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[yankeefan]

I am probably assuming that we don't have the data to derive the steady state yet, with too few cases reaching the ultimate (defaulting) state out of those in the penultimate (4+) state.

Of course, I could be wrong (again)

[cubbiesnextyr]

Well, isn't a property of a Markov chain the ability to derive the steady state equilibrium and determine the % of loans that ultimately end up in each absorbing state?

Sometimes when I'm reading things I don't understand, there's one defining sentence that I can point to and say "That one, I don't understand a single word of it." 

For this discussion, this would be that line.

[lenderguy]

Sometimes when I'm reading things I don't understand, there's one defining sentence that I can point to and say "That one, I don't understand a single word of it." 

For this discussion, this would be that line.

I had my money on mtnchick not understanding a single word of it.  Shoulda guessed it would be "debits on the left credits on the right and life is good" guy.  (Despite the fact that if your assets equal your liabilities, mathematically, it doesn't really matter what side they go on as long as you're consistent.)

Here's the excutive summary: The English version of that sentence you don't understand is, "Given our ability to predict next month's loan performance, shouldn't we be able to determine the long run % of loans that will bk, default, pay off early, or payoff on time?"

The explanation is as follows, for those who desire a more in-depth explanation:

A discrete-time (things change at specific time points) Markov chain is something that can be modeled where something is in a specific state and has constant probability of moving to another state.  In this example, loans more or less change states on a monthly basis.  You can figure out the odds of a loan staying current or going late.  Then from going late to really late, and eventually default.  The current and all forms of late states are called transient states, because over the long run, a loan will never stay there.  Eventually, loans get paid off, default, or file bk.

An absorbing state is a state that once entered is never left.  This would be the paid off state, defaults, or bk.  See that chart I have above?  See how all of the rows have a percentage that is less than 1 (or less than 100%) for each combination?  Well, mathematically, I would extend the rows of the table so that everything in the columns appears in the rows.  The rows with Early PIF, FT PIF, BK, and Default would have a "1" in the corresponding column.  That means once a loan enters the payoff, bk, and default states it stays that way.  You might say "well duh" but that does have some important mathematical properties.

Once we have that matrix filled out, we now have a square matrix (# of rows = # of columns).  We can multiply square matrices together (excel has an mmult function).  If I multiplied that matrix by itself, I could then find the odds of the third payment being in a given state (current/late/bk/whatever) given its initial state.  Multiply the matrix by itself again, and I can find the odds that a fourth payment will be in a specific state given an initial state.  I can also find the oddds that something that is <15 days late will become current on the 10th payment.  Follow me so far?

Well, if you keep multiplying that matrix by itself enough times, you get to the point where two things happen.  First, ALL loans will eventually bk, pay off, or default.  The entries in all of the other columns will be 0.  Second, the entries in each column are homogenous within the column -- that is, they're all the same.  In the long run, we simply ask, "what are the odds of a loan going bk, default, early payoff or payoff on time?"  It doesn't matter if we look at the loan being <15 , late, or even 4+ right now.  All responses would be the same.  When I get around to fine tuning my chart, I might be able to show you some of that.  The problem right now is that we don't have any full term loans paid off, so I don't know how to incorporate that into the analysis.

[ira01]

An absorbing state is a state that once entered is never left.  This would be the paid off state, defaults, or bk.  . . . That means once a loan enters the payoff, bk, and default states it stays that way.  You might say "well duh" but that does have some important mathematical properties.

Just so you know, this isn't ncessarily true for BK.  It is possible for a loan to be marked BK, and then later come out of BK status.  This would happen if the court dismissed the borrower's BK, for example, or if Prosper ever took the steps to get a fraudulently obtained loan declared non-dischargeable, or if the BK was a Chapter 13 and the Borrower paid off the loan as part of his/her Plan (although I am not sure whether Prosper would remove the BK status as the borrower made the payments, or just at the end once all payments were made).  In addition, theoretically a BK borrower could decide to reaffirm the Prosper debt, which would also take it out of BK.  As far as I know, the second possibility has never happened (although it should), and the fourth is highly unlikely (except, perhaps, in a special case like the Malama Ohana BK).  But the first and third are quite real possibilities, that may well have already happened.

