An actuarial approach to pricing Mortgage Insurance considering simultaneously mortgage default and prepayment

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1 An actuarial aroach to ricing Mortgage Insurance considering simultaneously mortgage default and reayment Jesús Alan Elizondo Flores Comisión Nacional Bancaria y de Valores aelizondo@cnbv.gob.mx Valeria Álvarez Navarro Sociedad Hiotecaria Federal valvarez@shf.gob.mx Israel Sergio Valladares Cedillo Sociedad Hiotecaria Federal ivalladares@shf.gob.mx ICA 2 49 TRACK E - Financial Riss (AFIR) Abstract Bans are exosed to otential losses originated from credit and reayment ris embedded on mortgages. Both reayment and credit ris are of oosing nature, not indeendent among them and generally lead to the termination of the mortgage contract. Bans commonly tae into consideration these riss for ricing, reserving and caitalization uroses. They may also acquire mortgage insurance which generates a way to alleviate and diversify credit ris on their ortfolios by ceding art of the credit ris to a third arty. However mortgage insurance does not rovide coverage against reayment ris and usually benefits the rovider of rotection as exected credit related losses diminish. This wor rovides an actuarial framewor to rice and reserve mortgage insurance considering credit and reayment ris as cometing riss. By establishing the aroriate analogy, multile decrement standard actuarial terminology is alied to mortgage insurance. Aroriate formulas for ricing and reserving mortgage insurance are develoed which consider simultaneously credit and reayment ris. These formulas are subsequently comared to corresonding formulas that consider solely credit ris and hence the marginal effect of reayment ris is derived on the final roduct rice. The aer further elaborates on the estimation of arameters needed to rice mortgage insurance by using 23 to 28 default and reayment ris exerience from the Mexican mortgage sector. Statistical relationshis between a set of exlanatory variables and default and reayment exerience are established through the use of searate standard logistic regressions. These relationshis are further estimated simultaneously by using

2 2 multinomial logistic regression which is shown to refine credit and reayment ris arameters while showing the cometing nature of both riss. Finally, mortgage insurance rices are calculated for a set of credit and reayment arameters, mortgage and mortgage insurance roduct characteristics and differences are analytically exlained by using the theoretical framewor established in the document. Key Words: default, reayment, cometing riss, mortgage insurance, ricing.

3 3. Introduction Managing the ris of mortgage ortfolios requires analytical tools that are able to consider a variety of factors that affect the cash flows of the loans. With adequate analytical tools, bans can establish aroriate reserves against credit losses, rice their roducts to reflect the underlying riss and comare the riss and returns associated with ortfolio segments. Mortgage Insurance (MI) reresents an imortant tool to manage the ris associated with the ortfolio of mortgages and allows the ban to diversify the inherent credit ris. The rice of the MI roduct is a ey factor for bans to evaluate the cost and benefits of the roduct as the adotion of the insurance affects the ris and reward balance of the credit ortfolio. Simultaneously, the rice of the MI roduct reresents the unique source of income for the mortgage insurance comany. The ricing of such contracts is a challenging tas even when data is available. The tas is generally more challenging in the case of emerging marets, where, in the case of borrower default, the rocess of reossession of loan collateral may last a variable number of years and where data on ayment behavior is either unavailable or of oor quality. It is a well documented fact that borrowers face two mutually exclusive alternatives when aying a mortgage, either to reay the loan or to default on it. From a lender and insurer oint of view, these are cometing riss in the sense that they are mutually exclusive and the realization of one ris recludes the other. For the MI rovider, reayment means cessation of cash flows in the form of any mortgage insurance remium and thereafter eliminates any ossibility of default. Conversely, defaulting imlies incurring in costs associated with the MI contract. Thus, one cannot calculate accurately the economic value of the cost of the insurance without considering default and reayment. There is a vast amount of literature related to mortgage ricing. The contingent claims model develoed by Merton (973) rovides a motivation for borrower behavior using otions theory. At the beginning most of the studies on default and reayment used this aroach to rice mortgages; however they focused on default and reayment as individual riss. For instance, Cunningham and Hendershott (984) alied Blac and Scholes (973) otion ricing model to rice default ris considering default as a ut otion sold by the FHA and urchased by the homebuyer for the default rotection of the lender. Schwartz and Torous (989) and Quigley and Van Order (99) rovided emirical estimates of otion-based reayment models consistently with the contingent claims model. Kau and Kennan (995) also rovided a theoretical framewor to rice mortgages as derivative assets in a stochastic economic environment. They described reayment and default as an American call otion and a Euroean comound ut otion resectively. Kau, Keenan and Muller III, emhasized the imortance of the jointness of reayment and default otions in the calculation of insurance rices for mortgages. The hyothesis resented is that a borrower that decides to default today is giving u his otion to default in the future, but also is giving u the otion to reay the mortgage (Deng, Quigley, Van Order, 2). Deng, Quigley, and Van Order (996) resented a joint model for default and reayment in a roortional hazard framewor using an otion based aroach. They also emhasized the imortance of non economic trigger events in affecting default and reayment behavior and

