Scenario Generation for Market Risk Models Using Generative Neural Networks
Abstract
:1. Introduction
 Expand scenario generation by a GAN to a complete market risk calculation for Solvency 2 purposes in insurance companies;
 Compare the results of a GANbased ESG to ESG approaches implemented in regulatoryapproved market risk models in Europe.
2. Background
2.1. Market Risk Calculation under Solvency 2
 The underlying data in the financial market are publicly available and equal for all insurers;
 Market risk forms a major part of the SCR of an insurance company (EIOPA (2021b, p. 22), which states that market risk accounts for 53% of the net solvency capital requirement before diversification benefits; this varies between life (59%) and nonlife (43%) insurers);
 A comprehensive benchmark exercise, called “market and credit risk comparison study” MCRCS conducted by EIOPA, is available for a comparison of the results.
2.2. Introduction to the MCRCS Study
 Assetonly benchmark portfolios: BMP1, BMP2, …, BMP10;
 Liabilityonly benchmark portfolios: L1 and L2;
 Combined portfolios: BMP1+L1, BMP3+L1, BMP7+L1, BMP9+L1, BMP10+L1, BMP1+L2, BMP3+L2, BMP7+L2, BMP9+L2 and BMP10+L2.
2.3. Generative Adversarial Networks
Algorithm 1 Algorithm for GAN training with SGD (stochastic gradient descent) as an optimizer; see Goodfellow et al. (2014, Chapter 4, Algorithm 1) 
The discriminator is trained k times more often than the generator, the dimension of the latent space Z is ${N}^{Z}$, $M\in \mathbb{N}$ is the batch size. All are hyperparameters of the GAN. The learning rates of the SGD algorithm are ${\gamma}_{D}$ and ${\gamma}_{G}$.

3. Results of a GANBased Internal Model
3.1. Workflow of a GANBased Internal Model
3.2. Comparison of GAN Results with the Results of the MCRCS Study
3.3. Comparison on RiskFactor Level
3.4. Comparison on the Portfolio Level
3.5. Comparison of the COVID19 Backtesting Results
3.6. Comparison of Joint Quantile Exceedance Results
3.7. Stability of GAN Results
4. Methodology and Data
4.1. Data Selection
4.2. Data Preparation
4.3. Implementation of a GANBased ESG
 4 layers for discriminator and generator;
 400 neurons per layer in the discriminator and 200 in the generator;
 $k=10$ training iterations for the generator in each discriminator training;
 Batch size is $M=200$;
 Dimension of the latent space is 200, and the distribution of Z is a multivariate normal with mean = 0 and std = $0.02$;
 Initialization of the generator and discriminator using multivariate normal distribution with mean = 0 and std = $0.02$;
 We use LeakyReLu as activation functions except for the output layers, which use sigmoid (for discriminator) and linear (for generator) activation functions. We use the Adam optimizer and regulation technique batch normalization after each hidden layer in the network. The loss function is binary crossentropy.
4.4. Valuation of Financial Instruments and Portfolio Aggregation
 (1).
 Zerocoupon bond valuation
 (2).
 Equity and property instrument valuation
 (3).
 Valuation of the liabilities
 (4).
 Portfolio aggregation
5. Conclusions and Discussion of Results
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A. Table of Risk Factors and Data Sources Used for the MCRCS Study
Asset Class  Subtype  Maturity  Bloomberg Ticker  CQV, 0.5perc.  CQV, 99.5perc. 

