Low Default Portfolio (LDP) modelling Probability of Default (PD) Calibration Conundrum 3 th August 213
Introductions Thomas Clifford Alexander Marianski Krisztian Sebestyen Tom is a Senior Manager in Deloitte s Financial Services Advisory Group. He specialises in credit risk modelling across the banking sector, having implemented, reviewed and applied credit risk models across the full spectrum of Retail, Commercial, Corporate and Wholesale lending operations. Tom has a Masters degree in Physics, an Honours degree in Financial Services and is a qualified Prince2 practitioner. Alexander is a Manager in Deloitte s Financial Services Advisory Group. He specialises in credit risk measurement and modelling for the banking sector. Before joining Deloitte, Alexander worked in the international wholesale risk measurement team at a large UK bank where he worked on the development and rollout of corporate credit risk models mainly in emerging markets portfolios. He was also involved with credit process & policy, pricing, capital, impairment and stress testing. Alexander holds a MEng degree in Engineering. Krisztian is an Assistant Manager in Deloitte s Financial Services Advisory Group. He specialises in Basel II credit risk and operational risk modelling. Krisztian joined Deloitte in 212 from a Hungarian consulting company where he worked as a consultant and trainer in Basel II operational risk and credit risk modelling. Krisztian holds a Masters degree in Financial Mathematics and is a qualified Financial Risk Manager (FRM). We would like to extend our sincere thanks to Edward Venter whose hard work and commitment whilst on secondment with Deloitte played a significant role in generating the results produced in this analysis and producing this presentation. 2
Agenda 1. Aim & Conclusion 2. The significance of LDP modelling 3. Approaches to LDP modelling 4. Portfolios used in PD calibration 5. Calibration Results 6. Sensitivity Results 7. Questions 3
Aim Analyse two different LDP Probability of Default (PD) calibration methodologies and apply these on three sample portfolios to evaluate the model risks. Low Default Portfolios (LDP) account for a large share of total bank lending. Due to the scarcity of default observations and subsequent need for numerous assumptions, model calibrations introduce significant model risk. This is usually absorbed by applying high level conservatism. The aim of this presentation is to compare the Pluto/Tasche (25) confidence based methodology with the Tasche (211) Bayesian methodology, applying different prior distributions. PD calibrations and sensitivities to assumptions are compared on three simulated portfolios. 4
Conclusion Choice of methodology and assumptions (prior distributions and correlations) have a significant impact on PD estimates which can lead to significant variation in capital requirements and provisions. Expert based prior distributions can be an alternative to the proposed conservative and uniform distributions, producing comparable PD estimates with the other methodologies. Expert based prior distribution show lower sensitivity to correlation inputs The PD estimates produced by both methodologies can not be backtested due to data constraints, but can be benchmarked against estimates from different methodologies Regulatory expectations of RWA floors may override capital calculated using LDP PD estimates for some portfolios. However, a growing list of business applications require appropriate estimates of PDs. 5
FI excl insurance and pension funds Financial intermediation Insurance and pension funds Health & Social Work Electricity, Gas and Water Supply Education Public admin Subtotal LDP Real estate & professional services Construction Retail and Wholesale trade Manufacturing Transport, storage and comms Accommodation and food Recreational, personal, community Subtotal wholesale Individuals Total all lending million Significance of LDP modelling At least 5% of commercial banking book assets are in portfolios which have LDP characteristics: 2,5, 2,, 1,5, Breakdown of UK bank lending 55bn 5% = Large P&L or capital impact 1,, 5, Asset Class PD Central Tendency has a significant impact on overall capital requirement (both Pillar 1 and 2). The recent BIS survey reported significant differences in PD methodologies used by banks for the same LDPs, leading to significant differences in PD estimates and Pillar 1 capital. 6 Source: Bank Of England
