Rationales of Mortgage Insurance Premium Structures

JOURNAL OF REAL ESTATE RESEARCH Raionales of Morgage Insurance Premium Srucures Barry Dennis* Chionglong Kuo* Tyler T. Yang* Absrac. This sudy examines he raionales for he design of morgage insurance premium srucures. The acuarially sound premium prices of several widely used srucures are formally derived. Two ypes of cross-subsidizaion are idenified in differen srucures: () subsidizaion across erminaion years and () exra-subsidizaion of defaulers by nondefaulers. Because hese wo ypes of subsidizaion exis o differen degree among he srucures, a borrower may self-selec ino cerain srucures o maximize (minimize) he benefis (losses) of cross-subsidies. Adverse selecion arises when he borrower s characerisics canno be compleely observed by he insurer. The acuarially sound premium prices should be adjused for such adverse selecion behaviors. Numerical examples are provided o illusrae such adjusmens. Inroducion Currenly, differen premium srucures are used by various insurance/guaranee agencies, such as he Federal Housing Adminisraion (FHA), Veeran s Adminisraion (VA), Federal Naional Morgage Associaion (Fannie Mae), Federal Home Loan Morgage Corporaion (Freddie Mae), and privae morgage insurance companies (PMIs). These insurance programs charge combinaions of upfron and annual premiums, and premium refunds are provided when a loan is prepaid wihin a relaively shor period of ime. Given hese differen srucures, he insurer will realize differen revenue paerns over ime. In addiion, oal premiums incurred by a borrower who prepays or defauls vary by he premium srucure and he ime of defaul/prepaymen. The raionales for using differen premium srucures from eiher he borrowers or he insurers perspecives have no ye been sudied in a rigorous manner. Numerous morgage defaul pricing aricles have been published over he las wo decades. Von Fursenberg (969), Vandell (978), Jackson and Kaseman (98), and Swan (98) sudy he value of morgage defaul risk using economeric models. Campbell and Dierich (983) and Vandell and Thibodeau (985) address he pricing of morgage defaul risk and morgage insurance wih a uiliy maximizaion approach. Foser and Van Order (984), Epperson, Kau, Keenan, and Muller (985), and Cunningham and Hendersho (984) use coningen claim pricing approaches o price morgage defaul risk. Kau, Keenan, Muller, and Epperson (99), sudy he value of defaul risk when prepaymens and defauls are inerrelaed. Cunningham and Capone (99), Ambrose and Capone (996), and Deng and Calhoun (997) focus heir sudies on he esimaion of morgage erminaion raes wih hisorical daa. Deng, Quigley and Van Order (994) develop a condiional hazard model o esimae he defaul funcion. 997 Real Esae Finance award, sponsored by he Fannie Mae Foundaion. *Price Waerhouse LLP, 66 N. For Myer Drive, Arlingon, Virginia 9. 359

36 JOURNAL OF REAL ESTATE RESEARCH Mos of hese papers focus on he valuaion of defaul risk as a lump sum (upfron premium). However, lieraure ha goes beyond he valuaion of upfron premiums o sudy he effecs of using differen premium srucures o cover defaul risk is almos nonexisen. Morgage insurance differs from oher ypes of insurance in several respecs. These differences make i difficul o adop echniques developed elsewhere in he insurance indusry. Firs, casualy insurance covers a single period, so he hisorical performance of a paricular policy can be used in deermining he premium o be charged in subsequen periods. This informaion canno be used in deermining morgage insurance premiums, because morgage insurance covers muliple periods, and he premium for he life of he morgage is defined a he originaion dae. Second, in conras o life insurance, morgage insurance has a definie erminaion dae and he claim risk decreases raher han increases over ime due o he amorizaion schedule. Third, wih proper geographic diversificaion, oher ypes of insurers can usually reduce risk exposure o a minimum; however, because he prepaymen and defaul raes of morgages are highly dependen on macroeconomic variables such as ineres raes, house price growh raes, and household income (or unemploymen raes), subsanial sysemaic risk is involved in morgage insurance. Finally, when morgage insurance is mandaory i covers he risk o he lender raher han he risk o he borrower. The borrower has no righ o swich insurance companies or o be emporarily uninsured wihou erminaing he exising morgage loan. Given hese unique feaures, he design of premium srucures and he calculaion of morgage insurance premiums deserve careful sudy. This research sudies he raionales of he design of morgage insurance premium srucures. We develop a framework o calculae morgage premiums for differen srucures based on exogenous erminaion probabiliies. The insurer s objecive is o collec premiums o cover he expeced claim coss and earn economic profi. Throughou his sudy, economic value refers o he reurn commensurae wih he level of risk being aken by he insurer. Differen premium srucures ha combine upfron premiums, annual premiums, and/or premium refunds can be designed o fulfill such requiremens. Insurers do no have informaion, however, abou he enure plans and defaul risks of individual borrowers. Borrowers who differ in hese respecs incur differen oal insurance premiums under differen srucures and, herefore, have an incenive o choose one srucure over anoher. The FHA and PMI srucures are used o illusrae his effec. We find ha when no consrained by underwriing guidelines, borrowers wih shorer enure plans have an incenive o choose he FHA srucure over he PMI srucure. Properly designed premium srucures can reduce he problem caused by informaion asymmery. We find ha wo ypes of subsidizaion inheren wihin morgage insurance pricing srucures play a key role in he effec of hese premium srucures. These forms of subsidy are () erminaion year subsidizaion and () exra defauler subsidizaion. The nex secion describes several exising premium srucures. Secion hree develops he framework for deermining insurance premium srucures. Secion four analyzes he ex-pos performance of various srucures and discusses he effec of heerogeneous borrowers wih asymmeric informaion. The fifh secion provides numerical examples of exising premium srucures, he incenive effec on he behavior of borrowers, and he effec of differen pricing approaches on he issuers. The las secion draws conclusions abou our findings and heir implicaions. VOLUME 4, NUMBER 3, 997

