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The efficient frontier was first defined by Harry Markowitz in his groundbreaking (1952) paper that launched portfolio theory. That theory considers a universe of risky investments and explores what might be an optimal portfolio based upon those possible investments.

Consider an interval of time. It starts today. It can be any length, but one-year is typically assumed. Today’s values for all available risky investments are known. Their accumulated values (reflecting price changes, coupon payments, dividends, stock splits, etc.) at the end of the horizon are random. As random quantities, we may assign them expected returns and volatilities. We may also assign a correlation to each pair of returns. We can use these inputs to calculate the expected return and volatility of any portfolio that can be constructed using the available instruments.

The notion of “optimal portfolio” can be defined in one of two ways:

1. For any level of volatility, consider all the portfolios which have that volatility. From among them all, select the one which has the highest expected return.
2. For any expected return, consider all the portfolios which have that expected return. From among them all, select the one which has the lowest volatility.

Each definition produces a set of optimal portfolios. Definition (1) produces an optimal portfolio for each possible level of risk. Definition (2) produces an optimal portfolio for each expected return. There is a set of portfolios that are optimal under both definitions. That set of portfolios is called the efficient frontier. This is illustrated in Exhibit 1:

Exhibit 1: For every point in the achievable region, there will be at least one portfolio that can be constructed from available risky assets that has the volatility and expected return corresponding to that point. The efficient frontier is the orange curve that runs along the top of the achievable region. Portfolios on the efficient frontier are optimal in both the sense that they offer maximal expected return for some given level of risk and minimal risk for some given level of expected return. For example, the indicated portfolio is optimal in the sense of having the maximum expected return possible at 15% volatility. It is also optimal in the sense of having the minimum volatility possible for 9% expected return.

In Exhibit 1, the green region on the right corresponds to the achievable region in risk-return space. For every point in that region, there will be at least one portfolio constructible from available risky investments that has the volatility and expected return corresponding to that point. Outside the achievable region is the unachievable region. No portfolios can be constructed from available risky assets corresponding to points in that region.

The orange curve running along the top of the achievable region is the efficient frontier. The portfolios that correspond to points on that curve are optimal according to both definitions (1) and (2) above.

Typically, the portfolios which comprise the efficient frontier are the ones which are most highly diversified. Less diversified portfolios tend to be closer to the middle of the achievable region.

## References

• Markowitz, Harry M. (1952). Portfolio Selection, Journal of Finance, 7 (1), 77-91.