Credit risk of a loan portfolio (Credit Value at Risk) Esa Jokivuolle Bank of Finland erivatives and Risk Management 208
Background Credit risk is typically the biggest risk of banks Major banking crises usually caused by excessive credit, extended by banks, which eventually lead to large credit losses (or fear of them) Banks Credit Value at Risk models developed since 990 s for risk management and to help in capital budgeting Banks regulatory capital requirements (Basel requirements) have made use of CVaR techniques since 2004 (the Internal Ratings Based Model) 2
Modelling credit risk 3
Binomial model of default of a (corporate) loan ebt amount = Probability of default = P Loss Given efaut = LG Assume int. rate = 0 -P P (-LG)* 0 T Expected payoff= (-P)* + P*(-LG)* = - P*LG* Expected loss 4
Merton s model of default V istribution of terminal asset value V T V 0 efault probability P = f(v 0,, volatility of V, drift of V, T) T Assume V is normally distributed 5
Probability of default Assuming V is normally distributed: P = f(v 0,, volatility of V, drift of V, T=) = f(0,,, 0, ) with standardized normal distribution such that V has zero mean and variance equal to. (In what follows, a bank s P estimate for a customer determines for that customer) 6
Estimating probability of default (of a loan customer) in practice Banks assign internal credit ratings to their (corporate) customers Rating process involves Balance sheet analysis Expert judgement Rating scale Compare with agency ratings (AAA, AA, A, BBB, BB, B, ) Can estimate (average) probability of default within a given rating class from historical rating and default information 7
Example of empirical annual probabilities of default per (internal) rating class Source: Kiema-Jokivuolle 204 8
Loan portfolio model Building a portfolio model for corporate loans Using elements from both the binomial model and the Merton model In a portfolio setting, we need to model correlations between default events Assume each credit customer s asset value, V, is driven by a systematic risk factor, X, and a customer-specific idiosyncratic risk factor, e customer, which are identically and independently distributed with one another 9
V V < Loss Loss = 0 = LG *.. Pairwise asset correlations, ρ ( V i, V j ). Asset values standard normal, i s calibrated according to statistically estimated P i s, asset values correlated via exposure to common risk factor, X, so that defaults are correlated V V < Loss Loss = 0 = LG * o. of customers, n, in portfolio: n=,, n= Loss n 0
Like a binomial model V V < Loss Loss = 0 = LG *.. Pairwise asset correlations, ρ ( V i, V j ). Asset values standard normal, i s calibrated according to statistically estimated P i s, asset values correlated via exposure to common risk factor, X, so that defaults are correlated V V < Loss Loss = 0 = LG * o. of customers, n, in portfolio: n=,, n= Loss n
Solving the model One way to compute the probability distribution of portfolio credit losses is to do a Monte Carlo simulation of the previous model by generating random realizations for the V :s determine loss events by the thresholds, :s, respectively E.g. 0 000 simulation rounds should give a good approximation This is Credit Value-at-Risk [Obs. The model could also be extended to calculate the VaR of a corporate bond portfolio (see CreditMetrics Technical ocument)] 2
3 Portfolio comprises 500 loans, each of size 2. Each loan s P=0.5% and LG=20%. Asset correlation of each loan customer ρ=20%.
Probability distribution of loan portfolio loss (sketch): skewed and fat-tailed This amount of equity capital protects bank from bankruptcy with 99.9% probability (if bank only does lending) Expected loss Unexpected loss Loss Loss at chosen percentile, say, 99.9% (i.e., Credit VaR at 99.9%) 4
Credit Value at Risk and capital requirements for banks loan books 5
Using the model for regulation Closed-form solution? (that can be easily calculated in Excel) How to measure the contribution of a single credit customer to the credit portfolio s risk? Assumption: Credit portfolio is fully diversified (no. of customer credits approaches infinity and each is infinitesimally small) ot so bad an assumption for, say, mortgage and SME portfolios; care must be taken with large corporate portfolio 6
International minimum capital requirements (The Basel framework) Risk-weighted Assets (RWA) (* Capital RiskWeight i Exposure i 8% Basel II (2004): Risk-weights based on a Credit VaR model became possible The risk-weight of a loan is the contribution of that loan to the loan portfolio s Credit VaR at a chosen percentile; Basel uses 99.9% 7
Internal Ratings Based Approach (IRBA) Inputs needed Bank s own internal customer ratings Customer default probabilities (P) estimated per rating class Loss Given efault (LG) per loan eed also an assumption of the asset correlation between loan customers (provided as part of the regulatory model) Banks need the supervisor s approval to use the IRBA to calculate risk-weights (otherwise there is a simpler alternative to determine riskweights) 8
IRBA risk weight formula Recall, each credit customer s asset value, V, is driven by a systematic risk factor, X, and a customer-specific idiosyncratic risk factor (e) The conditional expected portfolio loss is i= i * LG i * p ( X = i x) in which p i (X=x) is conditional probability of default, given X=x For a fully diversified portfolio this is the conditional portfolio loss (not only expected) because the effect of idiosyncratic risks, the e:s, are diversified away (i.e., they cancel out one another) 9
IRBA risk weight formula A customer s risk contribution is the customer s part of the previous sum: * LG * p ( X x) i i i = Conditional probability of default, p i (X=x), when X is standard normal (see e.g. Kiema-Jokivuolle 204, JBF): p i ( X = x) = ( P i ) + x ρ ρ where P i is the unconditional prob. of default and ρ is the customer s asset correlation with factor X. (.) is std. normal cumulative density function and - (.) is its inverse function 20
IRBA-formula Conditional prob. of default ( Pi ) + 3.090 ρ i Ki = [ LGi ρi P LG ] " Maturity Adjustment" i Expected loss i - x = X 99.9% = 3.09 - Risk-weight = K i *(/0.08) - Capital requirement=8%*k i *(/0.08)* i =K i * i - ote: capital requirement is measured in terms of the unexpected loss - In the Basel framework, ρ i is set as a function of firm size 2
300 Risk weight, % Illustrative IRBA risk weights 250 200 2 50 3 00 Basel 50 0 0 2 4 6 8 0 2 4 6 8 20 Probability of default, % LG=45%, S=50 2 LG=45%, S=5 3 LG=25%, S=50 Source: Based on Basel Committee on Banking Supervision (2004) S=firm size 22
Literature CreditMetrics Technical ocument (RiskMetrics Group) Basel II and III documents and publications; see www.bis.org Vasicek, O.A. 997: The loan loss distribution. Technical report. KMV Corporation (obs. KMV is now part of Moody s) Gordy, M.B. 2003: A risk-factor model foundation for ratings-based bank capital rules, Journal of Financial Intermediation 2: 3,99 232 23