RATIONALES OF MORTGAGE INSURANCE PREMIUM STRUCTURES 36 The Exising Premium Srucures The Fannie Mae and Freddie Mac morgage guaranee programs charge borrowers fixed premium raes of abou 5 basis poins of he remaining balance hroughou he life of he morgage. While premium raes can vary by lenders, hey do no vary among morgages wih differen characerisics nor do hey vary by he age of he morgage. The value of premium income under such a srucure may no mach he expeced ne presen value of poenial losses faced by a lender during differen sages of a morgage. For example, he defaul risk during he early years of a morgage is usually much higher han ha during he laer years. Furhermore, he dollar amoun of loss o he lender is usually lower during he laer years of he morgage due o amorizaion. A raional premium srucure should reflec hese paerns. Tha is, a any poin in ime, he value of morgage insurance premium income should mach he expeced ne presen value of he poenial loss over he remaining life of he morgage. The premium raes of morgage insurance provided by FHA vary wih he loan-ovalue (LTV) raio, he ype of morgage (purchase or refinance), and he erm of he morgage. Loans wih higher LTV raio or longer erms are charged higher premium raes. The premiums for sreamline refinance loans are lower han hose for purchase loans. Beween Sepember, 983 and Sepember 3, 99, FHA charged an upfron premium of 3.8% wihou any annual premiums. Since 99, FHA premiums have been srucured so ha boh upfron premiums and annual premiums are colleced. Currenly, hiry-year FHA morgage insurance has an upfron premium rae of.5% and an annual premium rae of.5% of he remaining balance for five, eigh and en years for loans wih LTV raios below 9%, beween 9% and 95%, and above 95%, respecively. When an FHA borrower prepays before he end of he sevenh year of he morgage s life, he FHA will refund a porion of he upfron premium. The percenage of he refund decreases from 98% in he firs year afer originaion o 8% in he sevenh year. An FHA hiry-year sreamlined refinancing has an upfron premium rae of.5% and annual premium raes of.5% for seven years. Since no appraisal is required in FHA refinancing, no LTV informaion is available and premiums vary only wih he erm of he morgage. Privae morgage insurers offer a variey of insurance programs. The premiums hey charge are funcions of morgage balance, LTV, loan ype such as fixed-rae morgage (FRM) or adjusable-rae morgage (ARM), he erm lengh, he cap rae of he ARM, he buydown provision, he coverage of he loss, and he frequency of paymens. Unlike FHA insurance, which charges he same premium raes for FRMs and ARMs, PMIs charge higher premiums for ARMs han for FRMs. The premium raes for FRMs increase wih he erm of he morgage and hose for ARMs increase wih he magniude of he cap rae. Temporary buydowns have he same premium raes as he ARMs wih a % cap rae. According o he frequency of he paymens, ypical PMI programs can be classified as monhly premiums, level annuals, sandard annuals, and super singles (see descripions below). For monhly and annual premium paymens, borrowers can choose beween consan renewal and amorized (or declining) renewal. In consan renewal programs, premium raes are muliplied by he original loan amoun o calculae he paymen, while in amorized renewal, premium raes higher han hose in he consan renewal are applied o he remaining balance. The amorized renewal raes ypically remain he same

36 JOURNAL OF REAL ESTATE RESEARCH hrough he life of he morgage. The consan renewal raes are normally he same as he amorized renewal raes in he firs en years afer he originaion and adjused downward for he period from he elevenh year o erm. Because he raes charged by differen privae insurers are eiher he same or very similar, we repor he raes based upon informaion from a ypical insurer. In he monhly premium program, he monhly premiums are calculaed according o annualized premium raes. The annualized premium raes based upon amorized loan balances are he same for he firs year and subsequen renewal years. For a hiry-year FRM wih LTV beween 9.% and 95% and loss coverage of 3%, he amorized renewal rae is.78%. For a % ARM, he comparable rae is.9%. In he sandard annuals program, he premium rae for he firs year is greaer han he renewal rae. For a hiry-year FRM wih LTV from 9.5 o 95% and loss coverage of 3%, he firs year premium rae is.45%, he amorized renewal rae is.49%, and he consan renewal rae for years hrough expiraion is.5%. In he level annuals program, he raes for he firs year are he same as he amorized renewal raes. For a hiry-year FRM wih he same LTV and loss coverage, he amorized level annual premium rae is.74% and he consan renewal rae for years hrough erm is.%. In he single plans and super single programs, premiums are paid up fron. Single plans provide coverage for he firs hree, five, seven, or en years of he life of he morgage; super single programs insure agains defaul unil he loan is paid in full. For hiry-year loans, single plans are refundable if he conracs are canceled before he expiraion of he insurance plan. The refund schedule for super singles vary wih he erm of he loans and may also vary wih he originaion LTV. For example, hiry-year loans use en-year schedules, and fifeen-year loans use five-year refund schedules. The refundable upfron premium rae is 4.8% percen for hiry-year FRMs and.9% for fifeen/weny-year FRMs. Noe ha he monhly and annual PMI premiums are refundable a prepaymen for he fracion of he monh or he year ha has no passed. However, his ype of refund differs from he upfron premium refund ha could occur more han one year afer he premium paymen. The refund of he laes monhly or annual premium is of limied ineres and is no applicable in he discree ime model discussed in he nex secion. Thus, we address only he refund of he upfron premium afer he firs year of originaion. The Model A Framework for Deermining Insurance Premium Srucures In his secion, we presen a framework under which feasible insurance premium srucures can be consruced. A feasible premium srucure is defined as one such ha he presen value of he expeced loss (plus a gross margin) for he insurer is equal o ha of he expeced premium revenues. Assume he erm of he morgage is T periods wih paymens o be made a ime,..., T. Morgage loans are originaed a he curren period, ime. In each period, a borrower deermines wheher o defaul, prepay, or make he paymen. The condiional defaul and prepaymen raes a ime are given by d and p. The condiional probabiliy ha a borrower will say curren a ime is c d p. When a borrower defauls, he insurer is assumed o incur a loss ha is VOLUME 4, NUMBER 3, 997

