I can calculate the motion of heavenly bodies but not the madness of people. -Isaac Newton

## A SAS Macro Calculating PDO

In the development of credit scorecards, the model developer usually will scale the predicted probability of default / delinquent into a range of discrete score points for the purpose of operational convenience. While there are multiple ways to perform scaling, the most popular one in the credit risk arena is to scale the predicted probability logarithmically such that the odds, the ratio between goods and bads, will be doubled / halved after the increase / decrease of a certain number of score points, which is also called PDO (points to double the odds) in the industry.

In practice, PDO along the time carries strong implications in credit policies and strategies and is often used as an effectiveness measure for a scorecard. Chances are that a scorecard could still maintain a satisfactory predictiveness, e.g. K-S statistic or AUC, but might lose its effectiveness, e.g. increase in PDO. For instance, if a credit strategy was originally developed to take effect at credit score = 700 by design, it might require credit score = 720 later after the increase in PDO, which will significantly discount the effectiveness of such strategy. As a result, it is critically important to monitor PDO of a scorecard in the production environment.

Below is a sas macro to calculate PDO of a scorecard.

options nocenter nonumber nodate mprint mlogic symbolgen
orientation = landscape ls = 125 formchar = "|----|+|---+=|-/\<>*";

%macro get_pdo(data = , score = , y = , wt = NONE, ref_score = , target_odds = , target_pdo = );
**********************************************************************;
* THIS MACRO IS TO CALCULATE OBSERVED ODDS AND PDO FOR ANY SCORECARD *;
* AND COMPUTE ALIGNMENT BETAS TO REACH TARGET ODDS AND PDO           *;
* ------------------------------------------------------------------ *;
* PARAMETERS:                                                        *;
*  DATA       : INPUT DATASET                                        *;
*  SCORE      : SCORECARD VARIABLE                                   *;
*  Y          : RESPONSE VARIABLE IN (0, 1)                          *;
*  WT         : WEIGHT VARIABLE IN POSITIVE INTEGER                  *;
*  REF_SCORE  : REFERENCE SCORE POINT FOR TARGET ODDS AND PDO        *;
*  TARGET_ODDS: TARGET ODDS AT REFERENCE SCORE OF SCORECARD          *;
*  TARGET_PDO : TARGET POINTS TO DOUBLE ODDS OF SCARECARD            *;
* ------------------------------------------------------------------ *;
* OUTPUTS:                                                           *;
*  REPORT : PDO REPORT WITH THE CALIBRATION FORMULA IN HTML FORMAT   *;
* ------------------------------------------------------------------ *;
* AUTHOR: WENSUI.LIU@53.COM                                          *;
**********************************************************************;

options nonumber nodate orientation = landscape nocenter;

*** CHECK IF THERE IS WEIGHT VARIABLE ***;
%if %upcase(&wt) = NONE %then %do;
data _tmp1 (keep = &y &score _wt);
set &data;
where &y in (1, 0)  and
&score ~= .;
_wt = 1;
&score = round(&score., 1);
run;
%end;
%else %do;
data _tmp1 (keep = &y &score _wt);
set &data;
where &y in (1, 0)        and
&score ~= .         and
round(&wt., 1) > 0;
_wt = round(&wt., 1);
&score = round(&score., 1);
run;
%end;

proc logistic data = _tmp1 desc outest = _est1 noprint;
model &y = &score;
freq _wt;
run;

proc sql noprint;
select round(min(&score), 0.01) into :min_score from _tmp1;

select round(max(&score), 0.01) into :max_score from _tmp1;
quit;

data _est2;
set _est1 (keep = intercept &score rename = (&score = slope));

adjust_beta0 = &ref_score - (&target_pdo * log(&target_odds) / log(2)) - intercept * &target_pdo / log(2);
adjust_beta1 = -1 * (&target_pdo * slope / log(2));

