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