4 4 their relevance in the exercise of otions to default or reay on mortgages in the fully rational way redicted by finance theory. Since the Cox Proortional Hazard (CPH) model was introduced by Green and Shoven (986) to analyze mortgage termination by refinance, these has been a recurrent technique to model mortgage termination. Deng, Quigley and Van Order (2) modeled the cometing riss of mortgage termination emirically in a roortional hazard framewor which allows correlated cometing riss. However they concluded that unobserved borrower heterogeneity is imortant for redicting borrower behavior and CPH does not consider the heterogeneity among borrowers. Recent research on mortgage borrower behavior has roosed several models for the cometing riss of mortgage termination by reayment or default. Deng, Quigley and Van Order (2); Deng, Calhoun (22) resent default and reayment as discrete choices where a Multinomial Logistic Model (MNL) would be an aroriate technique and an alternative to the Cox Proortional Hazard Model (CPH). The MNL avoids the roortional imact of the covariates in the exonential comonent imlicit in CPH and allows direct cometition among the choices: An increase in one termination robability must be offset by a decline in robability for one or more of the alternatives. However, the MNL cannot allow correlations among the termination riss, and one must assume indeendence from the alternatives. In the resent wor, the theoretical and emirical analysis to estimate the cometing riss of mortgage reayment and default is based uon a discrete-time multinomial logistic model of mortgage termination. The emirical analysis is based uon a set of mortgage lending data from a housing government ban which concentrates information of most of the financial intermediaries that oerated mortgage credit in the eriod The dataset allows to study mortgage borrower reayment and default behavior at loan level. The availability of loan-level data allowed to account for the dynamic of mortgage behavior based on ayment history of individual mortgage loans. Logistic regression analysis was alied to establish the relationshi between individual riss (default or reayment) and exlanatory variables. Secondly, the MNL simultaneously established the relationshi between both riss and a set of exlanatory variables. The resulting arameters illustrated the cometing nature of riss. Finally, once the conditional robabilities of default and reayment were obtained, an actuarial aroach for MI ricing is resented using single and multile decrement (default and reayment) tables. Pricing differences when considering only credit ris as comared to the simultaneous default and reayment riss aroach are exlained and interreted for a series of mortgage credit ortfolio segments. The remainder of this aer is organized as follows, section 2 resents the multinomial logistic model for default and reayment riss. Section 3 resents Mexican exerience and the estimated value of relevant arameters. Section 4 develos rice and reserve formulas within an actuarial framewor as alied to mortgage insurance. Section 5 discusses the differences between rices considering a single source of riss versus cometing riss as alied to Mexican exerience. Finally, section 6 resents conclusions. 2. Mortgage default and reayment model Mortgage borrowers face different alternatives to ay their mortgage obligations at each ayment date. Three mutually exclusive choices are considered in this document, i) failure

5 5 from the borrower to ay his loan obligations subsequently referred as default, ii) the comletion of the loan obligations rior to the date originally set in the contract subsequently referred as reayment and iii) the successful comletion of the contract under the terms originally set. It is imortant to note that the three events lead to contract termination. These choices can be modeled by using multinomial logistic regression in which a joint model for the robability of discrete choices can be estimated. Conditional robabilities of reayment and default are also liely to deend on common factors so that there is some inherent simultaneity in the decisions of borrowers. Assume there exist covariates and a constant term. We denote the two roosed functions as: g ( x,...,x ) β β x j j ( x,...,x ) ( x,...,x ) P Y ln P Y + β x β j x And g 2 ( x,...,x ) β 2 β x j ( 2 x,...,x ) ( x,...,x ) P Y ln P Y + β x β 2 j 2 2 j 2 x It follows that the conditional robabilities of each outcome category given the covariate vector are: Default P( Y x,...,x ) Preayment P( Y 2 x,...,x ) + e + e g g g( x,...,x ) e ( x,...,x ) g2 ( x,...,x ) + e g2 ( x,...,x ) e ( x,...,x ) g2 ( x,...,x ) + e Successful comletion ( 3 x,...,x ) g ( x,...,x ) g ( x,..., ) P Y + e + e 2 x Covariates X are described in annex and were constructed considering the following categories:

6 6. Financial Characteristics of the loan 2. Borrower information 3. Mortgage Behavior 4. Financial environment 5. Payment history Term Currency Loan to Value at origination LTV Subsidy House Value Emloyment Payment to Income (PTI) Monthly Income Mortgage age Loan Size Current Loan to Value (CLTV) Mortgage Premium Burnout Delinquency Outstanding Balance 3. Estimation of mortgage default and reayment MNL model with Mexican exerience The information used for the estimation of the model is organized in blocs conformed by 25 months of data divided in, i) a eriod of 2 months ( t 2,t ) called the observation eriod, ii) a second eriod of 2 months ( t,t2 ) called the erformance eriod, and iii) T which is considered the time at observation. The observation eriod facilitates the analysis of the characteristics of the loan and is used to build the exlanatory variables that describe the future behavior of the loan. The erformance eriod allows the observation of either default or reayment from the borrower. The following definitions were considered for the observation of default and reayment during the erformance eriod. Default is observed when the obligor is ast due more than 9 days on any material credit obligation to the creditor during the erformance eriod. Preayment is defined as the comletion of the totality of the loan obligations rior to the date originally set in the contract during the erformance eriod. Loans that either defaulted or reaid a t are not considered for the estimation of the model. The database used for the estimation covers the eriod January 24 to august 27 and reflects the ayment behavior of 3,667 mortgages.

7 7 Loan's distribution by origination date 6.% 5.% 4.% 3.% 2.%.%.% Origination date Figure 3.: Mortgage s origination distribution in the total oulation database. A random samle of 69,772 loans was taen for the estimation of the model. Figure 3.2 reresent the time series for default and reayment for the chosen samle and the oulation. Default Probability by observation window Preayment Probability by observation window DP% 4% 2% % 8% 6% 4% 2% % PP% 8% 6% 4% 2% % 8% 6% 4% 2% % observation window Total oulation Samle observation window Total oulation Figure 3.2: Default and reayment robabilities by observation window total oulation vs samle. For some variables the cometing nature of both riss is made aarent through exloratory analysis. CLTV, Mortgage remium, num2_ (number of months that the loan has ast due ayments in the last 2 months) and Burnout show a contrasting tendency towards default and reayment. Samle Annual % 4% 2% % 8% 6% 4% 2% % Default and Preayment robabilities (% - 6%] (6% - 8%] (8% - +] CLTV DP% PP% Annual % 6% 4% 2% % 8% 6% 4% 2% % Default and Preayment robabilities ( - %] ( - %] (% - +) Mortgage remium DP% PP%

8 8 Annual % 3% 25% 2% 5% % 5% % Default and Preayment robabilities Num2_ DP% PP% Annual % 8% 6% 4% 2% % 8% 6% 4% 2% % Defautl and Preayment robabilities Burnout DP% PP% Figure 3.3: Default and reayment observed rates. The estimation of the model leads to the following selected significant variables: Current Loan to Value Mortgage Premium Number of months with ayments due in the last 2 months (num2_) Mortgage Premium Current Loan to Value Analysis of Maximum Lielihood Estimates Parameter Function Estimate Standard Error Pr > ChiSq Intercet Default <. Preayment <. ( - %] ( - %] Default <. Preayment <. (% - +) Default <. Preayment <. (% - 6%] (6% - 8%] Default <. Preayment (8% - mayor] Default <. Preayment <. Num2_ Default <. Preayment.87.3 <. Table : Multinomial logistic regression estimates. In order to assess the imact of the simultaneity of the estimation searate individual logit models for default and reayment were run to comare the arameters involved in the estimation.