do i = -5 to 5;
old_pdo = round(-log(2) / slope, 0.01);
old_ref = &ref_score + (i) * old_pdo;
old_odd = exp(-(slope * old_ref + intercept));
if old_ref >= &min_score and old_ref <= &max_score then output;
end;
run;

data _tmp2;
set _tmp1;

if _n_ = 1 then do;
set _est2(obs = 1);
end;

run;

proc logistic data = _tmp2 desc noprint outest = _est3;
freq _wt;
run;

data _est4;

adjust_beta0 = &ref_score - (&target_pdo * log(&target_odds) / log(2)) - intercept * &target_pdo / log(2);
adjust_beta1 = -1 * (&target_pdo * slope / log(2));

do i = -5 to 5;
new_pdo = round(-log(2) / slope, 0.01);
new_ref = &ref_score + (i) * new_pdo;
new_odd = exp(-(slope * new_ref + intercept));
if new_ref >= &min_score and new_ref <= &max_score then output;
end;
run;

proc sql noprint;
create table
_final as
select
&target_pdo            as target_pdo,
&target_odds           as target_odds,
a.old_pdo              as pdo1,
a.old_ref              as ref1,
a.old_odd              as odd1,
log(a.old_odd)         as ln_odd1,
b.new_pdo              as pdo2,
b.new_ref              as ref2,
b.new_odd              as odd2,
log(b.new_odd)         as ln_odd2
from
_est2 as a inner join _est4 as b
on
a.i = b.i;

select round(pdo1, 1) into :pdo1 from _final;

select put(max(pdo1 / pdo2 - 1, 0), percent10.2) into :compare from _final;

select case when pdo1 > pdo2 then 1 else 0 end into :flag from _final;

select put(adjust_beta0, 12.8) into :beta0 from _final;

select put(adjust_beta1, 12.8) into :beta1 from _final;
quit;

%put &compare;
ods html file = "%upcase(%trim(&score))_PDO_SUMMARY.html" style = sasweb;
title;
proc report data  = _final box spacing = 1 split = "/"
style(header) = [font_face = "courier new"] style(column) = [font_face = "courier new"]
style(lines) = [font_face = "courier new" font_size = 2] style(report) = [font_face = "courier new"];

column("/SUMMARY OF POINTS TO DOUBLE ODDS FOR %upcase(&score) WEIGHTED BY %upcase(&wt) IN DATA %upcase(&data)
/( TARGET PDO = &target_pdo, TARGET ODDS = &target_odds AT REFERENCE SCORE &ref_score ) / "
pdo1 ref1 odd1 ln_odd1 pdo2 ref2 odd2 ln_odd2);

define pdo1    / "OBSERVED/SCORE PDO"   width = 10 format = 4.   center;
define ref1    / "OBSERVED/REF. SCORE"  width = 15 format = 5.   center order order = data;
define odd1    / "OBSERVED/ODDS"        width = 15 format = 14.4 center;
define ln_odd1 / "OBSERVED/LOG ODDS"    width = 15 format = 8.2  center;
define pdo2    / "ADJUSTED/SCORE PDO"   width = 10 format = 4.   center;
define ref2    / "ADJUSTED/REF. SCORE"  width = 15 format = 5.   center;
define odd2    / "ADJUSTED/ODDS"        width = 15 format = 14.4 center;
define ln_odd2 / "ADJUSTED/LOG ODDS"    width = 15 format = 8.2  center;

compute after;
%if &flag = 1 %then %do;
line @15 "THE SCORE ODDS IS DETERIORATED BY %trim(&compare).";
line @15 "CALIBRATION FORMULA: ADJUSTED SCORE = %trim(&beta0) + %trim(&beta1) * %trim(%upcase(&score)).";
%end;
%else %do;
line @25 "THERE IS NO DETERIORATION IN THE SCORE ODDS.";
%end;
endcomp;
run;;
ods html close;

*************************************************;
*              END OF THE MACRO                 *;
*************************************************;
%mend get_pdo;