RATIONALES OF MORTGAGE INSURANCE PREMIUM STRUCTURES 363 proporional o he unpaid morgage balance. Denoe he unpaid morgage balance a he insan afer he ime paymen by B and he raio of he loss o he unpaid principal balance by L R. The loss raio L R is assumed consan hroughou he morgage erm. The presen value of he expeced accumulaed loss for he lender from he presen o ime, denoed by EAL, is: EAL = EL + EL +... + EL = d LR B R + cd LR BR +... + cc... c d LR B R s s = d LR B R + c d s LR B s R =,,..., T, s= l= () where R is defined as one plus he discoun rae used for calculaing he presen value of loss and revenue and EL is he presen value of he expeced loss arising from ime defaul: EL = cc... c d LR B R =,,..., T. () Denoe he raio of ime insurance premium paymen o B by a and he raio of refund o B by f. Denoe he presen value of he expeced accumulaed revenue from he presen o ime by EAR and he presen value of he expeced revenue a ime by ER. Then: ( ) = ER = c... c c a B p f B R,,..., T, (3) and: EAR = ER + ER +... + ER = a B + c a B p f B R + c c a B p f B R... c... c c a B p f B R = a B + c a B p f B R + ( ) ( ) + + ( ) ( ) s= s l= s c ( c s a s B s p s f s B ) R =,,..., T. (4) We assume ha he presen value of he expeced revenue is equal o ( q) imes he presen value of he expeced accumulaed loss, i.e., he insurers earn a gross profi margin of q percenage. Noe ha his does no necessarily mean ha he gross margin is consan and equal o q in each period. Depending upon he premium srucure, he insurers may be expeced o lose or make money a any specific period. However, across he morgage life, he insurers are expeced o earn a profi even hough hey may lose money in some periods. The equilibrium condiion for lenders is: ( ) EAR = + T q EALT. (5)

364 JOURNAL OF REAL ESTATE RESEARCH In wha follows, we discuss alernaive premium srucures ha saisfy lender equilibrium condiions. A premium srucure is characerized by he combinaion of premium raes (a, a,..., a T ) and refund raes (f, f,..., f T ). Case : Level annual premium rae, no upfron premium, no refund. In Case, he premium colleced a ime is proporional o he ime balance B. Because he defaul raes for a pool of morgages do no remain consan over he life of he porfolio, he premium paid a ime is no equal o he expeced loss arising from he defaul risk a ime. Thus, cross-subsidies exis among borrowers who erminae morgages during differen policy years. The equilibrium condiion is EAR T ( q)eal T subjec o a a a... a T and f f... f T =, which implies: a= ( ) ( + + + ) T + q LR d B R cd BR... cct d T BT R ( T ) ( B + cbr + cc B R +... + c... ct BT R ). (6) Case : Upfron premium only, no annual premium, no refund. In Case, only upfron premiums are colleced bu none of he premium is o be refunded when a borrower prepays. The equilibrium premium ha he insurer requires is ( q)eal T. Case 3: Upfron premium only, unused premium refunded when borrowers prepay, no annual premium. Given he condiional defaul and prepaymen raes, we wan o deermine he upfron premium and he refund schedule according o which he unused premiums are refunded. The unused premium is defined as he porion of he upfron premium covering he ime periods from afer he prepaymen o he end of he erm. Thus, he earlier a borrower prepays, he larger he proporion of he upfron premium o be refunded. The oal upfron premium can be considered he sum of he premiums paid in advance for insuring agains he defaul risk in each period wih he condiion of refunding unused premiums upon prepaymen. Le g be such ha g B is he porion of he refundable upfron premium for insuring agains he ime defaul risk. The raio of he refundable upfron premium o he original loan amoun, denoed by a, can be wrien as: a = g + g +... + g T. (7) The premium o be refunded a ime is equal o f B, where: ( )( ) f = g + + g + +... + g T v, (8) VOLUME 4, NUMBER 3, 997

RATIONALES OF MORTGAGE INSURANCE PREMIUM STRUCTURES 365 and v is one plus he ineres rae used for calculaing he ineres paid on he premium o be refunded. Subsiuing equaion 7 and equaion 8 ino equaion 4 and leing a for >, we obain: ( ) + ( ) ( ) EART = g B + p ( v R ) g B p v R c p v R g B + / / / 3... T + p ( v R ) c p ( v R ) c c ct pt ( v R ) g / /...... / T B = ERD + ERD +... + ERD T, (9) where: ERD = p ( v R ) c p ( v R ) c c c p ( v R ) g B / /...... / () is he expeced premium revenue from insuring agains he ime defaul risk. ERD can be beer undersood by seing v R. In his case, i is obvious ha ERD is he produc of he porion of he upfron premium for insuring agains ime defaul and he probabiliy ha he borrower will no prepay before ime so ha he insurer will keep his premium. The probabiliy ha he insurer will no refund he ime porion of he upfron premium is equal o one minus he uncondiional probabiliy ha he borrower will prepay before ime. In general, v R, and equaion indicaes ha he expeced ne premium revenue from insuring agains he ime defaul risk is equal o he refundable upfron premium for ime defaul minus he presen value of he expeced premium refund in all he periods before ime. By imposing he condiion ha equaes he expeced loss and expeced premium revenue derived from insuring agains he ime defaul risk, we have: ERD ( ) = + q EL. () Subsiuing equaion and equaion 9 ino equaion and solving for g yields: g = ( ) + q c c... c d L B R R p v R c p v R c c p v R / /...... / B ( ) ( ) ( ). () Case 4: Upfron premium financed, no refund, no annual premium When he upfron premium is financed and herefore amorized, he borrowers pay a fixed amoun of premium in each period unil he morgage is erminaed. When a borrower prepays, he unpaid premium balance will be due immediaely. However, he unpaid premium balance a he insan before he prepaymen is equal o he presen value of he fuure premium paymens if he borrower did no prepay. The presen value of he premium paymen for he prepayers does no depend on when he prepaymen occurs. The accumulaed expeced premium revenue for Case 4 is:

366 JOURNAL OF REAL ESTATE RESEARCH ( ) + ( ) EAR = m+ d mr d c d mr T ( ) +... + ( T ) d c d... c... c d mr, T T (3) where m is he premium paymen in each period. From equaion 5 and equaion 3, we solve m o be: ( + q ) EALT m= + ( d ) R + ( d c d ) R +... + ( d c d... c... c d ) R T T ( T ). The presen value of he premium paymens for a non-defauler is: UFP m R T = R. (4) (5) Comparison of Alernaive Premium Srucures Algebraic Approach Given a premium srucure discussed in he above secion, he ex-pos premium paymen made by a borrower depends upon he premium srucure and he behavior of he borrower. We firs compare he ex-pos paymens for borrowers under he same premium srucure bu wih differen behaviors. Then we compare he ex-pos paymens for one borrower under differen premium srucures. The implicaion of his analysis is ha borrowers wih differen expecaions of mobiliy and defaul will have differen preferences for specific premium srucures. We compare he ex-pos paymen for hree ypes of borrowers: borrowers who hold he morgage unil mauriy (accumulaed premium paymens denoed by NDNP), borrowers who defaul a ime and s (accumulaed premium paymens denoed by DF and DF s, > s), and borrowers who prepay a ime and s (accumulaed premium paymens denoed by PP and PP s). Comparison wihin he Same Premium Srucure Case : Annual Premium Only. Since here is no refund, borrowers who defaul and prepay a he same ime pay he same insurance premiums. The earlier borrowers erminae he morgage, he smaller he premium paymens are. Thus he borrowers who hold he morgage unil he mauriy dae pay he highes premiums. NDNP> DF = PP > DF s = PPs > s. (6) Case : Upfron Premium Only, No Refund. Because here is no refund and no annual premiums, he premium paymens for all ypes of borrowers are he same. NDNP= DF = PP = DF s = PPs > s. (7) VOLUME 4, NUMBER 3, 997

RATIONALES OF MORTGAGE INSURANCE PREMIUM STRUCTURES 367 Case 3: Upfron Premium wih Refund, No Annual Premiums. In he upfron wih refund case, defaulers pay he same premiums as borrowers holding he morgage o mauriy because neiher receives a refund. The premium paymens for prepayers are less han hose for he defaulers, and he earlier he borrowers prepay, he lower are he oal paymens. NDNP= DF = DF s > PP > PPs > s. (8) Case 4: Upfron Premium wih Financing, No Refund, No Annual Premiums. In he upfron wih financing case, defaulers pay he leas amoun of premiums and prepayers pay he same amoun of premiums as borrowers holding he morgages o mauriy. The earlier a borrower defauls, he less he oal premium paymen is. NDNP= PP = PPs > DF > DF s > s. (9) Comparison of Alernaive Premium Srucures The comparison of he presen value of ex-pos premium paymens is based upon he premium schedules obained from a se of defaul and prepaymen probabiliies and a fixed gross margin. We separaely compare paymens for defaulers, prepayers and borrowers holding morgages o mauriy. We use superscrips AN, UF, RF, FG o UF represen Cases 4, respecively. For example, DF represens he ex-pos paymen for ime defaulers under he upfron premium srucure (Case ). The definiions and abbreviaed names for hese four cases are summarized as below: Case Descripion Abbreviaed Name Annual premium only, no upfron premium Annual case Upfron premium only, no refund, no financing Upfron case 3 Upfron premium wih refund, no financing Refund case 4 Upfron premium wih financing, no refund Financing case Borrowers Holding Morgages o Mauriy. For borrowers holding morgages o mauriy, he paymen in he Upfron case is lower han ha in he Annual case because in he Annual case boh defaulers and prepayers are paying less han borrowers holding he morgages o erm. In comparison wih he Upfron case, under he Annual case borrowers holding he morgages o mauriy, as well as he lae defaulers and prepayers, mus pay higher premiums o make up for he lower premiums paid by he early defaulers and prepayers. The premium paymen in he Refund case is greaer han ha in he Upfron case because he Refund case prepayers are refunded for he unused premiums while he Upfron case prepayers are no. The premium paymens made by early defaulers are lower in he Annual case han hose in he Refund case, implying ha borrowers holding morgages o mauriy will pay more in he Annual case han under he Refund case. The relaionship of he accumulaed paymens for borrowers holding morgages o mauriy is:

368 JOURNAL OF REAL ESTATE RESEARCH NDNP > NDNP > NDNP NDNP AN RF UF FG UF > NDNP. () Defaulers. The relaionship of paymens made by defaulers under differen premium srucures is a funcion of he ime of defaul. For he Annual case and he Upfron case, here exiss a ime of defaul ( CC ) such ha: UF DF > DF if < AN C C UF DF < DF if > AN C C. () Similarly, for he Annual case and he Refund case, here exiss a ime of defaul CC3 ( CC3 > CC ) such ha: RF DF > DF if < AN C C 3 RF DF < DF if > AN C C 3. () The paymens for defaulers in he Refund case are higher han hose in he Upfron case, i.e., DF RF > DF UF. The difference beween he paymen made by a defauler in he Upfron case and ha in he Refund case is posiively correlaed wih prepaymen raes. A defauler s premium paymens in he Refund case reduce o hose in he Upfron case when he raes of prepaymen are zero. Under normal circumsances when he prepaymen raes are no expeced o be close o zero, a defauler s paymen in he Financing case is always below ha in he Annual case. However, when prepaymen raes are expeced o be very low, he relaive size of he premium paymen beween he Annual case and he Financing case depends upon he ime of defaul. Early defaulers pay lower premium in he Financing case han in he Annual case bu he lae defaulers pay higher premium in he Financing case han in he Annual case. Prepayers. Borrower premium paymens in he Refund case (PP RF ) are lower han hose in he Annual case (PP AN ) if he prepaymen occurs in he early and lae policy years, bu may become larger han hose in he Annual case if he prepaymen occurs during he periods of high defaul probabiliy. PP RF is also lower han DF RF and approaches i as approaches T. The prepayers always pay more in he Financing case han in he Upfron case because insurers lose he unamorized premium balance in he Financing case once defaul occurs. The premium paymens in he Annual case and he Refund case are lower han hose in he Upfron case and he Financing case for early prepayers bu higher for lae prepayers. Numerical Resuls To illusrae he above framework and compare he alernaive premium programs, we consruced numerical examples. The underlying condiional prepaymen and defaul raes used o deermine he fair insurance premium raes for he four programs are repored in Price Waerhouse LLP (997). The acual FHA defaul and prepaymen experience is compued for policy years o and he remaining years are esimaed wih grouped logi models. The raes are esimaed by aking a simple average across all hiry-year fixed-rae morgages for each policy year. The resuling defaul and prepaymen raes are displayed in Exhibi. VOLUME 4, NUMBER 3, 997

