Portfolio diversification reduces risk by including various investments that are not perfectly correlated. The standard deviation is commonly used to quantify risk and measure how concentrated or diversified a portfolio is. Modern portfolio theory holds that investors can construct an efficient portfolio that optimizes the risk-return tradeoff by balancing different assets. Mathematical tools like the variance-covariance matrix and Lagrange multipliers can be used to calculate the minimum-variance or optimal portfolio given expected returns, variances and correlations of constituent assets.