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A Cash Flow Based-Approach To Estimate Default Probabilities

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.

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A Cash Flow Based-Approach To Estimate Default Probabilities

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  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 Mcash sources and consider Ntime periods We denote as Xi (i=1, …, M) the vector associated with the cash flows generated by source iat times t1, …, tN. Thus, Xi = (xi1, …, xiN)t. [ xij refers to the cash flow generated by source i at time tj. ] We assume that the vectors Xi follow multi-normal distributions, Xi ∼MN(μi, Ci), where μi represents the vector of expected values and Ci denotes the corresponding correlation matrix

  6. Example: 3 Cash Sources and 4 Time Periods X11 X12 X13 X14 [ 1 ] 1 2 3 4 X21 X22 X23 X24 [ 2 ] 1 2 3 4 X31 X32 X33 X34 [ 3 ] 1 2 3 4 xij∼N(μij, σij) = N(μij, λijμij)

  7. Example: 3 Cash Sources and 4 Time Periods ρ1 ρ1 ρ1 X11 X12 X13 X14 [ 1 ] 1 2 3 4 X21 X22 X23 X24 [ 2 ] 1 2 3 4 X31 X32 X33 X34 [ 3 ] 1 2 3 4 xij∼N(μij, σij) = N(μij, λijμij)

  8. Example: 3 Cash Sources and 4 Time Periods ρ1 ρ1 ρ1 X11 X12 X13 X14 [ 1 ] 1 2 3 4 ρ12 X21 X22 X23 X24 [ 2 ] 1 2 3 4 ρ23 X31 X32 X33 X34 [ 3 ] 1 2 3 4 xij∼N(μij, σij) = N(μij, λijμij)

  9. Example: Structure Of Correlation Matrix

  10. X11 X12 X13 X14 [ 1 ] 1 2 3 4 X21 X22 X23 X24 [ 2 ] 1 2 3 4 X31 X32 X33 X34 [ 3 ] 1 2 3 4 d1 d3 d4 d2 DEBT [ D ] 1 2 3 4 D=(d1, …, dN)t is deterministic. Z= (z1, …, zN)t where zi= x1i +x2i+ … + xMi(i=1, …, N) represents the total cash flow (from all sources) If for any j (j=1, …, N) dj> zjthe company defaults

  11. Example: 3 Cash Sources and 4 Time Periods X11 X12 X13 X14 [ 1 ] 1 2 3 4 X21 X22 X23 X24 [ 2 ] 1 2 3 4 X31 X32 X33 X34 [ 3 ] 1 2 3 4 Z=(z1, z2, z3 ,z4)t = (x11+x21+ x31, x12+x22+ x32, x13+x23+ x33, x14+x24+ x34)t X*= (x1*, … , x12*)t = (x11, x12, x13, x14, x21, x22, x23, x24, x31, x32, x33, x34)t X*∼MN(μ*, C*)

  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(x1i) = μ1i = 25; λ1i=0.3; and ρ1= 0.3 Source 2. For i=1, …, 10; E(x2i) = μ2i = 10; λ2i=0.5; and ρ2= 0.3 Source 3. For i=1, …, 10; E(x3i) = μ3i = 60; λ3i=0.05; and ρ3= 0.3 In addition we assume that ρ13 = ρ12 = ρ23 = 0.2. Debt Payments di= 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

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