9 9 Variable Mortgage Premium Current Loan to Value Reference β binomial Categories Function β joint model Category models Intersection Default Preayment ( - %] ( - %] Default Preayment (% - +) Default Preayment (% - 6%] Default (6% - 8%] Preayment Default (8% - mayor] Preayment Num2_ Default Preayment Table 2: Comarison between multinomial vs. binomial logistic regression models. As can be seen models rovide very similar estimation values, however in some segments lie mortgage remium and num2_ a larger value of default arameter is accomanied to a corresonding smaller estimate for reayment arameter which suggests that whenever a segment of the oulation shows more tendency to default the joint model diminishes the robability to reay reflecting the contrasting nature of events. The following figures illustrate the comarison between robability estimates using a binomial logistic model versus a multinomial logistic model. %DP / %PP 9% 8% 7% 6% 5% 4% 3% 2% % % Default Probabilities for CLTV (% - 6%] (6% - 8%] (8% - mayor] CLTV Binomial default robability Multinomial default robability %DP / %PP Preayment Probailities for CLTV 2% % 8% 6% 4% 2% % (% - 6%] (6% - 8%] (8% - mayor] CLTV Binomial reayment robability Multinomial reayment robability Default robabilities for Mortgage Premium Preayment robabilities for Mortgage Premium 6% 6% %DP / %PP 5% 4% 3% 2% % % ( - %] ( - %] (% - +) Mortgage Premium Binomial default robability Multinomial default robability %DP / %PP 4% 2% % 8% 6% 4% 2% % ( - %] ( - %] (% - +) Mortgage Premium Binomial reayment robability Multinomial reayment robability

10 %DP / %PP 4% 35% 3% 25% 2% 5% % 5% % Default robabilities for num2_ num2_ Binomial default robability Multinomial default robability %DP / %PP Preayment robabilities for num2_ 4% 2% % 8% 6% 4% 2% % num2_ Binomial reayment robability Multinomial reayment robability Figure 3.4: Comarison between binomial vs. multinomial default and reayment robabilities As can be observed, robability estimates do not differ significantly and MLN estimates aear to be higher than its logistic regression counterarts. It is interesting to note that in no case the sign or the interretation of each covariate arameter is changed due to the estimation method used. Sensitivity analysis multinomial model Sensitivity analysis multinomial model %DP / %PP 2% % 8% 6% 4% 2% % (% - 6%] (6% - 8%] (8% - mayor] CLTV Default Preayment %DP / %PP 4% 35% 3% 25% 2% 5% % 5% % num2_ Default Preayment Figure 3.5: Sensitivity analysis for covariates in the model. 4. Actuarial framewor for ricing and reserving mortgage insurance 4. Mortgage insurance and life insurance %DP / %PP 6% 4% 2% % 8% 6% 4% 2% % Sensitivity analysis multinomial model ( - %] ( - %] (% - +) Mortgage Premium Default Preayment Life insurance is a contract between the insurer and the olicy owner whereby a benefit is aid to the designated beneficiaries if an insured event occurs. The olicy owner agrees to ay a stiulated amount called a remium. In mortgage insurance the olicy owner is the financial institution that rovides the mortgage loan, the benefit is an amount of money exressed as a ercentage of the outstanding amount

11 of the loan at the moment of default and the insured event is the failure from the borrower to ay his loan obligations. The olicy owner agrees to ay a remium that is aid either as a fixed amount at the beginning of the loan or eriodically during the life time of the loan. 4.2 Terminology The ricing and reserving framewor for mortgage insurance benefits from the extensive literature develoed for life insurance such as Bowers (986) To develo the corresonding formulas, the following definitions are necessary: l number of loans that have not incurred in the insured event eriods after origination. d number of loans that incur in the insured event when they reach eriods after origination. By definition: d l l+ robability that a loan does not incur in the insured event from eriod to + after origination. l l + q robability that a loan that has not incurred in the insured event in the first eriods after origination, incurs in the next. q d l l l l + l l + robability that a loan at origination does not incur in the insured event in the next eriods. qx+ robability that a loan at origination does not incur in the insured event in the next eriods and incurs one eriod after eriod. 4.3 Contract termination In life insurance the oulation of new born individuals incurs with certainty in the insured event (death) during a finite eriod (e years) which means that e q. In mortgage insurance where default triggers the ayment of the insured amount, the oulation of loans not necessarily incurs in the insured event over the life time of the contract since only a roortion fails to ay his loan obligations. However, a number of events may lead to the termination of the mortgage contract which in turn may reresent costs to the olicy owner. In what follows the definition of insured event will tae the form of the events defined in section 2 (default, reayment and successful comletion of the contract).

