The Vasicek Distribution

The Vasicek Distribution Dirk Tasche Lloyds TSB Bank Corporate Markets Rating Systems dirk.tasche@gmx.net Bristol / London, August 2008 The opinions expressed in this presentation are those of the author and do not necessarily reflect views of Lloyds TSB Bank. i

Outline What is special about the Vasicek distribution? (slide 1) Properties of the Vasicek distribution (slide 8) Modelling default rates (slide 17) Related topics (not covered) (slide 30) ii

What is special about the Vasicek distribution? (1) The Vasicek distribution is a two-parametric (0 < p < 1, 0 < ϱ < 1) continuous distribution on (0, 1) with density f p,ϱ (x) = 1 ϱ ϱ exp ( 1 2 { Φ 1 (x) 2 ( 1 ϱ Φ 1 (x) Φ 1 (p) ϱ ) 2 }). (1.1) Other two-parametric continuous distributions on (0, 1) include the beta-distribution with parameters α, β > 0 and density f α,β (x) = Γ(α+β) Γ(α) Γ(β) xα 1 (1 x) β 1, the Kumaraswamy distribution with parameters a, b > 0 and density f a,b (x) = a b x a 1 (1 x a ) b 1. 1

What is special about the Vasicek distribution? (2) Examples of densities 3.0 Mean 0.25, Stddev 0.137 2.5 Density value 2.0 1.5 1.0 Vasicek Beta Kumaraswamy 0.5 0.0 0% 25% 50% 75% 100% Loss percentage 2

What is special about the Vasicek distribution? (3) Examples of densities 5 Mean 0.25, Stddev 0.354 4 Density value 3 2 Vasicek Beta Kumaraswamy 1 0 0% 25% 50% 75% 100% Loss percentage 3

What is special about the Vasicek distribution? (4) Observations: When fitted to the same mean and standard deviation, the Vasicek, beta, and Kumaraswamy densities do not differ much. Vasicek: Density looks sophisticated, moment matching of correlation parameter ϱ requires numerical root-finding. Kumaraswamy: Density looks simple, but moment matching requires numerical solution of two-dimensional optimisation problem. beta: Density looks relatively simple, moment matching is easy. Why not just working with the beta distribution? 4

What is special about the Vasicek distribution? (5) Kumaraswamy: Similar to beta but more efficient for simulation. beta: http://en.wikipedia.org/wiki/beta distribution Appears naturally as distribution of order statistics: X 1,..., X n i.i.d. uniformly distributed, X [k] = min I {1,...,n}, I =k max{x i : i I} (k-th smallest) X [k] β(k, n k + 1), i.e. P[X [k] x] = Γ(n) Γ(k) Γ(n k+1) x 0 uk 1 (1 u) n k du. (5.1) Dirichlet distribution is multi-variate extension of beta distribution. However, there is no economic motivation of the beta distribution. 5

What is special about the Vasicek distribution? (6) It s the economy, stupid Vasicek distribution can be interpreted in context of a trigger mechanism that is useful for modeling credit risk. Denote by A the change in value of an obligor s assets in a fixed time period (say 1 year). Denote by t the default threshold. The obligor s default is triggered if A t. Then P D = P[A t] is the unconditional probability of default. 6

What is special about the Vasicek distribution? (7) Oldrich Vasicek (1987) : Assume that A = ϱ S + 1 ϱ ξ where S is the systematic and ξ the idiosyncratic risk factor. S and ξ are independent and standard normal. Consider the conditional (on the systematic factor) probability of default P D(S) : P D(S) = P[A t S] = Φ ( t ϱ S ) 1 ϱ (7.1) Then P D(S) is a Vasicek-distributed random variable with p = Φ(t). Probability of Loss on Loan Portfolio, http://www.moodyskmv.com/research/portfoliocreditrisk wp.html Φ denotes the standard normal distribution function: Φ(x) = e u2 /2 2 π du. 7

