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.                                                 |
-----------------------------------------------------------------------------------------------------------------------
*/
```