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riskratioprog

Contents

riskratioprog subroutine maximizes ratio of risk functions with linear constraints. For instance, this subroutine can maximize a "Sharpe-like" ratio of mean access return (linear function) and CVaR deviation.

•	Mathematical Problem Statement

•

Syntax

•	Description

•	Intput Arguments

•	Output Arguments

•

Options

•

Examples

•	Case Studies with riskratioprog

•

See also

•	List of PSG functions for riskratioprog

Mathematical Problem Statement

The subroutine solver the following optimization problem:

subject to

where

A, and Aeq are matrices;

d, and x are column vectors;

b, beq, lb, and ub are column vectors or scalars.

risk1 and risk2 are PSG risk functions or PSG deterministic functions (see List of risk functions for riskratioprog).

Syntax

xout = riskratioprog (risk1,risk2, w1, H1, c1, p1, w2, H2, c2, p2, d, A, b)

xout = riskratioprog (risk1,risk2, w1, H1, c1, p1, w2, H2, c2, p2, d, A, b, Aeq, beq)

xout = riskratioprog (risk1,risk2, w1, H1, c1, p1, w2, H2, c2, p2, d, A, b, Aeq, beq, lb, ub)

xout = riskratioprog (risk1,risk2, w1, H1, c1, p1, w2, H2, c2, p2, d, A, b, Aeq, beq, lb, ub, x0)

xout = riskratioprog (risk1,risk2, w1, H1, c1, p1, w2, H2, c2, p2, d, A, b, Aeq, beq, lb, ub, x0, options)

[xout, fval] = riskratioprog (…)

[xout, fval, status] = riskratioprog (…)

[xout, fval, status, output] = riskratioprog (…)

Remarks

Omitted parameters should be substituted by square brackets “[]”.

Description

xout = riskratioprog (risk1,risk2, w1, H1, c1, p1, w2, H2, c2, p2, d, A, b) returns a vector x that maximizes subject to . The PSG functions risk1 and risk2 are specified by w1, H1, c1, p1, w2, H2, c2, and p2.

xout = riskratioprog (risk1,risk2, w1, H1, c1, p1, w2, H2, c2, p2, d, A, b, Aeq, beq) solves the problem additionally satisfying the equality constraints .

xout = riskratioprog (risk1,risk2, w1, H1, c1, p1, w2, H2, c2, p2, d, A, b, Aeq, beq, lb, ub) solves the problem with bounds on decision variable x, so that the solution is in the range .

xout = riskratioprog (risk1,risk2, w1, H1, c1, p1, w2, H2, c2, p2, d, A, b, Aeq, beq, lb, ub, x0) sets the starting point x0 (see Initial Point).

xout = riskratioprog (risk1,risk2, w1, H1, c1, p1, w2, H2, c2, p2, d, A, b, Aeq, beq, lb, ub, x0, options) maximizes with the optimization options specified in the structure options.

[xout, fval] = riskratioprog (…) returns the value of the objective function on point x: . The PSG functions risk1, and risk2 are specified by w1, H1, c1, p1, w2, H2, c2, and p2.

[xout, fval, status] = riskratioprog (…) returns a problem status ={ optimal | feasible | unbounded | infeasible } describing the exit status of riskratioprog.

[xout, fval, status, output] = riskratioprog (…) returns a structure output containing information about the optimization with riskratioprog.

