PORTFOLIO OPTIMIZATION USING SECOND ORDER CONIC PROGRAMMING APPROACH

In this paper, we examine the framework to estimate financial risk called conditional-value-at-risk (CVaR) and examine models to optimize portfolios by minimizing CVaR. We note that total risk can be a function of multiple risk factors combined in a linear or nonlinear forms. We demonstrate that, when using CVaR, several common nonlinear models can be expressed as second order cone programming problems and therefore efficiently solved using modern algorithms. This property is not shared with the more classical estimation of financial risk based on value-at-risk.


INTRODUCTION
Financial institutes make investments in different assets to grow their business but the variability of returns gives rise to risk. So risk, along with returns, is a major consideration for capital budgeting decisions. Thus, it becomes pertinent to measure risk.
transformed into an SOCP in Section 3. Finally, in Section 4 conclusion is made and future directions are discussed.

DESCRIPTION OF THE MODEL
Assuming N different assets, the investment decision vector x is expressed as: where x i , i = 1, 2, ..., N represents the proportion of budget to invest in i th asset. These investment proportions x i are assumed to be non-negative and must satisfy the unity budget constraint The returns of each asset are random and so are the losses (being negative of returns).
Our goal is to minimize total risk. There are several types of risk in market: interest rate risk, equity risk, currency risk, commodity risk, credit spread risk, default risk, operational risk etc [13]. Each asset can have a distinct combination of risk factors and these risk factors may follow different distributions of losses. Here we consider m distinct risk factors, so total risk is the function of these risk factors.
where γ k is the distribution of losses for k th risk factor. We classify the assets into three categories as follows where (1) The collection γ k , k ∈ A represents the distributions of losses for the linear risk factors.
(2) The collection γ k , k ∈ B represents the distributions of losses for the risk factors which have non-linear contribution to the total risk and their non-linearity is defined by the l 2 norm.
(3) The collection γ k , k ∈ C represents distributions of losses for the risk factors which have non-linear contribution to the total risk and their non-linearity is defined by the l 2 norm squared.
(4) The distribution of losses for the k th risk factor γ k , across the N assets is written as where γ i j represents the j th scenario of the i th asset while considering Q k number of historical scenarios for k th risk factor.
To minimize the total risk, there is need to calculate the risk in all the distributions of losses for different risk factors. The random variable X(x, γ k ) represents the losses of each asset for k th risk factor. These depend on the random losses and the investment in each asset. The risk measure CVaR α k (X(x, γ k )), for a given parameter 0 < α k < 1, can be calculated as [22] (6) CVaR α k (X(x, γ k )) = min = min where (a) + = maximum{0, a} and l k represents VaR that is obtained as a by product while optimizing CVaR [22]. The function E[·] represents the expected loss/risk of the loss distribution with joint probability distribution function p(γ 1 j , γ 2 j , ..., γ N j )) = ρ j , j ∈ {1, 2, ..., Q k } for k th risk factor.

Risk Estimation across the Linear Risk Factors.
The set A provides indices of the risk factors that are treated linearly. The risk across these risk factors is calculated as a linear combination as follows: where β k , k ∈ A are assumed to be any non-negative real numbers.

Risk Estimation across the Non-linear Risk Factors. The sets B and C provide indices
that are treated non-linearly. Set B corresponds to risk factors that are combined through an l 2 norm. For each k ∈ B, we generate a set B k that links the appropriate risk measures together.
The resulting risk across risk factors in B is calculated as: where β k , k ∈ B are assumed to be any non-negative real numbers.
Set C corresponds to risk factors that are combined through an l 2 norm squared. For each k ∈ C, we generate a set C k that links the appropriate risk measures together. The resulting risk across risk factors in C is calculated as: where β k , k ∈ C are assumed to be any non-negative real numbers.
The total risk across all the linear and non-linear risk factors can be defined as follows where β k , 1 ≤ k ≤ m and β k , 1 ≤ k ≤ m are any non-negative real numbers.

MODEL FORMULATION AND TRANSFORMATION
The model to minimize total risk (9) subject to constraints (2) is formulated as follows Now, we transform this non-linear model (10) into an equivalent SOCP. Assuming r k = CVaR α k (X(x, γ k )) for 1 ≤ k ≤ m, the model (10) is transformed into Using equation (7), this can be rewritten as subject to The non-linear constraints of problem (12) can be transformed into linear constraints [22] using subject to , k ∈ C, the problem (13) can be transformed into the following SOCP Minimize The quadratic constrains ∑ k∈B k (14) represents the Lorentz cones, so the model can be rewritten as follows The constraint ∑ k∈B k r 2 k ≤ t 2 k , k ∈ B, is equivalent to ∑ k∈B k r 2 k = t 2 k , k ∈ B due to the fact that we are minimizing The model (15) is an SOCP. As such, a number of algorithms exist that guarantee converging to a solution to this problem [3,15,16,24]. For example, it can be solved by using any of the solvers: QUADPROG, SeDuMi, CPLEX, Gurobi, or MOSEK.
Remark 1. If set B = φ and C = φ as well, then Problem (15) will be a linear model that can also be solved by using Simplex algorithm.

CONCLUSION AND FUTURE DIRECTIONS
In portfolio design, there generally exist many risk factors across the assets. The total risk depends on all the risk factors simultaneously. This total risk, being a function of different risk factors, can include linear or non-linear forms. We have studied the problem by considering total risk as a non-linear function where some risk factors are assumed to be linear, some follow the l 2 norm and others follow the l 2 norm squared. We demonstrate that, when risk is measured using CVaR, the resulting model is an SOCP and therefore solvable by a number of modern algorithms.
In this paper we have presented the methodology to transform a specific non-linear model into a tractable SOCP. Our next step is to perform some experiments and check the applicability of the proposed method by implementing it to some real life data. Moreover, this method is applicable to a specific class of problems where non-linearity can be defined as a quadratic form.
However, total risk could have any kind of non-linearity so there is still scope for advancements in the proposed method that can support the more general forms of non-linearity.