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![Page 1: 1 Modeling and Calibration Errors in Measures of Portfolio Credit Risk Nikola Tarashev and Haibin Zhu Bank for International Settlements April 2007 The.](https://reader038.fdocuments.us/reader038/viewer/2022110403/56649e855503460f94b8733d/html5/thumbnails/1.jpg)
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Modeling and Calibration Errors in
Measures of Portfolio Credit Risk
Nikola Tarashev and Haibin Zhu
Bank for International Settlements
April 2007
The views expressed in this paper need not represent those of the BIS.
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Motivation ASRF model: a well-known model of portfolio credit risk Main reason for the model’s popularity: “portfolio invariance” of capital
• Assumption 1: perfect granularity of the portfolio
• Assumption 2: single common factor of credit risk
Does ASRF allow for “bottom-up” calculation of economic capital ?
Not really: estimating ρi requires a global approach !!!
To take full advantage of portfolio invariance off-the-shelf values for ρi:
Pillar 1 of Basel II: ρi = ρIRB( PDi )
ASRF-based capital measures are subject to:
• misspecification errors: ie, violated assumptions of the model
• calibration errors: eg, off-the-shelf values for ρi.
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Related literature (BCBS WP No 15)
Misspecification of the ASRF model:
• Granularity assumption:• Martin and Wilde (2002), Vasicek (2002)
Emmer and Tasche (2003), Gordy and Luetkebohmert (2006)
• Sector concentration and the number of common factors• Pykhtin (2004), Duellmann (2006), Garcia Cespedes et al (2006)
Duellmann and Masschelein (2006)
• Both assumptions:
• Heitfield et al (2006), Duellman et al (2006)
Flawed calibration:
• Loeffler (2003), Morinaga and Shiina (2005)
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This paper
Decomposes the wedge between target and shortcut capital measures.
Exhaustive and non-overlapping components due to:
Misspecification of the ASRF model
• Multifactor effect
• Granularity effect Flawed calibration the ASRF model
• Correlation dispersion effect
• Correlation level effect
• Plausible estimation errors
Takes seriously the overall distribution of risk factors Gaussian versus fatter-tailed distributions
The paper does not:
(i) Discuss calibration of PDs or LGDs; (ii) address time variation in risk parameters
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Main results
Applying ASRF model
Large deviations from target capital for realistic bank portfolios
The main drivers of these deviations are calibration errors
• noise in the estimated value of ρi
• wrong distributional assumptions
Misspecification of the ASRF model has a much smaller impact
• exception: granularity effect in small portfolios
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Roadmap
The ASRF model (overview)
Alternative sources of error in capital measures (intuition)
Empirical methodology
Findings
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The ASRF model (an overview)
PDi, LGDi, ρi and weights wi are all known
Assets (driven by a single common factor) drive defaults
M ~ F (0,1), Z ~ G (0,1), V ~ H (0,1): weak restrictions
Perfectly fine granularity:
Idiosyncratic risk is diversified away (low wi & many exposures)
Pairwise correlations: ρi ρk
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To attain solvency with probability (1- α):
The ASRF model (implications)
Portfolio invariance
If V, M, Z are all normal and α = 0.001:
which underpins the IRB approach of Basel II
.999
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Errors in calculated capital charges
The granularity effect (specification error 1)
• Stylized homogeneous portfolio: PD=1%, LGD=45%, ρ2 = 10%
• Granularity ASRF capital undershoots target capital
target capital
ASRF capital
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The multi-factor effect (specification error 2)
• Stylized homogeneous portfolio: PD=1%, LGD=45%• Two sectors, weight = ω. Intra-sector correlation = 0.2; inter-sector = 0
• Owing to its single-CF structure, ASRF model undershoots target
Errors in calculated capital charges (cont’d)
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Correlation level effect (calibration error 1)
• Stylized homogeneous portfolio: PD=1%, LGD=45%
• Higher correlation raises the capital measure
Errors in calculated capital charges (cont’d)
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Correlation dispersion effect (calibration error 2)
• Start with a homogeneous portfolio: PD=1%, LGD=45%
ρ varies within
• Then, let PDs vary too, within [0.5%,1.5%]
Errors in calculated capital charges (cont’d)
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Methodology. Decomposing the gap
Select a portfolio:
• Realistic distribution of exposures across industrial sectors
• representative portfolio of large US banks, Heitfield et al. (2006)
• Size
• Large: 1000 exposures (homogeneous weights)
• Small: 200 exposures (homogeneous weights)
For each portfolio, quantify the difference between two extremes:
• Target capital
• Shortcut capital, implied by off-the-shelf calibration of ASRF model
Three additional capital calculations bring up the
(i) multifactor (ii) granularity
(iii) correlation-level (iv) correlation-dispersion effects
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Target capitalN firms, (σ) + Monte Carlo simulations
Shortcut capitalN = , + ASRF model
Methodology: decomposing the four effects
N firms, R() (1-factor best fit) + copula
Multi-factor effect
For all measures:homogeneous portfolios,
the same {PDi}iN and LGD,
Gaussian distributions
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R() is the “one-factor approximation” of (σ)
