Best Execution in Mortgage Secondary Markets

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1 Best Exeution in ortgage Seondary arkets hung-jui Wang and Stan Uryav 2 ESEAH EPOT # isk anageent and Finanial Engineering Lab Departent of Industrial and Systes Engineering University of Florida, Gainesville, FL 326 Version: arh 4, 2005 orrespondene should be addresd to: Stan Uryav Abstrat A signifiant task faed by ortgage bankers attepting to profit fro the ortgage ondary arket is the best exeution proble. This paper built a foral odel to perfor the best exeution analysis. The odel offers ondary arketing funtionality inluding the loan-level best exeution, guarantee fee buy-up/buy-down, and ba/exess rviing fee. The odel is forulated as a ixed integer prograing proble. The a study shows that a realisti large-sale ixed-integer proble an be solved in an aeptable tie (5 onds) by PLEX-90 solver on a P. eywords: best exeution, ondary ortgage arket, ortgage-baked urity (BS), Fannie ae, BS oupon rate, guarantee fee, guarantee fee buy-up/buy-down, rviing fee, ixed integer prograing. University of Florida, ISE departent, P.O. Box 6595, 303 Weil Hall, Gainesville, FL ; 2 University of Florida, ISE departent, P.O. Box 6595, 303 Weil Hall, Gainesville, FL ; UL:

2 . Introdution ortgage originators underwrite ortgages in the priary arket. The ortgages bring fixed inoe ash flows to their owners. Besides keeping the ortgages as a part of the portfolio, a ortgage banker ay ll the ortgages to other ortgage buyers or uritize the ortgages by pooling the into a ortgage-baked urity (BS) in the ondary arket. The proess that optiizes the ortgage assignents in the ondary arket is known as the Best Exeution strategy. Three governent-sponsored enterpris (GSEs) (Fannie ae, Freddie ae, and Ginnie ae) provide different types of BS swap progras in whih ortgage bankers pool their ortgages into an appropriate BS. ortgages ust eet ertain standards to be eligible for pooling into a BS. Usually, ortgage bankers prefer to pool ortgages into a BS to get higher revenue. They ll ortgages as a whole loan only beau tho ortgages do not qualify for uritization. GSEs who issue the ortgage-baked urities (BSs) provide insurane against the default risk and harge a guarantee fee. This paper onsiders that eah ortgage ay be either sold as a whole loan or pooled into a BS. We fous on the pass-through BS swap progras provided by Fannie ae. In partiular, if a ortgage is sold as a whole loan, then the ortgage banker gets the prie of the ortgage fro the sale. On the other hand, if a ortgage is pooled into a BS, the ortgage is uritized. The BS investors, who buy the BS, pay the prie of the BS to the ortgage banker. When a ortgage is pooled into a BS, besides assigning the ortgage into the BS with the proper oupon rate, ortgage bankers onsider the guarantee fee buy-up/buy-down and ortgage rviing ll/retain features to axiize their total revenue. We built a odel to solve the best exeution proble. The odel is quite flexible and an be adjusted for different requireents of the ortgage uritization progra. A a study shows that a realisti large-sale best exeution proble an be solved in aeptable tie by PLEX-90 solver on a P. 2. Proble desription Best exeution is a entral proble of ortgage ondary arketing. ortgage bankers originate ortgages in the priary arket and dispo of the ortgages in the ondary arket to axiize their revenue. In the ondary arket, eah ortgage an be exeuted in two ways, either pooled into a BS, or sold as a whole loan. ortgage bankers ay ll ortgages at the prie higher than the par value 3 and get revenue fro 3 ortgage bankers underwrite ortgages at a ertain ortgage note rate. The par value is the value of the ortgage when the disount interest rate equals the ortgage note rate. In other word, the par value of a ortgage is its initial loan balane. 2

