Christopher C. Finger Austrian Workshop on Credit Risk Management Vienna February 2, 2001

1. Enhancing Monte Carlo Techniquesfor Economic Capital Estimation Christopher C. Finger Austrian Workshop on Credit Risk ManagementViennaFebruary 2, 2001

2. Outline • Introduction • Quantities of interest -- portfolio capital, marginal capital • Troubles with direct Monte Carlo • Dealing with size -- portfolio compression • Dealing with the model – importance sampling • Dealing with the model – analytic marginal capital • Conclusions

3. Model-based risk capital • A natural way to define risk capital is as a level required to guarantee solvency with some (high) degree of confidence. • Once capital is established for the portfolio, examine the contribution to capital of new positions or increased lines. • Applications of capital contributions • Internal credit charges or capital allocation, giving hurdle rates of return or capital budgets • Pricing -- charge for addition to capital, not just expected loss

4. Risk capital at level p Expected horizon value or total notional value Worst case horizon value at level p 5500 5600 5700 5800 5900 6000 VaR as risk capital Portfolio value at horizon Holding risk capital in this way assures that the likelihood of bankruptcy-causing losses is p.

5. Risk capital for new portfolio Distribution of base portfolio plus new exposure Risk capital for base portfolio 5500 5600 5700 5800 5900 6000 Contribution of a new exposure to portfolio risk capital Portfolio value at horizon Increase in capital comes from an increase in the total portfolio value and a decrease in the worst case level.

6. VaR as capital • Analytic shortcuts are available for marginal standard deviation, but marginal VaR is more difficult. • A common practice is to define capital as a multiple of standard deviation and use the previous results. • An established but little used result: • The derivative of VaR with respect to a single exposure weight is the conditional expectation, given that the realized loss is VaR, of the loss on the exposure in question. • This leads to the base capital term, but the size penalty is more difficult to obtain.

7. The trouble with standard Monte Carlo • The model presents a tough problem: • Small default probabilities • Discrete exposure distributions • Portfolio distribution smoothes out very slowly • Typical applications make things harder • Large portfolios • Economic capital = extreme portfolio percentiles • Focus on capital contributions; values can be comparable to portfolio MC error • “Going faster when you’re lost don’t help a bit.”

8. Outline • Introduction • Quantities of interest -- portfolio capital, marginal capital • Troubles with direct Monte Carlo • Dealing with size -- portfolio compression • Dealing with the model – importance sampling • Dealing with the model – analytic marginal capital • Conclusions

9. Naïve compression on ISDA benchmark portfolio • Initial portfolio • $30 billion total value, 1680 exposures • Investment grade, average rating of A • Diversified across nine industries, average correlation of 43% • Compressed portfolio • Maintain largest 2.3% of exposures • Bucket remaining exposures homogeneously by industry/rating • Resulting portfolio has 478 (28% of 1680) exposures

10. Compressed portfoliois almost homogeneous Compressed Size distribution of original and compressed portfolios 20% of exposures account for 70% of portfolio size 100% 80% 60% Cumulative size 40% 20% Original 0 0 20% 40% 60% 80% 100% Exposures

11. Near perfect fit in middle of distribution Very good fit in tails as well Comparison of portfolio results 100% Original 80% Compressed 60% Cum prob 40% 20% 0 32.3 32.5 32.7 32.9 Portfolio value ($B) 1.0% 0.8% 0.6% Cum prob 0.4% 0.2% 0 27 28 29 30 31 32 Portfolio value ($B)

12. Comparison of portfolio results • Mean ($B) • St. Dev. (bp) • 5% loss (bp) • 1% loss (bp) • 0.1% loss (bp) Original 32.7 45.9 11.2 150 633 Compressed 32.7 46.2 11.4 150 679 % Diff 0 0.6 1.8 -0.2 7.6 All simulation estimates are within one standard error.

