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The portfolio construction problem discussed in this section is the simplest form of mean-variance optimization. The universe of risky investments available to the portfolio manager consists of N asset classes. The single-period total rate of return on the risky asset i is denoted by R_i, for i = 1, … N. We assume that asset zero is riskless, and its rate of return is given by R₀. The weight of asset i in the portfolio is given by w_i. Therefore, the rate of return on a portfolio of the N + 1 risky and riskless asset can be expressed as:

(1.13)

(1.14)

For now, we do not impose any short-sale restriction, and therefore the weights could assume negative values.

From Equation 1.14, we can see that . If this is substituted in Equation 1.13 and terms are collected, the rate of return on the portfolio can be expressed as:

(1.15)

The advantage of writing the portfolio's rate of return in this form is that we no longer need to be concerned that the weights appearing in Equation 1.15 will add up to one. Once the weights of the risky assets are determined, the weight of the riskless asset will be such that all the weights would add up to one.

Next, we need to consider the risk of this portfolio. Suppose the covariance between asset i and asset j is given by σij. Using this, the variance-covariance of the N risky assets is given by:

(1.16)

The portfolio problem can be written in this form, where the weights are selected to maximize the objective function:

(1.17)

This turns out to have a simple and well-known solution:

(1.18)

The solution requires one to obtain an estimate of the variance-covariance matrix of returns on risky assets. Then the inverse of this matrix will be multiplied into a vector of expected excess returns on the N risky assets. It is instructive to notice the role of the degree of risk aversion. As the level of risk aversion (λ) increases, the portfolio weights of risky assets decline. In addition, those assets with large expected excess returns tend to have the largest weights in the portfolio.