In the following example, I called the macro to calculate PDO of bureau_score in a test data. Parameter inputs of “ref_score”, “target_odds”, and “target_pdo” could be acquired from the development or the benchmark sample. As shown in the output, the observed PDO is higher than the target PDO, implying 2.93% deterioration in the score effectiveness. In addition to PDO calculation, the sas macro also generated a formulation to calibrate the bureau_score back to the target PDO.

data tmp1;
set data.accepts;
where bureau_score ~= .;
run;

%get_pdo(data = tmp1, score = bureau_score, y = bad, wt = weight, ref_score = 680, target_odds = 20, target_pdo = 45);

/*
-----------------------------------------------------------------------------------------------------------------------
|                  SUMMARY OF POINTS TO DOUBLE ODDS FOR BUREAU_SCORE WEIGHTED BY WEIGHT IN DATA TMP1                  |
|                            ( TARGET PDO = 45, TARGET ODDS = 20 AT REFERENCE SCORE 680 )                             |
|                                                                                                                     |
|SCORE PDO    REF. SCORE         ODDS          LOG ODDS     SCORE PDO    REF. SCORE         ODDS          LOG ODDS    |
|---------------------------------------------------------------------------------------------------------------------|
|     46   |       448     |        0.6404 |      -0.45    |     45   |       455     |        0.6250 |      -0.47    |
|----------+---------------+---------------+---------------+----------+---------------+---------------+---------------|
|     46   |       495     |        1.2809 |       0.25    |     45   |       500     |        1.2500 |       0.22    |
|----------+---------------+---------------+---------------+----------+---------------+---------------+---------------|
|     46   |       541     |        2.5618 |       0.94    |     45   |       545     |        2.5000 |       0.92    |
|----------+---------------+---------------+---------------+----------+---------------+---------------+---------------|
|     46   |       587     |        5.1239 |       1.63    |     45   |       590     |        5.0000 |       1.61    |
|----------+---------------+---------------+---------------+----------+---------------+---------------+---------------|
|     46   |       634     |       10.2483 |       2.33    |     45   |       635     |       10.0000 |       2.30    |
|----------+---------------+---------------+---------------+----------+---------------+---------------+---------------|
|     46   |       680     |       20.4976 |       3.02    |     45   |       680     |       20.0000 |       3.00    |
|----------+---------------+---------------+---------------+----------+---------------+---------------+---------------|
|     46   |       726     |       40.9971 |       3.71    |     45   |       725     |       40.0000 |       3.69    |
|----------+---------------+---------------+---------------+----------+---------------+---------------+---------------|
|     46   |       773     |       81.9981 |       4.41    |     45   |       770     |       80.0000 |       4.38    |
|----------+---------------+---------------+---------------+----------+---------------+---------------+---------------|
|     46   |       819     |      164.0038 |       5.10    |     45   |       815     |      160.0000 |       5.08    |
|---------------------------------------------------------------------------------------------------------------------|
|              THE SCORE ODDS IS DETERIORATED BY 2.93%.                                                               |
|              CALIBRATION FORMULA: ADJUSTED SCORE = 20.92914838 + 0.97156812 * BUREAU_SCORE.                         |
-----------------------------------------------------------------------------------------------------------------------
*/

In the second example, I used the formulation from the above to generate a adjust_score from the bureau_score and then calculated PDO again. As shown in the output, after the calibration, the observed PDO of adjust_score now becomes equivalent to the target PDO of bureau_score. When the score increases every 45 points, e.g. from 680 to 725, the odds value would be doubled as anticipated, e.g. from 20 to 40.

data tmp2;
set tmp1;
adjust_score = 20.92914838 + 0.97156812 * bureau_score;
run;

%get_pdo(data = tmp2, score = adjust_score, y = bad, wt = weight, ref_score = 680, target_odds = 20, target_pdo = 45);

