1. A Cash Flow Based-Approach To Estimate Default Probabilities Arturo Cifuentes (*) Academic Director, CREM arturo.cifuentes@fen.uchile.cl (*) Joint Project with Francisco Hawas, Math Department, University of Chile Santiago, Chile--January 2014

2. Fact: Companies (Sometimes) Default On Their Debt Problem: What’s The Likelihood That Company XYZ Could Default? Importance: Very High Interested Parties: Many (Investors, Regulators, Finance Managers, Board Members) Bottom Line: It Would Be Very Useful To Have Reliable Methods To Estimate The Probability That A Company Could Default

3. Journal of Economics and Business (November 2011): After 40 years of research on this topic there is no reliable predictor So Far… Statistical Methods/ Use of Ratios Options-Based Methods (KMV, Merton) Neural-Networks Ratings CDS Spreads

4. Formal Statement Of The Problem: A company is an engine that produces cash flows over a certain period of time and often from multiple sources… These cash flows are uncertain, that is, stochastic in nature… Typically, the company debt is deterministic and the payments are spread out over a well-defined time span THEREFORE.. the problem consists of estimating the likelihood that the aggregate cash flows might not be sufficient to make the debt payments

5. A More Formal Statement Of The Problem: Company has M cash sources and consider N time periods We denote as X i (i=1, …, M) the vector associated with the cash flows generated by source i at times t 1, …, t N. Thus, X i = (x i1, …, x iN ) t. [ x ij refers to the cash flow generated by source i at time t j. ] We assume that the vectors X i follow multi-normal distributions, X i ∼ MN(μ i, C i ), where μ i represents the vector of expected values and C i denotes the corresponding correlation matrix

6. Example: 3 Cash Sources and 4 Time Periods 1 2 3 4 [ 1 ] 1 2 3 4 [ 2 ] 1 2 3 4 [ 3 ] X 11 X 12 X 13 X 14 X 21 X 22 X 23 X 24 X 31 X 32 X 33 X 34 x ij ∼ N(μ ij, σ ij ) = N(μ ij, λ ij μ ij )

7. Example: 3 Cash Sources and 4 Time Periods 1 2 3 4 [ 1 ] 1 2 3 4 [ 2 ] 1 2 3 4 [ 3 ] X 11 X 12 X 13 X 14 X 21 X 22 X 23 X 24 X 31 X 32 X 33 X 34 x ij ∼ N(μ ij, σ ij ) = N(μ ij, λ ij μ ij ) ρ1ρ1 ρ1ρ1 ρ1ρ1

8. Example: 3 Cash Sources and 4 Time Periods 1 2 3 4 [ 1 ] 1 2 3 4 [ 2 ] 1 2 3 4 [ 3 ] X 11 X 12 X 13 X 14 X 21 X 22 X 23 X 24 X 31 X 32 X 33 X 34 x ij ∼ N(μ ij, σ ij ) = N(μ ij, λ ij μ ij ) ρ1ρ1 ρ1ρ1 ρ1ρ1 ρ 12 ρ 23

9. Example: Structure Of Correlation Matrix

10. 1 2 3 4 [ 1 ] 1 2 3 4 [ 2 ] 1 2 3 4 [ 3 ] X 11 X 12 X 13 X 14 X 21 X 22 X 23 X 24 X 31 X 32 X 33 X 34 1 2 3 4 [ D ] d1d1 d2d2 d4d4 d3d3 D=(d 1, …, d N ) t is deterministic. Z= (z 1, …, z N ) t where z i = x 1i +x 2i + … + x Mi (i=1, …, N) represents the total cash flow (from all sources) If for any j (j=1, …, N) d j > z j the company defaults DEBT

11. Example: 3 Cash Sources and 4 Time Periods 1 2 3 4 [ 1 ] 1 2 3 4 [ 2 ] 1 2 3 4 [ 3 ] X 11 X 12 X 13 X 14 X 21 X 22 X 23 X 24 X 31 X 32 X 33 X 34 X*= (x 1 *, …, x 12 * ) t = (x 11, x 12, x 13, x 14, x 21, x 22, x 23, x 24, x 31, x 32, x 33, x 34 ) t X* ∼ MN(μ*, C*) Z=(z 1, z 2, z 3, z 4 ) t = (x 11 +x 21 + x 31, x 12 +x 22 + x 32, x 13 +x 23 + x 33, x 14 +x 24 + x 34 ) t

12. Solution: A Simulation Technique An efficient technique to tackle the problem at hand is via a Monte Carlo simulation approach. This technique reduces to generating a family of X* vectors satisfying the condition X* ∼ MN(μ*, C*) (Details regarding the algorithm, see paper/report)

13. Example of Application Simple company, M=3 and ten time periods (N=10). Source 1. For i=1, …, 10; E(x 1i ) = μ 1i = 25; λ 1i =0.3; and ρ 1 = 0.3 Source 2. For i=1, …, 10; E(x 2i ) = μ 2i = 10; λ 2i =0.5; and ρ 2 = 0.3 Source 3. For i=1, …, 10; E(x 3i ) = μ 3i = 60; λ 3i =0.05; and ρ 3 = 0.3 In addition we assume that ρ 13 = ρ 12 = ρ 23 = 0.2. Debt Payments d i = 100 β (i=1, …, 10) where 100 is a basic reference value and β represents a scaling factor. The goal is to examine for values of β between 0 and 1 the default likelihood of the company

14. Default probabilities: The table shows, for different values of β, the likelihood that the company might default on each period. P-of-Def is the overall default probability (the sum of the period-by-period default probabilities).

15. Sensitivity Analysis