PSG allows user to solve complex optimization problems without intensive programming. Some basic knowledge of calculus and optimization is desirable to better grasp the power of PSG; no previous programming knowledge, such as C++, is needed. Although PSG can be used for decision making in any area, mostly it is focused on applications involving uncertainties.

The Main Objects of Portfolio Safeguard (PSG) are Data, Function and Problem. A Data object includes the main data blocks for optimization and analysis. Data consists of three lower level objects: Point, Matrix, and Vector.  Data structures are associated to a function, the function is assigned to objects such as Objective and Constraint. Objective and Constraints are then associated to create a Problem. Optimization and Calculation Problems in PSG are formulated in a concise format, which makes problem structures transparent and easy to understand. This is achieved by representing objectives and constraints with a set of standardized functions with clear engineering interpretations. PSG provides extensive set of built-in functions.  PSG include so called, Scenario, Deterministic and Risk Functions; each function is an independent object with an associated data structure.

Deterministic functions are included in Linear, Nonlinear, Cardinality, and Deterministic Norms Groups. Linear Group includes Linear, Linear Multiple, and Variable functions.

Linear function is used for evaluation of various characteristics of models such as a portfolio return, budget, and for construction linear constraints on decision variables. For many linear constraints a Linear Multiple function is used instead. Linear Multiple function is a vector-function whose components are Linear functions.  Variable is a special case of a linear function with only one variable.

Functions from Nonlinear Group included in PSG have a wide coverage. For instance, Polynomial Absolute function is used in case study "Portfolio Optimization with Nonlinear Transaction Costs" to account for nonlinear transactions costs involving both short and long positions. Relative Entropy function  is used to find a probability distribution which is the most close to some “prior” probability distribution subject to available information about the distribution. Logarithms Exponents Sum function is used for maximization of  log-likelihood function in optimization formulation of the logistic regression problem in case study "Logistic Regression and Regularized Logistics Regression Applied to Estimating the Probability of Cesarean Section". Nonlinear functions CVaR Component Absolute, Polynomial Absolute, Maximum Component Absolute, and Quadratic functions are used for modeling various norms in case study "Projection on Polyhedron with Various Norms".  

Sophisticated functions from Cardinality Group allow to control decision variables and solve discrete optimization problems. For instance, the case study "Portfolio Optimization with Probabilistic Constraint and Fixed and Proportional Transaction Costs" uses cardinality function for assigning fixed transaction cost for instruments with non-zero positions (non-zero positions are defined by exceeding some small threshold). With cardinality functions there is no need to introduce additional discrete variables. In the case study "Portfolio Replication with Cardinality and Buyin Constraints" functions Cardinality Positive and Buyin Positive model the discontinuous performance of objective function.

2. PSG can deal with random characteristics (for example, portfolio return) having random outcomes. For modeling objects with  random characteristics PSG provides Scenario and Risk Functions.

The simplest type of Scenario Function  is  a random Linear Loss function, which  takes a finite number of values with some probabilities. For an advanced user, PSG provides complicated nonlinear Scenario functions, such as Recourse and Spline Sum functions.

Risk Functions are combined in the following Groups: in Average, CVaR, VaR, Maximum, Mean Absolute, Partial Moment, Probability, CDaR, Standard, Utilities, Error and Distance Between Distributions.

Average Loss, Average Gain, Average Recourse, and Average Gain Recourse  functions from the Average Group calculate average of random Scenario Functions (Loss, Gain,  Recourse, and Spline_Sum). For instance, the case study "Omega Portfolio Rebalancing" reduces portfolio optimization problem with the Omega performance function to maximizing Average Gain subject to a Partial Moment constraint and linear constraints. Average Recourse function is used, for instance, in the case studies "Stochastic Two Stage Linear Problem", and "Supply Chain Planning Problem".