Government bond  Austria  5 years  GTATS5Y Govt  9.2%  4.8% 
Austria  10 years  GTATS10Y Govt  4.3%  3.2%  
Belgium  5 years  GTBEF5Y Govt  4.1%  4.0%  
Belgium  10 years  GTBEF10Y Govt  5.5%  5.7%  
Germany  5 years  GTDEM5Y Govt  1.4%  4.8%  
Germany  10 years  GTDEM10Y Govt  3.3%  4.5%  
Spain  5 years  GTESP5Y Govt  10.0%  1.9%  
Spain  10 years  GTESP10Y Govt  2.5%  4.0%  
France  5 years  GTFRF5Y Govt  2.3%  9.5%  
France  10 years  GTFRF10Y Govt  3.6%  4.8%  
Ireland  5 years  GIGB5Y Index  2.7%  4.0%  
Ireland  10 years  GIGB10Y Index  2.2%  7.8%  
Italy  5 years  GTITL5Y Govt  4.0%  8.0%  
Italy  10 years  GTITL10Y Govt  2.0%  2.7%  
Netherlands  5 years  GTNLG5Y Govt  4.8%  1.4%  
Netherlands  10 years  GTNLG10Y Govt  3.1%  6.5%  
Portugal  5 years  GSPT5YR Index  3.1%  3.3%  
UK  5 years  C1105Y Index  5.6%  4.3%  
US  5 years  H15T5Y Index  3.0%  5.3%  
Covered bond  AArated issuer  5 years  C9235Y Index  2.6%  4.3% 
AArated issuer  10 years  C92310Y Index  3.2%  4.3%  
Corporate bond  bond, rating AA  5 years  C6675Y Index  1.9%  4.4% 
bond, rating AA  10 years  C66710Y Index  2.8%  1.6%  
bond, rating A  5 years  C6705Y Index  1.6%  3.2%  
bond, rating A  10 years  C67010Y Index  3.5%  2.5%  
bond, rating BBB  5 years  C6735Y Index  5.5%  5.0%  
bond, rating BBB  10 years  C67310Y Index  4.9%  3.4%  
high yield bonds  5 years  ML HP00 Swap Spread  1.4%  4.4%  
Interest rates, riskfree  EUR  1 year  S0045Z 1Y BLC2 Curncy  1.9%  0.5% 
EUR  3 years  S0045Z 3Y BLC2 Curncy  2.6%  2.0%  
EUR  5 years  S0045Z 5Y BLC2 Curncy  2.1%  1.0%  
EUR  7 years  S0045Z 7Y BLC2 Curncy  3.5%  1.0%  
EUR  10 years  S0045Z 10Y BLC2 Curncy  4.8%  2.6%  
EUR  15 years  S0045Z 15Y BLC2 Curncy  4.4%  3.2%  
EUR  20 years  S0045Z 20Y BLC2 Curncy  3.0%  1.6%  
EUR  25 years  S0045Z 25Y BLC2 Curncy  2.2%  3.3%  
EUR  30 years  S0045Z 30Y BLC2 Curncy  4.6%  2.4%  
EUR  40 years  S0045Z 40Y BLC2 Curncy  4.4%  2.5%  
EUR  50 years  S0045Z 50Y BLC2 Curncy  1.7%  3.2%  
USD  5 years  USSW5 Index  1.4%  2.2%  
GBP  5 years  BPSW5 Index  4.6%  3.3%  
Equity  EuroStoxx50    SX5T Index  6.1%  5.6% 
MSCI Europe    MSDEE15N Index  4.2%  3.0%  
FTSE100    TUKXG Index  4.1%  3.2%  
S&P500    SPTR500N Index  4.7%  6.8%  
Realestate  Europe, commercial    EXUK Index  2.8%  4.5% 
Appendix B. Table of Instruments Used for the MCRCS Study
Instrument  Maturity  Risk Factors Used for Valuation 

EUR riskfree interest rate  1 year  EUR swap rate, 1 year 
EUR riskfree interest rate  3 years  EUR swap rate, 3 years 
EUR riskfree interest rate  5 years  EUR swap rate, 5 years 
EUR riskfree interest rate  7 years  EUR swap rate, 7 years 
EUR riskfree interest rate  10 years  EUR swap rate, 10 years 
EUR riskfree interest rate  15 years  EUR swap rate, 15 years 
EUR riskfree interest rate  20 years  EUR swap rate, 20 years 
EUR riskfree interest rate  25 years  EUR swap rate, 25 years 
EUR riskfree interest rate  30 years  EUR swap rate, 30 years 
EUR riskfree interest rate  40 years  EUR swap rate, 40 years 
EUR riskfree interest rate  50 years  EUR swap rate, 50 years 
EUR riskfree interest rate  60 years  EUR swap rate, 60 years 
Austrian Sovereign bond  5 years  EUR interest rate, 5 years & AT_Spread, 5 years 
Austrian Sovereign bond  10 years  EUR interest rate, 10 years & AT_Spread, 10 years 
Austrian Sovereign bond  20 years  EUR interest rate, 20 years & AT_Spread, 10 years 
Belgium Sovereign bond  5 years  EUR interest rate, 5 years & BE_Spread, 5 years 
Belgium Sovereign bond  10 years  EUR interest rate, 10 years & BE_Spread, 10 years 
Belgium Sovereign bond  20 years  EUR interest rate, 20 years & BE_Spread, 10 years 
German Sovereign bond  5 years  EUR interest rate, 5 years & DE_Spread, 5 years 
German Sovereign bond  10 years  EUR interest rate, 10 years & DE_Spread, 10 years 
German Sovereign bond  20 years  EUR interest rate, 20 years & DE_Spread, 10 years 
Spain Sovereign bond  5 years  EUR interest rate, 5 years & ES_Spread, 5 years 
Spain Sovereign bond  10 years  EUR interest rate, 10 years & ES_Spread, 10 years 
Spain Sovereign bond  20 years  EUR interest rate, 20 years & ES_Spread, 10 years 
