How Stochastic Optimization Works

The CVaR of a portfolio is a conditional expectation. (For the definition of the CVaR function, see Risk Proxy.) Therefore, the CVaR portfolio optimization problem is a stochastic optimization problem. Given a sample of scenarios, the conditional expectation that defines the sample CVaR of the portfolio can be expressed as a finite sum, a weighted average of losses. The weights of the losses depend on their relative magnitude; for a confidence level α, only the worst (1 − α) x 100% losses get a positive weight. As a function of the portfolio weights, the CVaR of the portfolio is a convex function (see [48], [49] Rockafellar & Uryasev at Portfolio Optimization). It is also a nonsmooth function, but its edges are less sharp as the sample size increases.

There are reformulations of the CVaR portfolio optimization problem (see [48], [49] at Rockafellar & Uryasev) that result in a linear programming problem, which can be solved either with standard linear programming techniques or with stochastic programming solvers. The PortfolioCVaR object, however, does not reformulate the problem in such a manner. The PortfolioCVaR object computes the CVaR as a nonlinear function. The convexity of the CVaR, as a function of the portfolio weights and the dull edges when the number of scenarios is large, make the CVaR portfolio optimization problem tractable, in practice, for certain nonlinear programming solvers, such as fmincon from Optimization Toolbox™. The problem can also be solved using a cutting-plane method (see Kelley [45] at Portfolio Optimization). For more information, see Algorithms section of setSolver. To learn more about the workflow when using PortfolioCVaR objects, see PortfolioCVaR Object Workflow.

What Are Scenarios?

Since conditional value-at-risk portfolio optimization works with scenarios of asset returns to perform the optimization, several ways exist to specify and simulate scenarios. In many applications with CVaR portfolio optimization, asset returns may have distinctly nonnormal probability distributions with either multiple modes, binning of returns, truncation of distributions, and so forth. In other applications, asset returns are modeled as the result of various simulation methods that might include Monte-Carlo simulation, quasi-random simulation, and so forth. Often, the underlying probability distribution for risk factors may be multivariate normal but the resultant transformations are sufficiently nonlinear to result in distinctively nonnormal asset returns.

For example, this occurs with bonds and derivatives. In the case of bonds with a nonzero probability of default, such scenarios would likely include asset returns that are −100% to indicate default and some values slightly greater than −100% to indicate recovery rates.

Although the PortfolioCVaR object has functions to simulate multivariate normal scenarios from either data or moments (simulateNormalScenariosByData and simulateNormalScenariosByMoments), the usual approach is to specify scenarios directly from your own simulation functions. These scenarios are entered directly as a matrix with a sample for all assets across each row of the matrix and with samples for an asset down each column of the matrix. The architecture of the CVaR portfolio optimization tools references the scenarios through a function handle so scenarios that have been set cannot be accessed directly as a property of the PortfolioCVaR object.

Setting Scenarios Using the PortfolioCVaR Function

Suppose that you have a matrix of scenarios in the AssetScenarios variable. The scenarios are set through the PortfolioCVaR object with:

m = [ 0.05; 0.1; 0.12; 0.18 ];
C = [ 0.0064 0.00408 0.00192 0;
0.00408 0.0289 0.0204 0.0119;
0.00192 0.0204 0.0576 0.0336;
0 0.0119 0.0336 0.1225 ];

m = m/12;
C = C/12;

AssetScenarios = mvnrnd(m, C, 20000);

p = PortfolioCVaR('Scenarios', AssetScenarios);

disp(p.NumAssets)
disp(p.NumScenarios)

4

20000

Notice that the PortfolioCVaR object determines and fixes the number of assets in NumAssets and the number of scenarios in NumScenarios based on the scenario’s matrix. You can change the number of scenarios by calling the PortfolioCVaR object with a different scenario matrix. However, once the NumAssets property has been set in the object, you cannot enter a scenario matrix with a different number of assets. The getScenarios function lets you recover scenarios from a PortfolioCVaR object. You can also obtain the mean and covariance of your scenarios using estimateScenarioMoments.

