## Monday, January 14, 2013

### High level formula I use for calculating returns

1. Calculate an estimated loss rate for each of the different "Late" categories based on historical data.
2. Calculate the gross return of each individual loan.
3. Calculate the estimated rate of loss for each individual loan.
4. For all the loans in any particular query I then calculate the dollar weighted and time weighted average of both the gross return and annual loss.
5. Subtracting the time weighted average loss from the gross return to come up with an overall net return as a yearly APR.

There are a few nuances for each of these steps that people have asked about and so I am going to provide more details for each of these steps below.

### Calculation for loss rate of "Late" categories

To come up with an estimated loss rate I first found the percentage of loan value that has historically been lost during a default, this was about 85%.  Then I calculated how often a loan goes to charge off after going late.  I calculated this separately for each late value and rounded off the results for ease of use (since this is an estimate).  The charge off rates are about 60% for late, 85% for 1 month late, 90% for 2 months late, and 95% for 3+ months late.  I multiplied the two together to come up with a loss estimate.  For example, if you had a loan with \$10.00 in principal remaining and it was Late, I would estimate a loss of \$5.10 (\$10.00 * .6 * .85). The same loan at 3 Months late would be an estimated loss of \$8.08 (\$10.00 * .95 *.85).  Anything that has defaulted gets completely written off.

### Calculation for gross return and rate of loss

The annual rate of return and annual rate of loss for any loan is not as straight forward of a calculation as one would think.  First off, Prosper does not provide the actual return for any loan in the data export, what they do provide is a history of the principal balance.  Using this history I go back and calculate an estimate of how much interest was accumulated and payed out between each of the changes in principal balance.  I then take the collected interest and average daily balance to generate a gross return for the loan.  I do not take accrued interest into account in any of my calculations.  The calculation pretends like the most recent month doesn't exist until that month's payment is either made or missed.

The loss rate is calculated in a similar manner but the estimated loss in principal (based on the formula above in "Late" categories) is used against the average daily balance instead of the accumulated interest to come up with an estimated loss rate.

### Dollar weighted and time weighted averages

Dollar weighted returns are turned on by default, but the loan analysis tool has a check box to turn them off and I highly recommend you think about what it is you are trying to calculate/analyze to figure out if dollar weighting the return is appropriate for you.  Dollar weighting the returns will make it so that a \$10,000 loan will have more of an impact on the total return than a \$5,000 loan will.  This might be appropriate if you are looking at an individual investors portfolio and you want to see the return based on how much they actually invested in each loan.  However if you are trying to come up with an investing scheme and you plan to buy \$25 notes (or any dollar amount that is not a percentage of the total loan amount), then you might want to analyze the data without dollar weighted averages, since your note sizes will not be weighted by the size of the loan.

Time weighting is what converts the return and loss rate into a yearly rate.  This can have interesting effects for notes that are very young, especially when calculating a loss estimate.  Take the example of a note that never pays and defaults at month 4.  It has a 100% loss over 4 months that when converted to a yearly rate of loss would be 300%.  Obviously it is impossible to lose 300% on any individual note, so taken out of the larger context of calculating a return for a pool of loans, this might look like something is wrong.  However in the context of a larger pool of notes, this behavior is actually correct.  It is helping to estimate future losses that will occur in other notes before the year is out.  So if this note was added in to a portfolio of young HR notes that were all current, it balances out the estimated 30% return those notes have, to give you a more realistic view of what you can expect the returns for your portfolio to be.  As you can see, the accuracy level of the estimated yearly return can be quite suspect for extremely young loans or for a very small subset of loans, however the accuracy level becomes more reliable when the size and age of the loan pool being analyzed increases.

### The final net return (which should be looked at along with the loss estimate)

This is the easiest part of the whole calculation.  You just take the yearly gross return and subtract the loss estimate to get the net return.  Nothing magical happens here.  The only reason this step stands out as significant is because the intermediate loss estimate that goes into calculating the net return is a meaningful number that should be taken into account along with the net return when analyzing the data.   Consider a pool of loans returning 8% with a loss estimate of 0% and a pool of loans returning 8% with a loss estimate of 8%.  Which is better?  Both are netting you the same dollar amount each month, but they are getting there differently and you should understand how changes to the APR of new loans at Prosper will impact each of these portfolios differently.  Also if you are not in a tax sheltered account the way losses are handled on your taxes can make the after tax return for each of these two portfolios different.

I hope that all makes sense.  If you have any questions and/or would like clarification on anything I mentioned feel free to ask.

#### 1 comment:

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