France Sovereign bond  5 years  EUR interest rate, 5 years & FR_Spread, 5 years 
France Sovereign bond  10 years  EUR interest rate, 10 years & FR_Spread, 10 years 
France Sovereign bond  20 years  EUR interest rate, 20 years & FR_Spread, 10 years 
Ireland Sovereign bond  5 years  EUR interest rate, 5 years & IE_Spread, 5 years 
Ireland Sovereign bond  10 years  EUR interest rate, 10 years & IE_Spread, 10 years 
Ireland Sovereign bond  20 years  EUR interest rate, 20 years & IE_Spread, 10 years 
Italia Sovereign bond  5 years  EUR interest rate, 5 years & IT_Spread, 5 years 
Italia Sovereign bond  10 years  EUR interest rate, 10 years & IT_Spread, 10 years 
Italia Sovereign bond  20 years  EUR interest rate, 20 years & IT_Spread, 10 years 
Netherlands Sovereign bond  5 years  EUR interest rate, 5 years & NE_Spread, 5 years 
Netherlands Sovereign bond  10 years  EUR interest rate, 10 years & NE_Spread, 10 years 
Netherlands Sovereign bond  20 years  EUR interest rate, 20 years & NE_Spread, 10 years 
Portugal Sovereign bond  5 years  EUR interest rate, 5 years & PT_Spread, 5 years 
UK Sovereign bond  5 years  GBP interest rate, 5 years & UK_Spread, 5 years 
US Sovereign bond  5 years  USD interest rate, 5 years & US_Spread, 5 years 
Bond issued by ESM  10 years  EUR interest rate, 10 years & DE_Spread, 10 years 
Covered bond rated AAA  5 years  EUR interest rate, 5 years & COV_Spread, 5 years 
Covered bond rated AAA  10 years  EUR interest rate, 10 years & COV_Spread, 10 years 
Financial bond, rated AAA  5 years  EUR interest rate, 5 years & COV_Spread, 5 years 
Financial bond, rated AAA  10 years  EUR interest rate, 10 years & COV_Spread, 10 years 
Financial bond, rated AA  5 years  EUR interest rate, 5 years & AA_Spread, 5 years 
Financial bond, rated AA  10 years  EUR interest rate, 10 years & AA_Spread, 10 years 
Financial bond, rated A  5 years  EUR interest rate, 5 years & A_Spread, 5 years 
Financial bond, rated A  10 years  EUR interest rate, 10 years & A_Spread, 10 years 
Financial bond, rated BBB  5 years  EUR interest rate, 5 years & BBB_Spread, 5 years 
Financial bond, rated BBB  10 years  EUR interest rate, 10 years & BBB_Spread, 10 years 
Financial bond, rated BB  5 years  EUR interest rate, 5 years & HY_Spread, 5 years 
Financial bond, rated BB  10 years  EUR interest rate, 10 years & HY_Spread, 10 years 
NonFinancial bond, rated AAA  5 years  EUR interest rate, 5 years & COV_Spread, 5 years 
NonFinancial bond, rated AAA  10 years  EUR interest rate, 10 years & COV_Spread, 10 years 
NonFinancial bond, rated AA  5 years  EUR interest rate, 5 years & AA_Spread, 5 years 
NonFinancial bond, rated AA  10 years  EUR interest rate, 10 years & AA_Spread, 10 years 
NonFinancial bond, rated A  5 years  EUR interest rate, 5 years & A_Spread, 5 years 
NonFinancial bond, rated A  10 years  EUR interest rate, 10 years & A_Spread, 10 years 
NonFinancial bond, rated BBB  5 years  EUR interest rate, 5 years & BBB_Spread, 5 years 
NonFinancial bond, rated BBB  10 years  EUR interest rate, 10 years & BBB_Spread, 10 years 
NonFinancial bond, rated BB  5 years  EUR interest rate, 5 years & HY_Spread, 5 years 
NonFinancial bond, rated BB  10 years  EUR interest rate, 10 years & HY_Spread, 10 years 
Equity Index, Eurostoxx 50    Equity Index, Eurostoxx 50 
Equity Index, MSCI Europe    Equity Index, MSCI Europe 
Equity Index, FTSE100    Equity Index, FTSE100 
Equity Index, S&P500    Equity Index, S&P500 
Residential real estate in Netherlands    Diversified European REIT index 
Commercial real estate in France    Diversified European REIT index 
Commercial real estate in Germany    Diversified European REIT index 
Commercial real estate in UK    Diversified European REIT index 
Commercial real estate in Italy    Diversified European REIT index 
 As most participants in the study, we do not distinguish between different types of corporate bond spreads, i.e., financial and nonfinancial corporates are modeled with the same data. As written in EIOPA (2021a, p. 24), this is a simplification used by twothirds of participants.
