What an X% default rate should mean to you, part 3

Note from 3/30/06: We users have gotten mixed answers to questions addressed to Prosper.com re: default rate horizons -- that is, we've been told 3 years, but then a 2 year horizon was confirmed as correct. Throughout this exercise, I'm assuming 1 year horizons (as the most pessimistic possible). Clearly, if 19% of HR borrowers default every year, that's very different from saying that over 3 years, only 19% of HR borrowers default.

In fact, I do believe the 2 year horizon is the correct one (it seems consistent w/the 2-year horizon delinquency rates I got from Fair Isaac, inventors of FICO scores), but it doesn't hurt to double expected risk as a rule of thumb when dealing with uncharted territory. (end of note)


Note 2 from 3/31/06: Looks like I was right -- the 2 year terms were for annualized numbers, so my worst-case-scenario, is actually the correct one.(end of note)

An attempt at approximating 3 year default effects on returns:


1.one might assume (unrealistically, but easy-to-calulate-edly) that people's credit scores don't change over time, and that the age of a loan doesn't change probability of default.
2.Assume 3 year payments, w/monthly even payments throughout.
3.Assume E rating, 19% default, 36% interest.
4.Assume, even though ridiculous, that your reinvestment rate of the payments you receive before default occurs, is 36% as well.
5.Assume that once default occurs, there is 0% chance of you ever seeing your money again.

Even this crappy approximation will get fairly calculative -- not a problem, really -- but it makes it harder to explain every single step:

A)Let's generalize for any loan w/36% interest and 36 flat monthly payments of the same amount. To make this simple, make each payment $1. I'm going backwards to figure out "for what amount of money borrowed today at 36%, will one have to make 36 flat monthly payments of $1 each?)

If one discounts each of the payments at an effective 36% annual rate (which requires assumption 4, reinvestment at 36% as well), one gets a string of numbers of the present value of each payment as the following:

for month X,
$1 divided by {(1.36) raised to the (X over 12th) power}

1: $0.97
2: $0.95
3: $0.93
4: $0.90
5: $0.88
6: $0.86
7: $0.84
8: $0.81
9: $0.79
10: $0.77
11: $0.75
12: $0.74
13: $0.72
14: $0.70
15: $0.68
16: $0.66
17: $0.65
18: $0.63
19: $0.61
20: $0.60
21: $0.58
22: $0.57
23: $0.55
24: $0.54
25: $0.53
26: $0.51
27: $0.50
28: $0.49
29: $0.48
30: $0.46
31: $0.45
32: $0.44
33: $0.43
34: $0.42
35: $0.41
36: $0.40

which sum to $23.21. So, if one lends $23.21 at 36%, it should be repaid w/ 36 $1 monthly payments.

B)Sloppy assumption 2: Since we believe that every year there is a 19% chance of default, and everyone stays E rated forever, and past behavior is irrelevant to present behavior, what is the probability that someone doesn't default after 3 years?

81% raised to the third power, or (100%-19%)x(100%-19%)x(100%-19%)
=53%

So, the probability of default is 47%

C)Sloppy assumption 3: This is sloppy mathematically and conceptually, wheras previously I was just sloppy conceptually by making assumptions. For what I'm, about to do, I know there is an objective, correct (but painful to calculate) mathematical answer once I make the conceptual assumption -- but I'm not calculating it.

Still OK mathematically:
There's an 81% rate of survival over 1 year. What's the rate of survival over half a year? 90%. Why?

Say you have 100 identical loans, all uncorrelated, w/ 81% surviving over 1 year.
After the first half of the year, 90%, or 90, are left. Apply 90% survival rate to the remaining 90 over the next half year -- you get 90x90%= 81 left. So, of the original 100, 81 are left, giving you the 81% annual rate being equivalent to 90% half-yearly survival rate. What I implicitly did was take 0.81, or 81%, and take its square root to get 90%.

Similarly, if you want the monthly survival rate, you take the 12th root of 81%, which is 98.26%. If you really wanted, you could calculate at each step of the way how many of your 100 loans pay off, how many are gone, etc, and take the present value of it all.

About to make a mathematically incorrect assumption:
Instead, let me pretend that all the defaults occur precisely halfway through. I figure the early defaults balance out the late ones, etc, etc -- NOT true, but makes my life easier to pretend so.

D) Still with me? Ok, from part B, 47% default over the entire period. From part C, pretend they all default after 18 months.

So, after lending $23.21, you make
i)from defaulters, 18 $1 payments over the first 18 months. Multiply this by 47% (probabilty of default) to get 18 $0.47 payments.
ii)from nondefaulters, 36 $1 payments over all 36 months. Multiply by 53% (probability of nondefault) to get $0.53
iii)Add these 2 income streams together to get:
18 $1 payments over the first 18 months, and thereafter 18 more $0.53 payments.

Stick this string of numbers into a financial calculator or Microsoft Excel,
0 -$23.21 (initial outlay)
1 $1.00
2 $1.00
3 $1.00
4 $1.00
5 $1.00
6 $1.00
7 $1.00
8 $1.00
9 $1.00
10 $1.00
11 $1.00
12 $1.00
13 $1.00
14 $1.00
15 $1.00
16 $1.00
17 $1.00
18 $1.00
19 $0.53
20 $0.53
21 $0.53
22 $0.53
23 $0.53
24 $0.53
25 $0.53
26 $0.53
27 $0.53
28 $0.53
29 $0.53
30 $0.53
31 $0.53
32 $0.53
33 $0.53
34 $0.53
35 $0.53
36 $0.53

and ask it -- what % return does this represent? 1.13% or so per month. Annualized, you get a 14.5% return -- fairly respectable, though risk is also higher.

Part C), take 2: What if instead of making the (mathematically) incorrect assumption that all defaults occur exactly halfway through, lets go back to the monthly survival rate of 98.26%? What payments should we expect if we do the math properly?
after month 1: $1 x 98.26% =$0.9826
month 2: $0.9826x 98.26%=$0.9655
.
.
month n: $1 x [(98.26%) raised to the nth power]

Ask Excel what's my interest rate for the following cash flow:
0 -$23.21 (initial outlay)
1 $0.98
2 $0.97
3 $0.95
4 $0.93
5 $0.92
6 $0.90
7 $0.88
8 $0.87
9 $0.85
10 $0.84
11 $0.82
12 $0.81
13 $0.80
14 $0.78
15 $0.77
16 $0.76
17 $0.74
18 $0.73
19 $0.72
20 $0.70
21 $0.69
22 $0.68
23 $0.67
24 $0.66
25 $0.64
26 $0.63
27 $0.62
28 $0.61
29 $0.60
30 $0.59
31 $0.58
32 $0.57
33 $0.56
34 $0.55
35 $0.54
36 $0.53

and you get an interest rate of 0.81% per month, annualizing to a 10.16% return.

Now, one should probably inch this up because technically, default might mean something much milder than you never seeing another cent -- it might mean someone's 30 days late, it might mean you have to sell your nonperforming loan at a (huge) discount, etc. But even so, keep in mind the raw numbers as you bid...