[Urbi_et_Orbi]

Interesting stuff.  Keep it up.

[Mtnchick]

I had my money on mtnchick not understanding a single word of it. 

Oh, fear not good friend. You know me well. I don't understand a single word of it

[traveler505]

Sometimes when I'm reading things I don't understand, there's one defining sentence that I can point to and say "That one, I don't understand a single word of it." 

For this discussion, this would be that line.

I had my money on mtnchick not understanding a single word of it.  Shoulda guessed it would be "debits on the left credits on the right and life is good" guy.  (Despite the fact that if your assets equal your liabilities, mathematically, it doesn't really matter what side they go on as long as you're consistent.)

Here's the excutive summary: The English version of that sentence you don't understand is, "Given our ability to predict next month's loan performance, shouldn't we be able to determine the long run % of loans that will bk, default, pay off early, or payoff on time?"

The explanation is as follows, for those who desire a more in-depth explanation:

A discrete-time (things change at specific time points) Markov chain is something that can be modeled where something is in a specific state and has constant probability of moving to another state.  In this example, loans more or less change states on a monthly basis.  You can figure out the odds of a loan staying current or going late.  Then from going late to really late, and eventually default.  The current and all forms of late states are called transient states, because over the long run, a loan will never stay there.  Eventually, loans get paid off, default, or file bk.

An absorbing state is a state that once entered is never left.  This would be the paid off state, defaults, or bk.  See that chart I have above?  See how all of the rows have a percentage that is less than 1 (or less than 100%) for each combination?  Well, mathematically, I would extend the rows of the table so that everything in the columns appears in the rows.  The rows with Early PIF, FT PIF, BK, and Default would have a "1" in the corresponding column.  That means once a loan enters the payoff, bk, and default states it stays that way.  You might say "well duh" but that does have some important mathematical properties.

Once we have that matrix filled out, we now have a square matrix (# of rows = # of columns).  We can multiply square matrices together (excel has an mmult function).  If I multiplied that matrix by itself, I could then find the odds of the third payment being in a given state (current/late/bk/whatever) given its initial state.  Multiply the matrix by itself again, and I can find the odds that a fourth payment will be in a specific state given an initial state.  I can also find the oddds that something that is <15 days late will become current on the 10th payment.  Follow me so far?

Well, if you keep multiplying that matrix by itself enough times, you get to the point where two things happen.  First, ALL loans will eventually bk, pay off, or default.  The entries in all of the other columns will be 0.  Second, the entries in each column are homogenous within the column -- that is, they're all the same.  In the long run, we simply ask, "what are the odds of a loan going bk, default, early payoff or payoff on time?"  It doesn't matter if we look at the loan being <15 , late, or even 4+ right now.  All responses would be the same.  When I get around to fine tuning my chart, I might be able to show you some of that.  The problem right now is that we don't have any full term loans paid off, so I don't know how to incorporate that into the analysis.

Sometimes when I'm reading things I don't understand, there are four defining paragraphs that I can point to and say "That one, I don't understand a single word of it." 

[lenderguy]

Sometimes when I'm reading things I don't understand, there's one defining sentence that I can point to and say "That one, I don't understand a single word of it." 

For this discussion, this would be that line.

I had my money on mtnchick not understanding a single word of it.  Shoulda guessed it would be "debits on the left credits on the right and life is good" guy.  (Despite the fact that if your assets equal your liabilities, mathematically, it doesn't really matter what side they go on as long as you're consistent.)

Here's the excutive summary: The English version of that sentence you don't understand is, "Given our ability to predict next month's loan performance, shouldn't we be able to determine the long run % of loans that will bk, default, pay off early, or payoff on time?"

