Consider a set of

assets with variable returns

, and suppose that the mean return of asset

is

and the covariance of assets

and

is

. The volatility of asset

is its standard deviation

. A risk-free asset has a positive mean return but zero volatility and it has zero covariance with every other asset. A portfolio consisting of the

assets with random value

, where

, has mean return

and variance

. An investor may choose the weights

according to various criteria, but modern portfolio theory (MPT), introduced by Markowitz in 1952, which earned him the 1990 Nobel Memorial Prize in Economic Sciences, chooses the weights to minimize the variance (or volatility) of the portfolio. That is, MPT solves the quadratic program

Minimize

subject to

.

The investor may optionally add a constraint that the mean return must meet or exceed some level

so that

, and the investor may exclude the possibility of short sales by requiring

for all

. The set of MPT portfolios for different values of

constitute the efficient frontier, which is shown in red in the plot. A characteristic of the efficient frontier is that the mean return and volatility increase in tandem.

Snapshot 1: short sales have been excluded, so all of the assets appear in green, and

. That is, the portfolio is expected to have a 20% return on average.

Snapshot 2: AAPL and MSFT have been removed from the portfolio, and that causes a strong curve in the efficient frontier as the curve bends around DIS to end at GE.

Snapshot 3: the risk-free asset has been removed from the portfolio and

. Short sales are also permitted, and BA is short-sold.