Transcript Comments on Kiefer, Small Probabilities… Some Entropic Extensions

Default Modeling :
Some Entropic Comments
With Reference to Talks By
Choi and Glennon
Mike Stutzer
Prof. of Finance
University of Colorado
Example: Standard Default Loss Model with
Independent Defaults
d  BinaryDefaultIndicator
r  # of Defaults =  di

i
Probability of a Single Default
n!
r
nr
 (1   ) i.e. Bernoulli Default Trials
p(r |  ) 
r ! n  r !
k ck ( )
Post ( r ) : p(r  ) prior ( ); with prior ( ) ~ e k
•In some of his work, Kiefer used MaxEnt to constrain prior to be consistent with
Expert Partial Info. on probabilities, expressed as expectations of indicator
functions of 
•Or, one could assume a “base” parametric prior (e.g. a Beta distribution with
posited parameter values), and mimimize over parameters to find constrained
relative entropy distribution closest to it. Because a Dirichlet Distribution is sortof a multivariate Beta distribution, this is reminiscent of Prof. Choi’s idea using
the infinite dimensional generalization: Dirichlet Process Models.
Note that:
1
n!
Prob[# Defaults  r ]  
 r (1   ) n r Post ( )d
r ! n  r  !
0
The calculation requires a continuum of binomials.
Another Example: The 1-Factor Model
Vi 
 S  1   Zi , i  1,, n
1. Debtor defaults when cash flow Vi < K (similar to Choi)
2. S is a N(0,1) latent state of nature
3. Zi is a N(0,1) idiosyncratic shock
 ( s )  Prob Vi  K | S  s 

 K  s 
K  s
 Prob  Z i 

 =  
1   

 1  

n!
Prob[r ]  
 ( s)r (1   ( s))nr d ( s)
 r !(n  m)!
 K  s 
n!



 r !(n  r )! 
 1  

r

 K  s 
1   
 

 1   

nr
d ( s)
So here, there is an infinite number of states s, each indexing
a possible individual default probability  ( s )
•This calculation also requires a continuum of binomials.
•Generates fat-tailed default distribution (via subexponential integral)
I did a Google search to find the internet’s verdict:
“Blaming Quants for Dummies:
The Formula that Killed Wall Street”
“It's all David Li's fault apparently. That bastard!”
Another Well-Known Mixture Model: Regime Switching
•m-latent states
s1...sm index m-normally distributed variates Yi  N (i ,  i 2 )
• Markov Process determines evolution of states over time
• Again, default occurs when Y < K
1. Classical Approach: Max. Likelihood to jointly determine Markov
transition matrix and parameters of normal distributions
2. Bayesian Approach: Described in Geweke’s (2005) text, pp.233-244
• Uses conjugate priors for normal distribution’s parameters, as
Prof. Choi does by adopting the Dirichlet Process Model
• While analyst commits to a finite number of states m, specification
search proceeds by trying m =2, then m = 3, etc.
Prof. Choi’s paper:
Based on the Infinite Mixture Normal DPM
• Is the extra flexibility enabled by an infinite mixture needed?
•If so, then Choi’s entropic method of incorporating expert
prior information is relevant to practice, in the spirit of
Kiefer’s work presented at an earlier InfoMetrics Conference
•But note that Geweke student G. Chang found that m = 5
states is adequate to represent the S&P 500 index. So it
would be interesting to do something similar with the stock
portfolio in Prof. Choi’s paper.
A Well-Known Alternative: Logit Defaults
t  h
X t
e

X t
1 e
Where the vector Xt contains observable individual borrower correlates
(e.g. FICO score), as well macroeconomic state variables correlated
with default (e.g. unemployment rate)
Glennon, et.al. usefully extend the model to time-varying
coefficients via Kernel estimation:
ˆ t  arg max  
1 T  s t 
k
ls

T s 1  Th 
where k  . is the one-sided kernel function, used to smooth coefficient
evolution over time while increasing the effective number of observations
at each time (h is the bandwidth and ls is the log likelihood)
A Challenge for Authors of Both Papers
• Consider incorporating information from financial
markets (e.g. credit default swap prices, prediction
futures markets ) into the estimation process.
– Markets aggregate seemingly expert opinions into
prices, and subsequently parameters by “inversion”
• Expert opinion may already be doing that. So incorporating
some experts’ opinion into priors is an ad-hoc approach.
– This accounts for the popularity of the 1-factor model
• but maybe that didn’t work so well with subprime loans…
– Breaks the reliance on what may not be abundant
nor accurate nor relevant historical information.