Other functions from Average Group, such as, Average Max, are used in the following, more complicated setup. There are M  Linear Loss scenario functions (every Linear Loss  scenario function is defined by a Matrix of Scenarios). A new Maximum Loss scenarios function is calculated by maximizing losses over Linear Loss functions (over M functions for every scenario).  Average Max is calculated by averaging Maximum Loss scenarios. For instance, Average Max for Gain  is used in the case study "Optimal Position Liquidation",  and Average Max  is used in the case study "Stochastic Utility Problem".

 

CVaR, CVaR for Gain, CVaR Deviation, and CVaR Deviation for Gain are the simplest functions from CVaR Group. These functions are applied to Linear Loss (Gain) scenario functions having discrete probability distributions defined by a set of scenarios.

CVaR Group has special functions with

CVaR Group contains also risk functions applied to  nonlinear scenario functions, for instance:

Examples of functions from CVaR Group are in case studies: "Portfolio Management with Basel Accord" (CVaR Deviation) , "Cash Matching with CVaR Constraints" (CVaR Max), and "Credit-Risk Optimization Modeled by Scenarios and Mixtures of Normal Distributions" (Average CVaR  Normal Independent).

 

VaR is a percentile of a distribution. VaR Group is similar to CVaR Group. Examples of  functions from VaR Group are in case studies: "Mortgage Pipeline Hedging" (VaR Deviation), "VaR Optimization Retail Portfolio of Bonds"  (VaR Deviation),  "Portfolio Management with Basel Accord" (VaR),  "VaR vs Probability Constraints" (VaR),  "Credit-Risk Optimization Modeled by Scenarios and Mixtures of Normal Distributions" (Average VaR Normal Independent), and "Portfolio Optimization with Mixed CVaR and Mixed VaR Profiles" (VaR).

Maximum Group applies Maximum operation to linear loss functions: Maximum (for Loss) Maximum for Gain, Maximum Deviation, and Maximum Deviation for Gain.

Function Maximum CVaR (from Maximum Group). There are M  Linear  Loss  scenario functions (every Linear Loss scenario function is defined by a Matrix of Scenarios). M new CVaR functions are calculated (for every Loss scenario function).  Maximum CVaR is calculated by taking Maximum over M CVaR functions. The following functions from Maximum Group have a similar setup: Maximum CVaR, Maximum CVaR for Gain, Maximum CVaR Deviation, Maximum CVaR Deviation for Gain, Maximum VaR, Maximum VaR for Gain, Maximum VaR Deviation,  and Maximum VaR Deviation for Gain.

Maximum  Recourse  (and Maximum for Gain Recourse, Maximum Deviation Recourse, and Maximum Deviation for Gain Recourse Functions). These functions are applied to nonlinear Recourses scenario functions obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Examples of functions from Maximum Group are in case studies: "Stochastic Utility Problem" (Maximum), "Data Envelopment Analysis" (Maximum), "Cash Matching Bond Portfolio" (Maximum), "Portfolio Replication with Cardinality and Buyin Constraints" (Maximum), "Optimal Position Liquidation" (Maximum), and "Portfolio Optimization with Mixed CVaR and Mixed VaR Profiles" (Maximum).

 

Mean Absolute Group includes functions similar to the Mean Absolute Deviation which is an alternative to Variance.

Functions L1 Norm, Mean Absolute Risk, Mean Absolute Risk for Gain, Mean Absolute Deviation (from Mean Absolute Group) are applied to Linear Loss (Gain) scenario functions.

Mean Absolute Group has special functions with

L1 Norm for Recourse  (and Mean Absolute Risk Recourse, Mean Absolute Risk for Gain Recourse, Mean Absolute Deviation Recourse Functions). These functions are applied to nonlinear Recourses scenario functions obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Examples of functions from Mean Absolute Group are in case studies: "Optimal Hedging of CDO Book"  (Mean Absolute Deviation, L1 Norm), "Mortgage Pipeline Hedging" (Mean Absolute Deviation), "Portfolio Replication with Risk Constraint" (L1 Norm), "Calibrating Risk Preferences" (L1 Norm), and "Spline Approximation" (L1 Norm),

 

Partial Moment is expected  access over some fixed threshold of a function depending on scenario function (linear or nonlinear). For instance, Partial Moment function is expected  access of  Linear Loss  over some fixed threshold; Partial Moment Two function is expected  squared Linear Loss  in access of of some fixed threshold.