RATIONALES OF MORTGAGE INSURANCE PREMIUM STRUCTURES 369 Given he condiional prepaymen and defaul raes, we calculaed he premium schedule for each srucure case assuming zero ne presen values for he insurers and a claim loss rae of 4% of he remaining principal balance. Expressed as a percenage of he iniial loan amoun, he premium raes are.55% per annum for he Annual case, 3.36% for he Upfron case, 4.4% for he Refund case, and 3.76% for he Financing case. Borrower s Perspecive: Comparison of Alernaive Premium Paid by Premium Srucures Exhibis and 3 display he presen value of he ex-pos premium paymens for prepayers and defaulers, respecively, by year of erminaion. The dramaic difference in oal premiums paid, depending upon year of erminaion and premium program, is apparen in hese exhibis. These differences in oal premiums paid provide incenives for borrowers o selec he mos financially advanageous program, i.e., he program wih lowes oal cos. Which program is mos aracive o a prepayer depends upon he year in which erminaion occurs. The relaionship beween oal premiums paid under he differen premium programs and by erminaion year is based largely upon he exen o which eiher of wo ypes of subsidizaion is implici in he programs: () subsidizaion across erminaion years, and () subsidizaion of defaulers by non-defaulers. To he exen premiums are charged each year equal o he presen value of he expeced fuure loss in he respecive year, no subsidy across erminaion years will exis. However, for a given book of business, i.e., morgage loans originaed during he same year, his program would require an annual premium ha varies by year. In addiion, he premium paern would be differen for each book of business due o differences in expeced loss paerns by book. For simpliciy, Exhibi FHA Condiional Prepaymen and Defaul Raes Condiional Raes (%) 8 6 4 Defaul Prepaymen 4 7 3 6 9 5 8 Policy Year These condiional prepaymen and defaul raes are based upon Price Waerhouse LLP (997). The raes are esimaed by aking simple averages across all hiry-year fixed-rae morgages for each policy year. Boh hisorical raes and fuure forecased raes are used in calculaion of he average raes.

37 JOURNAL OF REAL ESTATE RESEARCH Exhibi Ex-Pos Presen Value of Premiums Paid by Borrowers Who Prepay a Differen Poin in Time Percen PV (Premium Paid) 7 6 5 4 3 3 6 9 5 8 4 7 3 Policy Year Annual Only Upfron Only Financing Upfron Upfron Refund The numbers are presened as percenages of iniial loan amouns. The premium paymens are based upon he raes ha generae zero ne presen values under he FHA prepaymen and defaul raes shown in Exhibi. The raes are.55% per annum for he annual-only case, 3.36% for he upfron-only case, 4.4% for he upfron wih refund case, and 3.76% for he financing upfron case. Exhibi 3 Ex-Pos Presen Value of Premiums Paid by Borrowers ha Defaul a Differen Poin in Time Percen PV (Premium Paid) 7 6 5 4 3 3 6 9 5 8 4 7 3 Policy Year Annual Only Upfron Only Financing Upfron Upfron Refund The numbers are presened as percenages of iniial loan amouns. The premium paymens are based upon he raes ha generae zero ne presen values under he FHA prepaymen and defaul raes shown in Exhibi. The raes are.55% per annum for he annual only case, 3.36% for he upfron only case, 4.4% for he upfron wih refund case, and 3.76% for he financing upfron case. VOLUME 4, NUMBER 3, 997

RATIONALES OF MORTGAGE INSURANCE PREMIUM STRUCTURES 37 mos annual premium programs charge a consan premium rae ha resuls in premiums being less han fuure expeced losses during he early years of he policy and premiums being greaer han fuure expeced losses during laer years of he policy. Thus, borrowers who do no prepay or prepay laer in he life of he loan subsidize borrowers who prepay early in he life of heir loan. Even if an acuarially sound premium is charged ha maches he iming of he premium wih he paern of expeced fuure losses, non-defaulers will subsidize defaulers (ha is he naure of insurance). However, he srucure of some premium programs resul in defaulers no even paying an acuarially sound oal premium, while non-defaulers pay a greaer han acuarially sound oal premium amoun. In such cases, non-defaulers are exra-subsidizing defaulers. I is possible o generally caegorize premium srucures as being erminaion year subsidizers or defauler exra-subsidizers. However, he same premium program may no have he same subsidizaion implicaions for boh prepayers and defaulers. For example, for prepayers, he Upfron case and he Financing case are heavy erminaion year subsidizer programs. In boh cases, he presen value of he oal premium paid is invarian wih regard o erminaion year, so early prepayers pay a premium ha is larger han he presen value of expeced fuure losses and lae prepayers pay a oal premium ha is lower han fuure expeced losses. The Annual cases and he Refund case have less implici erminaion year subsidizaion because he presen value of oal premiums paid varies by erminaion year. Given an acuarially sound refund schedule, he Refund case would mach ne premiums paid o he presen value of expeced fuure losses, and hus would have no implici erminaion year subsidizaion. Referring o Exhibi, borrowers prepaying in year 6 or earlier financially prefer he Annual case or he Refund case o eiher he Upfron premium case or he Financing case, while borrowers prepaying afer six years have he opposie preference. This preference paern resuls from he greaer amoun of erminaion year subsidizaion inheren in he wo upfron premium wihou refund cases (he Upfron case and he Financing case) relaive o he Annual case and he Refund case. The leas preferred premium program for lae prepayers (afer year ) is he Annual case. While boh he Upfron case and he Financing case have idenical degrees of erminaion year subsidizaion, he oal premium cos o he Financing case prepayers is greaer han ha o he Upfron case prepayers because he Financing case has a greaer amoun of exra defauler subsidizaion han he Upfron case. In he Upfron case boh prepayers and defaulers pay he same upfron premium, none of which is refunded no maer wha he iming or reason for erminaion. In conras, under he Financing case, he premium is included in he loan principal amoun and amorized over he life of he loan. Early prepayers mus pay he unpaid principal balance and he unpaid premium. However, when a borrower defauls he defauls boh on he unpaid principal balance and on he unpaid premium. Thus, he presen value of he amorized premium equals he amoun of he upfron premium (as long as he discoun rae equals he morgage ineres rae) for prepayers, bu is less han he amoun of he upfron premium for defaulers. Therefore, non-defaulers mus pay a higher premium o make up for he nonpaymen by defaulers, resuling in exra defauler subsidizaion. Afer year, oal premiums in he Refund case become increasingly lower han hose in he Annual case. This occurs because he Refund case has significanly less exra defauler subsidizaion han he Annual case does afer year. While boh he Annual

