Andrei Iulian Andreescu Application of Value-at-Risk Methods for Measuring of the Financial Risk Andrei Iulian Andreescu

Agenda Introduction Definition of the financial risk Financial risk measurement techniques The VaR concept Application of VaR methods Conclusion

Introduction The need for an effective risk measurement system for companies and institutions Worst case scenarios unlikely to happen, but very dramatical consequences for real economy Most recently: Financial crisis from 2008 Negative effects: Spill over on the real economy 03.12.2018 Andrei Andreescu

Risk Measurement Techniques Year Instrument 1938 Bond duration 1952 Markowitz mean-variance framework 1963 Sharpe‘s CAPM 1966 Multiple factors models 1973 Black-Scholes option pricing model „Greeks“ 1988 Risk-weighted assets for banks (Basel) 1993 Value at Risk 1994 Risk metrics 1997 Credit metrics 1998 Integration of credit and market risk Risk budgeting … 03.12.2018 Andrei Andreescu

The VaR concept: Definition The maximum expected loss for a portfolio for a specific time frame with a certain probability - Market price risiko model. Parameters: holding period (1-30 days) and confidence level (90-99%) 03.12.2018 Andrei Andreescu

The elements of a VaR system Source: Jorion (2003), p. 256. Model Mapping Risk factors Portfolio Historical data Portfolio positions VaR method Distribution of risk factors Exposures Result 03.12.2018 Andrei Andreescu

Overview of the VaR methods Local valuation : Linear models, Full covariance matrix, Factor models, Diagonal models Non linear models: Gama, Convexity Full valuation: Historical Simulation Monte Carlo Simulation 03.12.2018 Andrei Andreescu

Application of VaR methods A hypothetical portfolio consisting of assets in five different currencies USD, GBP, CHY, CHF, PLZ. German exports in 2010 in non Euro zone Source: German Statistical Bureau (2010), p. 2.  Countries Mio. € % USA 65570 25% 25 Great Britain 59487 23% 23 China 53636 21% 21 Switzerland 41711 16% 16 Poland 38053 15% 15 Total 258457 100% 100 03.12.2018 Andrei Andreescu

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Time frame from 01.01.2007 until 19.04.2011 Portfolio value 100 Mio. € Source: Own calculations Time frame from 01.01.2007 until 19.04.2011 Portfolio value 100 Mio. € Confidence level 90-99% Time horizon 1-10 days 03.12.2018 Andrei Andreescu

Delta-Normal method 1/2 Variance-covariance method Portfolio return Portfolio variance Value at Risk 03.12.2018 Andrei Andreescu

Delta-Normal method 2/2 Step 1: Collect the historical data Step 2: Potential mapping of the security positions Step 3: Calculate weights, standard deviation, and correlation coefficient. Step 4: Estimate the VaR by multiplying the portfolio standard deviation by the portfolio value and by the parameter specifically for the desired confidence level. Pros: Easy to implement, small number of necessary data Cons: Normal distribution of the returns, non suitable for non linear instruments 03.12.2018 Andrei Andreescu

Comparison of VaR using Delta-Normal method Time horizon   Confidence level 1 day 2 days 3 days 4 days 5 days 10 days 90% 0,46% 0,68% 0,85% 0,99% 1,12% 1,66% 95% 0,58% 0,87% 1,09% 1,27% 1,44% 2,14% 99% 0,83% 1,23% 1,54% 1,80% 2,03% 3,02% 03.12.2018 Andrei Andreescu

Historical Simulation 1/2 No assumptions about the distribution of the risk parameters Portfolio return Step 1: Calculation of the daily returns Step 2: Compute the market value of the assets and revaluating the portfolio. Step 3: Sort the series of simulated profit and losses. Step 4: For the desired confidence level, read the value of the simulated VaR. 03.12.2018 Andrei Andreescu

Historical Simulation 2/2 Pros: Easy to use and to communicate to the interested outsiders. No estimation of volatility or correlation is necessary. In the most case, the necessary past data are available. Since there are no assumptions made regarding the distribution, tail, skewness and different non-normal features can be considered. Cons: If this time frame does not consider some likely market movements, than the calculated risk will probably be biased. A very long data set leads to aged data and distortion of future results due to past events which are not longer likely to recur. The news will have an underestimated impact on the VaR, due to their relatively low weight. The volatility through time is assumed to be constant. 03.12.2018 Andrei Andreescu

Comparison of VaR using historical simulation Time horizon    Confidence level 1 day 2 days 3 days 4 days 5 days 10 days 90% 0,44% 0,61% 0,81% 0,93% 1,07% 1,55% 95% 0,67% 0,85% 1,13% 1,24% 1,39% 2,08% 99% 2,15% 2,92% 4,07% 4,14% 4,21% 4,97% 03.12.2018 Andrei Andreescu