The explanation is as follows, for those who desire a more in-depth explanation:

A discrete-time (things change at specific time points) Markov chain is something that can be modeled where something is in a specific state and has constant probability of moving to another state.  In this example, loans more or less change states on a monthly basis.  You can figure out the odds of a loan staying current or going late.  Then from going late to really late, and eventually default.  The current and all forms of late states are called transient states, because over the long run, a loan will never stay there.  Eventually, loans get paid off, default, or file bk.

An absorbing state is a state that once entered is never left.  This would be the paid off state, defaults, or bk.  See that chart I have above?  See how all of the rows have a percentage that is less than 1 (or less than 100%) for each combination?  Well, mathematically, I would extend the rows of the table so that everything in the columns appears in the rows.  The rows with Early PIF, FT PIF, BK, and Default would have a "1" in the corresponding column.  That means once a loan enters the payoff, bk, and default states it stays that way.  You might say "well duh" but that does have some important mathematical properties.

Once we have that matrix filled out, we now have a square matrix (# of rows = # of columns).  We can multiply square matrices together (excel has an mmult function).  If I multiplied that matrix by itself, I could then find the odds of the third payment being in a given state (current/late/bk/whatever) given its initial state.  Multiply the matrix by itself again, and I can find the odds that a fourth payment will be in a specific state given an initial state.  I can also find the oddds that something that is <15 days late will become current on the 10th payment.  Follow me so far?

Well, if you keep multiplying that matrix by itself enough times, you get to the point where two things happen.  First, ALL loans will eventually bk, pay off, or default.  The entries in all of the other columns will be 0.  Second, the entries in each column are homogenous within the column -- that is, they're all the same.  In the long run, we simply ask, "what are the odds of a loan going bk, default, early payoff or payoff on time?"  It doesn't matter if we look at the loan being <15 , late, or even 4+ right now.  All responses would be the same.  When I get around to fine tuning my chart, I might be able to show you some of that.  The problem right now is that we don't have any full term loans paid off, so I don't know how to incorporate that into the analysis.

Sometimes when I'm reading things I don't understand, there are four defining paragraphs that I can point to and say "That one, I don't understand a single word of it." 

Cubbies was kind enough to highlight the single sentence that stymied him.  You, OTOH, have to make me guess which four?  Thanks

[Fred93]

I captured state changes as they occurred.  In that regard, I errored when I said these are monthly roll rates,

I think that may be a problem.  I think to use this matrix to analyze loan performance we need it to be based on a fixed time scale, such as monthly.  If it were based on data sampled monthly, I could for example raise the matrix to the 36th power to see what happens at the end of a loan.

Actually I have to do more than that, because a loan can default in up to about the 45th month, given the way prosper handles transitions to the default state.  I can't simply raise the matrix to the 45th power, however, because after the 36th month, loans will never make some of the transitions such as from 1 month late to current.  A loan in the 38th month will never transition to 15 days late.  Unfortunately, the matrix contains these transitions, so does not describe what happens after the 36th month.  You can get to 36 months by multiplying the matrix, but after 36 months you have to multiply by a modified matrix. 

I ignored these details for a first try.  I added some rows for the terminal states to make the matrix square, and raised it to the 36th power.  Several things don't look right in the result.  For example the fraction of loans ending in default looks too low.  I'm not sure of the cause, but I found your statement above which seems to indicate that the data is not sampled on a fixed interval, and that could be part of the problem, presuming that I understand what you said.

The next difficulty is that the problem is not set up in a way that makes the trivial sort of analysis work right.  You can't just raise this matrix to a high power and expect a meaningful asymptotic result.  If you do that all the loans will default.  The setup of the problem contains no understanding of the fact that loans have finite duration, and that after they end they can no longer go late or default.

This problem can be fixed various ways.  One way is to raise the matrix to a finite power representing the length of the loan and then doing something different for the extension of a few months while prosper waits around for that last payment. 

