Rationales of mortgage insurance premium structures

Rationales of Mortgage Insurance Premium Structurest

Abstract. This study examines the rationales for the design of mortgage insurance premium structures. The actuarially sound premium prices of several widely used structures are formally derived. Two types of cross-subsidization are identified in different structures: (1) subsidization across termination years and (2) extra-subsidization of defaulters by nondefaulters. Because these two types of subsidization exist to different degree among the structures, a borrower may self-select into certain structures to maximize (minimize) the benefits (losses) of cross-subsidies. Adverse selection arises when the borrower's characteristics cannot be completely observed by the insurer. The actuarially sound premium prices should be adjusted for such adverse selection behaviors. Numerical examples are provided to illustrate such adjustments.

Introduction

Currently, different premium structures are used by various insurance/guarantee agencies, such as the Federal Housing Administration (FHA), Veteran's Administration (VA), Federal National Mortgage Association (Fannie Mae), Federal Home Loan Mortgage Corporation (Freddie Mae), and private mortgage insurance companies (PMIs). These insurance programs charge combinations of upfront and annual premiums, and premium refunds are provided when a loan is prepaid within a relatively short period of time. Given these different structures, the insurer will realize different revenue patterns over time. In addition, total premiums incurred by a borrower who prepays or defaults vary by the premium structure and the time of default/prepayment. The rationales for using different premium structures from either the borrowers' or the insurers' perspectives have not yet been studied in a rigorous manner.

Numerous mortgage default pricing articles have been published over the last two decades. Von Furstenberg (1969), Vandell (1978), Jackson and Kaseman (1980), and Swan (1982) study the value of mortgage default risk using econometric models. Campbell and Dietrich (1983) and Vandell and Thibodeau (1985) address the pricing of mortgage default risk and mortgage insurance with a utility maximization approach. Foster and Van Order (1984), Epperson, Kau, Keenan, and Muller (1985), and Cunningham and Hendershott (1984) use contingent claim pricing approaches to price mortgage default risk. Kau, Keenan, Muller, and Epperson (1992), study the value of default risk when prepayments and defaults are interrelated. Cunningham and Capone (1990), Ambrose and Capone (1996), and Deng and Calhoun (1997) focus their studies on the estimation of mortgage termination rates with historical data. Deng, Quigley and Van Order (1994) develop a conditional hazard model to estimate the default function.

Most of these papers focus on the valuation of default risk as a lump sum (upfront premium). However, literature that goes beyond the valuation of upfront premiums to study the effects of using different premium structures to cover default risk is almost nonexistent.

Mortgage insurance differs from other types of insurance in several respects. These differences make it difficult to adopt techniques developed elsewhere in the insurance industry. First, casualty insurance covers a single period, so the historical performance of a particular policy can be used in determining the premium to be charged in subsequent periods. This information cannot be used in determining mortgage insurance premiums, because mortgage insurance covers multiple periods, and the premium for the life of the mortgage is defined at the origination date. Second, in contrast to life insurance, mortgage insurance has a definite termination date and the claim risk decreases rather than increases over time due to the amortization schedule. Third, with proper geographic diversification, other types of insurers can usually reduce risk exposure to a minimum; however, because the prepayment and default rates of mortgages are highly dependent on macroeconomic variables such as interest rates, house price growth rates, and household income (or unemployment rates), substantial systematic risk is involved in mortgage insurance. Finally, when mortgage insurance is mandatory it covers the risk to the lender rather than the risk to the borrower. The borrower has no right to switch insurance companies or to be temporarily uninsured without terminating the existing mortgage loan. Given these unique features, the design of premium structures and the calculation of mortgage insurance premiums deserve careful study.

This research studies the rationales of the design of mortgage insurance premium structures. We develop a framework to calculate mortgage premiums for different structures based on exogenous termination probabilities. The insurer's objective is to collect premiums to cover the expected claim costs and earn economic profit.

Throughout this study, economic value refers to the return commensurate with the level of risk being taken by the insurer. Different premium structures that combine upfront premiums, annual premiums, and/or premium refunds can be designed to fulfill such requirements. Insurers do not have information, however, about the tenure plans and default risks of individual borrowers. Borrowers who differ in these respects incur different total insurance premiums under different structures and, therefore, have an incentive to choose one structure over another. The FHA and PMI structures are used to illustrate this effect. We find that when not constrained by underwriting guidelines, borrowers with shorter tenure plans have an incentive to choose the FHA structure over the PMI structure. Properly designed premium structures can reduce the problem caused by information asymmetry. We find that two types of subsidization inherent within mortgage insurance pricing structures play a key role in the effect of these premium structures. These forms of subsidy are (I) termination year subsidization and (2) extra defaulter subsidization.