Intput Arguments

risk1	name of PSG function (see List of risk functions for riskratioprog);
risk2	name of PSG function (see List of risk functions for riskratioprog);
w1	parameter (real) of the PSG function risk1. If parameter is not present, then substitute parameter by square brackets “[]” );
w2	parameter (real) of the PSG function risk2. If parameter is not present, then substitute parameter by square brackets “[]” );
H1	matrix for PSG function risk1;
H2	matrix for PSG function risk2
c1	vector of benchmark for PSG function risk1. If the benchmark is not used, then substitute it by square brackets “[]”;
c2	vector of benchmark for PSG function risk2. If the benchmark is not used, then substitute it by square brackets “[]”;
p1	vector of probabilities for PSG function risk1. If all probabilities are equal, you can substitute p1 by square brackets “[]”);
p2	vector of probabilities for PSG function risk2. If all probabilities are equal, you can substitute p2 by square brackets “[]”);
d	column vectors for linear component of objective;
A	matrices for linear inequality constraint;
b	vectors or scalars for linear inequality constraint;
Aeq	matrices for linear equality constraint;
beq	vectors for linear equality constraint;
lb	vectors of lower bounds;
ub	vectors of upper bounds;
x0	initial point for x (see Initial Point);
options	structure that contains optimization options (see Options);

Output Arguments

Strucure output contains information about the optimization. The fields of the structure are:

xout	optimal point;
fval	optimal value of objective function ;
frval1	optimal values of PSG functions risk1;
frval2	optimal values of PSG functions defined in risk constraint risk2;
fdval	optimal values of linear component of objective;
fAval	optimal values of left hand sides of linear inequality constraints;
rAval	residual of linear inequality constraints;
fAeqval	optimal values of left hand sides of linear equality constraints;
rAeqval	residual of linear equality constraints;
data_loading_time	data loading time;
preprocessing_time	preprocessing time;
solving_time	solving time.

Options

Structure options contains optimization options used by riskratioprog. The fields of the structure are:

Solver	choose optimization PSG solvers from a list: VAN, CAR, BULDOZER, TANK (for more details see Optimization Solvers);
Precision	number of digits that solver tries to obtain in objective and constraints;
Time_Limit	time in seconds restricting the duration of the optimization process;
Stages	number of stages of the optimization process (this parameter should be specified for VaR, Probability, and Cardinality groups of functions).
Linearization	controls internal representation of risk function, which can speed up the optimization process (used with CAR and TANK solvers). 'On' or 'Off' (default) (see Linearizing);
Path_to_Export_to_Text	string with path to the folder for storing problem in General (Text) Format of PSG). If Path_to_Export_to_Text=‘’ (default) than the problem is not exported in General (Text) Format of PSG;
Path_to_Export_to_MAT	string with path to the folder for storing problem as mat-file. If Path_to_Export_to_MAT=‘’ (default) than the problem is not exported as mat-file. This option is intended for converting the problem to the PSG MATLAB Data Structure;
Display	key specifying if messages that may appear when you run this subroutine should be suppressed (Display = 'Off') or not (by the default: Display ='On') (see PSG Messages);
Warnings	key specifying if warnings that may appear when you run this subroutine should be suppressed (Warnings = 'On') or not (by the default: Warnings ='Off'); (see PSG Messages)
Types	specifies the variables types of a problem. If Types is defined as column-vector, it should include as many components as number of variables Problem includes. The components of column-vector can possess the values "0" - for variables of type real, or 1- for variables of type boolean, or 2 - for variables of type integer. If Types is defined as one number (0, or 1, or 2) than all variables types real, or boolean, or integer respectively.

Example

Find maximizing

subject to

First, consider parameters, specifying the PSG function risk1 from mathematical problem statement of riskratioprog subroutine:

risk1='pm_pen';

w1=0.01;

Matrix:

benchmark vector is

p1=[];

Next, consider parameters, specifying the risk function risk2.

Since Average_Gain = - Average_Loss, then

risk2 = 'avg';

H2=H1;

c2=c1;

p2=[];

Run the following example from ./Aorda/PSG/MATLAB/Examples/Problems/Example_riskratioprog.m or type the next PSG problem in MATLAB:

Define input data:

H1 = [0.11 0.04 -0.12 0.01 -0.05; ...

-0.01 -0.1 0.03 0.21 0.01; ...

0.05 0.2 0.07 -0.03 -0.12; ...

-0.1 0.14 0.04 -0.1 0.09; ...

