# Minimum variance portfolio

Minimum variance portfolio |
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A **minimum variance portfolio** is a portfolio that consists of a set of assets that have the minimum amount of risk when compared with other portfolios with similar expected returns. The goal is to maximize the return for a given level of risk.

The portfolio can be constructed using a formula that minimizes the variance of the portfolio, also known as the portfolio risk.

In conclusion, the minimum variance portfolio is a portfolio that consists of a set of assets that have the minimum amount of risk when compared with other portfolios with similar expected returns. The portfolio is constructed using a formula that minimizes the variance of the portfolio.

## Example of Minimum variance portfolio

A minimum variance portfolio can be constructed using stocks and bonds. The portfolio might include stocks from different sectors and types of bonds such as corporate bonds, government bonds and high-yield bonds. Each asset in the portfolio is assigned a weight that is optimized to minimize the portfolio risk. For example, a portfolio might include 20% stocks from the technology sector, 30% stocks from the financial sector, 20% corporate bonds, 20% government bonds and 10% high-yield bonds.

In conclusion, a minimum variance portfolio is a portfolio that is constructed using stocks and bonds from different sectors and types. Each asset in the portfolio is assigned a weight that is optimized to minimize the portfolio risk.

## Formula of Minimum variance portfolio

The formula is given below:

<math>\begin{equation*} \sigma_P^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + \dots + w_n^2 \sigma_n^2 + 2w_1w_2cov_{12}+\dots +2w_{n-1}w_ncov_{n-1,n} \end{equation*}<math>

In this equation, σ_{P} is the variance of the portfolio, $\sigma_i$ is the variance of the asset i, w_{i} is the weight of the asset i, and cov_{ij} is the covariance between assets i and j. To find the minimum variance portfolio, the formula is solved so that the weights w_{i} are optimized to minimize the variance of the portfolio.

## When to use Minimum variance portfolio

- When the investor has limited resources and cannot invest in a large number of assets, a minimum variance portfolio is a good choice as it minimizes risk while still achieving a good return.
- When the investor is risk averse and wants to minimize the risk of their portfolio, a minimum variance portfolio can be used.
- When the investor wants to maximize their return while minimizing the risk of their portfolio, a minimum variance portfolio can be used.

## Types of Minimum variance portfolio

**Equally Weighted Portfolio**: This type of portfolio assigns the same weight to each asset in the portfolio. The risk of the portfolio is minimized by diversifying the assets.**Market-Weighted Portfolio**: This type of portfolio assigns the weights based on the market value of the assets. This type of portfolio reduces the risk by investing more in the assets with the highest market value.**Risk Parity Portfolio**: This type of portfolio assigns the weights based on the risk of the asset. This type of portfolio reduces risk by investing more in the assets with the lowest risk.

## Steps of Minimum variance portfolio calculation

**Step 1**: Calculate the expected return for each asset. This should be done by looking at the historical data of each asset and estimating the expected return based on past performance.**Step 2**: Calculate the covariance matrix of the assets. The covariance matrix is a table of values that shows the correlation between each pair of assets. This helps to understand the level of risk associated with a portfolio.**Step 3**: Calculate the portfolio weights. The weights of the assets need to be calculated in order to construct the portfolio. This can be done by solving the formula given above.**Step 4**: Calculate the expected return and variance of the portfolio. Once the weights have been calculated, the expected return and variance of the portfolio can be calculated.

## Advantages of Minimum variance portfolio

**Diversification**: The minimum variance portfolio helps to diversify risk by allocating capital to different assets and helps to reduce the overall portfolio risk.**Cost Effectiveness**: The minimum variance portfolio is more cost effective as it seeks to maximize returns for a given level of risk, while minimizing the costs associated with trading.**Liquidity**: The minimum variance portfolio also provides liquidity, as the portfolio is invested in many different assets and can be easily liquidated in case of a need.

## Limitations of Minimum variance portfolio

There are several limitations of the minimum variance portfolio that should be considered when constructing a portfolio. These limitations include:

- The portfolio assumes that the expected return and covariance between assets remain constant over time, which may not be the case in reality.
- The portfolio also assumes that the assets are normally distributed, which might not be true in practice.
- The portfolio assumes that the weights are known ahead of time, which may not be the case.

**Markowitz Portfolio Theory**: Also known as Modern Portfolio Theory, Markowitz Portfolio Theory is a mathematical framework for constructing a portfolio that maximizes returns for a given level of risk. It takes into account the correlation between different assets and focuses on balance and diversification.**Risk Parity Portfolio**: Risk parity portfolios are portfolios that have a balanced and diversified level of risk across different asset classes. This means that each asset class has an equal share of the total risk, and that the portfolio as a whole is equally balanced across different risk levels.**Maximum Diversification Portfolio**: This portfolio is constructed to achieve maximum diversification by investing in a variety of asset classes. The portfolio is balanced in such a way that it minimizes the correlation between different asset classes, which reduces the overall risk of the portfolio.

In conclusion, there are a few different approaches related to the minimum variance portfolio, such as Markowitz Portfolio Theory, Risk Parity Portfolio, and Maximum Diversification Portfolio. Each of these approaches have a different focus and use different techniques to construct a portfolio that maximizes returns while minimizing risk.

## Suggested literature

- Clarke, R., De Silva, H., & Thorley, S. (2011).
*Minimum-variance portfolio composition*. The Journal of Portfolio Management, 37(2), 31-45. - Clarke, R. G., De Silva, H., & Thorley, S. (2006).
*Minimum-variance portfolios in the Us equity market*. The journal of portfolio management, 33(1), 10-24. - Kempf, A., & Memmel, C. (2006).
*Estimating the global minimum variance portfolio*. Schmalenbach Business Review, 58, 332-348.