The next section describes several existing premium structures. Section three develops the framework for determining insurance premium structures. Section four analyzes the ex-post performance of various structures and discusses the effect of heterogeneous borrowers with asymmetric information. The fifth section provides numerical examples of existing premium structures, the incentive effect on the behavior of borrowers, and the effect of different pricing approaches on the issuers. The last section draws conclusions about our findings and their implications.

The Existing Premium Structures

The Fannie Mae and Freddie Mac mortgage guarantee programs charge borrowers fixed premium rates of about 25 basis points of the remaining balance throughout the life of the mortgage. While premium rates can vary by lenders, they do not vary among mortgages with different characteristics nor do they vary by the age of the mortgage. The value of premium income under such a structure may not match the expected net present value of potential losses faced by a lender during different stages of a mortgage. For example, the default risk during the early years of a mortgage is usually much higher than that during the later years. Furthermore, the dollar amount of loss to the lender is usually lower during the later years of the mortgage due to amortization. A rational premium structure should reflect these patterns. That is, at any point in time, the value of mortgage insurance premium income should match the expected net present value of the potential loss over the remaining life of the mortgage.

The premium rates of mortgage insurance provided by FHA vary with the loan-tovalue (LTV) ratio, the type of mortgage (purchase or refinance), and the term of the mortgage. Loans with higher LTV ratio or longer terms are charged higher premium rates. The premiums for streamline refinance loans are lower than those for purchase loans. Between September 1, 1983 and September 30, 1991, FHA charged an upfront premium of 3.8% without any annual premiums. Since 1992, FHA premiums have been structured so that both upfront premiums and annual premiums are collected.

Currently, thirty-year FHA mortgage insurance has an upfront premium rate of 2.25% and an annual premium rate of 0.5% of the remaining balance for five, eight and ten years for loans with LTV ratios below 90%, between 90% and 95%, and above 95%, respectively. When an FHA borrower prepays before the end of the seventh year of the mortgage's life, the FHA will refund a portion of the upfront premium. The percentage of the refund decreases from 98% in the first year after origination to 8% in the seventh year. An FHA thirty-year streamlined refinancing has an upfront premium rate of 2.25% and annual premium rates of 0. So/o for seven years. Since no appraisal is required in FHA refinancing, no LTV information is available and premiums vary only with the term of the mortgage.

Private mortgage insurers offer a variety of insurance programs. The premiums they charge are functions of mortgage balance, LTV, loan type such as fixed-rate mortgage (FRM) or adjustable-rate mortgage (ARM), the term length, the cap rate of the ARM, the buydown provision, the coverage of the loss, and the frequency of payments. Unlike FHA insurance, which charges the same premium rates for FRMs and ARMs, PMIs charge higher premiums for ARMs than for FRMs. The premium rates for FRMs increase with the term of the mortgage and those for ARMs increase with the magnitude of the cap rate. Temporary buydowns have the same premium rates as the ARMs with a 1% cap rate.

According to the frequency of the payments, typical PMI programs can be classified as monthly premiums, level annuals, standard annuals, and super singles (see descriptions below). For monthly and annual premium payments, borrowers can choose between constant renewal and amortized (or declining) renewal. In constant renewal programs, premium rates are multiplied by the original loan amount to calculate the payment, while in amortized renewal, premium rates higher than those in the constant renewal are applied to the remaining balance. The amortized renewal rates typically remain the same through the life of the mortgage. The constant renewal rates are normally the same as the amortized renewal rates in the first ten years after the origination and adjusted downward for the period from the eleventh year to term.

Because the rates charged by different private insurers are either the same or very similar, we report the rates based upon information from a typical insurer. In the monthly premium program, the monthly premiums are calculated according to annualized premium rates. The annualized premium rates based upon amortized loan balances are the same for the first year and subsequent renewal years. For a thirty-year FRM with LTV between 90.01% and 95% and loss coverage of 30%, the amortized renewal rate is 0.78%. For a 2% ARM, the comparable rate is 0.92%.

In the standard annuals program, the premium rate for the first year is greater than the renewal rate. For a thirty-year FRM with LTV from 90.015 to 95% and loss coverage of 30%, the first year premium rate is 1.45%, the amortized renewal rate is 0.49%, and the constant renewal rate for years 11 through expiration is 0.25%. In the level annuals program, the rates for the first year are the same as the amortized renewal rates. For a thirty-year FRM with the same LTV and loss coverage, the amortized level annual premium rate is 0.74% and the constant renewal rate for years 11 through term is 0.20%.