Although not recommended for the casual user, an alternative way exists to recover scenarios by working with the function handle that points to scenarios in the PortfolioCVaR object. To access some or all the scenarios from a PortfolioCVaR object, the hidden property localScenarioHandle is a function handle that points to a function to obtain scenarios that have already been set. To get scenarios directly from a PortfolioCVaR object p, use

scenarios = p.localScenarioHandle([], []);

scenarios = p.localScenarioHandle(startrow, endrow);

Setting Scenarios Using the setScenarios Function

You can also set scenarios using setScenarios. For example, given the mean and covariance of asset returns in the variables m and C, the asset moment properties can be set:

m = [ 0.05; 0.1; 0.12; 0.18 ];
C = [ 0.0064 0.00408 0.00192 0;
0.00408 0.0289 0.0204 0.0119;
0.00192 0.0204 0.0576 0.0336;
0 0.0119 0.0336 0.1225 ];

m = m/12;
C = C/12;

AssetScenarios = mvnrnd(m, C, 20000);

p = PortfolioCVaR;
p = setScenarios(p, AssetScenarios);

disp(p.NumAssets)
disp(p.NumScenarios)

4

20000

Estimating the Mean and Covariance of Scenarios

The estimateScenarioMoments function obtains estimates for the mean and covariance of scenarios in a PortfolioCVaR object.

m = [ 0.05; 0.1; 0.12; 0.18 ];
C = [ 0.0064 0.00408 0.00192 0;
0.00408 0.0289 0.0204 0.0119;
0.00192 0.0204 0.0576 0.0336;
0 0.0119 0.0336 0.1225 ];

m = m/12;
C = C/12;

AssetScenarios = mvnrnd(m, C, 20000);

p = PortfolioCVaR;
p = setScenarios(p, AssetScenarios);
[mean, covar] = estimateScenarioMoments(p)

mean =

    0.0043
    0.0085
    0.0098
    0.0153


covar =

    0.0005    0.0003    0.0002    0.0000
    0.0003    0.0024    0.0017    0.0010
    0.0002    0.0017    0.0049    0.0029
    0.0000    0.0010    0.0029    0.0102

Simulating Normal Scenarios

As a convenience, the two functions (simulateNormalScenariosByData and simulateNormalScenariosByMoments) exist to simulate scenarios from data or moments under an assumption that they are distributed as multivariate normal random asset returns.

Simulating Normal Scenarios from Returns or Prices

Given either return or price data, use the function simulateNormalScenariosByData to simulate multivariate normal scenarios. Either returns or prices are stored as matrices with samples going down the rows and assets going across the columns. In addition, returns or prices can be stored in a table or timetable (see Simulating Normal Scenarios from Time Series Data). To illustrate using simulateNormalScenariosByData, generate random samples of 120 observations of asset returns for four assets from the mean and covariance of asset returns in the variables m and C with portsim. The default behavior of portsim creates simulated data with estimated mean and covariance identical to the input moments m and C. In addition to a return series created by portsim in the variable X, a price series is created in the variable Y:

m = [ 0.0042; 0.0083; 0.01; 0.15 ];
C = [ 0.005333 0.00034 0.00016 0;
0.00034 0.002408 0.0017 0.000992;
0.00016 0.0017 0.0048 0.0028;
0 0.000992 0.0028 0.010208 ];

X = portsim(m', C, 120);
Y = ret2tick(X);

Given asset returns and prices in variables X and Y from above, this sequence of examples demonstrates equivalent ways to simulate multivariate normal scenarios for the PortfolioCVaR object. Assume a PortfolioCVaR object created in p that uses the asset returns in X uses simulateNormalScenariosByData:

p = PortfolioCVaR;
p = simulateNormalScenariosByData(p, X, 20000);