37 JOURNAL OF REAL ESTATE RESEARCH case and he Refund case mach expeced fuure losses fairly closely hrough year, a ha poin in ime defauls decline markedly. This decline in fuure defauls is refleced in he refund schedule of he Refund case, bu is no refleced in he Annual case which coninues o remain unchanged. Under he Annual case, defaulers avoid paying annual premiums for he remainder of he loan period. However, since defaulers lose heir refund under he Refund case, hey effecively sill pay he premium ha covers he remainder of he loan erm. Thus, he Refund case has a reduced exra defauler subsidizaion. Exhibi 3 displays similar informaion for defaulers. The Financing case is he mos preferred for all bu he laes defaulers. This general preference for he Financing case is due o he heavy implici exra defauler subsidizaion. The Upfron case and he Refund case have subsanial implici erminaion year subsidizaion for defaulers, while he Annual case and he Financing case have less. Thus, for boh prepayers and defaulers, upfron premiums have subsanial erminaion year subsidizaion. While for prepayers, he Financing case has subsanial erminaion year subsidizaion, for defaulers his case has much less erminaion year subsidizaion. This occurs because, since defaulers defaul on he unpaid amorized premium, he laer hey defaul he more oal premium hey pay. Non-defaulers pay he same oal presen value premium regardless of when hey prepay. In conras, he Refund case has subsanial erminaion year subsidizaion for defaulers since hey give up heir refund upon defaul and herefore pay he same oal premium regardless of when hey defaul. The Refund case has he leas erminaion year subsidizaion for prepayers. In summary, early prepayers (hrough year 7) will end o choose eiher he Annual case or he Refund case, while laer prepayers will prefer he Upfron case. All bu he laes defaulers (afer year ) will end o choose he Financing case. The laes defaulers, like prepayers, will choose he Upfron case. Insurer s Perspecive: Alernaive Premium Srucure and Adverse Selecion The financial effec of he differen premium srucures on insurers will depend on he revenues generaed relaive o he claim coss incurred. While he claim coss are independen of he premium srucure (absen ancillary or feedback effecs), he revenues depend upon he combinaion of he prepaymen and defaul paerns, and he premium srucure. To evaluae he financial effec, we consruc a se of nine ypes of prepaymen/defaul paerns based upon he FHA porfolio. The nine ypes comprise combinaions of low, medium and high raes for boh prepaymens and defauls. Medium raes are FHA average condiional prepaymen and defaul raes. High raes are 5% of medium, and low raes are 5% of medium. The same mix of borrowers is used for each premium program. Premium levels are calculaed such ha oal premium paymens generae zero ne (of claim coss) presen value under he average FHA prepaymen and defaul raes (he medium prepaymen/medium defaul case). Exhibi 4 presens revenues (expressed as a percen of original principal balance) for each of he erminaion combinaions and for each premium program, as well as claim coss for each erminaion combinaion. The means and sandard deviaions are also presened. Exhibi 5 presens he ne presen value of he differen premium programs by erminaion combinaion. Because premiums were se based upon average defauls and prepaymens, he ne presen value of he medium defaul/medium prepaymen (M Def M Prep) case is zero for all premium programs. If borrowers are evenly disribued among VOLUME 4, NUMBER 3, 997

RATIONALES OF MORTGAGE INSURANCE PREMIUM STRUCTURES 373 Exhibi 4 Presen Value under Differen Prepaymen and Defaul Paerns Revenues Annual Upfron Upfron Financing Claim Only Only wih Refund Upfron Cos H Def H Prep.84 3.36 3.3 3.37 4.8 H Def M Prep 3.7 3.36 3.39 3.6 4.9 H Def L Prep 3.9 3.36 3.64 3.8 5.85 M Def H Prep.9 3.36 3. 3.46.87 M Def M Prep 3.36 3.36 3.36 3.36 3.36 M Def L Prep 4.5 3.36 3.6 3. 4.4 L Def H Prep.98 3.36 3.7 3.56.47 L Def M Prep 3.46 3.36 3.34 3.48.74 L Def L Prep 4. 3.36 3.6 3.3.9 Mean 3.44 3.36 3.39 3.34 3.39 Sd Dev..48..7.4.4 The numbers are presened as percenages of iniial loan amouns. M represens he FHA mean condiional raes; H represens.5 imes he FHA mean condiional raes; and L represens.5 imes he FHA mean condiional raes. The premium paymens are based upon he raes ha generae zero ne presen values under he FHA prepaymen and defaul raes as shown in Exhibi. he nine erminaion rae caegories, he nonlineariy of revenues wih regard o he erminaion profiles resuls in average ne presen value across all erminaion combinaions differen from zero. In oher words, pricing based upon he mean only, no aking ino accoun he variance, resuls in biased pricing. In fac, as illusraed in Exhibis and 3, borrowers have incenives o selec he premium program mos financially beneficial o hem. If his self-selecion is no aken ino accoun in pricing he programs, insurers can experience serious financial consequences. To ake his self-selecion ino accoun, we assign a disribuion of borrowers characerized by combinaions of high, medium and low defaul and prepaymen raes for each premium program, based upon he incenives embedded in he oal presen value premium paymens. We assume ha he marginal disribuion of borrowers in erms of boh he defaul and he prepaymen rae caegories is one half, one hird and one sixh. Tha is, depending upon he premium program, one half of he borrowers are assumed o fall in he mos likely defaul (or prepaymen) rae caegory (which could be high, medium, or low), one hird in he nex likely caegory, and he res in he leas likely caegory. We furher assume ha he disribuion of borrowers in erms of defaul raes and in erms of prepaymen raes is independen. Thus, he borrower percenage in each of he nine combined erminaion rae caegories is he produc of he wo borrower percenages in he corresponding defaul rae and prepaymen rae caegories. For he Annual case, he mos likely borrowers are hose wih high defaul and high prepaymen raes. The borrower disribuion in erms of defaul raes is one half of he borrowers wih high defaul raes (.5 imes FHA condiional raes), one hird wih

