Credit risk assessment of fixed income portfolios: an analytical approach (*)

1. 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 January 2014 (*) Based on Credit Risk Assessment of Fixed Income Portfolios Using Explicit Expressions, Finance Research Letters, forthcoming.

2. A Brief History of an Interesting Problem • Regulatory Implications

3. Portfolio of Risky Assets • 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 assets Default Probability, p Correlation, ρ

4. Example Assume that the total notional amount is $ 100 each default results in a loss of $ 100/ 40 = $ 2.5 N = 50 p = 27% ρ = 18.36% $ 100 How risky is this portfolio ?

5. The naïve approach (assume no correlation) Corre (Yi, Yj) = 0 For all i, j 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

6. Probability Number of Defaults E(X) = Np = 13.5 defaults Var(X) = N p (1-p) = 9.85

7. Other approaches (1) 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… N = 50 p = 27% ρ = 18.36% E(X) = Np = 16.2 defaults Var(X) = N p (1-p) = 10.89

8. Other approaches (2) N = 50 p = 27% ρ = 18.36% DS = 5 p = 27% ρ = 0 ≈ Replace the original portfolio with a portfolio that has zero correlation but a lower number of bonds (5 instead of 50 in this case)

9. Defaults Using A Normal Distribution Default Probability Default Index Assume P = 30% I = 0 I = 1

10. Monte Carlo Simulations [see Ref. 4]

11. Probability Number of Defaults The fat tails thing…

12. Finally: The Golden Formula if i=0 then δ = (1-p) ρ If i=N then δ = p ρ otherwise δ = 0 ρ = Corre(Yi, Yj) For all i, j E(X) = Np Var(X) = p (1-p) (N + ρ N (N-1))

13. Almost 5% Probability Number of Defaults It’s Not The Fat Tails Stupid !!! It’s The Bump At The End !!!

14. Probabilities Correct (Analytical) Distribution Number of Defaults Monte Carlo (with Correlation)

15. A Brief History of an Interesting Problem • Regulatory Implications

16. Example: A Typical Securitization Structure Cash flow allocation Assets Liabilities $ 70 $ 10 $ 20 $ 100 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

17. Issue # 1: St Deviation matters !!! Cash flow allocation Assets Liabilities $ 70 $ 10 $ 20 Senior $ 100 Mezzanine Equity QUESTION: If you are going to buy the senior tranche, would you prefer portfolio (A) or (B) as collateral?

18. QUESTION: If you are going to buy the senior tranche, would you prefer portfolio (A) or (B) as collateral?

19. Issue # 2: Correlation is tricky !!! Is Correlation Good or Bad??

20. Issue # 3: Subordination does not always help !!! Portfolio A, Probability of each default scenario Probability Number of Defaults

21. Probabilities Very Low Probability Scenarios Number of Defaults $ 70 $ 10 $ 20 Senior 21 defaults; Loss= 21x $1.5= $31.5 Mezzanine Equity 14 defaults; Loss= 14x $1.5= $21