[passetmean, passetcovar] = estimateScenarioMoments(p)

passetmean =

    0.0043
    0.0083
    0.0102
    0.1507


passetcovar =

    0.0053    0.0003    0.0002    0.0000
    0.0003    0.0024    0.0017    0.0010
    0.0002    0.0017    0.0049    0.0028
    0.0000    0.0010    0.0028    0.0101

The default behavior of simulateNormalScenariosByData is to work with asset returns. If, instead, you have asset prices as in the variable Y, simulateNormalScenariosByData accepts a name-value pair argument name 'DataFormat' with a corresponding value set to 'prices' to indicate that the input to the function is in the form of asset prices and not returns (the default value for the 'DataFormat' argument is 'returns'). This example simulates scenarios with the asset price data in Y for the PortfolioCVaR object q:

p = PortfolioCVaR;
p = simulateNormalScenariosByData(p, Y, 20000, 'dataformat', 'prices');

[passetmean, passetcovar] = estimateScenarioMoments(p)

passetmean =

    0.0043
    0.0084
    0.0094
    0.1490


passetcovar =

    0.0054    0.0004    0.0001   -0.0000
    0.0004    0.0024    0.0016    0.0009
    0.0001    0.0016    0.0048    0.0028
   -0.0000    0.0009    0.0028    0.0100

Simulating Normal Scenarios with Missing Data

Often when working with multiple assets, you have missing data indicated by NaN values in your return or price data. Although Multivariate Normal Regression goes into detail about regression with missing data, the simulateNormalScenariosByData function has a name-value pair argument name 'MissingData' that indicates with a Boolean value whether to use the missing data capabilities of Financial Toolbox™. The default value for 'MissingData' is false which removes all samples with NaN values. If, however, 'MissingData' is set to true, simulateNormalScenariosByData uses the ECM algorithm to estimate asset moments. This example shows how this works on price data with missing values:

m = [ 0.0042; 0.0083; 0.01; 0.15 ];
C = [ 0.005333 0.00034 0.00016 0;
0.00034 0.002408 0.0017 0.000992;
0.00016 0.0017 0.0048 0.0028;
0 0.000992 0.0028 0.010208 ];

X = portsim(m', C, 120);
Y = ret2tick(X);
Y(1:20,1) = NaN;
Y(1:12,4) = NaN;

Notice that the prices above in Y have missing values in the first and fourth series.

p = PortfolioCVaR;
p = simulateNormalScenariosByData(p, Y, 20000, 'dataformat', 'prices');

q = PortfolioCVaR;
q = simulateNormalScenariosByData(q, Y, 20000, 'dataformat', 'prices', 'missingdata', true);

[passetmean, passetcovar] = estimateScenarioMoments(p)
[qassetmean, qassetcovar] = estimateScenarioMoments(q)

passetmean =

    0.0020
    0.0074
    0.0078
    0.1476


passetcovar =

    0.0055    0.0003   -0.0001   -0.0003
    0.0003    0.0024    0.0019    0.0012
   -0.0001    0.0019    0.0050    0.0028
   -0.0003    0.0012    0.0028    0.0101


qassetmean =

    0.0024
    0.0085
    0.0106
    0.1482


qassetcovar =

    0.0071    0.0004   -0.0001   -0.0004
    0.0004    0.0032    0.0022    0.0012
   -0.0001    0.0022    0.0063    0.0034
   -0.0004    0.0012    0.0034    0.0127

Simulating Normal Scenarios from Time Series Data

The simulateNormalScenariosByData function implicitly works with matrices of data or data in a table or timetable object using the same rules for whether the data are returns or prices. To illustrate, use array2timetable to create a timetable for 14 assets from CAPMuniverse and the use the timetable to simulate scenarios for PortfolioCVaR.

load CAPMuniverse
time = datetime(Dates,'ConvertFrom','datenum');
stockTT = array2timetable(Data,'RowTimes',time, 'VariableNames', Assets);
stockTT.Properties
% Notice that GOOG has missing data, because it was not listed before Aug 2004
head(stockTT, 5)

ans = 

  TimetableProperties with properties:

             Description: ''
                UserData: []
          DimensionNames: {'Time'  'Variables'}
           VariableNames: {'AAPL'  'AMZN'  'CSCO'  'DELL'  'EBAY'  'GOOG'  'HPQ'  'IBM'  'INTC'  'MSFT'  'ORCL'  'YHOO'  'MARKET'  'CASH'}
    VariableDescriptions: {}
           VariableUnits: {}
      VariableContinuity: []
                RowTimes: [1471×1 datetime]
               StartTime: 03-Jan-2000
              SampleRate: NaN
                TimeStep: NaN
        CustomProperties: No custom properties are set.
      Use addprop and rmprop to modify CustomProperties.

ans =

  5×14 timetable

       Time          AAPL         AMZN         CSCO         DELL         EBAY       GOOG       HPQ          IBM         INTC         MSFT         ORCL         YHOO        MARKET         CASH   
    ___________    _________    _________    _________    _________    _________    ____    _________    _________    _________    _________    _________    _________    _________    __________

    03-Jan-2000     0.088805       0.1742     0.008775    -0.002353      0.12829    NaN       0.03244     0.075368      0.05698    -0.001627     0.054078     0.097784    -0.012143    0.00020522
    04-Jan-2000    -0.084331     -0.08324     -0.05608     -0.08353    -0.093805    NaN     -0.075613    -0.033966    -0.046667    -0.033802      -0.0883    -0.067368     -0.03166    0.00020339
    05-Jan-2000     0.014634     -0.14877    -0.003039     0.070984     0.066875    NaN     -0.006356      0.03516     0.008199     0.010567    -0.052837    -0.073363     0.011443    0.00020376
    06-Jan-2000    -0.086538    -0.060072    -0.016619    -0.038847    -0.012302    NaN     -0.063688    -0.017241     -0.05824    -0.033477    -0.058824     -0.10307     0.011743    0.00020266
    07-Jan-2000     0.047368     0.061013       0.0587    -0.037708    -0.000964    NaN      0.028416    -0.004386      0.04127     0.013091     0.076771      0.10609      0.02393    0.00020157

Use the 'MissingData' option offered by PortfolioCVaR to account for the missing data.

p = PortfolioCVaR;
p = simulateNormalScenariosByData(p, stockTT, 20000 ,'missingdata',true);
[passetmean, passetcovar] = estimateScenarioMoments(p)

passetmean =

    0.0012
    0.0007
   -0.0005
   -0.0000
    0.0016
    0.0043
   -0.0001
    0.0000
    0.0001
   -0.0002
    0.0000
    0.0004
    0.0001
    0.0001


passetcovar =

    0.0013    0.0005    0.0006    0.0005    0.0006    0.0003    0.0005    0.0003    0.0006    0.0004    0.0005    0.0006    0.0002   -0.0000
    0.0005    0.0024    0.0007    0.0005    0.0010    0.0005    0.0005    0.0003    0.0006    0.0004    0.0006    0.0011    0.0002   -0.0000
    0.0006    0.0007    0.0013    0.0006    0.0007    0.0004    0.0006    0.0004    0.0008    0.0005    0.0008    0.0008    0.0002   -0.0000
    0.0005    0.0005    0.0006    0.0009    0.0006    0.0002    0.0005    0.0003    0.0006    0.0004    0.0005    0.0006    0.0002   -0.0000
    0.0006    0.0010    0.0007    0.0006    0.0018    0.0007    0.0005    0.0003    0.0006    0.0005    0.0007    0.0011    0.0002   -0.0000
    0.0003    0.0005    0.0004    0.0002    0.0007    0.0013    0.0002    0.0002    0.0002    0.0002    0.0003    0.0011    0.0001   -0.0000
    0.0005    0.0005    0.0006    0.0005    0.0005    0.0002    0.0010    0.0003    0.0005    0.0003    0.0005    0.0006    0.0002   -0.0000
    0.0003    0.0003    0.0004    0.0003    0.0003    0.0002    0.0003    0.0005    0.0004    0.0002    0.0004    0.0004    0.0002    0.0000
    0.0006    0.0006    0.0008    0.0006    0.0006    0.0002    0.0005    0.0004    0.0011    0.0005    0.0007    0.0007    0.0002   -0.0000
    0.0004    0.0004    0.0005    0.0004    0.0005    0.0002    0.0003    0.0002    0.0005    0.0006    0.0004    0.0005    0.0002   -0.0000
    0.0005    0.0006    0.0008    0.0005    0.0007    0.0003    0.0005    0.0004    0.0007    0.0004    0.0014    0.0007    0.0002   -0.0000
    0.0006    0.0011    0.0008    0.0006    0.0011    0.0011    0.0006    0.0004    0.0007    0.0005    0.0007    0.0020    0.0002   -0.0000
    0.0002    0.0002    0.0002    0.0002    0.0002    0.0001    0.0002    0.0002    0.0002    0.0002    0.0002    0.0002    0.0001   -0.0000
   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000    0.0000   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000    0.0000