Functions Partial Moment, Partial Moment for Gain, Partial Moment  Deviation, Partial Moment Gain Deviation, Partial Moment Two, Partial Moment Two for Gain, Partial Moment Two Deviation for Loss, and Partial Moment Two Deviation for Gain from Partial Moment Group are applied to Linear Loss (Gain) scenario functions having discrete probability distributions defined by a set of scenarios.

Partial Moment Group has special functions with

Partial Moment Group contains also risk functions applied to  nonlinear scenario functions, for instance:

Examples of functions from Partial Moment Group are in case studies: "Credit-Risk Optimization Modeled by Scenarios and Mixtures of Normal Distributions" (Average Partial Moment Normal Independent), "Omega Portfolio Rebalancing" (Partial Moment), "Structuring step up CDO" (Partial Moment),  "Portfolio Optimization with Exponential, Logarithmic, and Linear-Quadratic Utilities" (Partial Moment Two), "Style Classification with Quantile Regression" (Partial Moment, Partial Moment for Gain).

 

Probability  Group includes two subgroups: one dimensional probability functions, and multidimensional probability functions.

Probability of Exceedance, Probability of Exceedance for Gain, Probability of Exceedance Deviation, and Probability of Exceedance Deviation for Gain are the simplest one dimensional probability functions from Probability  Group. These functions are applied to Linear Loss (Gain) scenario functions having discrete probability distributions defined by a set of scenarios.

One dimensional probability functions subgroup has special functions with

One dimensional probability functions subgroup contains also risk functions applied to nonlinear scenario functions, for instance:

Probability of Exceedance Multiple, Probability of Exceedance for Gain Multiple, Probability of Exceedance Deviation  Multiple, and Probability of Exceedance Deviation for Gain Multiple are the simplest multidimensional probability functions from Probability  Group. These functions are applied to multiple Linear Loss (Gain) scenario functions having discrete probability distributions defined by a set of scenarios.

Multidimensional probability functions subgroup has special functions with

Examples of functions from Partial Moment Group are in case studies:  "Credit-Risk Optimization Modeled by Scenarios and Mixtures of Normal Distributions" (Average Probability of Exceedance Normal Independent), "Structuring step up CDO" (Probability of Exceedance, Probability of Exceedance Multiple), "Portfolio Optimization with Probabilistic Constraint and Fixed and Proportional Transaction Costs" (Probability of Exceedance), "Classification by Maximizing Area Under ROC Curve (AUC)" (Probability of Exceedance), "Optimal Crop Production and Insurance Coverage" (Probability of Exceedance), "Optimal Tests Selection" (Probability of Exceedance for Gain Multiple Normal Independent), "Stochastic Multicommodity Network Flow Problem" (Probability of Exceedance Multiple).

Functions from CDaR Group calculate drawdown (underwater curve)  for a portfolio.

CDaR Group includes two subgroups. Functions from the first subgroup (one dimensional) are based on a single portfolio return sample-path, defined by a single matrix of scenarios. Functions from the second subgroup (multidimensional) are based on multiple portfolio return sample-paths, defined by multiple matrices of scenarios.

Functions from the first subgroup apply  risk functions to  drawdowns calculated on a single portfolio return sample-path, for instance:

CDaR. For every time moment, j=1,...J ,  portfolio  drawdown = d(j) = maxn ((uncompounded cumulative portfolio return at time moment n ) - (uncompounded cumulative portfolio return at time moment j )). CDaR  = CVaR Component Positive of vector (d(1), ..., d(J)) = average of the largest (1-α)%  components of the vector (d(1), ..., d(J)), where 0≤α≤1 . The following functions have a similar setup: CDaR for Gain, Drawdown  Maximum, Drawdown  Maximum for Gain, Drawdown  Average, Drawdown  Average for Gain.