13. Comparison of marginal statistics • Add an additional exposure in the most concentrated industry • for each investment grade rating • “small” exposure • average size in the base portfolio ($18M, 0.25%) • “large” exposure • maximum size in the base portfolio ($74M, 0.06%) • Capital statistics • increase in portfolio standard deviation • increase in 0.1% loss • Report increase as percentage of the new exposure size

14. St. Dev., large exposure St. Dev., small exposure 1.0% 1.0% 0.8% 0.8% 0.6% 0.6% 0.4% 0.4% 0.2% 0.2% 0 0 Aaa Aa A Baa Aaa Aa A Baa 0.1% loss, small exposure 0.1% loss, large exposure 25% 25% 20% 20% 15% 15% 10% 10% 5% 5% 0 0 Aaa Aa A Baa Aaa Aa A Baa Comparison of marginal statistics Original Compressed Overestimation of capital with compressed portfolio Underestimation of capital with compressed portfolio The real problem is the lack of convergence with any method.

15. Outline • Introduction • Quantities of interest -- portfolio capital, marginal capital • Troubles with direct Monte Carlo • Dealing with size -- portfolio compression • Dealing with the model – importance sampling • Dealing with the model – analytic marginal capital • Conclusions

16. BBB Current state Possible states at horizon AAA AA A BBB BB B CCC Default 0.00% 0.11% 5.28% 86.71% 6.12% 1.27% 0.23% 0.28% Probabilities (determined exogenous to model) 100.9% 100.8% 100.7% Par 97.5% 95.8% 83.2% Rec. Instrument value Overview of CreditMetricsSingle exposures follow a discrete distribution

17. Second threshold so next region contains CCC probability. Set first threshold so tail contains default probability. Z Z Z Z Z Z Z BB BBB CCC Def B A AA Overview of CreditMetricsCorrelations driven by asset value distributions • Assume a connection between asset value and credit rating. Asset return over one year • Transition probabilities give us asset return “thresholds”.

18. Similar asset returns produce joint defaults Opposite asset returns produce different credit moves. Overview of CreditMetricsOne correlation parameter gives all joint probabilities Obligor 1 Obligor 2

19. BankingIndex First obligor depends strongly on its industry... … somewhat on specificmovements … and mostly on specificmovements Obligor 1 Obligor 2 Industries are correlated Idio-syncratic Idio-syncratic Second obligor depends weakly on its industry... BeverageIndex Equity factor model gives obligor correlationsbased on mappings to industry indices.

20. Asset distribution conditional on down factor move Area is conditional default probability -3 -2 -1 0 1 2 3 With few factors, conditioning on market move makes many calculations easier. Unconditional asset distribution Conditional on factor move, all rating changes are independent.

21. To be more specific, at least in a simple case … • Default condition -- obligor assets less than default threshold • Represent obligors through regressions on a common index. • Given a value for the index, conditional default condition leads to conditional default probability • Given X, defaults are conditionally independent and portfolio follows a binomial distribution.

22. Example portfolio • Proxy a large bank lending book by two homogeneous groups (and one common actor): • A-rated -- 1700 exposures representing 75% of holdings, 20% asset correlation within group • BB-rated -- 400 exposures representing 25% of holdings, 50% asset correlation within group • 32% asset correlation between groups • Total notional of $60B, “unit exposure” of $25M • Higher concentration in investment grade, but lower grade exposures are larger and more highly correlated.

23. Many factor scenarios where value does not change Value changes the most where we do not simulate much Biggest issue is high sensitivity to extreme factor moves 4450 4440 Portfolio value 4430 4420 4410 -3 -2 -1 0 1 2 3 Factor return

24. 4450 4430 4410 4390 4370 4350 3 -5 -4 -3 -2 -1 0 1 2 3 -5 -4 -3 -2 -1 0 1 2 Importance sampling involves “cheating” and forcing the scenarios where they are most interesting Shift factor scenarios into region where portfolio is more sensitive New scenarios capture sensitive area better

25. Importance sampling stated in mathematical terms • Goal is to estimate the expectation of V=v(X) where X~f Straight MC Imp Sampling Integral expression Random variates Estimate V • Trick is to choose g to reduce the variance of the estimate

