In financial engineering, we often work with asset returns instead of asset prices. Given a financial instrument's price S_t where t denotes the period, the discrete return r_{t + 1} between t and t + 1 is defined as:

The global return of a portfolio of assets R_t, is defined as the sum of the weighted return of each asset, expressed as follows:

with w_ti being the weight of asset i during period t expressed as a value between [0.0, 1.0] and r_ti being the return of asset i during period t.

Following Markowitz we define the problem of portfolio selection as follows:

The formula expresses that we minimize the variance - covariance risk σ̄ , where the matrix Σ̂ is an estimate of the covariance of the assets. The vector ω denotes the individual investments subject to the condition ω^T 1 = 1 that the available capital is fully invested. The expected or target return r̄ is expressed by the condition ω^T μ̂ = r̄ , where the p-dimensional vector μ̂ estimates the expected mean of the assets.

Markowitz' portfolio modell has a unique solution:

where

with

The corresponding standard deviation σ̄ for the optimal portfolio with weights ω^* is

The locus of this set in the { σ̄ , r̄ }-space are hyperbolas. The set inside the hyperbola is the feasible set of mean/standard deviation portfolios, and the borders are the efficient frontier (upper border), and the minimum variance locus (lower border). Here, r^* is the return of the minimum variance portfolio.

The point with the smallest risk on the efficient frontier is called the global minimum variance portfolio, MVP. The MVP represents just the minimum risk point on the efficient frontier.The set of weight is:

In contrast to minimum risk portfolios, where we minimize the risk for a given target return, maximum return portfolios work the opposite way: Maximize the return for a given target risk.

Note that now we are concerned with a linear programming problem and quadratic constraints. This can be solved in the Portfolio optimization (15) dynamic form using the available linear solver (15.4.5).

In general, minimizing CVaR_α and VaR_α are not equivalent. Since the definition of CVaR_α involves the VaR_α function explicitly, it is difficult to optimize or work with this function. Instead, we consider the following simpler auxiliary function:

Alternatively, we can write F_α(ω, γ) as follows:

where z⁺ = max(z, 0). This final function of γ has the following important properties that make it useful for the computation of CVaR_α and VaR_α:

As a consequence, we deduce that CVaR_α can be optimized via optimization of the function F_α(ω, γ) with respect to the weights w and VaR g. If the loss function f (ω, r) is a convex function of the portfolio variables w, then F_α(ω, γ) is also a convex function of ω. In this case, provided the feasible portfolio set ω is also convex, the optimization problems are smooth convex optimization problems that can be solved using well-known optimization techniques for such problems.

Figure Figure 17.1, “Efficient mean-variance frontier [8]” shows the Efficient mean-variance frontier with the global minimum variance portfolio, the global minimum Value at Risk (5%) portfolio and the global minimum Conditional Value at Risk (5%) portfolio. The efficient frontiers under the various measures, are the subset of boundaries above the corresponding minimum global risk portfolios. We see that under 5% VaR and 5% CVaR the set of efficient portfolios is reduced with respect to the variance.

Figure 17.1. Efficient mean-variance frontier [8]

Often it is not possible or desirable to compute/determine the joint density function p(r) of the random events in our formulation. Instead, we may have a number of scenarios, say r_s for s = 1, ... , S, which may represent some historical values of the returns. In this case, we obtain the following approximation to the function F_α(ω, γ) by using the empirical distribution of the random returns based on the available scenarios:

Now, the problem min_ω∈W CVaR_α(ω) can be approximated by

To solve this optimization problem, we introduce artificial variables z_s to replace (f(ω, r_s) - γ)⁺. This is achieved by imposing the constraints z_s ≥ f(ω, r_s) - γ and z_s ≥ 0.

Note that the constraints z_s ≥ f(ω, r_s) - γ and z_s ≥ 0 alone cannot ensure that z_s = (f(ω, r_s) - γ)⁺, since z_s can be larger than both right-hand term and still be feasible. However, since we are minimizing the objective function, which involves a positive multiple of z_s, it will never be optimal to assign z_s a value larger than the maximum of the two quantities f(ω, r_s) - γ and 0, and therefore, in an optimal solution z_s will be precisely (f(ω, r_s) - γ)⁺, thus justifying our substitution.

Figure Figure 17.2, “Mean-VaR(5%)-boundary [8]” shows the Mean-VaR(5%)-boundary with the global minimum variance portfolio. Portfolios on the mean-VaR(5%)-boundary between the global minimum VaR( 5%) portfolio and the global minimum variance portfolio, are mean-variance efficient. The VaR constraint (vertical line) could force mean-variance investors with high variance to reduce the variance, and mean-variance investors with low variance to increase the variance, in order to be on the left side of the VaR constraint.

Figure 17.2. Mean-VaR(5%)-boundary [8]

In the case that f(ω, r_s) - γ is linear in ω , all the expressions z_s ≥ f(ω, r_s) - γ represent linear constraints and therefore the optimization problem becomes a linear programming problem that can be solved using the simplex method or alternative linear programming algorithms.