12 2 All three events lead to the termination of the contract which comletes the analogy with life insurance whereby e q Each of the insured events leads to a different insured amount for the insurance comany. Each cause of contract termination is considered a decrement which shows a different robability of occurrence. Let be the age of the loan and define T() and J() two random variables such that: T() is the future life time of the loan aged or alternatively the future life time while the loan does not incur in a contract termination cause and J() the random variable describing the decrement cause for a loan aged. 4.4 Future life time of the loan Let us define the density function of the random variable future life time of the loan T() as follows: T( ) t.q + t f q Where suer index (τ ) denotes all decrement causes described above. The future life time of the loan is a function of the age of the loan and rovides us with the robability that a loan aged months terminates the contract in the interval defined by ages (+t, +t+). t The survival function of the variable future life time of the loan is: t Pr(T > t ) Pr(T t ) t i i q q t 4.5 Joint distribution of future life time of the loan considering multile decrements The joint distribution of J() y T() is defined as follows: ( ) τ T,J ( t, j ) f ( t ).( j,t ) t.q T + t f The term t denotes the robability that the contract remains active for t eriods subject to all ossible causes of termination at age. q + t.( j t ) is interreted as the robability that a contract aged is terminated due to cause j at time t given that the contract has not been terminated reviously. This is denoted as: Substituting: q ( j ) + t q + t. ( j t ). ( j t ( j ) f T,J ( t, j ) Pr(T( ) t, J( ) j ) t. q+ t )

13 3 The joint distribution function of J and T is therefore defined as: T,J 4.6 Mortgage insurance ricing t i i j + F ( t, j ) Pr(T( ) t,j( ) j ).q q The exected value of losses (EL) to the insurance comany under mortgage insurance subject to multile causes of termination of the contract is calculated as follow: t ( j ) EL J term j B j j q v Where: - j B Benefit aid at age to the olicy owner under termination of the contract j. - v discount factor calculated as v. (+ i ) - i funding interest rate - j number of causes of contract termination - term denotes the legal term of the mortgage at origination As mentioned before in this document J is roosed to be equal to 3 where j is the event of default, j 2 is the event of reayment of the loan and j 3 is the successful comletion of the contract. The benefit for the olicy owner for each j is: B for first loss mortgage insurance is generally an amount equal to a ercentage of the outstanding balance of the loan at the moment of default, therefore: B c* OB Where: - c is the ercentage of the outstanding balance of the mortgage agreed to be aid in the event of default by the insurance comany - OB Outstanding balance of the loan at age. For j 2 (reayment) and 3 (successful comletion of the mortgage) the insurance comany generally ays no benefit as mortgage insurance covers default and hence B 2 and B3 for all. EL reresents the actuarial value of the ufront remium to be charged by the insurance comany to the olicy owner for the mortgage insurance contract.

14 4 The exected value of income for the insurance comany under mortgage insurance subject to multile causes of termination of the contract when the remium is exressed as a fixed ercentage of the outstanding balance of the mortgage is calculated as follow: EI term s OB v where: - s reresents the actuarial value of the remium exressed as a ercentage of the outstanding balance of the loan. It is imortant to note that j < (j ) which imlies that for the same remium s, EI < EI which imlies that if other causes of termination different than default are not considered, the insurance comany may overestimate EI which in turn would lead to subsequent losses for the mortgage insurance roduct. s is estimated by deriving its value from the equation EL EI. 4.7 Mortgage insurance reserving Reserves are defined as: MIR EL EI Where: - EL is the exected loss of the mortgage insurance contract at age. - EI is the exected income of the mortgage insurance contract at age. Reserves may be estimated for each termination of the contract cause by estimating: Where: ( j ) ( j ) ( t ) MIR EL EI EL ( j ) term i B j i i j qi 4.8 Concet of a mortality table for default and reayment The values of l and v j d which reresent the number of mortgages that has not terminated the contract for any cause and the number of contracts that terminates by cause j resectively can be observed from ayment exerience from a mortgage ortfolio. The robabilities of termination of the contract are therefore estimated as: i