Properties of the Vasicek distribution (1) Important observation: If X has a Vasicek distribution with parameters p and ϱ, then by Slide 7 X can be represented as X = Φ ( t ϱ S 1 ϱ ), (8.1) where t = Φ 1 (p) and S is a standard normal random variable. Consequences: Distribution function: F p,ϱ (x) = P[X x] = Φ ( 1 ϱ Φ 1 ) (x) t ϱ Density: f p,ϱ (x) = x F p,ϱ(x) (see (1.1)) Quantiles: Fp,ϱ 1 (α) = Φ ( t+ ϱ Φ 1 (α) ) 1 ϱ 8

Properties of the Vasicek distribution (2) Moments Let n be a positive integer and ξ 1,..., ξ n i.i.d. standard normal. If X is Vasicek-distributed with parameters p = Φ(t) and ϱ, then E[X n ] = E [ Φ ( t ϱ S ) n ] 1 ϱ = E [ n i=1 P[ ϱ S + 1 ϱ ξ i t S] ] = E [ P[ ϱ S + 1 ϱ ξ 1 t,..., ϱ S + 1 ϱ ξ n t S] ] = P[ ϱ S + 1 ϱ ξ 1 t,..., ϱ S + 1 ϱ ξ n t] = P[Y 1 t,..., Y n t], (9.1) where (Y 1,..., Y n ) is a multi-variate normal vector with E[Y i ] = 0, var[y i ] = 1, and corr[y i, Y j ] = ϱ, i j. 9

Properties of the Vasicek distribution (3) Special cases: E[X] = P[Y t] = p var[x] = E[X 2 ] E[X] 2 = P[Y 1 t, Y 2 t] p 2 (10.1) Define Φ 2 (s, t; ϱ) = P[Y 1 s, Y 2 t], where (Y 1, Y 2 ) is a bivariate normal vector with E[Y i ] = 0, var[y i ] = 1, and corr[y 1, Y 2 ] = ϱ. Estimating p and ϱ from a sample x 1,..., x m (0, 1) by momentmatching: ˆp = 1 m m x i i=1 ˆϱ is unique solution of Φ 2 (t, t; ϱ) = 1 m m x 2 i i=1 10

Properties of the Vasicek distribution (4) Indirect moment matching Observation: If X is Vasicek-distributed with parameters p and ϱ, then Y = Φ 1 (X) is normally distributed with E[Y ] = µ = Φ 1 (p) 1 ϱ, var[y ] = σ 2 = ϱ 1 ϱ (11.1) Estimate µ and σ by ˆµ = 1 m m i=1 Φ 1 (x i ), ˆσ 2 = 1 m m i=1 Φ 1 (x i ) 2 ˆµ 2 (11.2) Solve then (11.1) for p and ϱ: p = Φ ( ) ˆµ 1+ˆσ 2, ϱ = ˆσ 2 1+ˆσ 2 (11.3) ( p, ϱ) is also the maximum likelihood estimator of (p, ϱ). 11

Properties of the Vasicek distribution (5) Quantile-based estimators Use again observation (11.1) on the transformation of the sample into a normally distributed sample. Choose probabilities 0 < α 1 < α 2 < 1 and calculate the empirical quantiles ˆq(α 1 ) < ˆq(α 2 ) of the transformed sample. Determine estimators ˆµ and ˆσ from the equations ˆq(α i ) = ˆµ + ˆσ Φ 1 (α i ), i = 1, 2 (12.1) Determine estimator p of p and ϱ of ϱ by solving (11.3) for p and ϱ. 12

Properties of the Vasicek distribution (6) Simulation study of estimation efficiency Simulate 1000 times a Vasicek sample of length 5, 10, 25, and 100, for parameter values p = 0.1 and ϱ = 0.25. Compare results for direct moment matching (DMM) indirect moment matching (= maximum likelihood estimation, MLE) quantile-based estimation (QBE) Tabulate estimators (Est.), standard deviations (SD) of estimators, and standard errors (SE). 13

