A LINEAR PROGRAM TO COMPARE MULTIPLE GROSS CREDIT LOSS FORECASTS Dr. Derald E. Wentzen, Wesley College, (302) 736-2574, wentzde@wesley.edu ABSTRACT A lnear programmng model s developed and used to compare alternatve forecasts for a sngle varable. An nherent problem assocated wth generatng multple forecasts s the queston of how to ncorporate the results nto a sngle forecast. The lnear programmng model s desgned to assgn weghts to the dfferent forecastng methodologes such that the mean absolute devaton between actual and forecast results s mnmzed. Senor executves can use the weghts to analyze the accuracy of each forecast and ncorporate each of the dfferent forecasts nto one set of numbers. The model formulaton, an example of an applcaton usng the forecast of gross credt losses for a credt card company, and a dscusson of how the weghts can be used to assess the dfferent methodologes are presented. The four methodologes nclude a regresson-based roll-rate forecast, a collecton roll-rate forecast, a rsk analyss nvaraton-based forecast, and a fnance vntage-based roll-rate forecast. The model evaluates the accuracy of each forecast usng an analytcal approach. The nformaton s then used to suggest how a sngle forecast can be selected usng the best features of the dfferent methodologes. BACKGROUND Credt card companes must plan and prepare for credt losses. It s mperatve that a relable credt loss forecast be avalable for plannng purposes. The process often begns when dfferent departments prepare separate forecasts. The numbers are presented to senor management at a monthly forecast meetng. The senor executves evaluate the nformaton, ask questons, and then select the offcal best forecast. Some of the common methodologes to forecast credt losses nclude roll-rate models, vntage loss rate curves, credt bureau score loss rate models, and Markov chans. A dscusson of these methodologes can be found on the FDIC webste [1]. Credt management technques ncludng credt loss forecastng usng Markov chans are analyzed and dscussed by Resenberg and Glet [2]. The forecast of gross credt losses used n ths analyss were produced each month from November to June usng varatons of roll-rate and vntage based methodologes. Each forecast ncluded monthly gross credt losses for a rollng twelve-month tme perod. Comparsons to actual gross credt losses reveal the accuracy of each methodology. Table 1 shows the absolute value of the forecast errors assocated for the forecast of gross credt losses one-month out. The one-month mean absolute devaton (MAD) values were $2,749 (thousand) for the regresson-based methodology, $790 (thousand) for the collecton roll-rate methodology, $2,440 (thousand) for the rsk analyss nvaraton methodology, and $1,752 (thousand) for the fnance vntage roll-rate methodology. The forecast produced by the collecton department had the smallest one-month MAD and was always selected as the offcal forecast at the monthly forecast meetngs.
Table 1 One-month Forecast Error (n thousands) Date Actual Regresson Collecton Rsk Analyss Fnance GCL Forecast Error Forecast Error Forecast Error Forecast Error Nov $83,512 $83,674 $162 $85,895 $2,383 $89,109 $5,597 $84,424 $912 Dec $92,248 $87,444 $4,804 $92,937 $689 $88,660 $3,588 $94,859 $2,611 Jan $90,237 $89,768 $469 $90,873 $636 $90,329 $92 $90,710 $473 Feb $98,319 $95,565 $2,754 $98,563 $244 $93,831 $4,488 $94,744 $3,575 Mar $96,162 $98,871 $2,709 $95,437 $725 $95,280 $882 $94,082 $2,080 Apr $98,980 $102,835 $3,855 $99,516 $536 $99,915 $935 $100,122 $1,142 May $98,609 $101,723 $3,114 $98,035 $574 $98,303 $306 $99,193 $584 Jun $94,470 $98,594 $4,124 $95,005 $535 $98,102 $3,632 $97,107 $2,637 MAD $2,749 $790 $2,440 $1,752 The Collecton Department was drectly responsble for workng the accounts and t was therefore not surprsng that the forecast prepared by that department had the smallest MAD. However, the queston of whether the other forecast methodologes can be used to provde addtonal nformaton or mprove the offcal forecast s not answered usng ths procedure. The lnear programmng model presented n ths paper wll help determne f and how the nformaton from the other forecasts methodologes provde addtonal nformaton for senor management. The lnear program assgns weghts to the dfferent methodologes to help address ths ssue. The weghts wll reveal whch methodologes should be used to mnmze the MAD. MODEL FORMULATION The followng lnear program s desgned to mnmze the MAD of the four alternatve forecast methodologes. Although n ths case there were four dfferent methodologes, ths model can be used when there s any gven number of competng methodologes. In addton, ths model s not lmted to the forecast of gross credt losses. Ths model can be appled to the forecast of any gven value of nterest. The model s stated as follows: Mn z = m n j= 1 = 1 w F j A j Subject to: n = 1 w w = 1 0