26. 6000 6000 Direct Monte Carlo Importance Sampling 5000 5000 4000 4000 Loss Loss 3000 3000 2000 2000 1000 1000 0 0 0.1 % 5 % 1 % 0.1 % Percentile level Result is greater precision with fewer scenarios required, particularly at extreme loss levels 2,000 simulations,optimized for 5% loss 10,000 simulations for Direct MC,100 for Importance Sampling

27. Outline • Introduction • Quantities of interest -- portfolio capital, marginal capital • Troubles with direct Monte Carlo • Dealing with size -- portfolio compression • Dealing with the model – importance sampling • Dealing with the model – analytic marginal capital • Conclusions

28. Suppose 1000 scenarios to estimate 1% VaR • Losses in each scenarios (in descending order) 1 2 3 … 9 10 11 Pos. 137 35 29 32 31 28 Pos. 212 39 27 10 31 23 … Pos. N60 57 58 … 62 54 53 Total 2500 2312 2297 … 1689 1500 1476 • A small position change will not change the ordering, so VaR will change by amount that position loss changes in scenario 10 Intuition for the conditional loss result comes from considering Monte Carlo estimation of VaR

29. Examine the conditional portfolio distribution, given the factor return The factor is the greatest determinant of portfolio value 6000 5950 5900 Portfolio value ($M) 5850 5800 Cond SD bands MC scenarios 5750 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Factor return

30. 5860 5870 5880 5890 5900 5910 Portfolio value ($M) Since exposures are conditionally independent, the conditional portfolio distribution is close to normal

31. Practical assumptions on the conditional distribution • Assumptions • One factor drives all correlations • Factor is normally distributed • Portfolio is conditionally normal • Exposures are independent given factor returns • First is not necessary, though results are only practical for a reduced set of factors • Second is part of CreditMetrics assumptions, but can be relaxed • Third is not essential (results require small modification for arbitrary standardized distribution) • Fourth is crucial to the analysis

32. 6000 5950 5900 5850 5800 5750 -2 -1.5 -1 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Size penalty analytic results • Notation: • L - portfolio loss, lq - portfolio VaR at level q • l(Z) - conditional portfolio loss, li(Z) - conditional exposure loss • (Z) - conditional portfolio SD, i (Z) - conditional exposure SD • Capital estimates are expectations over the conditional distribution of the factor, given that VaR is realized

33. Base capital (conditional expected loss on unit exposure) Conditionalvariance of newexposure Variance ofconditional mean • Size penalty Positive contribution if factor move is less than expected, negative otherwise Variance contribution of new exposure Size penalty analytic results

34. 2 4 6 8 10 Why is this useful?Capital at 50bp for additional investment grade exposure 160 140 120 100 80 Capital (bp) 60 40 20 0 -20 Unit exposure

35. 10bp loss 50bp loss 1% loss 5% loss Capital and size penalty for additional investment grade exposure 140 120 100 80 Capital (bp) 60 40 20 0 0 20 40 60 80 100 Unit exposure

36. 10bp loss 50bp loss 1% loss 5% loss Capital and size penalty for additional speculative grade exposure 50 45 40 35 30 25 Capital (%) 20 15 10 5 0 0 20 40 60 80 100 Unit exposure

37. Express capital charges as a grid • Base capital (basis points) 5% 1% 50bp 10bp • Inv grade 26.8 47.7 57.3 90.8 • Spec grade 695 1070 1220 1650 • Incremental capital (basis points) per unit exposure 5% 1% 50bp 10bp • Inv grade 0.15 0.23 0.27 0.43 • Spec grade 22.2 26.2 27.5 30.9

38. Conclusions • Inherent features of a direct Monte Carlo approach will cause convergence problems, particularly with capital calculations • Practical assumptions go a long way, regardless of the model • Portfolio capital based on compressed portfolio • Capital contribution based on generic new exposures rather than for each unique exposure in the portfolio • For CreditMetrics particularly, a reduced factor approach allows for variance reduction and hybrid techniques for the most difficult quantities to obtain through Monte Carlo