3 lling the ortgages. They ay also pool the ortgages into BSs. The partiipants in the BS arket an be ategorized into five groups: borrowers, ortgage bankers, ortgage rviers, BS insurers, and ortgage investors. The relationship and the ash flow between the five partiipants in the pass-through BS arket is shown in Figure. Figure : The partiipants and ash flows in the pass-through BS arket. The ortgage borrowers pay a ortgage note rate. ortgage rviers harge a rviing fee for the rvie. Fannie ae harges a guarantee fee for the BS insurane. Both the rviing fee and guarantee fee are defined as a perentage of the outstanding balane of ortgages. After the two fees, the pass-through BS oupon rate reeived by investors equals ortgage note rate rviing fee guarantee fee. Borrowers (pay ortgage note rate) Loan ortgage banker ortgage payent BS swap BS prie ortgage rviers (harge rviing fee) ortgage payent after rviing fee BS Issuer Fannie ae (harges guarantee fee) BS oupon payent = ortgage note rate - rviing fee - guarantee fee BS Investors Suppo ortgage bankers pool ortgages into a BS. Borrowers pay the onthly payents in a fixed interest rate known as ortgage note rate. ortgage rviers ollet the onthly payents and forward the proeeds to the BS investors. They obtain their revenue ainly fro a rviing fee, whih is a fixed perentage of the outstanding ortgage balane, and delines over tie as the ortgage balane aortizes. ortgage bankers ay ll the ortgage rviing with a ba rviing fee to the ortgage rvier and reeive a payent. The bankers ay hoo to retain the rviing fee and provide the ortgage rviing. Fannie ae, one of the governent-sponsored enterpris that issue BSs, provides BS insurane, proteting the BS investors against loss in the event of default of a borrower, and harges a guarantee fee. The BS investors pay the prie of the BS and get the BS with the oupon rate equal to the ortgage note rate inus the su of the rviing fee and the guarantee fee. ortgage bankers get the payents fro BS investors. The payents an be ud to originate ore ortgages in the priary arket. 3

4 2. The BS progra of Fannie ae In this tion, we introdue the BS produts of Fannie ae. See Jess Lederan (997) for ore resoures about BS progras. Fannie ae purhas and swaps ore than 50 types of ortgages on the basis of standard ters. This paper fous on pass-through BS swaps of 0-, 5-, 20- and 30-year fixed-rate ortgages. ortgages ust eet ertain standards to be eligible for sale or swapping. Fannie ae s BS progra has its own requireents, but provides flexibility in athing loan originations proessing requireents. ortgages ust be pooled parately by the tie to aturity. For instane, 30-year fixed-rate ortgages are parated fro 5-year fixed-rate ones. The pass-through rate, or BS oupon rate, generally trades on the half perent (4.5%, 5.0%, 5.5%, et.) and ortgage lenders take advantage of pooling their loans to possible BS oupon rate options. The ortgage note rate in eah pool ust support the pass-through rate plus iniu rviing fee plus the guarantee fee required by Fannie ae. Therefore, the pass-through rate ust satisfy the following equality: ortgage note rate = Serviing Fee + Guarantee Fee + BS oupon rate. Guarantee fee ortgage bankers negotiate the guarantee fee with Fannie ae. The guarantee fee is a fixed perentage (generally 25 ba point (bp) to 35 bp bad on the type of produt) of the outstanding ortgage balane. ortgage lenders have the opportunity to buy down or buy up the guarantee fee. Buy down eans lenders redue the spread required for the guarantee fee and pay an equivalent payent to Fannie ae. On the other hand, buy up eans lenders inrea the spread of guarantee fee and reeive a payent fro Fannie ae. For exaple, if a lender wants to inlude a 7.875% ortgage in a 7.5% pass-through BS (Figure 2), he an buy down the guarantee fee to 0.25% fro 0.25% by paying Fannie ae the prent value of equivalent of 0.25% differene and aintaining the 0.25% iniu rviing fee. If a lender hoos to inlude an 8.25% ortgage in the 7.5% pass-through BS (Figure 3), instead of keeping the 0.375% rviing fee, he an ll the exess 0.25% rviing fee to Fannie ae in return for a prent value equivalent of 0.25%. The buy-down and buy-up guarantee fee features allow lenders to axiize the prent worth of revenue. Figure 2: Guarantee fee buy-down. A lender ay inlude a 7.875% ortgage in a 7.5% pass-through BS by buying down the guarantee fee to 0.25% fro 0.25% and paying Fannie ae the prent value of equivalent of the 0.25% differene and aintaining the 0.25% iniu rviing fee. Serviing Fee Guarantee Fee BS oupon ate 0.25% 0.25% 7.5% ortgage Note ate 7.875% 4