15 5 j d q l Similarly the subsequent survival robabilities are estimated accordingly: j j d q l Where no ayment exerience is available for the life time of the mortgage, robabilities of default and reayment can be adjusted to mortgage maret conventions such as de standard default assumtion (SDA) and ublic securities association reayment assumtion (PSA). 4.9 Mortgage default and reayment mortality conventions 4.9. Preayment measures Conditional Preayment Rate (CPR) The conditional constant reayment rate (CPR) is a constant robability of reayment of a mortgage ortfolio. The CPR is established in terms of the ercentage of the exected outstanding balance of the ool at the end of the year. The CPR is an annualized rate; however the mortgage ayments are received monthly so it can be exressed as a monthly rate nown as Single Monthly Mortality (SMM). The SMM is the ercentage of the loan s outstanding balance at the beginning of month assumed to reay in full during the month. It can be calculated as follows: j Total SMM ayments,including reayment s - Scheduled interest Unaid rincial balance - Scheduled ayment - Scheduled rincial ayment rincial ayment The SMM and the CPR have the following relationshi: ( CPR ) ( SMM ) 2 SMM ( CPR ) / 2 One of the advantages of CPR is its easy calculation; however it is a constant rate that does not reflect accurately the behavior of mortgages in ractice as it assigns a higher reayment seed at the beginning of the mortgage.

16 6 monthly amount $6 $5 $4 $3 $2 $ $ Preayment amount Preayment Amounts SMM.%.8%.6%.4%.2%.% Age Months Preayments CPR6% Figure 4.: Preayment behavior with a 6% CPR PSA standard The PSA standard is a benchmar develoed by the Public Securities Association. It is built through a series of annualized reayment rates changing uon the mortgage age in months. The rates are CPRs. The PSA assumes that reayment will slowly increase at the beginning of the amortization term then increase at a constant rate until month 3. After month 3, it will remain constant until maturity. The assumtion is that reayment rates are low for new originated mortgages and that it will increase as the mortgages become seasoned. PSA can be described as follows (Fabozzi, Bhattacharya, Berliner, 27): - A CPR of.2% for the first month, increased by.2% er year er month for the next 29 months when it reaches 6% er year and 6% CPR for the remaining years. CPR % PSA curve 6% 4% 2% % 8% 6% 4% 2% % 25% PSA % PSA 5% PSA Age (Months) Figure 4.2: PSA curve

17 7 Other reayment curves can be exressed through the PSA Standard. For examle, a mortgage ool reays in month 5 at 25%PSA meaning that it has an annual reayment rate of 5% 6% * Default measures Conditional default rate (CDR) CDR reresents the monthly default rate exressed as annual ercentage. It is an annualized value of the outstanding balance of the loans that defaulted in the current month as a ercentage of the outstanding balance of the mortgage ortfolio (Fabozzi, Bhattacharya, Berliner, 27). Mathematically it is calculated as follows: MDR t Default loan balance month t Beginning Balance Scheduled Pr incial Payment month t month t To obtain de CDR, the MDR is annualized: CDR ( MDR t ) 2 The CDR is not a historic descrition of defaults behavior in the mortgage ortfolio neither a redictive measure of the exected default rate, also it does not catures default variations over time. To assume a constant CDR may not be consistent with default behavior over the ool, however it is reference measure such as CPR. Standard Default Assumtion (SDA) The SDA curve refers to a standard measure develoed by The Bond Maret Association it is a reference measure to default in a mortgage ool of 3 years fixed rate loans, uon the age of the loan. It is exressed in terms of the CDR. The SDA assumes an initial CDR of.2% over the outstanding balance of the mortgage increasing in.2% monthly until month 3 where reaches a.6% CDR. After month 3, the CDR remains constant till month 6 (fifth year), month in which it begins to decrease lineally in.95% each month until it reaches.3% in month 2. From age 2 the CDR remains constant until maturity (age 36).