Monte Carlo Simulation 1/2 A random generator of numbers for estimation of risk Based on assumptions about distribution of the risk parameters Step 1: A stochastic process including the distribution and the parameters of the portfolio positions must be chosen. Step 2: A pseudo-sequence of variable must be generated. These means that pseudo future returns will be generated. Step 3: The value of the portfolio will be calculated using the prices for the target time frame. Step 4: Step 2 and 3 will be repeated for many times Step 5: Rank the simulated prices and calculate VaR 03.12.2018 Andrei Andreescu

Monte Carlo Simulation 2/2 Pros: A wide range of market behavior can be captured and analyzed. Complicated financial instrument such as mortgages, credit derivates, and nonlinear paths can also be successfully analyzed. The impact of extreme scenarios can also be simulated. Finally, extreme events do not influence the result as much as using the Historical Simulation method. Cons: The long computational time. It is also costly to implement the necessary operational infrastructure, both in terms of hardware and in terms of employing highly skilled operators. Finding a proper fitting of the joint distribution takes a lot of effort. The stochastic models used for the risk modeling can be wrong. 03.12.2018 Andrei Andreescu

Comparison of VaR using Monte Carlo Simulation Time horizon   Confidence level 1 day 2 days 3 days 4 days 5 days 10 days 90% 0,35% 0,49% 0,68% 0,71% 0,81% 1,19% 95% 0,47% 0,67% 0,86% 0,87% 1,03% 1,57% 99% 1,09% 1,40% 2,17% 1,98% 2,01% 2,89% @RISK Correlations UK USA Swiss Zloty Yuan 1   0,441634308 0,067829365 0,349116699 0,124994782 -0,287355187 -0,24837175 0,43048075 0,944002396 0,378542218 -0,278261063 03.12.2018 Andrei Andreescu

Inputs (Monte Carlo Simulation for the 10 day Rate of Return Dataset #19 hs AF1136   RiskLogistic(0,0020744;0,010633;RiskName("Dataset #19");RiskCorrmat(NewMatrix7;1)) -∞ 0,0020744 +∞ Dataset #20 AG1136 RiskLogistic(0,0014885;0,012789;RiskName("Dataset #20");RiskCorrmat(NewMatrix7;2)) 0,0014885 Dataset #21 AH1136 RiskLogistic(-0,0012962;0,0075775;RiskName("Dataset #21");RiskCorrmat(NewMatrix7;3)) -0,0012962 Dataset #22 AI1136 RiskLoglogistic(-0,15731;0,15597;14,007;RiskName("Dataset #22");RiskCorrmat(NewMatrix7;4)) -24,61102 Dataset #23 AJ1136 RiskLogistic(-0,00025645;0,012073;RiskName("Dataset #23");RiskCorrmat(NewMatrix7;5)) -2,5645 Distribution Fitting (Dataset #19) Correlation Matrix – 10 day Rate of Return @RISK Correlations Dataset #19 in $AF$1135 Dataset #20 in $AG$1135 Dataset #21 in $AH$1135 Dataset #22 in $AI$1135 Dataset #23 in $AJ$1135 1   0,441634308 0,067829365 0,349116699 0,124994782 -0,287355187 -0,24837175 0,43048075 0,944002396 0,378542218 -0,278261063

Criticism of VaR Pros: Cons: Increased the quality of risk and corporate management / bank supervisory Common risk measurement for different risk positions Considers correlations between risk factors Cons: It is not subadditive, therefore no coherent It is only quantitative and not qualitative Inconsistent result by using different methods 03.12.2018 Andrei Andreescu

Results of the VaR calculation VaR depends positively on the confidence level and on the time horizon The choice of the VaR method depends on many factors: Legal constrains Software tools and distribution assumptions Accuracy of the calculation Necessary effort and available data 03.12.2018 Andrei Andreescu

Market Risk Amendment and VaR The MRA specifies the dependence of the required capital on VaR Standard calculation of VaR: 10 holding days, 99% confidence level The multiplicator k depends on the time that bank daily losses exceed the estimated VaR in the last 250 trading days. Wrong estimation of daily losses Model accuracy and k value ≤ 4 Model is accurate, k=3 5≤9 Model is accurate, k increases ≥10 Model needs revision, k=4 03.12.2018 Andrei Andreescu

Conclusion Definition, history of risk Presentation of typical risk metrics Popular VaR methods Delta-Normal Historical Simulation Monte Carlo Simulation Empirical part: Different VaR result using different VaR methods, consequences for the capital holding requirements 03.12.2018 Andrei Andreescu

Thank you for your attention! andrei_andreescu@yahoo.de 03.12.2018 Andrei Andreescu