5 Figure 3: Guarantee fee buy-up. A lender ay inlude an 8.25% ortgage in a 7.5% pass-through BS by buying up the guarantee fee to 0.375% fro 0.25% and reeiving a prent value equivalent of the 0.25% differene fro Fannie ae. Serviing Fee Guarantee Fee BS oupon ate 0.25% 0.375% 7.5% ortgage Note ate 8.25% 2.2 The rviing fee plan The ortgage rviing fee onsists of the ba rviing fee and the exess rviing fee. Ba rviing fee ortgage rviers retain the ba rviing fee eah onth as opensation for the olletion and reittane of borrowers onthly ortgage payents. The ba rviing fee is a fixed perentage (generally 25 bp) of the outstanding ortgage balane, and delines over tie as the ortgage balane aortizes. ortgage bankers ay retain the ba rviing fee and provide ortgage rvies. They ay also ll the rviing to ortgage rviers and get revenue fro lling the rviing. Suppo the ortgage rviing is sold, the ortgage rvier will get the ba rviing fee and provide the rviing. Exess rviing fee If a ortgage is pooled into a BS with a ertain oupon rate, it has exess rviing fee equal to the ortgage note rate inus the BS oupon rate inus the guarantee fee inus the ba rviing fee. exess rviing fee = ortgage note rate - BS oupon rate- guarantee fee- ba rviing fee In the exaple shown in Figure 2, the exess rviing fee (spread) ay be sold to Fannie ae by buying up the guarantee fee. Another option for ortgage bankers is to retain the exess rviing fee in the portfolio, thereby generating fixed inoe ash flows during the life of the ortgage. Siilar to the guarantee fee buy-up/buy-down features, the exess rviing fee allows lenders to axiize the prent value of revenue. 3. odel building The best exeution proble an be forulated as an assignent proble. Eah ortgage in the odel an be assigned either into a BS with a speifi oupon rate or to be sold as a whole loan. 5

6 When ortgages are assigned into a BS, we onsider the guarantee fees buy-up/buy-down feature and the rviing retain/ll feature in our odel to axiize the total revenue. We forulate the best exeution proble as a ixed integer prograing proble. 3. odel desription The objetive of the odel is to axiize the revenue. Four soures of revenue are inluded in the odel: () evenue fro ortgages pooled into a BS or sold as a whole loan: where L ( P z ) + Pwhole z w, = = total nuber of ortgages, = index of ortgages ( =, 2,..., ), L = index of BS oupon rate, P = prie of BS with oupon rate index, P z z = = loan aount of ortgage, whole w = nuber of possible BS oupon rates of ortgage, = prie of ortgage that is sold as a whole loan,, if ortgage is pooled into BS with oupon rate, = 0, otherwi,, if ortgage is sold as a whole loan, = 0, otherwi. Sine eah ortgage an be either pooled into a BS or sold as a whole loan, we have the onstraint = z + z =. ( ) w If ortgage is pooled into a BS with oupon rate index ĉ, then z, = 0 for all ˆ = z ˆ, and z = 0, and the revenue fro ortgage equals L P z. On the other hand, if w ˆ ˆ ortgage is sold as a whole loan, then z = 0 for all, and =, and the revenue fro z w ortgage equals P whole z. The total revenue is the suation of revenues fro the w ortgages: L ( P ). z + Pwhole zw = = 6

7 (2) evenue fro ba rviing fee of ortgages pooled into a BS: where = ( L B zsbo + L sb sr zsbr ), B z sbo z sb sr sbr = ba rviing value of ortgage,, if the rviing of ortgage is sold out, = 0, otherwi, = ba rviing fee of ortgage, = retained rviing ultiplier of ortgage,, if the ba rviing of ortgage is retained, = 0, otherwi. Sine the rviing of eah ortgage an either be sold or retained, and the revenue fro ortgage rviing exists only if the ortgage is pooled into a BS instead of being sold as a whole loan, we ipo the onstraint z + z + z =. ( 2) w sbo sbr If ortgage is sold as a whole loan, then z =, z sbo z = 0, and the revenue fro rviing equals zero. On the other hand, if ortgage is pooled into a BS and the rviing of ortgage is sold, then z =, z = 0, =, and the revenue equals sbo sbr w = sbr otherwi z =, z = 0, = 0, and the revenue equals sbr sbo z w zw revenue fro the ortgages equals ( L B zsbo + L sb sr zsbr ). (3) evenue fro exess rviing fee of ortgages pooled into a BS: where r r sr = = ( L sr rr), = retained exess rviing fee of ortgage, = retained rviing fee ultiplier of ortgage. sb sr s L B z sbo ; L z br. The total If ortgage is pooled into a BS, the exess rviing fee generates revenue L r r fro retaining the exess rviing fee. The total revenue fro the ortgages equals = ( L sr rr). 7 sr