Properties of the Vasicek distribution (7) Results for estimation of p Sample size 5 10 Est. p SD SE Est. p SD SE DMM 0.1017 0.0455 0.0455 0.1000 0.0312 0.0312 MLE 0.1009 0.0446 0.0446 0.0996 0.0308 0.0308 QBE 0.1220 0.0603 0.0642 0.1012 0.0399 0.0399 Sample size 25 100 Est. p SD SE Est. p SD SE DMM 0.1004 0.0191 0.0191 0.1000 0.0099 0.0099 MLE 0.1002 0.0189 0.0189 0.0999 0.0098 0.0098 QBE 0.1005 0.0257 0.0257 0.1003 0.0133 0.0133 14

Properties of the Vasicek distribution (8) Results for estimation of ϱ Sample size 5 10 Est. ϱ SD SE Est. ϱ SD SE DMM 0.2091 0.1306 0.1368 0.2242 0.1004 0.1036 MLE 0.1972 0.1071 0.1194 0.2223 0.0785 0.0832 QBE 0.2964 0.2025 0.2076 0.2081 0.1469 0.1526 Sample size 25 100 Est. ϱ SD SE Est. ϱ SD SE DMM 0.2393 0.0714 0.0722 0.2461 0.0376 0.0378 MLE 0.2385 0.0514 0.0527 0.2468 0.0271 0.0272 QBE 0.2308 0.1052 0.1069 0.2459 0.0569 0.0570 15

Properties of the Vasicek distribution (9) Comments on the simulation study Estimation of p: Little difference between Direct Moment Matching (DMM) and Maximum Likelihood Estimation (MLE). Quantile-Based Estimation (QBE) is clearly less efficient (higher Standard Error). All estimators seem nearly unbiased (i.e. the mean of many estimates is close to the true value) for sample size 10 or greater. Estimation of ϱ: MLE is most efficient (as expected from statistical theory), DMM is second, QBE is worst. None of the estimators is unbiased. In average, they all underestimate the correlation parameter, even for relatively large sample size 100. 16

Modelling default rates (1) Future default rates of a portfolio cannot be predicted with certainty. Based on appropriate probabilistic models, however, probabilities of interesting future events can be calculated (e.g. the probability to observe a default rate of more than 1%). Models should be able to explain clustering of defaults (i.e. very high observed rates in some years, very low rates in other years). Common approach: create default correlation by dependence on one or more systematic factors. 17

Modelling default rates (2) Binomial approach n independent borrowers Default event: D i = { Borrower i defaults } Default indicator: 1 Di = 1, if D i occurs 0, otherwise Identical default probability: P D = P[D i ] = P[1 Di = 1] Default rate: DR = 1 n n i=1 1 Di 18

Modelling default rates (3) Calculating the default rate distribution Exact binomial probability by incomplete beta integrals: ( ) k n = P D l (1 P D) n l l [ P DR k ] n Normal approximation: = l=0 1 P D 1 0 u k (1 u) n k 1 du u k (1 u) n k 1 du P [ DR k n ] Φ ( ) k n P D n P D (1 P D) (19.1) Poisson approximation: P [ DR k n ] e n P D k l=0 (n P D) l l! 19

Modelling default rates (4) Comments on binomial calculations On principle, there is no reason to use approximations for binomial distributions. Calculations are fast (even in MSExcel) when the incomplete beta representation is used. For PDs up to 10% the Poisson approximation is better than the normal approximation. The Poisson approximation is useful for algebraic calculations with default or loss rates (actuarial approach, CreditRisk + ). 20

Modelling default rates (5) General observations on the binomial approach E[DR] = P D, var[dr] = 1 n P D (1 P D) Hence lim var[dr] = 0 n Conclusion: If default events were independent, for large pools of borrowers with similar credit-worthiness the variation of observed default rates over time would be small. Contradiction to empirically observed credit cycles. Problem: Independence of default events P[Borrowers i and j default] = P[D i D j ] = P D 2 (21.1) 21

