# Efficient frontier

*This article is about a financial mathematical concept. For other frontiers described as efficient, see Production possibilities frontier and Pareto frontier.*

In modern portfolio theory, the **efficient frontier** (or **portfolio frontier**) is an investment portfolio which occupies the 'efficient' parts of the risk-return spectrum. Formally, it is the set of portfolios which satisfy the condition that no other portfolio exists with a higher expected return but with the same standard deviation of return.^{[1]}^{[2]} The efficient frontier was first formulated by Harry Markowitz in 1952.^{[3]}

## Overview[edit]

A combination of assets, i.e. a portfolio, is referred to as "efficient" if it has the best possible expected level of return for its level of risk (which is represented by the standard deviation of the portfolio's return).^{[4]} Here, every possible combination of risky assets can be plotted in risk–expected return space, and the collection of all such possible portfolios defines a region in this space. In the absence of the opportunity to hold a risk-free asset, this region is the opportunity set (the feasible set). The positively sloped (upward-sloped) top boundary of this region is a portion of a hyperbola^{[5]} and is called the "efficient frontier."

If a risk-free asset is also available, the opportunity set is larger, and its upper boundary, the efficient frontier, is a straight line segment emanating from the vertical axis at the value of the risk-free asset's return and tangent to the risky-assets-only opportunity set. All portfolios between the risk-free asset and the tangency portfolio are portfolios composed of risk-free assets and the tangency portfolio, while all portfolios on the linear frontier above and to the right of the tangency portfolio are generated by borrowing at the risk-free rate and investing the proceeds into the tangency portfolio.

## See also[edit]

- Modern portfolio theory
- Critical line method, an optimization algorithm developed by Markowitz for this problem

## References[edit]

**^**"Efficient Frontier".*Investopedia*. investopedia.com. Retrieved 15 May 2017.**^**"Markowitz efficient frontier".*NASDAQ*. nasdaq.com. Retrieved 15 May 2017.**^**Markowitz, H.M. (March 1952). "Portfolio Selection".*The Journal of Finance*.**7**(1): 77–91. doi:10.2307/2975974. JSTOR 2975974.**^**Edwin J. Elton and Martin J. Gruber (2011).*Investments and Portfolio Performance*. World Scientific. pp. 382–383. ISBN 978-981-4335-39-3.**^**Merton, Robert. "An analytic derivation of the efficient portfolio frontier,"*Journal of Financial and Quantitative Analysis*7, September 1972, 1851-1872.

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