/*
-----------------------------------------------------------------------------------------------------------------------
|                  SUMMARY OF POINTS TO DOUBLE ODDS FOR ADJUST_SCORE WEIGHTED BY WEIGHT IN DATA TMP2                  |
|                            ( TARGET PDO = 45, TARGET ODDS = 20 AT REFERENCE SCORE 680 )                             |
|                                                                                                                     |
|SCORE PDO    REF. SCORE         ODDS          LOG ODDS     SCORE PDO    REF. SCORE         ODDS          LOG ODDS    |
|---------------------------------------------------------------------------------------------------------------------|
|     45   |       455     |        0.6249 |      -0.47    |     45   |       455     |        0.6250 |      -0.47    |
|----------+---------------+---------------+---------------+----------+---------------+---------------+---------------|
|     45   |       500     |        1.2498 |       0.22    |     45   |       500     |        1.2500 |       0.22    |
|----------+---------------+---------------+---------------+----------+---------------+---------------+---------------|
|     45   |       545     |        2.4996 |       0.92    |     45   |       545     |        2.5000 |       0.92    |
|----------+---------------+---------------+---------------+----------+---------------+---------------+---------------|
|     45   |       590     |        4.9991 |       1.61    |     45   |       590     |        5.0000 |       1.61    |
|----------+---------------+---------------+---------------+----------+---------------+---------------+---------------|
|     45   |       635     |        9.9980 |       2.30    |     45   |       635     |       10.0000 |       2.30    |
|----------+---------------+---------------+---------------+----------+---------------+---------------+---------------|
|     45   |       680     |       19.9956 |       3.00    |     45   |       680     |       20.0000 |       3.00    |
|----------+---------------+---------------+---------------+----------+---------------+---------------+---------------|
|     45   |       725     |       39.9905 |       3.69    |     45   |       725     |       40.0000 |       3.69    |
|----------+---------------+---------------+---------------+----------+---------------+---------------+---------------|
|     45   |       770     |       79.9796 |       4.38    |     45   |       770     |       80.0000 |       4.38    |
|----------+---------------+---------------+---------------+----------+---------------+---------------+---------------|
|     45   |       815     |      159.9565 |       5.07    |     45   |       815     |      160.0000 |       5.08    |
|---------------------------------------------------------------------------------------------------------------------|
|                        THERE IS NO DETERIORATION IN THE SCORE ODDS.                                                 |
-----------------------------------------------------------------------------------------------------------------------
*/

Written by statcompute

April 27, 2013 at 8:13 pm

Posted in Credit Risk, SAS, Scorecard

## Disaggregating Annual Losses into Each Quarter

In loss forecasting, it is often necessary to disaggregate annual losses into each quarter. The most simple method to convert low frequency to high frequency time series is interpolation, such as the one implemented in EXPAND procedure of SAS/ETS. In the example below, there is a series of annual loss projections from 2013 through 2016. An interpolation by the natural spline is used to convert the annual losses into quarterly ones.
SAS Code:

data annual;
input loss year mmddyy8.;
format year mmddyy8.;
datalines;
19270175 12/31/13
18043897 12/31/14
17111193 12/31/15
17011107 12/31/16
;
run;

proc expand data = annual out = quarterly from = year to = quarter;
id year;
convert loss / observed = total method = spline(natural);
run;

proc sql;
select
year(year) as year,
sum(case when qtr(year) = 1 then loss else 0 end) as qtr1,
sum(case when qtr(year) = 2 then loss else 0 end) as qtr2,
sum(case when qtr(year) = 3 then loss else 0 end) as qtr3,
sum(case when qtr(year) = 4 then loss else 0 end) as qtr4,
sum(loss) as total
from
quarterly
group by
calculated year;
quit;

Output:

year      qtr1      qtr2      qtr3      qtr4     total

2013   4868536   4844486   4818223   4738931  19270175
2014   4560049   4535549   4510106   4438194  18043897
2015   4279674   4276480   4287373   4267666  17111193
2016   4215505   4220260   4279095   4296247  17011107