Approaches to LDP modelling Confidence level based approach (Method A) The idea: setting asset correlation and intertemporal correlation assumptions, the number of defaults follows a correlated binomial distribution. Choosing a confidence level gives us an exact estimate of PD (subject to simulation variations). 1 γ M Where: r = 1 M ( n k j=1 k= N number of observations M number of simulations r - number of defaults T - number of years γ confidence level T (1 (1 G(PD, γ, St)) ) k ( (1 G(PD, γ, St)) ) n k ) t=1 G(PD, γ, St)= Φ( Φ 1 PD +y ρ ) probability of default in a given 1 ρ year, where y ~ N(,1) T t=1 Bayesian approach (Method B) The idea: there is a prior belief on the possible values of PD this can be represented in probabilistic terms. This prior belief is updated by the observations, using the prior distribution as a weighting function. Tasche (211) suggested taking the mean of the posteriori distribution as the estimate. P(θ PD observe k defaults)= P observe k defaults θ p(θ PD ) P(observe k defaults) Where: Θ is the prior distribution k is the number of defaults observed P observe k defaults θ = n k (1 T t=1 (1 G(θ, ρ, St)) ) k T ( t=1 (1 G(θ, ρ, St)) ) n k The prior distribution specifies the probability of a given long run average PD. Both the Confidence based and Bayesian methods can be applied to estimate and validate the central tendency (long run average) PD for a LDP. 7
Probability Probability Probability Flexibility of the Bayesian prior distribution Conservative prior distribution Assumes greater probability of higher PDs: Π Θ < λ = 1 (1 λ). Applicable if there is no prior assumption about the PD, but the objective is to generate a conservative estimates which can inform extreme expectations. Uniform prior distribution Assumes all PDs are equally possible Π Θ < λ Prior distributions considered = λ Less conservative approach and there is no specific belief about the distribution. This reflects a position where there is not expectations about the PD distribution. Expert distribution Experts can use expectations of the PD distribution to inform the prior distribution. In this example, a triangle distribution is used with expert judgment used to specify the: minimum PD; Maximum PD; and Mode (most likely) PD Min Expert distributions can incorporate stakeholder views on PD, influencing the result, which makes it more acceptable decreasing the black-box effect. Mode 1 1 Max PD PD PD Strengths and Weaknesses Produces conservative estimates Provides a cap to all other estimates Produces estimates close to conservative prior Simple to explain to senior managers to understand Incorporate management expectations Increased buyin of estimates Can be linked to industry benchmarks High correlation assumptions produce overly conservative estimates Assumes 99% PD most likely outcome High correlation assumptions produce overly conservative estimates. Assumes observed default rate as likely as 99% PD Subjectivity of estimates May produce less conservative results Major validation and documentation requirements 8
Objective Analyse the different PD calibration methodologies and evaluate the model risks introduced. 9
Example Portfolios Portfolio 1: Sovereign portfolio Default is defined according to the S&P definition as the failure to meet a principle or interest payment on the due date contained in the original terms of the debt issue. Sovereign portfolio (size 25 billion to 55 counterparties in 212) distributed across investment and sub-investment grade. Four defaults between 22 and 212 with the observed default rate over the 11 year period (.74%). Since there is either one or nil observed defaults per year, PD estimation cannot be completed using regression. Recently published BIS paper* highlighted that different PD methodologies used by banks lead to significant variances in PD estimates and RWA and recommended harmonisation of methodologies or publication of supervisory benchmarks. Sovereign Exposure: None %-2.5% 2.5%-5% >5% 16 14 12 1 8 6 4 2 Rating grade distribution for 212 8, 7, 6, 5, 4, 3, 2, 1, Counterparties (Primary) Defaults (Primary) Exposure (Secondary) Historic portfolio defaults Year Sovereigns Defaults Defaulting Country 22 43 23 46 1 Uruguay 24 47 25 48 1 Dominican Republic 26 5 27 51 28 52 1 Ecuador 29 53 21 53 211 54 212 55 1 Greece 1 *Analysis of risk-weighted assets for credit risk in the banking book http://www.bis.org/publ/bcbs256.pdf
AAA AA+ AA A+ A A- BBB+ BBB BB+ BB AA- BBB- BB- B+ B B- CCC/C 22 23 24 25 26 27 28 29 21 211 212 Example Portfolios Portfolio 2: Corporate portfolio Corporate portfolio (size 18 billion) of 1,292 (212) credit exposures to a global portfolio of major national corporates which has grown from 29 customers (22). 68% of the portfolio is provided to corporates operating primarily in 2 countries within Europe, with the remaining 32% mainly in the United States and China. 77defaults during the period with an annualised long run average observed default rate of 1.4%, although the maximum number of defaults to corporates in any single jurisdiction was ten. There was a spike in the default rate from 27 29 as a result of increased bankruptcy volumes following the global financial crisis. Regulatory expectation for separate PD calibrations for respective countries will be challenging given reduced volume of defaults. European investment proportions: >1% 5%-1% %-5% None Rating grade distribution for 212 Historic portfolio data 18 16 14 12 1 8 6 4 2 14, 12, 1, 8, 6, 4, 2, 16 14 12 1 8 6 4 2 2 3 4 5 5 7 9 9 9 1 14 1,4 1,2 1, 8 6 4 2 Defaults (Primary) Counterparties (Primary) Exposure (Secondary) Defaults (Primary) Counterparties (Secondary) 11
22 23 24 25 26 27 28 29 21 211 212 Example Portfolios Portfolio 3: Growing regional mortgage portfolio Credit risk is managed using manual underwriting, supported by a rating scorecard with defaults defined as 9 days past due. Mortgage portfolio has grown from 47m to 798 during a 1 year period, with customer volumes growing (from 47 to 4,7) and the average mortgage size increasing (from 1K to 171k). 51 defaults were observed during the period an annualised long run average observed default rate of.36%. The default spike (27 28) was followed by reduced defaults due to low interest rates and forbearance, with lending accelerating from 21. Low observed default rates make PD calibration for capital requirements challenging. The recent PRA exercise to assess capitalisation of 8 UK banks and building societies used a 15% RWA floor on residential mortgages which provides a benchmark for minimum expectations despite low default rates. Proportion of counterparties: >25% 1%-15% %-1% None Rating grade distribution for 212 Historic portfolio data 14 28, 12 6, 12 1 24, 2, 1 8 11 1 5, 4, 8 6 4 2 1 2 3 4 5 6 7 8 9 1 16, 12, 8, 4, 6 4 2 1 1 3 4 3 4 5 3 6 3, 2, 1, Defaults (Primary) Counterparties (Primary) Exposure (Secondary) Defaults (Primary) Counterparties (Secondary) 12
Probability Calibration results Sovereign portfolio Inputs: (Source) Expert PD distribution - Asset Correlation: (Basel) 24% - Intertemporal Correlation: (Expert) 7% - Confidence Level: (Industry Benchmark) 75% - Assumed LGD 45% % 1.5% 4% PD PD RWA Observed Default Rate.72% 86% Method A - Confidence Based PD Estimate @ 75% confidence level @ 9% confidence level Method B - Bayesian Mean PD Estimate 2.19% 3.53% 125% 143% Conservative Prior PD distribution 3.79% 146% Uniform Prior PD distribution 3.64% 144% Expert Prior PD distribution 1.85% 119% Due to low counterparty numbers, all estimates are significantly high compared to the observed default rate Expert prior based PD is comparable to the Confidence based approach result. Conservative and uniform prior distributions produce more conservative results, which are consistent with 9% confidence level for Method A Conservative and uniform prior distributions produce conservative estimates which are equivalent to applying a 9% confidence level in the Confidence Based Approach. 13