374 JOURNAL OF REAL ESTATE RESEARCH Exhibi 5 Ne Presen Value o he Insurer of Differen Premium Srucures Annual Upfron Upfron Financing Only Only wih Refund Upfron H Def H Prep.34.8.96.8 H Def M Prep.6.53.5.64 H Def L Prep.93.49..77 M Def H Prep.4.5.33.59 M Def M Prep.... M Def L Prep..67.4.84 L Def H Prep.5.89.7.8 L Def M Prep.73.63.6.74 L Def L Prep..8.5.3 Mean.6...5 Sd Dev..4.4.34.5 The numbers are presened as percenages of iniial loan amouns. M represens he FHA condiional raes; H represens.5 imes of FHA condiional raes; and L represens.5 imes of FHA condiional raes. The premium paymens are based upon he raes ha generae zero ne presen values under he FHA prepaymen and defaul raes as shown in Exhibi. medium defaul raes (FHA condiional raes), and one sixh wih low defaul raes (one half of FHA raes). The Annual case borrowers disribuion in erms of prepaymen raes is one half wih high prepaymen raes, one hird wih medium prepaymen raes, and one sixh wih low prepaymen raes. Therefore, among all borrowers in he insurance pool, 5% ( / /) have high defaul and high prepaymen raes, 6.67% ( / /3) have high defaul and medium prepaymen raes, 8.33% ( / /6) have high defaul and low prepaymen raes,.78% ( /6 /6) have low defaul and low prepaymen raes, and so on. The borrowers disribuion for he oher hree insurance premium cases can be derived similarly given borrowers self-selecion behavior. For he Upfron case, he mos likely borrowers are hose wih low defaul and low prepaymen raes; for he Refund case, hose wih low defaul and high prepaymen raes; for he Financing case, hose wih high defaul and low prepaymen raes. The borrower disribuions in he nine combinaions of defaul and prepaymen raes are presened in Exhibi 6. Based upon he borrower disribuion as shown in Exhibi 6, we can calculae he fair premiums for each program ha ake borrowers self-selecion behavior ino accoun. We disinguish wo ypes of premium pricing: deerminisic pricing and behavioral pricing. In he deerminisic pricing, he zero ne presen value premiums are calculaed assuming borrowers defaul and prepaymen paerns follow FHA experience; in he behavioral pricing, he premiums are deermined assuming borrowers self-selec ino he premium programs ha are beneficial o hem and he borrower disribuion follows hose repored in Exhibi 6. We also disinguish wo ypes of borrower choice oucomes: saic oucome and behavioral oucome. The saic oucome is he scenario ha borrowers are uniformly disribued among he nine combined erminaion rae caegories, while he behavioral oucome is he one ha borrowers self-selec so ha heir disribuion is he same as hose repored in Exhibi 6. In Exhibi 7, we presen he premium raes and he ne presen values for he four ses of insurer pricing / borrower selecion scenarios: () deerminisic VOLUME 4, NUMBER 3, 997

RATIONALES OF MORTGAGE INSURANCE PREMIUM STRUCTURES 375 Exhibi 6 Probabiliy Disribuion Assumpions Reflecing Borrowers Self-Selecion Behavior Annual Upfron Upfron Financing Only Only wih Refund Upfron H Def H Prep 5..78 8.33 8.33 H Def M Prep 6.67 5.56 5.56 6.67 H Def L Prep 8.33 8.33.78 5. M Def H Prep 6.67 5.56 6.67 5.56 M Def M Prep.... M Def L Prep 5.56 6.67 5.56 6.67 L Def H Prep 8.33 8.33 5..78 L Def M Prep 5.56 6.67 6.67 5.56 L Def L Prep.78 5. 8.33 8.33 The numbers represen he disribuion of borrowers in each of he four premium programs. For example, in he upfron wih refund program, 8.33% of he borrowers have high defaul and high prepaymen raes, 5.56% of he borrowers have high defaul and medium prepaymen raes, and so on. The consrucion of he borrower disribuion is described in an earlier secion of his research, Insurer s Perspecive, ha discusses he alernaive premium srucures and adverse selecion from he insurer s perspecive. pricing / saic oucome, () deerminisic pricing / behavioral oucome, (3) behavioral pricing / saic oucome, and (4) behavioral pricing / behavioral oucome. If pricing does no ake self-selecion ino accoun, and esimaion of he resuling ne presen values ignores self-selecion as well, he resuls indicae ha premiums jus abou cover claim coss for each program, wih he Upfron program and he Financing program being slighly negaive and he oher wo programs being slighly posiive (see he op bank of numbers). If he effec of deerminisic pricing is measured properly, aking ino accoun ha borrowers will self-selec, i becomes apparen ha some programs will be significan losers (have negaive ne presen value) and ohers may be significan gainers (second from op bank of numbers). As expeced, he wo programs preferred by he majoriy of defaulers (he Annual program and he Financing program) are he programs a mos financial risk due o adverse selecion, having -.48 and -.88 ne presen values. Of course, if prices are se aking ino accoun he self-selecion of borrowers, and he measuremen of he resuls also akes self-selecion ino accoun, hen each program s prices exacly cover oal claims cos, and he mean ne presen value is zero for all programs. Exhibis 8 and 9 display he effec of incorporaing self-selecion in pricing on he oal presen value of premium paymens by non-defaulers and defaulers by erminaion year (he equivalen of Exhibis and 3). As expeced, he profile of he oal premium paid curves is no changed by he premium adjusmen, since he adjusmen is essenially a level adjusmen and no a ime paern adjusmen. For boh non-defaulers and defaulers, he oal premium curves for he programs preferred by defaulers (he Annual case and he Financing case) shif upward, reflecing he higher premium ha is charged for hese programs when self-selecion is aken ino accoun. Similarly, he curves for he programs preferred by non-defaulers shif downwards relaive o pricing ha does no ake self-selecion ino accoun.

