credit risk assessment of fixed income portfolios: an analytical approach (*)
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Credit risk assessment of fixed income portfolios: an analytical approach (*). Bernardo PAGNONCELLI Business School Universidad Adolfo Ibanez Santiago, CHILE. Arturo CIFUENTES CREM/ FEN University of CHILE Santiago, CHILE. Primera Jornada de Regulación y Estabilidad Macrofinanciera - PowerPoint PPT PresentationTRANSCRIPT
Credit risk assessment of fixed income portfolios:
an analytical approach (*)Bernardo PAGNONCELLIBusiness SchoolUniversidad Adolfo IbanezSantiago, CHILE
ArturoCIFUENTESCREM/ FENUniversity of CHILESantiago, CHILE
Primera Jornada de Regulación y Estabilidad Macrofinanciera
January 2014(*) Based on Credit Risk Assessment of Fixed Income Portfolios Using Explicit Expressions, Finance Research Letters, forthcoming.
• A Brief History of an Interesting Problem
• Regulatory Implications
Portfolio of Risky Assets
N assets
Default Probability, p
Correlation, ρ
Issues:• How risky is this
pool?
• How much can I lose in a bad scenario?
• How much should I put aside to cover potential losses?
• Can it bring the company down?
• Systemic risk?
N = 50
p = 27%
ρ = 18.36%
Example
How risky is this portfolio ?
Assume that the total notional amount is $ 100
each default results in a loss of$ 100/ 40 = $ 2.5
$ 10
0
The naïve approach(assume no correlation) Yi (i=1, …, N) is 1 or 0 (1 = default; 0 = no default)
The number of defaults X is given by
X=Y1 + …+ YN.
X follows a binomial distribution with
E(X)= Np and Var(X)= N p (1-p).
The discrete probability density function is given by
Corre (Yi, Yj) = 0 For all i, j
Number of Defaults
Probability
E(X) = Np = 13.5 defaults Var(X) = N p (1-p) = 9.85
Other approaches (1)
N = 50
p = 27%
ρ = 18.36%
Still assume that ρ = 0 increase the value of p (more or less by pulling a number out of …), say by 20%and then hope that this trick will result in “conservative” results…
E(X) = Np = 16.2 defaults Var(X) = N p (1-p) = 10.89
Other approaches (2)
N = 50
p = 27%
ρ = 18.36%
Replace the original portfolio with a portfolio that has zero correlation but a lower number of bonds (5 instead of 50 in this case)
DS = 5
p = 27%
ρ = 0≈
9
Defaults Using A Normal Distribution
X
X* = -0.55 since Φ (-0.55) = p = 30%
30% 70%
NO defaultdefault
x < X* x > X*
Φ(x) < 30% Φ(x) > 30%
X
X* = -0.55 since Φ (-0.55) = p = 30%
30% 70%
NO defaultdefault
x < X* x > X*
Φ(x) < 30% Φ(x) > 30%
Assume P = 30%
Default Probability
I = 1 I = 0
Default Index
10
Monte Carlo Simulations
Z1 ~ N( 0,1)
(z1)1, (z1)2, (z1)3, …., (z1)L (z2)1, (z2)2, (z2)3, …., (z2)L
Z2 ~ N( 0,1)
One-Factor Gaussian Copula
ρA [ OR ρC ]Y*
I1 = (1, 0, ………..)
Y*
I2 = (1, …………, 0)
ρD
[1]
[2]
[3]
Uncorrelated, ρ = 0
(y1)1, (y1)2, …., (y1)L (y2)1, (y2)2,…., (y2)L
© A. Cifuentes & G. Katsaros
Z1 ~ N( 0,1)
(z1)1, (z1)2, (z1)3, …., (z1)L (z2)1, (z2)2, (z2)3, …., (z2)L
Z2 ~ N( 0,1)
One-Factor Gaussian Copula
ρA [ OR ρC ]Y*
I1 = (1, 0, ………..)
Y*
I2 = (1, …………, 0)
ρD
[1]
[2]
[3]
Uncorrelated, ρ = 0
(y1)1, (y1)2, …., (y1)L (y2)1, (y2)2,…., (y2)L
© A. Cifuentes & G. Katsaros[see Ref. 4]
Number of Defaults
Probability
The fat tails thing…
if i=0 then δ = (1-p) ρ
If i=N then δ = p ρ
otherwise δ = 0
Finally: The Golden Formula
E(X) = Np
Var(X) = p (1-p) (N + ρ N (N-1))
ρ = Corre (Yi, Yj) For all i, j
It’s Not The Fat Tails Stupid !!!
It’s The Bump At The End !!!
Probability
Number of Defaults
Almost 5%
Number of Defaults
Probabilities
Monte Carlo (with Correlation)
Correct (Analytical) Distribution
• A Brief History of an Interesting Problem
• Regulatory Implications
Example: A Typical Securitization Structure
$ 100$ 70
$ 10
$ 20
Assets Liabilities
Cash flow allocation
Portfolio A: p=12%; ρ=0.1; N=40 Recovery =40% each default = ($100/40) .6= a $1.5 loss
Portfolio B: p=43%; ρ=0; N=45 Recovery =40% each default = ($100/45) .6= a $1.335 loss
Issue # 1: St Deviation matters !!!
$ 100$ 70
$ 10
$ 20
Assets Liabilities
Cash flow allocation
Senior
Equity
Mezzanine
QUESTION: If you are going to buy the senior tranche, would you prefer portfolio (A) or (B) as collateral?
QUESTION: If you are going to buy the senior tranche, would you prefer portfolio (A) or (B) as collateral?
Issue # 2: Correlation is tricky !!!
Is Correlation Good or Bad??
Issue # 3: Subordination does not always help !!!
Portfolio A, Probability of each default scenario
Probability
Number of Defaults
$ 70
$ 10
$ 20
Senior
Equity
Mezzanine
Number of Defaults
Probabilities
14 defaults; Loss= 14x $1.5= $21
21 defaults; Loss= 21x $1.5= $31.5
Very Low Probability Scenarios