0.08 -0.02 -0.2 0.07 -0.01; ...

0.17 0.3 -0.12 -0.02 0.08

];

c1 = [0.06; -0.1; 0.12; 0.01; -0.06; 0.11];

A = [1 0 1 0 0; 0 1 0 1 1];

b = [0.6; 0.5];

Aeq = [1.0 1.0 1.0 1.0 1.0];

beq = 1.0;

lb = [0.0; 0.0; 0.3; 0.0; 0.0];

ub = [0.5; 0.5; 0.5; 0.5; 0.5];

Next, run the subroutine :

[x, fval, status, output] = riskratioprog('pm_pen', 'avg', 0.01, H1, c1, [], [], H1, c1, [], [], A, b, Aeq, beq, lb, ub)

You get the following output:

Optimal point:

xout =

0.2000

0.5000

0.3000

Objective value on the optimal point:

fval =

2.5000

Status of the solved optimization problem:

status =

optimal

Full report for solution of the optimization problem:

output =

xout: [5x1 double]

fval: 2.5000

fdval: []

frval1: 0.0073

frval2: -0.0183

fAval: [2x1 double]

rAval: -0.1000

fAeqval: 1

rAeqval: 0

data_loading_time: 0

preprocessing_time: 0

solving_time: 0

Case Studies with riskratioprog

•	Omega Portfolio Rebalancing.

List of risk functions for riskratioprog

Subroutine riskconstrprog works only with PSG functions described in this section.

Remarks

riskratioprog can be used when:

•	there should exists at least one feasible point on which numerator in the objective must be strongly positive;

•	denominator in the objective must be non-negative on all feasible points that satisfy additional inequality "numerator >=1".