376 JOURNAL OF REAL ESTATE RESEARCH Exhibi 7 NPV o Insurers When Premiums Are Designed wih and wihou Borrower Self-Selecion Behavior Annual Upfron Upfron Financing Only Only wih Refund Upfron Deerminisic Pricing Saic Oucome Premium Rae.55 3.36 4.4 3.76 Mean.6 -.. -.5 Sd Dev..4.4.34.5 Deerminisic Pricing Behavioral Oucome Premium Rae.55 3.36 4.4 3.76 Mean -.48.35.63 -.88 Sd Dev..39.3.5.7 Behavioral Pricing Saic Oucome Premium Rae.63 3. 3.8 4.77 Mean.57 -.37 -.63.86 Sd Dev..4.4.35.56 Behavioral Pricing Behavioral Oucome Premium Rae.63 3. 3.8 4.77 Mean.... Sd Dev..6.8.6.6 The numbers are presened as a percenage of he iniial loan amoun. Deerminisic pricing means premiums are deermined by he FHA experience. Behavioral pricing means premiums are calculaed o provide zero NPVs wih borrowers self-selecion behavior. Saic oucome means he mean and sandard deviaions are calculaed by equal weighs among he nine combined erminaion scenarios. Behavioral oucome means he mean and sandard deviaions are calculaed by having differen weighs for he nine scenarios o reflec borrowers preferences. Conclusions In his sudy, we develop a framework for deermining he premium srucure of morgage insurance. Morgage insurance has he unique feaures of being a conrac wih muli-period coverage, finie life, decreasing risk over ime, and high sysemaic (caasrophic) risk, and is a mandaory conrac o cover he risk o he lender (insead of he borrower who pays he premium). Because of hese unique feaures, he exising insurance heories and premium srucures in oher ypes of insurance may no be applicable. The framework we developed allows he insurer o calculae he premium amoun needed o cover he expeced claim cos and earn economic profi. The raionale of differen premium srucures is analyzed as a mechanism o address he problem of adverse selecion caused by he informaion asymmery beween he lender and borrower regarding he borrower s enure plan and level of defaul risk. Wih he combinaion of upfron premiums, annual premiums and premium refunds, i is possible o reduce he degree of cross-subsidy from sable borrowers o mobile borrowers and from low-risk VOLUME 4, NUMBER 3, 997

RATIONALES OF MORTGAGE INSURANCE PREMIUM STRUCTURES 377 Exhibi 8 Ex-Pos Presen Value of Premiums Paid by Non-Defaulers under Behavioral Pricing Percen PV (Premium Paid) 7 6 5 4 3 3 6 9 5 8 4 7 3 Policy Year Annual Only Upfron Only Financing Upfron Upfron Refund The numbers are presened as percenages of iniial loan amouns. The premium paymens are based upon he raes ha generae zero ne presen values under he FHA prepaymen and defaul raes shown in Exhibi. The raes are.55% per annum for he annual-only case, 3.36% for he upfron-only case, 4.4% for he upfron wih refund case, and 3.76% for he financing upfron case. Exhibi 9 Ex-Pos Presen Value of Premiums Paid by Defaulers under Behavioral Pricing Percen PV (Premium Paid) 7 6 5 4 3 3 6 9 5 8 4 7 3 Policy Year Annual Only Upfron Only Financing Upfron Upfron Refund The numbers are presened as percenages of iniial loan amouns. The premium paymens are based upon he raes ha generae zero ne presen values under he FHA prepaymen and defaul raes shown in Exhibi. The raes are.55% per annum for he annual-only case, 3.36% for he upfron-only case, 4.4% for he upfron wih refund case, and 3.76% for he financing upfron case.

378 JOURNAL OF REAL ESTATE RESEARCH borrowers o high-risk borrowers. However, upfron premiums end o increase homebuyer downpaymen burdens and decrease housing affordabiliy. For mos morgage insurers wih a social mission, he amoun of upfron premiums o charge would depend on he rade-off beween economic and social benefis. We find ha an imporan deerminan of he effec of premium srucures on borrowers and insurers is he exen o which eiher of he wo forms of subsidizaion is inheren in he srucure: () erminaion year subsidizaion and () exra defauler subsidizaion. Ineresingly, he same premium srucure may have very differen subsidizaion implicaion for prepayers and defaulers. References Ambrose, B. W. and C. A. Capone, Resoluion of Single-Family Borrower Defaul: Modeling he Condiional Probabiliy of Foreclosure, paper presened a 996 Mid-Year AREUEA Meeing, 996. Campbell, T. S. and J. K. Dierich, The Deerminans of Defaul on Insured Convenional Residenial Morgage Loans, Journal of Finance, 983, 38:5, 569 8. Cunningham, D. F. and C.A. Capone Jr., The Relaive Terminaion Experience of Adjusable o Fixed Rae Morgages, Journal of Finance, 99, 5:5, 687 73. Cunningham, D. F. and P. H. Hendersho, Pricing FHA Morgage Defaul Insurance, Housing Finance Review, Ocober 984, 3:4, 373 9. Deng, Y., J. M. Quigley and R. Van Order, Morgage Household Income, Equiy, and Morgage Defaul Risk, working paper, Universiy of California-Berkeley, 994. Deng, Y. and C. A. Calhoun, A Dynamic Analysis of Adjusable- and Fixed-Rae Morgage Terminaion, paper presened a 997 AREUEA Annual Meeing. Epperson, J. F., J. B. Kau, D. C. Keenan, and W. J. Muller, III, Pricing Defaul Risk in Morgages, AREUEA Journal, Fall 985, 3:3, 6 7. Foser, C. and R. Van Order, An Opion-Based Model of Morgage Defaul, Housing Finance Review, 984, 3:4, 35 7. Jackson, J. R. and D. L. Kaserman, Defaul Risk on Home Morgage Loans: A Tes of Compeing Hypoheses, Journal of Risk and Insurance, 98, 47:4, 678 9. Kau, J. B., D. C. Keenan, W. J. Muller, III, and J. F. Epperson, A Generalized Valuaion Model for Fixed-Rae Residenial Morgages, Journal of Money, Credi, and Banking, 99, 4:3, 79 99. Price Waerhouse LLP, An Acuarial Review for Fiscal Year 996 of he Federal Housing Adminisraion s Muual Morgage Insurance Fund, Arlingon, Va.: FHA, 997. Swan, C., Pricing Privae Morgage Insurance, AREUEA Journal, Fall 98, :3, 76 96. Vandell, K. D., Defaul Risk Under Alernaive Morgage Insurance Insrumens, Journal of Finance, December 978, 33:5, 79 96. and T. Thibodeau, Esimaion of Morgage Defauls Using Disaggregaed Loan Hisory Daa, AREUEA Journal, 985, 3:3, 9 36. von Fursenberg, G. M., Defaul Risk on FHA-Insured Home Morgages as a Funcion of he Terms of Financing: A Quaniaive Analysis, Journal of Finance, June 969, 4:3, 459 77. The conen and views expressed in his work are hose of he auhors, and do no reflec he posiion or views of Price Waerhouse LLP or is cliens. Quesions on he conen should be direced o he auhors. VOLUME 4, NUMBER 3, 997