8 (4) evenue fro buy-up/buy-down guarantee fee of ortgages pooled into a BS: where r gu r gd u d ( u L rgu) ( d L rgd) = = = guarantee fee buy-up spread of ortgage, = guarantee fee buy-down spread of ortgage, = guarantee fee buy-up ultiplier of ortgage, = guarantee fee buy-down ultiplier of ortgage., ultipliers (, u d ) transfer the fixed inoe value of the guarantee fee to the prent value. Buying up the guarantee fee of ortgage generates revenue, u L rgu. On the other hand, buying down the guarantee fee of ortgage generates negative revenue (ost),. The d L rg d ( ) ( total revenue fro the ortgages equals L r L r. u gu d = = We onsider the guarantee fee buy-up/buy-down and retain exess rviing fee only if ortgage gd ) is pooled into a BS. Therefore, when ortgage is sold as a whole loan, r,, and r should be zero. To define this ondition, we ipo the onstraint r gu r gd z + r + r + r. ( 3) w gu gd r onstraints Besides onstraints (), (2), and (3) listed above, the odel ontains other onstraints. The ost iportant one is the balane equation for ortgage pooled into a BS: ortgage note rate = BS oupon rate + rviing fee + guarantee fee We forulate this onstraint as where z + rgu rgd + rr n sb, gb = n sb gb = BS oupon rate related to index, = note rate of ortgage, = ba rviing fee of ortgage, = ba guarantee fee of ortgage. 8

9 For all ortgages pooled into a BS, the average exess rviing fee is liited by an upper a a bound. This requireent is forulated as L r ( ). Besides the L zw = = aggregate level onstraint, we ategorize ortgages into groups by their year to aturity and onsider the group-level average exess rviing fee liits j for eah group j as j L r ( ). The deision variables of the guarantee fee buy-up/buy-down L zw j j spreads and exess rviing fee are liited by their upper bounds 0 r 0 r gu, gu gd gd, and 0 r. Finally, we ipo non-negativity onstraints and binary onstraints: r, r, r 0, r z, z, z, z {0,} =, 2,..., whole sbr sbo r gu gd 3.2 odel Forulation This tion suarizes the odel desribed in the previous tion. See the list of notations in the Appendix. The objetive funtion and onstraints in the odel are linear and the odel ontains binary variables. Further, the best exeution proble is forulated as a ixed integer prograing proble. 9

10 ax st.. revenue fro ortgages pooled into ( ) L P z + Pwhole zw = = a BS or sold as a whole loan ( L B zsbo L sb sr zsbr ) + + = ( L sr rr ) + = revenue fro ba rviing fee of ortgages pooled into a BS revenue fro exess rviing fee of ortgages pooled into a BS revenue fro buy-up/buy-down guarantee + ( ) ( u L rgu d L rgd) = = fee of ortgages pooled into a BS = z + z = =,2,..., w z + z + z = =,2,..., w sbo sbr z + r + r + r =,2,..., w gu gd r = = ortgage ust be either pooled into BS or sold as a whole loan if ortgage is sold, there is no rviing; otherwi, rviing either be sold or retained if ortgage is sold, there is no guarantee fee and exess rviing fee if ortgage is pooled into BS, note rate= z + rgu rgd + rr n sb gb =,2,..., BS oupon rate+rviing fee+guarantee fee a the average exess rviing fee of all L r L ( zw ) = ortgages is liited by an upper bound. j L r L ( zw ) j =, 2,..., J j j 0 r =, 2,..., gu gu 0 r =,2,..., gd gd the average exess rviing fee of eah BS group j is liited by an upper bound for eah ortgage, the guarantee fee buy-up spread is liited by an upper bound for eah ortgage, the guarantee fee buy-down spread is liited by an upper bound for eah ortgage, the exess rviing fee is 0 rr =,2,..., liited by an upper bound z, z, z, z {0,} =,2,..., [ binary onstraints ] [ non-negativity onstraints ] whole sbr sbo r, r, r 0 r gu gd 4. a Study The a study was onduted with the ortgage datat provided by the Ohio Savings Bank. 4. Input data 0