While the mathematical interpolation is easy to implement, it might be difficult to justify and interpret from the business standpoint. In reality, there might be an assumption that the loss trend would follow the movement of macro-economy. Therefore, it might be advantageous to disaggregate annual losses into quarterly ones with the inclusion of one or more economic indicators. This approach can be implemented in tempdisagg package of R language. Below is a demo with the same loss data used above. However, disaggregation of annual losses is accomplished based upon a macro-economic indicator.
R Code:

library(tempdisagg)

loss <- c(19270175, 18043897, 17111193, 17011107)
loss.a <- ts(loss, frequency = 1, start = 2013)

econ <- c(7.74, 7.67, 7.62, 7.48, 7.32, 7.11, 6.88, 6.63, 6.41, 6.26, 6.12, 6.01, 5.93, 5.83, 5.72, 5.59)
econ.q <- ts(econ, frequency = 4, start = 2013)

summary(mdl <- td(loss.a ~ econ.q))
print(predict(mdl))

Output:

Call:
td(formula = loss.a ~ econ.q)

Residuals:
Time Series:
Start = 2013
End = 2016
Frequency = 1
[1]  199753 -234384 -199257  233888

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  2416610     359064   6.730   0.0214 *
econ.q        308226      53724   5.737   0.0291 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

'chow-lin-maxlog' disaggregation with 'sum' conversion
4 low-freq. obs. converted to 16 high-freq. obs.
Adjusted R-squared: 0.9141      AR1-Parameter:     0 (truncated)
Qtr1    Qtr2    Qtr3    Qtr4
2013 4852219 4830643 4815232 4772080
2014 4614230 4549503 4478611 4401554
2015 4342526 4296292 4253140 4219235
2016 4302864 4272041 4238136 4198067

In practice, if a simple and flexible solution is desired without the need of interpretation, then the mathematical interpolation might be a good choice. On the other hand, if there is a strong belief that the macro-economy might drive the loss trend, then the regression-based method implemented in tempdisagg package might be preferred. However, in our example, both methods generate extremely similar results.

Written by statcompute

April 23, 2013 at 11:35 pm

## How to Construct Piecewise Linear Spline in SAS

options nocenter;

data tmp1;
do i = 1 to 5000;
x = ranuni(1);
y = x + rannor(1) * 0.5;
if x >= 0.3 then y = y + 6 * (x - 0.3);
if x >= 0.6 then y = y - 10 * (x - 0.6);
output;
end;
run;

*** Manually Construct Piecewise Spline ***;
data tmp2;
set tmp1;
x1 = x;
x2 = max(x - 0.3, 0);
x3 = max(x - 0.6, 0);
run;

proc reg data = tmp2;
model y = x1 - x3;
run;
quit;
/*                   Parameter       Standard
Variable     DF       Estimate          Error    t Value    Pr > |t|

Intercept     1        0.02627        0.02432       1.08      0.2801
x1            1        0.81570        0.11576       7.05      <.0001
x2            1        6.29682        0.18477      34.08      <.0001
x3            1      -10.19025        0.14870     -68.53      <.0001
*/

*** Automatically Construct Piece Spline ***;
proc transreg data = tmp1 ss2;
model identity(y) = pspline(x / knots = 0.3 0.6 degree = 1);
run;
/*                                   Type II
Sum of       Mean
Variable        DF    Coefficient    Squares     Square    F Value    Pr > F    Label

Intercept        1       0.026272       0.28       0.28       1.17    0.2801    Intercept
Pspline.x_1      1       0.815702      12.12      12.12      49.65    <.0001    x 1
Pspline.x_2      1       6.296817     283.50     283.50    1161.34    <.0001    x 2
Pspline.x_3      1     -10.190247    1146.47    1146.47    4696.43    <.0001    x 3
*/

Written by statcompute

April 22, 2013 at 11:10 pm