# Mean-Variance Analysis

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## Definition of 'Mean-Variance Analysis'

Mean-variance analysis (MVA) is a portfolio selection model that seeks to maximize the expected return of a portfolio for a given level of risk. It is a popular method for portfolio optimization because it is relatively simple to implement and understand.

MVA is based on the assumption that the returns of individual assets are random variables and that the covariance between the returns of different assets can be estimated. The expected return of a portfolio is the weighted average of the expected returns of the individual assets, and the risk of a portfolio is measured by its variance or standard deviation.

The goal of MVA is to find the portfolio that has the highest expected return for a given level of risk. This is done by solving an optimization problem that minimizes the portfolio variance for a given target return.

The solution to the MVA optimization problem is the portfolio weights that maximize the expected return for a given level of risk. These weights can be used to construct an efficient frontier, which is a curve that shows the trade-off between expected return and risk.

MVA is a useful tool for portfolio optimization, but it has some limitations. First, it assumes that the returns of individual assets are random variables and that the covariance between the returns of different assets can be estimated. This may not always be the case, especially in the case of illiquid assets.

Second, MVA does not take into account the risk of the overall market. This can be a problem if the portfolio is exposed to a large market risk.

Third, MVA does not take into account the investor's risk tolerance. This can be a problem if the investor is not willing to accept the level of risk that is required to achieve the desired return.

Despite these limitations, MVA is a valuable tool for portfolio optimization. It is a relatively simple and easy-to-understand method that can be used to construct portfolios that meet the investor's risk and return objectives.

* MVA is based on the capital asset pricing model (CAPM), which is a theory of asset pricing. CAPM states that the expected return of an asset is equal to the risk-free rate plus a risk premium that is proportional to the asset's beta. Beta is a measure of the asset's volatility relative to the market.
* The risk of a portfolio is measured by its variance or standard deviation. Variance is the average squared deviation of the portfolio returns from their expected value. Standard deviation is the square root of variance.
* The goal of MVA is to find the portfolio that has the highest expected return for a given level of risk. This is done by solving an optimization problem that minimizes the portfolio variance for a given target return.
* The solution to the MVA optimization problem is the portfolio weights that maximize the expected return for a given level of risk. These weights can be used to construct an efficient frontier, which is a curve that shows the trade-off between expected return and risk.
* MVA is a useful tool for portfolio optimization, but it has some limitations. First, it assumes that the returns of individual assets are random variables and that the covariance between the returns of different assets can be estimated. This may not always be the case, especially in the case of illiquid assets.
* Second, MVA does not take into account the risk of the overall market. This can be a problem if the portfolio is exposed to a large market risk.
* Third, MVA does not take into account the investor's risk tolerance. This can be a problem if the investor is not willing to accept the level of risk that is required to achieve the desired return.

Despite these limitations, MVA is a valuable tool for portfolio optimization. It is a relatively simple and easy-to-understand method that can be used to construct portfolios that meet the investor's risk and return objectives.

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