### Minimum-Variance Portfolios

In theory, we could form a portfolio made up of all investable assets, however, this is not practical and we must find a way of filtering the investable universe. A risk-averse investor wants to find the combination of portfolio assets that minimizes risk for a given level of return.

## Minimum-variance Frontier

As we form a portfolio of assets, we can determine the portfolio return-risk characteristics that are a function of the characteristics of the underlying portfolio holdings and the correlation between the holdings. By varying the allocation to the underlying assets, we derive an investment opportunity set of different portfolio compositions. This is made up of the various combinations of risky assets that lead to specific portfolio risk-return characteristic which can be graphically plotted with portfolio expected return as the y-axis and portfolio standard deviation as the x-axis.

For each level of return, the portfolio with the minimum risk will be selected by a risk-averse investor. This minimization of risk for each level of return creates a minimum-variance frontier – a collection of all the minimum-variance (minimum-standard deviation) portfolios. At a point along this minimum-variance frontier curve, there exists a minimum-variance portfolio which produces the highest returns per unit of risk.

### Global Minimum-variance Portfolio

Along the minimum-variance frontier, the left-most point is a portfolio with minimum variance when compared to all possible portfolios of risky assets. This is known as the global minimum-variance portfolio. An investor cannot hold a portfolio of risky (note: risk-free assets are excluded at this point) assets with a lower risk than the global minimum-variance portfolio.

## Efficient Frontier

The portion of the minimum-variance curve that lies above and to the right of the global minimum variance portfolio is known as the Markowitz efficient frontier as it contains all portfolios that rational, risk-averse investors would choose. We can also monitor the slope of the efficient frontier, the change in units of return per units of risk. As we move to higher levels of risk, the resulting increase in return begins to diminish. The slope begins to flatten. This means we cannot achieve ever-increasing returns as we take on more risk, quite the opposite. Investors experience a diminishing increase in potential returns as portfolio risk is increased. ## Question

Which statement best describes the global minimum-variance portfolio?

A. The global minimum variance portfolio gives investors the highest levels of returns

B. The global minimum variance portfolio gives investors the lowest risk portfolio made up of risky assets

C. The global minimum variance portfolio lies to the right of the efficient frontier

Solution

The global minimum variance portfolio lies to the far left of the efficient frontier and is made up of a portfolio of risky assets that produces the minimum risk for an investor.

Describe and interpret the minimum-variance and efficient frontiers of risky assets and the global minimum-variance portfolio

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