Full Name	Brief Name	Short Description
Average Loss	avg	Average Loss obtained by averaging Linear Loss scenarios, i.e., it is a linear function with coefficients obtained by averaging coefficients of Linear Loss scenarios.
Average Gain	avg_g	Average Gain obtained by averaging -(Linear Loss ) scenarios, i.e., it is a linear function with coefficients obtained by averaging coefficients of -(Linear Loss) scenarios.
CDaR	cdar_dev	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 .
CDaR for Gain	cdar_dev_g	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 for Gain = 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 .
CVaR Component Negative	cvar_comp_neg	Average of the largest (1-α)% of components of a -( vector), where 0≤α≤1
CVaR Component Positive	cvar_comp_pos	Average of the largest (1-α)% of components of a vector, where 0≤α≤1
CVaR Deviation	cvar_dev	Conditional Value-at-Risk for (Linear Loss) - (Average over Linear Loss scenarios) , i.e., the average of largest (1-α)% of (Linear Loss) - (Average over Linear Loss scenarios) scenarios.
CVaR Deviation for Gain	cvar_dev_g	Conditional Value-at-Risk for -(Linear Loss ) + (Average over scenarios Linear Loss) , i.e., the average of largest (1-α)% of - (Linear Loss) + (Average over scenarios Linear Loss) scenarios.
CVaR	cvar_risk	Conditional Value-at-Risk for Linear Loss scenarios (also called Expected Shortfall and Tail VaR), i.e., the average of largest (1-α)% of Losses.
CVaR for Gain	cvar_risk_g	Conditional Value-at-Risk for -(Linear Loss ) scenarios (also called Expected Shortfall and Tail VaR), i.e., the average of largest (1-α)% of -(Losses).
Drawdown Average	drawdown_dev_avg	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 )). Drawdown Deviation = average of components of the vector (d(1), ..., d(J)) .
Drawdown Average for Gain	drawdown_dev_avg_g	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 )). Drawdown Average for Gain = average of components of the vector (-d(1), ...,- d(J)).
Drawdown Maximum	drawdown_dev_max	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 )). Drawdown Maximum = Maximum Component Positive of vector (d(1), ..., d(J)) = largest component of the vector (d(1), ..., d(J)).
Drawdown Maximum for Gain	drawdown_dev_max_g	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 )). Drawdown Maximum for Gain = Maximum Component Positive of vector (-d(1), ...,- d(J)) = largest component of the vector (-d(1), ..., -d(J)).
Linear	linear	Linear function with many variables
Maximum Component Negative	max_comp_neg	Maximal (largest) component of -(vector)
Maximum Component Positive	max_comp_pos	Maximal (largest) component of a vector
Maximum Deviation	max_dev	Maximum of ((Linear Loss ) - (Average over Linear Loss scenarios)) scenarios.
Maximum Deviation for Gain	max_dev_g	Maximum of (-(Linear Loss ) + (Average over Linear Loss scenarios)) scenarios..
Maximum	max_risk	Maximum of Linear Loss scenarios.
Maximum for Gain	max_risk_g	Maximum of -(Linear Loss ) scenarios.
Mean Absolute Deviation	meanabs_dev	Mean Absolute Deviation for Linear Loss scenarios.
Mean Absolute Error	meanabs_pen	Mean Absolute for Linear Loss scenarios function. Calculated by averaging over scenarios the absolute values of losses .
Mean Absolute Risk	meanabs_risk	Mean Absolute for Linear Loss scenarios. (Mean Absolute) =Average Loss + Mean Absolute Deviation.
Mean Absolute Risk for Gain	meanabs_risk_g	Mean Absolute for Gain for Linear Loss scenarios. (Mean Absolute for Gain) = -Average Loss + Mean Absolute Deviation.
Partial Moment Deviation	pm_dev	Expected access of ( (Loss ) - (Average Loss )) over some fixed threshold.
Partial Moment Gain Deviation	pm_dev_g	Expected access of (- (Loss ) + (Average Loss )) over some fixed threshold.
Partial Moment Deviation	pm_pen	Expected access of ( (Loss ) - (Average Loss )) over some fixed threshold.
Partial Moment for Gain	pm_pen_g	Expected access of -( Loss ) over some fixed threshold.
L1 Norm	polynom_abs	Sum of absolute values of components of a vector, i.e., L1 norm of a vector.
Standard Deviation	st_dev	Standard Deviation of Linear Loss scenarios calculated with Matrix of Scenarios.
Root Mean Squared Error	st_pen	Root Squared Error of Linear Loss scenarios calculated with Matrix of Scenarios or Expected Matrix of Products. By definition, it is an average of squared loss scenarios.
Standard Risk	st_risk	(Standard Deviation of Linear Loss scenarios)+(Average of Linear Loss scenarios). It is calculated with Matrix of Scenarios.
Standard Gain	st_risk_g	(Standard Deviation of Linear Loss scenarios)-(Average of Linear Loss scenarios). It is calculated with Matrix of Scenarios.
VaR Component Negative	var_comp_neg	αI -th value in the ordered ascending sequence of components of a -(I-dimensional vector), where 0≤α≤1; e.g., α=0.8, I=100, we order components of the -(vector) and take the element number αI = 80
Maximum Component Positive	var_comp_pos	Maximal (largest) component of a vector .
VaR Deviation	var_dev	Value-at-Risk for (Linear Loss ) - (Average over Linear Loss scenarios) , i.e., α% percentile of (Linear Loss) - (Average over Linear Loss scenarios) scenarios.
VaR Deviation for Gain	var_dev_g	Value-at-Risk for -(Linear Loss ) + (Average over Linear Loss scenarios) , i.e., α% percentile of -(Linear Loss) + (Average over Linear Loss scenarios) scenarios.
VaR	var_risk	Value-at-Risk for Linear Loss scenarios, i.e., α% percentile of Linear Loss scenarios.
VaR for Gain	var_risk_g	Value-at-Risk for -(Linear Loss ) scenarios, i.e., α% percentile of -(Linear Loss) scenarios.