18 8 CDR %.%.9%.8%.7%.6%.5%.4%.3%.2%.%.% SDA Curve 5% SDA % SDA Age (Months) Figure 4.3: SDA curve Both CPR and CDR can be translated to actuarial terminology assuming that loan amounts are the same as follows: d MDR q l 2 d SMM q l To illustrate the rocess consider the following mortality table that considers % SDA and % PSA standards. 4. Illustration of a multile decrement table Age () (2) (3) ( ) q Months q q q τ l ( ) (2) (3) ( ) d d d d τ ( τ) ( ) τ ( ) () ( τ) τ ( ). q τ ( ). q τ. q. q.%.%.%.%,.%.%.%.%.%.% 2.%.%.2%.2%, %.%.%.%.2%.2% 3.%.%.3%.4% 99, % 99.98%.%.%.3%.4% 4.%.%.5%.6% 99, % 99.94%.%.%.5%.6% 5.%.%.7%.7% 99, % 99.89%.%.%.7%.7% 6.%.%.8%.9% 99, % 99.82%.%.%.8%.9% 7.%.%.%.% 99,724.% 99.72%.%.%.%.% 8.%.%.2%.3% 99, % 99.6%.%.%.2%.3% 9.%.%.3%.5% 99, % 99.49%.%.%.3%.5%.%.2%.5%.7% 99, % 99.34%.%.%.5%.7%.%.2%.7%.8% 99, % 99.7%.%.2%.7%.8% 2.%.2%.9%.2% 98, % 98.99%.%.2%.8%.2% 3.%.2%.2%.22% 98, % 98.79%.%.2%.2%.22% 4.%.2%.22%.24% 98, % 98.57%.%.2%.22%.24% 5.%.2%.24%.26% 98, % 98.33%.%.2%.23%.26% 6.%.3%.25%.28% 98, % 98.8%.%.2%.25%.27% 7.%.3%.27%.3% 97, % 97.8%.%.3%.26%.29% 8.%.3%.29%.32% 97, % 97.5%.%.3%.28%.3% 9.%.3%.3%.34% 97, % 97.2%.%.3%.3%.33% 2.%.3%.32%.35% 96, % 96.88%.%.3%.3%.34% 2.%.3%.34%.37% 96, % 96.54%.%.3%.33%.36% 22.%.4%.36%.39% 96, % 96.7%.%.3%.34%.38% 23.%.4%.37%.4% 95, % 95.8%.%.4%.36%.39% 24.%.4%.39%.43% 95, % 95.4%.%.4%.37%.4% 25.%.4%.4%.45% 94, % 94.99%.%.4%.39%.43% %.%.5%.52% 6, % 6.63%.%.%.9%.9% 357.%.%.5%.52% 6, % 6.55%.%.%.9%.9% 358.%.%.5%.52% 6, % 6.46%.%.%.8%.9% 359.%.%.5%.52% 6, % 6.38%.%.%.8%.8% %.%.5%.% 6,293 6, , % 6.29% 6.2%.%.8% 6.29%.% 2 Table 4.: Multile decrement table. d / l

19 9 Finally for j which denotes the termination of the contract due to default, the robability of occurrence from age to age is: q ( ) q ( ) d l ( ).5% Probability of Default.4%.3%.2%.%.% Age Figure 4.4: Probability of default For j 2 which denotes the termination of the contract due to reayment the robability of occurrence from age to age is: q ( 2 ) q ( 2 ) d l ( 2 ).6% Probability of Preayment.5%.4%.3%.2%.%.% Age Figure 4.5: Probability of reayment The robability that the contract terminates for any cause is reresented as follows: q q d l

20 2.6% Probability of Multile Decrement.5%.4% %.3%.2%.%.% Age Figure 4.6:Probability of multile decrement 5. Price sensitivity 5. Probability estimation with Mexican exerience Default and reayment curves were obtained as a multile of the PSA and SDA original curves described in section 3. As default and reayment robabilities using the multinomial logistic model are given in terms of annual robabilities no transformations were needed to homologate them to CDR and CPR standards. The annual CDR used for % SDA is.6% and the annual CPR for % PSA is 6%. Finally, the default and reayment SDA and PSA curves for robabilities were introduced to the ricing model described in section 4 to obtain the ufront and level remiums for mortgage insurance. The remiums were differentiated according to different values of the exlanatory variables considered by the MNL model. The following tables show the default and reayment robabilities for a selected covariate and SDA and PSA factors associated to each of the bucets in which the variable was categorized. Current loan to value Current Loan to Value (CLTV) DP% PP% PSA Factor SDA Factor (% - 6%] 4.87% 3.2% (6% - 8%] 6.78% 2.8% (8% - +].99% 8.67% Total 8.74%.99% Table 5.: Default and Preayment robabilities and factors for CLTV. The loan to value ratio serves as an indicator of the current equity osition of the borrower in the roerty. Higher values of CLTV increase the default robability and lower the reayment robability. Mortgage Premium