11 Input data are lassified as follows. Data on ortgages: () loan aount of eah ortgage; (2) tie to aturity of eah ortgage; (3) note rate of eah ortgage; and (4) possible BS pools for eah ortgage assignent. Data on BS: () prie of BS (related to aturity and BS oupon rate) Data on ultipliers and paraeters: () ba rviing value; (2) guarantee fee buy-down/buy-up ultiplier; (3) retain rviing ultiplier; (4) upper bounds of guarantee fee buy-up and buy-down; (5) upper bounds of exess rviing fee in loan, group and aggregate levels. 4.2 Proble size The onsidered datat: 4,355 ortgages. Four different ties to aturity: 0, 5, 20, 30 years parating ortgages into four groups. 3 possible BS oupon rates fro 3.5% to 9.5% inreasing in inreents of 0.5%. The proble size: 3,065 real variables. 66,90 binary variables. 30,490 onstraints (binary and non-negativity onstraints are not inluded). 4.3 Solver We ud PLEX-90 to solve the large-sale proble on an Intel Pentiu 4, 2.8GHz P. The proble was solved in 5 onds with a solution gap 4 of less than 0.0% 4.4 Solution The total loan aount of the 4,355 ortgages is $756,426,037. We assued eah ortgage is sold at the prie of its loan aount. The total revenue equals the total loan aount ($756,426,037). For the a when eah ortgage either being pooled into a BS or sold as a whole loan for the prie of its loan aount, the optial revenue is $786,620,698, whih is $30,94,66 (4%) ore than the total loan aount. 4 Solution gap defines a relative tolerane on the gap between the best integer objetive and the objet of the best node reaining. When the value best node-best integer /(e-0 + best integer ) falls below this value, the IP optiization is stopped.

12 Table : Total revenue under different exeution strategies Exeution strategy ortgages are sold as a whole loan at the prie of loan aount Total revenue $756,426,037 (total loan aount) Best exeution strategy: ortgages are either pooled into a BS or sold as a whole loan at the prie of loan aount $786,620,698 If eah ortgage is sold at the prie of its loan aount, the total revenue equals $756,426,037. With the best exeution strategy suh that eah ortgage ould be either pooled into a BS or sold as a whole loan for the prie of its loan aount, the revenue is iproved to $786,620,698, whih is $30,95,04 (4%) ore than the total loan aount. 5. onlusion We built a foral odel to perfor the best exeution analysis. The odel inludes the loan-level best exeution for a BS/whole loan, guarantee fee buy-up/buy-down, and ba/exess rviing fee. The odel is forulated as a ixed integer prograing proble. The a study shows that a realisti large-sale ixed integer proble an be solved in aeptable tie (5 onds) by PLEX-90 solver on a P. Aknowledgeents We are grateful to Philip Laren, ihard rou, Sahin Goel, and Oleksandr Polishhuk fro the Ohio Savings Bank (OSB), for sharing their knowledge of the ortgage ondary arket and their guidane in establishing the proble fraework. We also want to thank OSB for providing us with the datat ud in this a study. eferenes. J. Lederan, Handbook of Seondary arketing, ortgage Bankers Assoiation of Aeria (997). 2. F. Fabozzi, The Handbook of ortgage-baked Seurities, 5 th edition, Graw-Hill (200). 3. F. Fabozzi, The Handbook of Fixed-Inoe Seurities, 6 th edition, Graw-Hill (2000). Appendix- Suary of notations for the odel The notation of the odel is parated in three groups as follow. Indies: = index of ortgages (,2,,) = total nuber of ortgages 2

13 j = index of ortgage groups (,2,,J) J = total nuber of groups = index of BS oupon rate (,2,, ) = nuber of possible BS oupon rates of ortgage Deision variables: z z w = binary variable of ortgage pooled into BS with oupon rate index = binary variable of ortgage sold as a whole loan z sbo = binary variable of the rviing of ortgage is sold z sbr r gu r gd r r = binary variable of the ba rviing fee of ortgage is retained = guarantee fee buy-up spreads of ortgage = guarantee fee buy-down spreads of ortgage = retained exess rvie fee of ortgage Input Data L P = loan aount of ortgage = prie of BS with oupon rate index P whol e = prie of ortgage that is sold as a whole loan = BS oupon rate related to index u d n = guarantee fee buy-up ultiplier of ortgage = guarantee fee buy-down ultiplier of ortgage = note rate of ortgage sb gu = ba rviing fee of ortgage = upper bound of buy-up guarantee fee of ortgage gd gb B = upper bound of buy-down guarantee fee of ortgage = ba guarantee fee of ortgage = ba rviing value of ortgage sr = retained rviing fee ultiplier of ortgage = upper bound of exess rviing fee of ortgage a = upper bound of average exess rviing fee of all ortgages pooled into BSs j = upper bound of average exess rviing fee of ortgages pooled into BSs in group j 3

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