Functions from the second subgroup apply  risk functions to portfolio drawdowns calculated on multiple  return sample-paths, for instance:

CDaR Multiple. Suppose we have k=1,..., K portfolio return sample-paths. For every sample-path k,  and every time moment, j=1,...J ,  portfolio drawdown = d(k,j) = maxn ((uncompounded cumulative portfolio return at time moment n on sample-path k ) - (uncompounded cumulative portfolio return at time moment j on sample-path k )). CDaR Multiple = CVaR Component Positive of the vector (d(1,1), ..., d(K,J)) = average of the largest (1-α)%  components of the vector  (d(1,1), ..., d(K,J)), where 0≤α≤1 . The following functions have a similar setup: CDaR for Gain Multiple, Drawdown  Maximum Multiple, Drawdown  Maximum for Gain Multiple, Drawdown  Average Multiple, Drawdown  Average for Gain Multiple.

Examples of application of functions from CDaR  Group can be found in the following case studies: "Portfolio Optimization with Drawdown Constraints on a Single Path" (Drawdown  Maximum, Drawdown  Average,  CDaR), "Portfolio Optimization with Drawdown Constraints on Multiple Paths" (Drawdown  Maximum Multiple, Drawdown  Average Multiple, CDaR Multiple), "Portfolio Optimization with Drawdown Constraints, Single Path vs Multiple Paths" (CDaR, CDaR Multiple).

 

Functions from Standard Group calculate the Standard Deviation-based measures of risk. This Group of functions may be defined by a Matrix of Scenarios or a symmetric matrix.

L2 Norm Defined on Matrix of Scenarios, Standard Risk, Standard Gain, Standard Deviation Defined on Matrix of Scenarios, Meansquare Error Defined on Matrix of Scenarios, and Variance functions from Standard Group are applied to Linear Loss (Gain) scenario functions having discrete probability distributions.

Standard Group has special functions with

L2 Norm Defined om Covariance Matrix, Standard Deviation Defined on Covariance Matrix, Meansquare Error Defined on Expected Matrix of Products, and Variance Defined on Covariance Matrix functions from Standard Group are defined on Symmetric Matrices.

Standard Group contains also risk functions applied to  nonlinear scenario functions:

L2 Norm Recourse (and  Standard Risk Recourse, Standard Gain Recourse, Standard Deviation Recourse, Meansquare Error Recourse, and Variance Recourse). These functions are applied to nonlinear Recourse Scenario functions, obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Examples of application of functions from Standard Group are in the following case studies:  "Mortgage Pipeline Hedging" (Standard Deviation Defined on Matrix of Scenarios), "Portfolio Optimization, CVaR vs. St_Dev" (Standard Deviation Defined on Matrix of Scenarios, Standard Deviation Defined on Covariance Matrix), "Convex-Concave-Concave Distributions in Application to CDO Pricing" (Meansquare Error Defined on Matrix of Scenarios, Variance), "Spline Approximation" (L2 Norm Defined on Matrix of Scenarios), "Support Vector Machines Based on Tail Risk Measures" (Quadratic), "Projection on Polyhedron with Various Norms" (Quadratic).

 

Utilities functions are used for description of decision maker's preferences.

Exponential Utility, Logarithmic Utility, and Power Utility risk functions from Utilities Group are applied to Linear Loss scenario functions having discrete probability distribution.

Utilities Group has special functions with

Example of  functions from Utilities Group is in the following case study:  "Portfolio Optimization with Exponential, Logarithmic, and Linear-Quadratic Utilities" (Exponential Utility, Logarithmic Utility).