 As for the required supranational paper issued by ESM (European Stability Mechanism), there is no long time series to be found, we use the approximation of the German spreads instead.
 There is no reliable daily data source for AAA and high yield bonds in Bloomberg. For AAArated bonds, as most participants in the study, we use the covered bond spreads, which are also rated AAA instead; see EIOPA (2021a, p. 24). The most frequent data for high yield bonds that we found can be derived from the Meryll Lynch spread index, which is a weekly index.
 For real estate, there is no direct transactionbased data available at high frequencies. The most frequent direct realestate data are available on a monthly basis. We will therefore use an index representing Real Estate Investment Trusts (REITs) and stocks from Real Estate Holding and Development Companies. As there is no index to be found that is geographyspecific for the real estate holdings in the study, we will use a diversified European index for all real estate instruments.
 As the liquidity of government bonds becomes thin with longer maturities, we use 10year spreads for 10 and 20year bonds in the study.
Appendix C. Optimization of GAN Architecture Using Wasserstein Distances
 The number of layers for generator and discriminator varying between 2, 4, 6 and 8;
 The number of neurons for generator and discriminator varying between 100, 200 and 400.
 Batch size is $M=200$
 $k=10$ training iterations for the generator in each discriminator training;
 Dimension of the latent space is 200, and distribution of Z is multivariate normal with mean = 0 and std = 0.02;
 Initialization of generator and discriminator using multivariate normal distribution with mean = 0 and std = 0.02.
 We use LeakyReLu with $\alpha =0.2$ as activation functions except for the output layers, which use Sigmoid (for discriminator) and linear (for generator) activation functions. Additionally, we apply the regulation technique batch normalization after each of the hidden layers in the network.
 We use the Adam optimizer with the parameters provided in Section 4.3 in Equation (2).
Number of Layers in D/G  2  4  6  8 

2  0.143  0.252  0.636  0.799 
4  1.036  0.118  0.235  0.435 
6  0.947  0.172  0.197  0.281 
8  0.807  0.188  0.171  0.178 
Number of Neurons per Layer in D/G  100  200  400 

100  0.180  0.117  0.125 
200  0.185  0.118  0.108 
400  0.197  0.116  0.124 
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Benchmark Portfolio  WorstCase(P)  ${\mathit{\alpha}}_{\mathit{P},\mathit{GAN}}$ 

BMP1  −2.8%  13.5% 
BMP2  −2.5%  19.2% 
BMP3  −2.8%  15.6% 
BMP4  −2.7%  19.6% 
BMP5  −2.4%  13.3% 
BMP6  −2.8%  10.9% 
BMP7  −7.8%  2.3% 
BMP8  −2.7%  16.2% 
BMP9  −3.7%  21.6% 
BMP10  −6.1%  5.9% 
L1  −15.2%  5.4% 
L2  −4.3%  15.8% 
BMP1+L1  −58.2%  3.5% 
BMP3+L1  −64.5%  0.9% 
BMP7+L1  −67.1%  6.8% 
BMP9+L1  −41.3%  7.1% 
BMP10+L1  −89.9%  3.4% 
BMP1+L2  −30.9%  3.8% 
BMP3+L2  −34.8%  2.5% 
BMP7+L2  −53.4%  3.9% 
BMP9+L2  −26.8%  13.6% 
BMP10+L2  −65.6%  3.6% 
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Flaig, S.; Junike, G. Scenario Generation for Market Risk Models Using Generative Neural Networks. Risks 2022, 10, 199. https://doi.org/10.3390/risks10110199
Flaig S, Junike G. Scenario Generation for Market Risk Models Using Generative Neural Networks. Risks. 2022; 10(11):199. https://doi.org/10.3390/risks10110199
Chicago/Turabian StyleFlaig, Solveig, and Gero Junike. 2022. "Scenario Generation for Market Risk Models Using Generative Neural Networks" Risks 10, no. 11: 199. https://doi.org/10.3390/risks10110199