Probability Calibration results Corporate portfolio Inputs: (Source) Expert PD distribution - Asset Correlation: (Basel) 24% - Intertemporal Correlation: (Expert) 7% - Confidence Level: (Industry Benchmark) 75% - Assumed LGD 45% 1% 1.5% 3% PD PD RWA Observed Default Rate 1.1% 98% Method A - Confidence Based PD Estimate @ 75% confidence level @ 9% confidence level Method B - Bayesian Mean PD Estimate 2.47% 3.84% 129% 146% Conservative Prior PD distribution 3.29% 14% Uniform Prior PD distribution 3.% 136% Expert Prior PD distribution 1.86% 119% Both the conservative and uniform priors produce high estimates compared to the confidence based approach due to the high correlation. Tight expert band range (1% minimum - 3% maximum PD) limits Bayesian estimates producing results which are conservative. Calculating the results per country significantly increases the total portfolio PD. Expert Bayesian distribution is less conservative than the confidence based approach due to the tight range of PD expectations. 14
Probability Calibration results Growing mortgage portfolio Inputs: (Source) Expert PD distribution - Asset Correlation: (Basel) 15% - Intertemporal Correlation: (Expert) 7% - Confidence Level: (Industry Benchmark) 75% - Assumed LGD 15%.1%.4% 4% PD PD RWA Observed Default Rate.36% 9.8% Method A - Confidence Based PD Estimate @ 75% confidence level @ 9% confidence level Method B - Bayesian Mean PD Estimate.78% 1.17% 16.9% 22.1% Conservative Prior PD distribution.98% 19.7% Uniform Prior PD distribution.97% 19.5% Expert Prior PD distribution.88% 18.3% Higher customer volumes significantly reduce the range of PDs produced by both methods. Bayesian estimates lie between 75% and 9% confidence based estimates. Conservative and uniform prior estimates are similar given the low asset correlation. Expert Bayesian estimate is close to 75% confidence level calculated using confidence based approach. All RWA results exceed 15% floor. Due to higher customer volumes and lower correlations, both methodologies produce comparable results but RWAs exceed 15% which could inform Pillar 2 capital estimates. 15
Model risk Summary of correlation sensitivity analyses results Method A Method B with Expert Distribution Min Base Max Sensitivity Min Base Max Sensitivity Sovereign Portfolio 1.29% 2.19% 7.7% Medium 1.2% 1.85% 1.99% Low Corporate Portfolio 1.38% 2.47% 7.97% Medium 1.5% 1.86% 2.1% Low Growing Mortgage Portfolio.49%.78% 4.29% High.56%.88% 1.92% Medium Range: Asset correlation: Minimum - 5%, Maximum - 5% Inter-temporal correlation : Minimum - 45%, Maximum - 9% Method A Confidence level based approach The confidence based method shows a very high sensitivity to correlation assumptions. The mortgage portfolio shows the highest sensitivity to correlation assumptions with PD estimates ranging by a factor of 9 (from.49% to 4.3%) of the default rate. Model risk is driven by the reliance on assumptions but the ability to set a confidence level provides an opportunity to link to risk appetite. Method B Bayesian approach Using a fixed interval expert distribution limits the range of PD produced by the methodology. Therefore, estimates are much less sensitive to correlation assumptions than the confidence based approach. The mortgage portfolio shows slightly higher sensitivity to correlations due the larger range of the expert prior distribution and high volumes. Model risk is driven by the expert choice of prior distribution as the sensitivity is low The expert distributions limits sensitivity to correlation assumptions, although introduces risk of subjectivity which could preclude unexpected outcomes being captured. 16
