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This example explores how to use a multifactor copula model to simulate correlated counterparty defaults. (Click “Read the original text” at the end for the complete code data).
Modeling Correlated Defaults with Copula
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Given the exposure to default risk, default probabilities, and loss given default information, estimate the potential loss of a counterparty portfolio. A Copula object is used for each debtor’s credit with a latent variable model. The latent variables consist of a series of weighted latent credit factors and each debtor’s specific credit factors. The latent variables are mapped to the default or non-default state of each debtor in each scenario based on their default probabilities. The Copula object supports portfolio risk measures, counterparty-level risk contributions, and simulation convergence information.
This example also explores the sensitivity of risk measures to the type of copula used for simulation (Gaussian copula vs. t copula).
Loading and Checking Portfolio Data
The portfolio contains 100 counterparties and their associated credit risk exposures (Exposure at Default – EAD), default probabilities (PD), and loss given default (LGD). Using the Copula object, you can simulate defaults and losses over a fixed time period (e.g., one year).
In this example, each counterparty is mapped to two underlying credit factors using a set of weights. The Weights2F variable is a matrix where each row contains the weights for a single counterparty. The first two columns are the weights for the two credit factors, and the last column is the specific weight for each counterparty. This example also provides the correlation matrix for the two underlying factors.
Loading Portfolio Information
Initialize the object using the portfolio information and factor correlations.
rng('default');
cc = creditDefaultCopula(EAD, PD, LGD, Weights2F, 'FactorCorrelation', FactorCorr2F);
cc.VaRLevel = 0.99;
DISP(cc)
creditDefaultCopula with properties:
FactorCorrelation: [2x2 double]
VaRLevel: 0.9900
PortfolioLosses: []
cc.Portfolio(1:5,:)
ans =
5x5 table
ID EAD PD LGD Weight
__ ______ _________ ____ ____________________
1 121.6270 0.0050 0.350 0.3500 0.65
2 23.2595 0.0601 0.350 0.4500 0.55
3 320.3910 0.1101 0.550 0.1500 0.85
4 43.7534 0.0020 0.250 0.3500 0.75
5 55.7193 0.0601 0.350 0.3500 0.65Simulating the Model and Plotting Potential Losses
Simulate the multifactor model. By default, the Gaussian copula is used. This function internally maps the implemented latent variables to default states and calculates the corresponding losses.
cc = simulate(cc, 1e5);
DISP(cc)
creditDefaultCopula with properties:
FactorCorrelation: [2x2 double]
VaRLevel: 0.9900
PortfolioLosses: [1x100000 double]The function returns risk measures and confidence intervals for the total portfolio loss distribution. The VaRLevel reports the Value at Risk (VaR) and Conditional Value at Risk (CVaR).
[pr, pr_ci] = portfolioRisk(cc);
fprintf('Portfolio risk metrics:\n');
DISP(pr);
fprintf('\n\nConfidence intervals for risk measures:\n');
DISP(pr_ci)
Portfolio risk metrics
EL Std VaR CVaR
______ ______ ______ ______
24.774 23.693 101.57 120.22
Confidence intervals for risk measures:
EL Std VaR CVaR
____________________ ________________ ________________ ________________
24.627 24.92 23.589 23.797 100.65 102.82 119.1 121.35Examine the distribution of portfolio losses. Expected Loss (EL), VaR, and CVaR are marked as vertical lines. The economic capital indicated by the difference between VaR and EL is shown as the shaded area between EL and VaR.
plotline = @(x, color) plot([x x], ylim, 'LineWidth', 2, 'Color', color);
cvarline = plotline(pr.CVaR, 'm');
% Shade the area for expected loss and economic capital.
plotband = @(x, color) patch([x fliplr(x)], [0 0 repmat(max(ylim), 1, 2)],...