Where, n = number of forecast methodologes m = number of observaton w = weght assgned to forecast methodology A j = Actual Gross Credt Charge-off n month j F j = Forecast of Gross Credt Charge-off n month j by methodology RESULTS The weghts assgned to the dfferent methodologes whch mnmze the one-month gross credt loss forecast error from the lnear program are presented n Table 2. The numbers n the table ndcate that the weghts stablzed when four or more months of observatons are ncluded n the model. The weght (0.8746) assgned to the collecton roll-rate methodology s relatvely large, valdatng the decson that t produces the best one-month forecast of gross credt losses. Ths ndcates that the collecton roll-rate methodology s accurate at predctng gross credt losses one-month n the future and that t wll be a major factor n determnng a combned forecast. An nterestng result and queston s why the model assgned a weght to regresson-based methodology when t was the methodology wth the largest MAD. Table 2 Assgned weghts for one-month gross credt losses Date Regresson Collecton Rsk Analyss Fnance Nov 1.000 0 0 0 Dec 0.3521 0 0 0.6479 Jan 0.3521 0 0 0.6479 Feb 0.1255 0.8745 0 0 Mar 0.1254 0.8746 0 0 Apr 0.1254 0.8746 0 0 May 0.1255 0.8745 0 0 Jun 0.1254 0.8746 0 0 The MAD comparsons between the regresson-based methodology, the collecton roll-rate methodology, and the combned forecast usng the weghts of the lnear programmng model are presented n Table 3. The combned forecast yelded a MAD of $635 (thousand) representng an average mprovement of $155 (thousand) per month over only usng the collecton roll-rate methodology. Snce the regresson-based roll-rate methodology had the largest MAD, an nterestng queston s what addtonal nformaton can be obtaned from the regresson-based methodology?
Table 3 One-month Forecast Error (n thousands) Date Actual Regresson Collecton Combned GCL Forecast Error Forecast Error Forecast Error Nov $83,512 $83,674 $162 $85,895 $2,383 $85,616 $2,104 Dec $92,248 $87,444 $4,804 $92,937 $689 $92,248 $0 Jan $90,237 $89,768 $469 $90,873 $636 $90,734 $497 Feb $98,319 $95,565 $2,754 $98,563 $244 $98,187 $132 Mar $96,162 $98,871 $2,709 $95,437 $725 $95,868 $294 Apr $98,980 $102,835 $3,855 $99,516 $536 $99,932 $952 May $98,609 $101,723 $3,114 $98,035 $574 $98,497 $112 Jun $94,470 $98,594 $4,124 $95,005 $535 $95,455 $985 MAD $2,749 $790 $635 An nvestgaton nto why the regresson methodology was assgned a weght was conducted. It was dscovered that on several occasons the regresson-based roll-rate methodology served as a check for the collecton roll-rate methodology. Specfcally, n sx of the eght months the regresson-based rollrate numbers were lower than the number produced by the collecton department when the collecton number was too hgh or hgher than the number produced by the collecton department when the collecton number was too low. Therefore, the regresson-based forecasts could serve as a check to the collecton roll-rate model. It could be used to ndcate when the collecton department was overestmatng the roll rates or underestmatng the roll rates. Although the forecast produced by the collecton department was always used as the offcal forecast, senor management looked at the regresson-based number to verfy the forecast. When large dfferences between the two methodologes were observed, members of the collecton department were asked to check ther numbers to see f there was somethng that they mssed or wanted to correct. CONCLUSION The lnear programmng model results can also be used to determne a combned forecast by calculatng a weghted average of the dfferent methodologes for each of the twelve months n the forecast horzon. Although the results are not avalable yet, weghts can be produced to answer the queston of how far nto the future can the collecton department accurately predct credt losses or do the other methodologes work better for longer tme horzons. Senor management executves can use the weghts of the lnear program to assess the effectveness of the alternatve forecast methodologes n the short-term. Ths methodology can also be used to develop medum-term and long-term forecasts. In ths example, the collecton roll-rate methodology performed the best (as defned by the largest weghts) n the short-term. Senor management used the collecton roll-rate forecast to predct the short-term gross credt losses wth the regresson-based methodology as a check. They were reluctant to use a combned forecast n the short term because they lked the relablty of the collecton roll-rate methodology and knew that the collecton department was ultmately responsble and accountable f the actual losses were much dfferent than forecast.
REFERENCES [1] http://www.fdc.gov/regulatons/examnatons/credt_card/ch12.html#sec5 [2] Rosenberg and Glet, Quanttatve Methods n Credt Management: A Survey, Journal of the Operatons Research Socety of Amerca, 1994, 4, 589-613