Modelling default rates (6) Vasicek approach: correlated binomial distribution Choose representation of default event D i such that defaults become dependent (cf. Slide 7): D i = { ϱ S + 1 ϱ ξ i t} (22.1) S is the systematic factor, ξ i is the idiosyncratic factor, and t = Φ 1 (P D). S and ξ i, i = 1,..., n, are independent and standard normal. Consequence: Defaults are no longer independent: P[Borrowers i and j default] = P[D i D j ] = Φ 2 (t, t; ϱ) > P D 2 (22.2) 22

Modelling default rates (7) Observations on the Vasicek approach E[DR] = P D, var[dr] = 1 n P D P D2 + n 1 n Φ 2(t, t; ϱ) Hence lim var[dr] = Φ n 2 (t, t; ϱ) P D 2 > 0 Conclusion: The Vasicek approach can depict cyclic variation of default rates even for very large portfolios. Expectation given systematic factor (from (7.1)) is Vasicek-distributed: E[DR S] = Φ ( t ϱ S ) 1 ϱ (23.1) Large portfolio behaviour (approximation by E[DR S]): lim n E[ (DR E[DR S]) 2] ( = lim 1 n n P D Φ2 (t, t; ϱ) ) = 0 (23.2) 23

Modelling default rates (8) Binomial and correlated binomial distributions Frequency 0.00 0.04 0.08 0.12 PD = 0.1, rho = 0.12, n = 100 0.0 0.1 0.2 0.3 0.4 0.5 Default rate Binomial Corr Binomial 24

Modelling default rates (9) Vasicek and correlated binomial distributions Frequency * n 0 2 4 6 8 PD = 0.1, rho = 0.12, n = 100 Corr binomial Vasicek 0.0 0.1 0.2 0.3 0.4 0.5 Default rate 25

Modelling default rates (10) Vasicek and correlated binomial distributions Frequency * n 0 2 4 6 8 PD = 0.1, rho = 0.12, n = 250 Corr binomial Vasicek 0.0 0.1 0.2 0.3 0.4 0.5 Default rate 26

Modelling default rates (11) Vasicek approach: Calculating the default rate distribution Exact probability by conditioning and integrating against density of systematic factor: P [ DR k n ] = E = [ P [ ϕ(s) DR k n k l=0 ]] S ( ) n Φ ( t ϱ s ) ( l l 1 ϱ 1 Φ ( t ϱ s ) ) n l 1 ϱ ds Alternative: by Monte-Carlo simulation (less accurate). Density of E[DR S]: by (1.1). ϕ denotes the standard normal density function: ϕ(s) = e s2 /2 2 π ds. (27.1) 27

Modelling default rates (12) Vasicek (2002) Fit the distribution of DR (correlated binomial) by a Vasicekdistributed variable X with parameters p = P D and ϱ such that Φ 2 (t, t; ϱ ) P D 2 = var[x] = var[dr] = 1 n P D P D2 + n 1 n Φ 2(t, t; ϱ) (28.1) Involves numerical root-finding and calculation of bivariate normal distribution. Kolmogorov-Smirnov distances for fitting correlated binomial distribution: (exact distribution by (27.1)) Approximation by Vasicek-distribution with P D, ϱ: 0.078 Approximation by Vasicek-distribution with P D, ϱ : 0.048 The distribution of loan portfolio value, RISK 15, 160-162 28

Modelling default rates (13) Vasicek fit and correlated binomial distributions Frequency * n 0 2 4 6 8 PD = 0.1, rho = 0.12, rho* = 0.143, n = 100 Corr binomial Vasicek fit Vasicek 0.0 0.1 0.2 0.3 0.4 0.5 Default rate 29

Related topics (not covered) Default rate modelling with negative binomial (conditional Poisson) distributions. Across-time default rate modelling. Multi-factor models. Portfolio-loss models (heterogeneity of exposures). Derivation of Basel II risk weight functions.... 30