The other thing one could do is to have more states so that the finite-state machine representation of the loan contained the knowledge of where you are in the loan.  It isn't obvious how to do that without an explosion in the number of states.  Obviously you'd have to expand "current" into 36 states, one for each month, but it seems to me that you'd also have to expand the late states into multiple states so that where you are in the loan's payment schedule is maintained even when loans are late.  Seems like if you did that then you could just raise the matrix to a high power and get a meaningful answer.  But would you want to do that?  Mathematically simple but gee there would be a lot of numbers.

I don't claim these comments are authoritative.

[Fred93]

Here's the 36th power of the original matrix:

Code: [Select]

         Current   <15   Late   1 mo   2 mo   3 mo   4 mo  earlyPIF  Ftpif    BK    default
Current  37.16%   3.17%  1.55%  1.19%  0.92%  0.82%  2.57%   28.85%  0.00%   1.06%  22.54%
<15      31.82%   2.71%  1.33%  1.02%  0.79%  0.70%  2.20%   23.01%  0.00%   1.62%  34.56%
Late     24.41%   2.08%  1.02%  0.78%  0.61%  0.54%  1.69%   17.10%  0.00%   2.30%  49.35%
1 mo     17.19%   1.47%  0.72%  0.55%  0.43%  0.38%  1.19%   11.61%  0.00%   2.96%  63.43%
2 mo     10.60%   0.90%  0.44%  0.34%  0.26%  0.23%  0.74%   6.79%   0.00%   3.54%  76.10%
3 mo      7.55%   0.64%  0.32%  0.24%  0.19%  0.17%  0.53%   4.70%   0.00%   3.69%  81.83%
4 mo      5.60%   0.48%  0.23%  0.18%  0.14%  0.12%  0.39%   3.47%   0.00%   3.41%  85.94%
earlyPIF  0.00%   0.00%  0.00%  0.00%  0.00%  0.00%  0.00% 100.00%   0.00%   0.00%   0.00%
Ftpif     0.00%   0.00%  0.00%  0.00%  0.00%  0.00%  0.00%   0.00% 100.00%   0.00%   0.00%
BK        0.00%   0.00%  0.00%  0.00%  0.00%  0.00%  0.00%   0.00%   0.00% 100.00%   0.00%
default   0.00%   0.00%  0.00%  0.00%  0.00%  0.00%  0.00%   0.00%   0.00%   0.00% 100.00%




[nonattender]

Multiple iterations of the calculation you describe will need to be normalized.

-t

[lenderguy]

The next difficulty is that the problem is not set up in a way that makes the trivial sort of analysis work right.  You can't just raise this matrix to a high power and expect a meaningful asymptotic result.  If you do that all the loans will default.  The setup of the problem contains no understanding of the fact that loans have finite duration, and that after they end they can no longer go late or default.

I'm not sure why all loans default.  Some should be in early payoff, some should BK, some should default, and some should pay off on schedule.  What this matrix also doesn't do is convey an understanding that some loans will successfully pay off at the 36th month.  I haven't quite figured out how to analyze that.

I'm not a big fan of reposting what people write to me privately, but in this case, I don't think yankeefan will mind.  He suggested that in his line of work, a lot of analysis for long term loans is done early in the loan life, and a lot of predictions are made.  So, it *is* proper to do an analysis at this stage of the game.  He suggested I consult published literature to see how to properly handle it.

The other thing one could do is to have more states so that the finite-state machine representation of the loan contained the knowledge of where you are in the loan.  It isn't obvious how to do that without an explosion in the number of states.  Obviously you'd have to expand "current" into 36 states, one for each month, but it seems to me that you'd also have to expand the late states into multiple states so that where you are in the loan's payment schedule is maintained even when loans are late.  Seems like if you did that then you could just raise the matrix to a high power and get a meaningful answer.  But would you want to do that?  Mathematically simple but gee there would be a lot of numbers.

I need to consult my prof and see what he suggests I do.  The more I think about it, the more I realize that I probably do have to include a whole bunch of states to model this properly.

