riskprog

risk

string with description of PSG function (objective include only one PSG function) or cell array with coefficients (real) and PSG functions (string) of the form: {coefficient 1, 'function 1', ... , coefficient K, 'function K'}. See, List of PSG functions for riskprog;

w

parameter (real) of the PSG function (for one PSG function in objective) or cell array of parameters (for linear combination of PSG functions; order of parameters are the same as the order of functions in risk). If parameter is not present, then substitute parameter by square brackets “[]” );

H

matrix for one PSG function in objective risk or cell array of matrices for linear combination of PSG functions risk (order of matrix are the same as the order of functions in risk);

c

vector of benchmark for one PSG function in objective risk or cell array of vectors of benchmarks for linear combination of PSG functions risk (order of vectors are the same as the order of functions in risk). If the benchmark is not used, then substitute it by square brackets “[]”;

p

vector of probabilities for one PSG function in objective risk or cell array of vectors of probabilities for linear combination of PSG functions risk (order of vectors are the same as the order of functions in risk). If all probabilities are equal, you can substitute p by square brackets “[]”);

d

column vectors for linear component of objective;

A

matrix for linear inequality constraint;

b

vector or scalars for linear inequality constraint;

Aeq

matrix for linear equality constraint;

beq

vector for linear equality constraint;

lb

vector of lower bounds for x;

ub

vector of upper bounds for x;

x0

initial point for x (see Initial Point);

options

structure that contains optimization options (see Options).

xout

optimal point;

fval

optimal value of objective function ;

frval

optimal values of PSG functions defined in objective;

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.

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.

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.

Buyin

buyin

For a finite dimension vector, buyin equals the number components satisfying two conditions: 1) absolute value of a component exceeds  threshold_1; 2) if a component is positive, then its absolute value is below  threshold_2, otherwise its absolute value is  below  threshold_3

Buyin Negative

buyin_neg

Number of components of -(vector) exceeding  threshold_1 and below  threshold_2

Buyin Positive

buyin_pos

Number of components of a vector exceeding  threshold_1 and below  threshold_2

Cardinality

cardn

Number of absolute values of components of a vector exceeding some threshold

Cardinality Negative

cardn_neg

Number of components of -(vector) exceeding some threshold

Cardinality Positive

cardn_pos

Number of components of a vector exceeding some threshold

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 Absolute

cvar_comp_abs

Scaled CVaR Norm = Average of the largest  (1-α)% of absolute values of components of a vector, 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)).

Relative Entropy

entropyr

Relative entropy where probabilities are variables.

Exponential Utility

exp_eut

Exponential Utility function for Linear Loss scenarios calculated with Matrix of Scenarios.

Fixed Charge

fxchg

Sum of fixed changes for absolute values of components of a vector exceeding some thresholds

Fixed Charge Negative

fxchg_neg

Sum of fixed changes for components of -(vector) exceeding some thresholds

Fixed Charge Positive

fxchg_pos

Sum of fixed changes for components of a vector exceeding some thresholds

Koenker and Basset Error

kb_err

Koenker and Bassett error of Linear Loss scenarios calculated with Matrix of Scenarios. Used for estimation of Value-at-Risk (i.e., percentile) in Linear Regression.

Linear

linear

Linear function with many variables

Logarithmic Utility

log_eut

Log Utility function for Linear Loss scenarios calculated with Matrix of Scenarios.

Logarithms Sum

log_sum

Linear combination of logarithms of components of a vector

Logarithms Exponents Sum

logexp_sum

Likelihood function in Logistic Regression

Maximum Component Absolute

max_comp_abs

L-infinity norm of a vector, i.e., the largest absolute value of components of the vector

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.

Mean Square Error

meansquare

Mean Square of Linear Loss scenarios calculated with Matrix of Scenarios or Expected Matrix of Products.

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.

Partial Moment Two Deviation for Loss

pm2_dev

Expected  squared access of  ((Loss ) - (Average Loss )) over some fixed threshold.

Partial Moment Two Deviation for Gain

pm2_dev_g

Expected  squared access of  (-(Loss ) + (Average Loss )) over some fixed threshold.

Partial Moment Two

pm2_pen

Expected  squared Linear Loss in access of of some fixed threshold.

Partial Moment Twofor Gain

pm2_pen_g

Expected  squared Linear -(Loss ) in access of of some fixed threshold.

Partial Moment Two Normal Dependent

pm2_pen_nd

Expected  squared Linear Loss in access of of some fixed threshold for the Loss with mutually dependent normally distributed random coefficients.

Partial Moment Two for Gain Normal Dependent

pm2_pen_nd_g

Expected  squared Linear -(Loss ) in access of of some fixed threshold for the Loss with mutually dependent normally distributed random coefficients.

Partial Moment Two  Normal Independent

pm2_pen_ni

Expected  squared Linear Loss in access of of some fixed threshold for Loss with independent normally distributed random coefficients.

Partial Moment Two for Gain Normal Independent

pm2_pen_ni_g

Expected  squared Linear -(Loss)  in access of of some fixed threshold for Loss with independent normally distributed random coefficients.

L1 Norm

polynom_abs

Sum of absolute values of components of a vector, i.e., L1 norm of a vector.

Power Utility

pow_eut

Power Utility function for Linear Loss scenarios calculated with Matrix of Scenarios.

Probability of Exceedance Deviation

pr_dev

Probability that (Loss)-(Average Loss) exceeds some fixed threshold.

Probability of Exceedance Deviation for Gain

pr_dev_g

Probability that -(Loss)+(Average Loss) exceeds some fixed threshold.

Probability of Exceedance

pr_pen

Probability that Linear Loss  exceeds some fixed threshold.

Probability of Exceedance for Gain

pr_pen_g

Probability that Linear -(Loss ) exceeds some fixed threshold.

Quadratic Function

quadratic

Quadratic function

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

α*I -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 α*I = 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.

Variance

variance

Variance of Linear Loss scenarios calculated with Matrix of Scenarios.