 

Error Group of functions may be defined by a matrix of scenarios or a symmetric matrix.

Meansquare Error Defined on Matrix of Scenarios, Koenker and Basset Error, and Rockafellar Error  functions from Error Group are applied to Linear Loss scenario functions having discrete probability distribution.

Error Group has special functions with

Meansquare Error Defined on Expected Matrix of Products  function from Error Group is defined on Symmetric Matrix.

Examples of  functions from Error Group are in the following case studies: "Convex-Concave-Concave Distributions in Application to CDO Pricing" (Meansquare Error Defined on Matrix of Scenarios),  "Support Vector Machines Based on Tail Risk Measures" (Quadratic), "Projection on Polyhedron with Various Norms" (Quadratic), "Style Classification with Quantile Regression" (Koenker and Basset Error), "Estimation of CVaR through Explanatory Factors with Mixed Quantile Regression" (Koenker and Basset Error, Rockafellar Error).

 

Kolmogorov-Smirnov Distance between Two Distributions, CVaR Kolmogorov-Smirnov Distance between Two Distributions, and Average Kolmogorov-Smirnov Distance between Two Distributions  functions from Distance Between Distributions Group apply  risk functions to difference between two distributions. It is supposed that at list one of these distributions is discrete.

Kolmogorov-Smirnov Distance from a Discrete Distribution to a Mixture of Normal Independent Distributions, and CVaR Kolmogorov-Smirnov Distance from a Discrete Distribution to a Mixture of Normal Independent Distributions functions from Distance Between Distributions Group apply  risk functions to difference between mixture of normal distributions and discrete distribution.

 

3. For building Linear and Nonlinear Deterministic functions, as well as Risk functions PSG Matrix is used.

PSG Matrix includes the following special type: Matrix of Scenarios

Matrix of scenarios is used for building random Linear Loss (Gain)  scenario function, which takes a finite number of values (scenarios) with some probabilities. The scenarios should be generated outside of PSG from historical data or with a statistical simulation. Scenarios of the Loss (Gain) Function and probabilities of scenarios are used to build a specific Risk Function. For building Risk Functions defined on  Linear Loss (Gain) .

Large sparse PSG Matrix can be packed with Packed Matrix  format, which stores only nonzero elements.  

 

4. Values of PSG Functions are calculated on some decision vector, which is called Point. PSG Matrix (or several matrices) together with Point completely specify PSG Function.

5. Linear combinations of PSG Functions (Linear, Nonlinear, and Risk Functions) are included in Objective, and Constraints  (see Mathematical Optimization Problem Statement ). A constraint on decision variables is an equality of inequality. Feasible decision vectors have to satisfy constraints.

6. Objective and Constraint are elements of Problem.  Problem may be of two types: Optimization Problem and Calculate Problem. Optimization problem is a minimization or a maximization problem. The minimization (maximization) problem minimizes (maximizes) an Objective function (linear combination of PSG Functions). Optimization Problem may include only one Objective and optionally several Constraints. Optimization Problem is interrelates functions, and constraints to find the best (optimal) decision Point. On Optimal Decision Point  the objective function attains a maximum (or a minimum) under condition that this point satisfies constraints. Calculate Problem calculates Objective,  Constraints and Functions at a specified Point. Problem may be formulated in Full or Short Problem (see Problem Statement Description).

 

An extensive set of documented case studies solved with PSG may be found in www.aorda.com.

6. PSG operates in four environments: Run-File, MATLAB, and C++.

In Shell Environment you can optimize problems with up to 255 decision variables and up to 1,000,000 scenarios.

Other environments can optimize problems with up to 10,000 decision variables and up to 1,000,000 scenarios.
Efficient algorithms for Relative Entropy are available (up to 1,000,000 decision variables and 10,000 constraints).
In all environments you can calculate values of PSG Functions and their sensitivities.

Below are specific characteristic of every environment: