From the lender's perspective:
However, this isn't immediate:
$$ \begin{align*} P_0 \utif^ T \ne c T \end{align*} $$The resolution is straightforward:
$P_0$ is lent at time 0.
$P_1$ = $\utif P_0 - c \qquad$ At time 1, $\utif P_0$ is owed and $c$ is paid.
Recursing,
$$ \begin{align*} P_k & = \utif P_{k-1} - c \\ & = \utif \left[ \utif P_{k-2} -c \right] - c \\ & = \utif^k P_0 - c \sum_{i=0}^{k-1} \utif^i \end{align*} $$At time $T$, $P_T = 0$: the loan is paid off.
$P_T = 0$ implies:
$$ \begin{equation*} \utif^T P_0 = c \sum_{i=0}^{T-1} \utif^i \end{equation*} $$Note, each payment must be reinvested.
$$ \begin{align*} \mathcal{F}(y) & = \prod_{i:t_i < y} \left( 1 - h_i \right) \end{align*} $$
Parametric modeling is often via the hazard function.
Add in censoring:
$d_i = 1$ indicates an observation, $y_i$, is an uncensored failure time; $d_i = 0$, a censored failure time
$$ \begin{align*} l(\theta) & \equiv \sum_{i \in \mathcal{U}} \log f(y_i; \theta) + \sum_{i \in \mathcal{C}} \log \mathcal{F}(y_i) \\ \end{align*} $$When failure is possible only at discrete times,
Set $\frac{\partial l(\theta)}{\partial h_k} = 0$ to recover Kaplan-Meier estimator.
If discretized in time:
For each $t_k$, a 2 by 2 contingency table:
Group A | Group B | Total | |
---|---|---|---|
failed | \begin{equation*} a_{k,1} \end{equation*} | $a_{k,2}$ | \begin{equation*} a_{k,1} + a_{k,2} \end{equation*} |
survived | \begin{equation*} r_{k,1} - a_{k,1} \end{equation*} | \begin{equation*} r_{k,2} - a_{k,2} \end{equation*} | \begin{equation*} (r_{k,1} + r_{k,2}) - (a_{k,1} + a_{k,2}) \end{equation*} |
total | \begin{equation*} r_{k,1} \end{equation*} | \begin{equation*} r_{k,2} \end{equation*} | \begin{equation*} r_{k,1} + r_{k,2} \end{equation*} |
Test statistic under null hypothesis is hypergeometric:
$$ A_{k,1} \sim \frac{ {{a_{k,1} + a_{k,2}} \choose a_{k,1}} {{(r_{k,1} + r_{k,2}) - (a_{k,1} + a_{k,2})} \choose {r_{k,1} - a_{k,1}}} } { r_{k,1} + r_{k,2} \choose r_{k,1} } $$
Log rank statistic: aggregate across the time index, $\mathcal{T}$: $$ Z = \frac{ \sum_{k \in \mathcal{T}} (A_{k,1} - \Ev{A_{k,1}}) }{ \sqrt{ \sum_{k \in \mathcal{T}} \Var{A_{k,1}} } } $$
For a cash flow with no risk of default, IRR is the $r$ that satisfies the earlier arbitrage condition:
$$ \begin{align*} \utif^T P_0 = c \sum_{i=0}^{T-1} \utif^i \end{align*} $$We write the present value of the cash flow as
$$ \begin{align*} P_0 = \frac{1}{\utif}c + \frac{1}{\utif^2}c + \cdots + \frac{1}{\utif^T}c \end{align*} $$and the idea is to appropriately discount each future installment.
Let $W_T^{(0)}$ be a portfolio with present value \$1; annualized interest rate, $r$; an installment of $f = c / P$, and $T$ payments remaining.
We can apply the above argument recursively:
$$ \begin{align*} 1 & = \Ev{ W_T^{(0)} } \\ & = d_{1,0} f + d_{2,1} d_{1,0} f + \cdots + \left( d_{T,T-1}\cdots d_{1,0} \right) f \end{align*} $$yielding the risk-neutral discount terms
$$ d_{k,0} = d_{k,k-1}\cdots d_{1,0} = \frac{ \prp{ T^{(0)} \geq k } }{ \utif^k } $$We can now meaningfully compute a risk-neutral IRR, $r^*$, for the weighted cash flow:
$$ \begin{align*} P_0 = \frac{\prp{ Y^{(0)} \geq 1 } }{\rnutif}c + \frac{\prp{ Y^{(0)} \geq 2}}{\rnutif^2}c + \cdots + \frac{\prp{ Y^{(0)} \geq T}}{\rnutif^T}c \end{align*} $$Trivial from KM curve, $\prp{ Y \geq T }$
Potentially useful as:
column | description |
---|---|
loan-amnt | loan amount requested |
funded-amnt | loan amount funded |
int-rate | interest rate on loan |
installment | monthly payment |
grade | loan quality grade |
sub-grade | loan quality subgrade |
purpose | loan category: DEBT-CONSOLIDATION, MEDICAL, ETC. |
emp-length | employment length |
home-ownership | home ownership status: RENT, OWN, MORTGAGE, OTHER |
annual-inc | self reported annual income |
is-inc-v | income verified |
fico | FICO score |
dti | debt to income ratio |
earliest-cr-line | date of earliest reported credit line |
open-acc | number of open credit lines |
revol-bal | total credit revolving balance |
revol-util | percent credit used |
total-acc | total number of credit lines in credit file |
delinq-2yrs | number of 30+ days past-due incidences of delinquency in credit |
inq-last-6mths | number of inquiries by creditors in the past 6 months |
mths-since-last-delinq | months since the borrower’s last delinquency |
column | description |
---|---|
loan-status | current status of the loan: Charged Off, Current, Fully Paid |
total-rec-int | interest received to date |
total-rec-prncp | principal received to date |
total-rec-late-fee | late fees received to date |
out-prncp | remaining outstanding principal for total amount funded |
total-pymnt | payments received to date for total amount funded |
last-pymnt-d | last date payment was received |
next-pymnt-d | next scheduled payment date |
pymnt-plan | indicates if a payment plan is in place for the loan |
The full cash flow is unavailable. For post-origination, on the day of data collection, we know only:
no visibility into timeliness or completeness of payments
Difficult to establish a time of default or assess prepayment risk
"""
Three estimators of loan length:
a. last recorded payment -- "dur_last_pymnt_date"
b. number of installments paid -- "dur_total_installment"
c. interest paid out -- "dur_total_interest"
"""
examples
charged off | prepayment | standard | malformed loan | |
---|---|---|---|---|
loan_amnt | 1800 | 20000 | 18000 | 7500 |
int_rate | 11.89 | 11.78 | 14.72 | 13.8 |
term | 36 | 36 | 36 | 36 |
installment | 59.7 | 662.19 | 621.52 | 236.86 |
issue_d | 2008-12-05 | 2008-12-05 | 2010-08-06 | 2008-09-26 |
last_pymnt_d | 2010-03-30 | 2010-09-08 | 2013-08-22 | 2011-09-29 |
total_pymnt | 673.21 | 23036.2 | 22368.3 | 8526.93 |
total_rec_prncp | 436.01 | 20000 | 18000 | 6950 |
full_interest | 349.2 | 3838.84 | 4374.72 | 1026.96 |
total_rec_int | 160.99 | 3003.13 | 4368.26 | 1576.93 |
loan_status | Charged Off | Fully Paid | Fully Paid | Fully Paid |
dur_last_pymnt_date | 15.7703 | 21.0928 | 36.5346 | 36.0747 |
dur_total_installment | 11.2765 | 34.7879 | 35.9896 | 35.9999 |
dur_total_interest | 10.134 | 19.9876 | 35.1976 | 24.7193 |
"""
ETL / data cleaning
"""
data = dataset()
# Loan parameters must be conformal; those that aren't destroy the KM curves
# *** 299 records removed ***
data['is_valid'] = False
for i,record in enumerate(data.itertuples()):
data.loc[record.Index,'is_valid'] = util.is_valid_loan(record)
data = data.loc[ data.is_valid ].copy()
# Only focus on 'Charged Off' and 'Fully Paid' loans
# *** 18 more records removed ***
data = data.loc[ data.loan_status.isin(['Charged Off', 'Fully Paid'])].copy()
# Any record where more principal is paid back than is borrowed is structurally wrong;
# *** 350 more records removed ***
data = data.loc[ data.total_rec_prncp <= data.loan_amnt ].copy()
# Any record where the totals don't add up to their components is badly formed;
# *** 642 more records removed ***
lit = data.total_rec_prncp + data.total_rec_int + data.total_rec_late_fee
agt = data.total_pymnt
data = data.loc[ np.abs(lit - agt) < 1 ].copy()
# Group the lower performing loans together
lgi = data.grade.isin(['E','F','G'])
data.loc[lgi,'grade'] = 'E+'
Take B bootstrap samples of size N.
For each bootstrap sample, build a partitioning tree, $T_b$:
Score a loan, $i$, via OOB tree set, $\mathcal{T}_i$:
Analogous to profile likelihood / parameter scan: