Aggregation and Credit Risk Measurement in Retail Banking
Ali Chabaane, BNP Paribas. Antoine Chouillou, BNP Paribas and Evry University. Jean-Paul Laurent, BNP Paribas and ISFA actuarial school.
Purpose of our study
!
Improve credit risk measurement !
!
Better assessment of correlation effects between credit portfolios.
Practical consequences of the choice of a risk measure !
VaR, expected shortfall,
!
Loss vs unexpected loss,
!
Sensitivity analysis.
Plan !
Credit loss modelling ! !
!
Basel II Extended approach.
A case study in retail banking ! ! !
Risk measures Computation of capital requirements Sensitivity analysis of risk measures
Default modelling : homogeneous portfolios !
In portfolio k, borrower i defaults with probability PDk when :
−1
Z k ,i = ρ k Ψk + 1 − ρ k ε k ,i < Φ ( PDk ) !
Common portfolio factor Ψk .
!
Specific independent factor ε k ,i for a borrower i.
!
Assumption : factors follow standard Gaussian distributions.
!
Gaussian cdf :
!
Correlation ρ k .
Φ.
Loss distribution : homogeneous portfolio !
Risk components for portfolio k: ! ! !
!
Marginal Probability of Default PDk . Marginal Loss Given Default LGDk . Portfolio Exposure At Default EADk .
Infinite granularity: ! !
Total loss Lk = sum of individual losses. When the number of borrowers is high, specific risk is diversified away (Gordy, 2000).
Lk (Ψk ) = EADk × LGDk × Φ (
−1
Φ ( PDk ) − ρ k Ψk 1 − ρk
)
Aggregation of homogeneous portfolios ! !
Homogeneous portfolios 1,…,K. Aggregate loss: K
L = ∑ EADk × LGDk × Φ (
Φ −1 ( PDk ) − ρ k Ψk
k =1
!
1− ρ k
Portfolio factors :
Ψk = ρη + 1 − ρη k
!
! !
and η follow standard Gaussian distributions. Systemic correlation between factors: ρ
(ηk )1≤ k ≤ K
)
Risk measures !
VaR of the loss distribution at the confidence level q :
VaRq ( L) = inf(l , P( L ≤ l ) ≥ q ). !
Expected Shortfall (the loss has a density) :
ES q ( L) = E ( L L ≥ VaRq ( L)) !
« Unexpected loss » :
ULq ( L) = VaRq ( L) − E ( L)
Our case study
!
!
Purpose of the case study !
Comparison of the regulatory model and its extended version,
!
Assessment of correlation effects,
!
Assessment of risk measure choice on capital allocation.
Input data !
14 credit lines, typical of retail banking.
Portfolio structure credit line 1 2 3 4 5 6 7 8 9 10 11 12 13 14
EAD 14% 20% 7% 10% 10% 7% 8% 2% 6% 1% 1% 5% 7% 3%
PD 0,1% 0,2% 0,2% 0,4% 0,6% 0,8% 1,4% 3,2% 3,2% 4,6% 7,2% 7,3% 16,0% 55,0%
LGD 60% 60% 60% 60% 60% 60% 60% 60% 60% 60% 60% 60% 60% 60%
correlation 16,7% 16,1% 15,8% 14,9% 14,2% 13,2% 11,1% 6,9% 6,8% 5,0% 3,2% 3,2% 2,1% 2,0%
Capital requirements !
Capital requirements:
VaR Basel, systemic correlation = 100% Multifactor, systemic correlation = 50% Absolute variation Relative variation
!
6,1% 4,6%
6,9% 5,0%
-1,5% -24,7%
-1,9% -27,6%
Basel II vs multifactor model: !
!
ES
Overestimation of capital of an order of magnitude of 25%, either with VaR or Expected Shortfall.
Expected Shortfall vs VaR: !
Expected Shortfall : 10% higher than VaR, in both setups.
Risk contributions based on total loss !
!
EADi : exposure of portfolio i. Risk contribution for subportfolio i: !
VaR based risk measure:
EADi !
∂VaRq ( L) ∂EADi
ES based risk measure:
EADi
∂ES q ( L) ∂EADi
Risk contributions based on total loss Risk contributions in the aggregated portfolio : the VaR case 1,2% 1,2% 1,0% 1,0% 0,8% 0,8%
!
0,6% 0,6%
The VaR case
multi multi Basel Basel
0,4% 0,4% 0,2% 0,2% 0,0% 0,0% 11 22 33 44 55 66 77 88 99 10 10 11 11 12 12 13 13 14 14
Risk contributions in the aggregated portfolio : the ES case 1,4% 1,4% 1,2% 1,2% 1,0% 1,0%
!
The Expected Shortfall case
0,8% 0,8% 0,6% 0,6%
multi multi Basel Basel
0,4% 0,4%
!
13 13
Basel : 100%, Multi : 50%.
77
99
11 11
!
0,0% 0,0%
55
Systemic correlation :
33
!
11
0,2% 0,2%
Risk contributions based on unexpected loss VaR risk contributions for a recentered loss distribution 0,6% 0,5% 0,4% 0,3%
multi Basel
0,2% 0,1% 0,0% 1 2 3 4 5
6
7
8
9 10 11 12 13 14
UL( L) = VaRq ( L) − E ( L)
!
Unexpected loss:
!
Risk contribution of portfolio i: EADi ×
!
Systemic correlation : 100% (Basel) and 50% (multi).
∂VaRq ( L) ∂EADi
− LGDi × PDi
Sensitivity analysis : systemic correlation VaR and ES as a function of systemic correlation 8% 7% 6% 5%
VaR ES
4% 3% 2% 1% 0% 0%
8%
15%
23%
30%
38%
45%
53%
60%
68%
systemic correlation
75%
83%
90%
98%
Summary !
!
!
!
Extension of the regulatory model,
Importance of risk diversification in an heterogeneous portfolio, Similitude between VaR and Expected Shortfall in the studied case, Taking into account expected loss…or not !
Annex : impact of low systemic correlation Risk contributions in the aggregated portfolio: the VaR case 1,2% 1,0% 0,8% 0,6%
multi Basel
0,4% 0,2% 0,0% 1 2 3 4 5 6 7 8
! !
9 10 11 12 13 14
Systemic correlation ρ : 100% (Basel) and 5% (multi). Computation with total loss L.