I'm also going to tweak my code to examine things at a fixed time interval. 

[Fred93]

I'm not sure why all loans default.  Some should be in early payoff, some should BK, some should default, and some should pay off on schedule. 

I said it wrong.  Let me try again.  If you raise your matrix to higher and higher powers, the fraction of loans that show default will keep going up (to a ridiculous degree).  That's 'cause the matrix doesn't know about the fact that loans become paid normally at 36 months.  If you go past the 36th power, the matrix will keep throwing more and more loans into the default state.

What this matrix also doesn't do is convey an understanding that some loans will successfully pay off at the 36th month.  I haven't quite figured out how to analyze that.

Precisely.

I'm not a big fan of reposting what people write to me privately, but in this case, I don't think yankeefan will mind.  He suggested that in his line of work, a lot of analysis for long term loans is done early in the loan life, and a lot of predictions are made.  So, it *is* proper to do an analysis at this stage of the game. 

I agree with yankeefan.  We have to do the analysis now, because the alternative is to wait many years!   

I need to consult my prof and see what he suggests I do.  The more I think about it, the more I realize that I probably do have to include a whole bunch of states to model this properly.

Not something to be done without some serious thinking.  For example, if you put in states to make the matrix understand where one is in the 36 month evolution of the loan, you'd have this big 'ol problem that your input data set has no data for the late months in the loan!   You're (intentionally) taking data from early in loans and you want the model to extrapolate from this, assuming that the state transition probabilities will be similar late in the loan.  You could do a fixup by using your (same) data to populate multiple entries in the matrix of course.  But if you're gonna do such fixups, maybe there's a better way.

I'm also going to tweak my code to examine things at a fixed time interval. 

Great!

The math that one should do using this matrix depends on what one is trying to get.

For example, if you want an answer to "What fraction of loans will default?", then you want the math to cycle thru the 36 months of the loan plus a few special cycles at the end (to handle prosper's delayed default), and then you look at the current row and default column.  That says if you start current (as all loans do) and cycle thru the entire loan, then what fraction default?

But consider another question ... one that comes up when evaluating things like "How much does a "late" devalue my portfolio? "  The right question here might be "Given that a loan goes 1 month late, what is the probability that it ends up in default?"  (I've used the collection agency stats to estimate that, but I seek affirmation from another source.)  That's harder, because I can't run the model the full 36+ cycles in this case, because loans don't go "1 month late" right at the start.  The answer for the probability of going from the "1 month late" state to the "default" state is DIFFERENT for different numbers of cycles of the model.  How the heck do I handle that?  I could get 40 or so different answers.  I don't know the answer.


 

[yankeefan]

I'm not a big fan of reposting what people write to me privately, but in this case, I don't think yankeefan will mind.  He suggested that in his line of work, a lot of analysis for long term loans is done early in the loan life, and a lot of predictions are made.  So, it *is* proper to do an analysis at this stage of the game.  He suggested I consult published literature to see how to properly handle it.

No problem.

Your fixed time interval tweak can also handle the fact that loans stay in the 4+ late state for different amounts of time, and (I think) exhibit different roll rates through that period.  At the least, as Fred93 notes, the defaults come along several months after going to 4+.

after understanding the data you have, you could introduce some assumed roll rates for use "in the field", like assuming that the 36month current to PIF is 100%, or 100% less the latest roll to late rate you have.

[lenderguy]

Your fixed time interval tweak can also handle the fact that loans stay in the 4+ late state for different amounts of time, and (I think) exhibit different roll rates through that period.  At the least, as Fred93 notes, the defaults come along several months after going to 4+.

FWIW, I'm not limited to Prosper's categories in my state definitions.  For instance, in the data dump, Prosper doesn't even use "<15 days late."  Likewise, I'm also not limited to using "4+" as my last late phase.  I can extend it to show "8 months late" if I want to.

I talked to someone in our finance department about the number of states I should use, and he told me I'm not wasting my time by developing a really large matrix that includes all of the states.