Use the name-value input 'DataFormat' to handle return or price data and 'MissingData' to ignore or use samples with missing values. In addition, simulateNormalScenariosByData extracts asset names or identifiers from a table or timetable if the argument 'GetAssetList' is set to true (the default value is false). If the 'GetAssetList' value is true, the identifiers are used to set the AssetList property of the PortfolioCVaR object. Thus, repeating the formation of the PortfolioCVaR object p from the previous example with the 'GetAssetList' flag set to true extracts the column names from the timetable object:

p = simulateNormalScenariosByData(p, stockTT, 20000 ,'missingdata',true, 'GetAssetList', true);
disp(p.AssetList)

 'AAPL'    'AMZN'    'CSCO'    'DELL'    'EBAY'    'GOOG'    'HPQ'    'IBM'    'INTC'    'MSFT'    'ORCL'    'YHOO'    'MARKET'    'CASH'

If you set the'GetAssetList' flag set to true and your input data is in a matrix, simulateNormalScenariosByData uses the default labeling scheme from setAssetList as described in Setting Up a List of Asset Identifiers.

Simulating Normal Scenarios with Mean and Covariance

Given the mean and covariance of asset returns, use the simulateNormalScenariosByMoments function to simulate multivariate normal scenarios. The mean can be either a row or column vector and the covariance matrix must be a symmetric positive-semidefinite matrix. Various rules for scalar expansion apply. To illustrate using simulateNormalScenariosByMoments, start with moments in m and C and generate 20,000 scenarios:

m = [ 0.0042; 0.0083; 0.01; 0.15 ];
C = [ 0.005333 0.00034 0.00016 0;
0.00034 0.002408 0.0017 0.000992;
0.00016 0.0017 0.0048 0.0028;
0 0.000992 0.0028 0.010208 ];

p = PortfolioCVaR;
p = simulateNormalScenariosByMoments(p, m, C, 20000);
[passetmean, passetcovar] = estimateScenarioMoments(p)

passetmean =

    0.0049
    0.0083
    0.0101
    0.1503


passetcovar =

    0.0053    0.0003    0.0002   -0.0000
    0.0003    0.0024    0.0017    0.0010
    0.0002    0.0017    0.0047    0.0028
   -0.0000    0.0010    0.0028    0.0101

simulateNormalScenariosByMoments performs scalar expansion on arguments for the moments of asset returns. If NumAssets has not already been set, a scalar argument is interpreted as a scalar with NumAssets set to 1. simulateNormalScenariosByMoments provides an additional optional argument to specify the number of assets so that scalar expansion works with the correct number of assets. In addition, if either a scalar or vector is input for the covariance of asset returns, a diagonal matrix is formed such that a scalar expands along the diagonal and a vector becomes the diagonal.