Loadings (i): allowed to differ across exposures
Approximation matches well average correlation Approximation could miss correlation dispersion
Fitting a single-factor structure
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Target capitalN firms, (σ) + Monte Carlo simulations
N firms, R() (one-factor best fit) + copula
Shortcut capitalN = , + ASRF model
Multi-factor effect
Methodology: decomposing the four effects
N = , R() + ASRF model
Granularity effect
N = , average + ASRF model
Correlation dispersion effectFor a
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Correlation level effect
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Data
10,891 non-financial firms worldwide:
• 40 industrial sectors
• mostly unrated firms
Risk parameters:
• from Moody’s KMV,
• for July 2006
• two sets of mutually consistent estimates:
• EDF: 1-year physical PD, at exposure level
• Global Correlation (GCORR) asset return correlations
Estimated based on a multi-factor loading structure
LGD = 45%
![Page 18: 1 Modeling and Calibration Errors in Measures of Portfolio Credit Risk Nikola Tarashev and Haibin Zhu Bank for International Settlements April 2007 The.](https://reader038.fdocuments.us/reader038/viewer/2022110403/56649e855503460f94b8733d/html5/thumbnails/18.jpg)
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Match sectoral distribution of the representative portfolio of US wholesale banks. Heitfield et al. (2006)
Two sizes: large (1000 exposures) and small (200 exposures)
3000 simulations (for each size): to average out sampling errors
mean (%)
Large Small
Average PD 2.42 2.28
Std dev of PD 5.16 5.05
Median PD 0.26 0.24
Average correlation 9.78 10.49
Std dev of loadings 9.33 10.54
Correl.(PD, loading) -20.0 -19.8
Simulated Portfolios
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Findings
Dissecting deviations from target capital:
Mean (%) Add-on (%) 95% interval (%)
Target capital
Multi-factor effect
Granularity effect
Correlation dispersion effect
Correlation level effect
Shortcut capital (correlation = 12%)
Total difference
2.95-0.03
-0.11
0.35
0.55
3.71
0.76
-1.02
-3.73
11.86
18.64
25.76
[2.64, 3.27]
[-0.09, 0]
[-0.14, -0.09]
[0.27, 0.43]
[0.44, 0.66]
[3.37, 4.06]
Correlation level effect if:
correlation = 6%
correlation = 18%
correlation = 24%
-0.96
2.01
3.47
-32.54
68.14
117.63
[-1.11, -0.83]
[1.84, 2.18]
[3.23, 3.72]
![Page 20: 1 Modeling and Calibration Errors in Measures of Portfolio Credit Risk Nikola Tarashev and Haibin Zhu Bank for International Settlements April 2007 The.](https://reader038.fdocuments.us/reader038/viewer/2022110403/56649e855503460f94b8733d/html5/thumbnails/20.jpg)
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For small portfolios: similar results, except for the granularity effect
Large Mean (%)
Small Mean (%)
Target capital
Multi-factor effect
Granularity effect
Correlation dispersion effect
Correlation level effect
Shortcut capital
Correlation level effect if
correlation = 6%
correlation = 18%
correlation = 24%
2.95
-0.03
-0.11
0.35
0.55
3.71
-0.96
2.01
3.47
3.35
-0.04
-0.53
0.38
0.36
3.52
-1.07
1.76
3.15
add-on: large -3.73%;small -15.8%
Findings, continued
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Findings, continued Discrepancy: shortcut minus target capital
Explaining the variation of the discrepancy across simulated portfolios
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Delving into calibration errors
Small-sample estimation errors in calibrated correlations
Conduct the following exercise
• Design true model: 2 = 9.78%, PD = 1%, Gaussian variables, N equal exposures
• Draw from the true model: T periods of asset returns
• Adopt the point of view of a user (has data, estimates parameters)
• Construct sample correlation matrix (PD, etc: known)
• Fit a one-factor model and calculate ASRF-implied capital
Repeat 1000 times for each (N,T)
![Page 23: 1 Modeling and Calibration Errors in Measures of Portfolio Credit Risk Nikola Tarashev and Haibin Zhu Bank for International Settlements April 2007 The.](https://reader038.fdocuments.us/reader038/viewer/2022110403/56649e855503460f94b8733d/html5/thumbnails/23.jpg)
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Small-sample estimation errors are significant
Meaningful reduction of errors requires unrealistic sample sizes
Delving into calibration errors, contd.
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Estimation errors affect substantially capital charges
• Bias
• Noise
Delving into calibration errors, contd.
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The importance of distributional assumptions
Stylized fact: fat tails of asset returns
Gaussian (thin tails) assumption too low capital measures
Quantify the bias:
• Student t distributions
• General ASRF formula:
![Page 26: 1 Modeling and Calibration Errors in Measures of Portfolio Credit Risk Nikola Tarashev and Haibin Zhu Bank for International Settlements April 2007 The.](https://reader038.fdocuments.us/reader038/viewer/2022110403/56649e855503460f94b8733d/html5/thumbnails/26.jpg)
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Alternative distributions
Gaussian assumption negative bias in measured capital
![Page 27: 1 Modeling and Calibration Errors in Measures of Portfolio Credit Risk Nikola Tarashev and Haibin Zhu Bank for International Settlements April 2007 The.](https://reader038.fdocuments.us/reader038/viewer/2022110403/56649e855503460f94b8733d/html5/thumbnails/27.jpg)
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Conclusions
Large deviations from target capital for realistic portfolios
The main drivers of the deviations:
• calibration errors
• not model misspecification
Challenges for risk managers and supervisors