color, 'FaceAlpha', 0.15);
elband = plotband([0 pr.EL], 'blue');
ulband = plotband([pr.EL pr.VaR], 'red');
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Identifying Concentration Risk of Counterparties
Use the riskContribution function to find the concentration risk in the portfolio. riskContribution returns each counterparty’s contribution to the portfolio’s EL and CVaR. These additional value contributions are summed with the corresponding total portfolio risk measures.
rc = riskContribution(cc);
% Report the percentage contribution of EL and CVaR.
rc(1:5,:)
ans =
5x5 table
ID EL Std VaR CVaR
__ _________ __________ _______ _________
1 10.0386 0.02495 0.10482 0.12868
2 20.0670 0.036472 0.17378 0.24527
3 1.2527 0.62684 2.0384 2.3103
4 0.0023253 0.00073407 0.0026274
5 0.11766 0.042185 0.27028 0.26223Identifying the Counterparties with the Highest Risk through CVaR Contribution
[rc_sorted, idx] = sortrows(rc, 'CVaR', 'descend');
rc_sorted(1:5,:)
ans =
5x5 table
ID EL Std VaR CVaR
__ _______ ______ ______ ______
1 89 2.261 2.2158 8.1095 9.2257
2 22 1.5672 1.8293 6.275 7.4602
3 66 0.85227 1.4063 6.3827 7.2691
4 16 1.6236 1.5011 5.8949 7.1083Plot the counterparty risk and CVaR contributions. Counterparties with the highest CVaR contributions are plotted in red and orange.
pointSize = 50;
colorVector = rc_sorted.CVaR;
scatter(cc.Portfolio(idx,:).EAD, rc_sorted.CVaR,...
pointSize, colorVector, 'filled')
colormap('jet')
Studying Simulation Convergence with Confidence Bands
Investigate the convergence of the simulation. By default, the CVaR confidence intervals are reported, but optional RiskMeasure parameters support confidence intervals for all risk measures.
cb = confidenceBands(cc);
% Confidence bands are stored in a table.
cb(1:5,:)
ans =
5x4 table
NumScenarios Lower CVaR Upper
______________ ______ ______ ______
1000 113.92 124.76 135.59
2000 111.02 117.74 124.45
3000 113.58 118.97 124.36
4000 113.06 117.44 121.81
5000 114.38 118.99 123.6Plot the confidence intervals to observe the speed of convergence of the estimates.
Obtain specific confidence intervals.
width = (cb.Upper - cb.Lower) ./ cb.CVaR;
plot(cb.NumScenarios, width * 100, 'LineWidth', 2);
% Find the confidence band at
% the 1% (two-sided) range of CVaR.
thresh = 0.02;
scenIdx = find(width <= thresh, 1, 'first');
scenValue = cb.NumScenarios(scenIdx);
widthValue = width(scenIdx);
Comparing Tail Risks of Gaussian and t Copulas
Using a t copula increases the default correlation between counterparties. This leads to a more severe tail distribution of portfolio losses and results in higher potential losses.
cc_t = simulate(cc, 1e5, 'Copula', 't');
pr_t = portfolioRisk(cc_t);
% Understand how portfolio risk changes with t copula.
% Portfolio risk with Gaussian copula:
EL Std VaR CVaR
______ ______ ______ ______
24.774 23.693 101.57 120.22
% Portfolio risk with t copula (dof = 5):
EL Std VaR CVaR
______ ______ ______ ______
24.924 38.982 186.33 251.38Compare the tail losses of each model.
Using a t copula with five degrees of freedom results in significantly higher tail risk measures for VaR and CVaR. The default correlation of t copulas is higher, leading to more instances of multiple counterparties defaulting. The number of degrees of freedom plays a crucial role. For very high degrees of freedom, the results using a t copula are similar to those using a Gaussian copula. For very low degrees of freedom, the results show significant differences. Furthermore, these results emphasize that the likelihood of extreme losses is highly sensitive to the choice of copula and the number of degrees of freedom.
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This article is excerpted from Using Copula for Simulating and Optimizing Market Risk Data VaR Analysis in MATLAB.
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