Comparison of the methodologies Evaluating strengths and weaknesses Strengths Weaknesses Application Confidence level based approach Produces conservative estimation, which can be appropriate for capital calculations Fast computation time Not required to justify a prior distribution Confidence level can be linked to defined model risk appetite. Can produce estimates which are too conservative and therefore hard to achieve buy-in from stakeholders. Very sensitive to asset correlation and intertemporal correlation assumptions as well as the confidence level setting Conservative PD estimation for portfolios where experts do not have additional knowledge of the data. Validation of Capital requirement estimates. For example: o Sovereigns o Growth portfolios in new markets Bayesian approach Flexible prior distributions can be used Stakeholder expectations and industry knowledge can be incorporated Less conservative estimations can be produced which may be applicable for provisioning or pricing. Expert estimations can be biased, which introduces a source of model risk. Using conservative and neutral priors, estimations become very sensitive to correlation assumptions. Computation time is significant. Less conservative PD estimations to reflect extra information or benchmark data which exists and can justify For example: o Mortgage portfolios o Special niche portfolios (Project Finance, Financial Institutions) The two methodologies can be applied in concert to benchmark PD results and prioritised for specific application in targeted portfolios. 17
Conclusions Choice of methodology and assumptions (priors and correlations) have a huge impact on PD estimates which can lead to significant variance in capital requirement. Expert based prior distributions can be an alternative to the conservative and uniform distributions, producing comparable PD estimates with the other methodologies. Expert distribution show lower sensitivity to correlation inputs The PD estimates produced by these methodologies can not be backtested due to data constraints, but can be benchmarked against estimates from different methodologies Regulatory expectations of RWA floors may override capital calculated using LDP PD estimates for some portfolios. However, Pillar 2 assessment and a growing list business requirements depend on appropriate estimates of PDs. 18
Appendix: Sensitivity Results 19
Correlation Correlation PD Estimate PD Estimate Appendix: Sensitivity analysis results Sovereign portfolio Method A, confidence level 75% Method B, expert distribution 8.% 7.% 2.4% 6.% 2.2% 5.% 2.% 4.% 3.% 2.% 1.%.% 45 5 55 6 65 7 75 5 8 85 9 2 35 5 1.8% 1.6% 1.4% 1.2% 1.% 45 5 55 6 65 7 75 5 8 85 9 2 35 5 Intertemporal Correlation Intertemporal Correlation.%-1.% 1.%-2.% 2.%-3.% 3.%-4.% 4.%-5.% 5.%-6.% 6.%-7.% 7.%-8.% 1.%-1.2% 1.2%-1.4% 1.4%-1.6% 1.6%-1.8% 1.8%-2.% 2.%-2.2% 2.2%-2.4% 2.4%-2.5% Due to the fixed range of the expert prior distribution, the sensitivity to correlation inputs is much lower. 2
Correlation Correlation Pd Estimate Pd Estimate Appendix: Sensitivity analysis results Corporate portfolio Method A, confidence level 75% Method B, expert distribution 8.% 2.5% 7.% 6.% 2.3% 5.% 2.1% 4.% 3.% 1.9% 2.% 1.%.% 45 5 55 6 65 7 75 5 8 85 9 2 35 5 1.7% 1.5% 5 55 6 65 7 75 8 85 9 5 2 35 5 Intertemporal Correlation Intertemporal Correlation.%-1.% 1.%-2.% 2.%-3.% 3.%-4.% 4.%-5.% 5.%-6.% 6.%-7.% 7.%-8.% 1.5%-1.7% 1.7%-1.9% 1.9%-2.1% 2.1%-2.3% 2.3%-2.5% Due to the fixed range of the expert prior distribution, the sensitivity to correlation inputs is much lower. 21
Correlation Correlation Pd Estimate PD Estimate Appendix: Sensitivity analysis results Growing mortgage portfolio Method A, confidence level 75% Method B, expert distribution 4.5% 4.% 3.5% 3.% 2.5% 2.% 1.5% 1.%.5%.% 45 5 55 6 65 7 5 75 8 85 9 2 35 5 2.% 1.8% 1.6% 1.4% 1.2% 1.%.8%.6%.4%.2%.% 45 5 55 6 65 7 75 8 85 9 5 2 35 5 Intertemporal Correlation Intertemporal Correlation.%-.5%.5%-1.% 1.%-1.5% 1.5%-2.% 2.%-2.5% 2.5%-3.% 3.%-3.5% 3.5%-4.% 4.%-4.5%.%-.2%.2%-.4%.4%-.6%.6%-.8%.8%-1.% 1.%-1.2% 1.2%-1.4% 1.4%-1.6% 1.6%-1.8% 1.8%-2.% Due to the fixed range of the expert prior distribution, the sensitivity to correlation inputs is much lower, however slightly higher than for the other two portfolios. 22
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