other investments fall. This reduced correlation can reduce the variance of a theoretical portfolio. In this sense, an individual investment's return is less important that its overall contribution to the portfolio, in terms of risk, return and diversification.
The level of risk in a portfolio is often measured using standard deviation, which is calculated as the square root of the variance. If data points are far away from the mean, the variance is high, and the overall level of risk in the portfolio is high, as well. Standard deviation is a key measure of risk used by portfolio managers, financial advisors and institutional investors. Asset managers routinely include standard deviation in their performance reports.
Hope this answer helps you.
Portfolio variance is a measurement of risk, of how the aggregate actual returns of a set of securities making up a portfolio fluctuate over time. This portfolio variance statistic is calculated using the standard deviations of each security in the portfolio as well as the correlations of each security pair in the portfolio.
The portfolio variance is equivalent to the portfolio standard deviation square .
Understanding Portfolio Variance
Portfolio variance looks at the covariance or correlation coefficients for the securities in the portfolio. Generally, a lower correlation between securities in a portfolio results in a lower portfolio variance.
Portfolio variance is calculated by multiplying the squared weight of each security by its corresponding variance and adding twice the weighted average weight multiplied by the covariance of all individual security pairs.
Modern portfolio theory says that portfolio variance can be reduced by choosing asset classes with a low or negative correlation, such as stocks and bonds, where the variance (or standard deviation) of the portfolio is the x-axis of the efficient frontier.
KEY TAKEAWAYS
Portfolio variance is a measure of a portfolio's overall risk, and is the portfolio's standard deviation squared.
Portfolio variance takes into account the weights and variances of each asset in a portfolio as well their covariances.
Portfolio variance (and standard deviation) define the risk-axis of the efficient frontier in Modern Portfolio Theory.
Equation for Portfolio Variance
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.
The equation for the portfolio variance of a two-asset portfolio, the simplest portfolio variance calculation, takes into account five variables:
w1 = the portfolio weight of the first asset
w2 = the portfolio weight of the second asset
σ1= the standard deviation of the first asset
σ2 = the standard deviation of the second asset
cov(1,2) = the covariance of the two assets, which can thus be expressed as:
p(1,2)σ1σ2, where p(1,2) is the correlation coefficient between the two assets.
As the number of assets in the portfolio grows, the terms in the formula for variance increase exponentially. For example, a three-asset portfolio has six terms in the variance calculation, while a five-asset portfolio has 15.
Two-Asset Portfolio Variance Example
For example, assume there is a portfolio that consists of two stocks. Stock A 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 A is 33.3% and 66.7% for Stock B. Plugging in this information 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 not a particularly easy statistic to interpret on its own, so most analysts calculate the standard deviation, which is simply the square root of variance. In this example, the square root of 1.64% is 12.82%.
Portfolio Variance and Modern Portfolio Theory
Modern Portfolio Theory is a framework for constructing an investment portfolio. MPT takes as its central premise the idea that rational investors want to maximize returns while also minimizing risk, sometimes measured using volatility. Investors seek what is called an efficient frontier, or the lowest level or risk and volatility at which a target return can be achieved.
Risk is lowered in MPT portfolios by investing in non-correlated assets. Assets that might be risky on their own can actually lower the overall risk of a portfolio by introducing an investment that will rise when other investments fall. This reduced correlation can reduce the variance of a theoretical portfolio. In this sense, an individual investment's return is less important that its overall contribution to the portfolio, in terms of risk, return and diversification.
The level of risk in a portfolio is often measured using standard deviation, which is calculated as the square root of the variance. If data points are far away from the mean, the variance is high, and the overall level of risk in the portfolio is high, as well. Standard deviation is a key measure of risk used by portfolio managers, financial advisors and institutional investors. Asset managers routinely include standard deviation in their performance reports.
Hope this answer helps you.
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