Mean Variance Analysis

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Harry W. Markowitz, the father of “Modern Portfolio theory”, developed the mean-variance analysis, which focuses on creating portfolios of assets that minimizes the variance of returns i.e. risk, given a level of desired return, or maximizes the returns given a level of risk tolerance. This theory aids the process of portfolio construction by providing a quantitative take on it. It integrates the field of quantitative analysis with portfolio management. Mean variance analysis has found wide applications both inside and outside financial economics. However it is based on certain assumptions which do not hold good in practice. Hence there have been certain revisions to it, so as to make it a more useful tool in portfolio management.
Mean
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All securities can be divided into parcels of any size.

Let R be the expected return of the assets:
R = (R1, R2,…..Rn)T
Let V be the variance/covariance matrix. It is assumed to be positive definite. σ ij = cov (Ri , Rj)
A portfolio X of asset weights is expressed as:
X = (x1, x2,……….xn)T.
[Note that HT denotes the transpose of a matrix H and xi denotes the weight on asset and ∑xi = 1, where i=1 to n]
The expected return of a portfolio is given by E[RX] = XT E[R], and the variance of the portfolio by σX = XT VX. Thus, the mean-return of the portfolio X satisfies: Mean = Rx= XT ¯R = ∑xi ¯RI, where i = 1 to n.
The variance Vx of portfolio return is given by: where σi2 is the variance of return i , and σij measures the covariance between returns i and j . We denote the standard deviation of X by σX. Often, it is not possible to be short on assets. In that case, we need to add a constraint that all portfolio weights shall be zero or above: xi ≥ 0.(Mz)

If we search for maximizing the expected return for a given variance, we have to solve the following optimization problem:

As explained by Markowitz, the portfolio selection problem can be formulated as a quadratic program. We can also search for minimizing the variance given a level of expected return. For a portfolio containing n assets, the minimum variance portfolio is a solution of: Where E* is the level of expected return.
Thus we get the efficient frontier as shown
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