In the single plans and super single programs, premiums are paid up front. Single plans provide coverage for the first three, five, seven, or ten years of the life of the mortgage; super single programs insure against default until the loan is paid in full. For thirty-year loans, single plans are refundable if the contracts are canceled before the expiration of the insurance plan. The refund schedule for super singles vary with the term of the loans and may also vary with the origination LTV. For example, thirty-year loans use ten-year schedules, and fifteen-year loans use five-year refund schedules. The refundable upfront premium rate is 4.80% percent for thirty-year FRMs and 2.9% for fifteen/twenty-year FRMs.

Note that the monthly and annual PMI premiums are refundable at prepayment for the fraction of the month or the year that has not passed. However, this type of refund differs from the upfront premium refund that could occur more than one year after the premium payment. The refund of the latest monthly or annual premium is of limited interest and is not applicable in the discrete time model discussed in the next section.

Thus, we address only the refund of the upfront premium after the first year of origination.

The Model

A Framework for Determining Insurance Premium Structures

In this section, we present a framework under which feasible insurance premium structures can be constructed. A feasible premium structure is defined as one such that the present value of the expected loss (plus a gross margin) for the insurer is equal to that of the expected premium revenues. Assume the term of the mortgage is T periods with payments to be made at time 1, . . ., T. Mortgage loans are originated at the current period, time 0. In each period, a borrower determines whether to default, prepay, or make the payment. The conditional default and prepayment rates at time t are given by d1 and p1. The conditional probability that a borrower will stay current at time t is ct=1-di-pt. When a borrower defaults, the insurer is assumed to incur a loss that is proportional to the unpaid mortgage balance. Denote the unpaid mortgage balance at the instant after the time t payment by B, and the ratio of the loss to the unpaid principal balance by LR. The loss ratio LR is assumed constant throughout the mortgage term. The present value of the expected accumulated loss for the lender from the present to time t, denoted by EAL,, is:

Case 2: Upfront premium only, no annual premium, no refund.

In Case 2, only upfront premiums are collected but none of the premium is to be refunded when a borrower prepays. The equilibrium premium that the insurer requires is (1 +q)EALT. Case 3: Upfront premium only, unused premium refunded when borrowers prepay, no annual premium.

Given the conditional default and prepayment rates, we want to determine the upfront premium and the refund schedule according to which the unused premiums are refunded.

The unused premium is defined as the portion of the upfront premium covering the time periods from after the prepayment to the end of the term. Thus, the earlier a borrower prepays, the larger the proportion of the upfront premium to be refunded. The total upfront premium can be considered the sum of the premiums paid in advance for insuring against the default risk in each period with the condition of refunding unused premiums upon prepayment. Let gt be such that g1B0 is the portion of the refundable upfront premium for insuring against the time t default risk. The ratio of the refundable upfront premium to the original loan amount, denoted by ao, can be written as:

Case 4: Upfront premium financed, no refund, no annual premium

When the upfront premium is financed and therefore amortized, the borrowers pay a fixed amount of premium in each period until the mortgage is terminated. When a borrower prepays, the unpaid premium balance will be due immediately. However, the unpaid premium balance at the instant before the prepayment is equal to the present value of the future premium payments if the borrower did not prepay. The present value of the premium payment for the prepayers does not depend on when the prepayment occurs. The accumulated expected premium revenue for Case 4 is:

Comparison of Alternative Premium Structures-Algebraic Approach

Given a premium structure discussed in the above section, the ex-post premium payment made by a borrower depends upon the premium structure and the behavior of the borrower. We first compare the ex-post payments for borrowers under the same premium structure but with different behaviors. Then we compare the ex-post payments for one borrower under different premium structures. The implication of this analysis is that borrowers with different expectations of mobility and default will have different preferences for specific premium structures.

We compare the ex-post payment for three types of borrowers: borrowers who hold the mortgage until maturity (accumulated premium payments denoted by NDNP), borrowers who default at time t and s (accumulated premium payments denoted by DFt and DFs, t > s), and borrowers who prepay at time t and s (accumulated premium payments denoted by PPt and PPs).

Comparison within the Same Premium Structure

Case 1: Annual Premium Only. Since there is no refund, borrowers who default and prepay at the same time pay the same insurance premiums. The earlier borrowers terminate the mortgage, the smaller the premium payments are. Thus the borrowers who hold the mortgage until the maturity date pay the highest premiums. Case 2: Upfront Premium Only, No Refund. Because there is no refund and no annual premiums, the premium payments for all types of borrowers are the same.

Case 3: Upfront Premium with Refund, No Annual Premiums. In the upfront with refund case, defaulters pay the same premiums as borrowers holding the mortgage to maturity because neither receives a refund. The premium payments for prepayers are less than those for the defaulters, and the earlier the borrowers prepay, the lower are the total payments.

Case 4: Upfront Premium with Financing, No Refund, No Annual Premiums. In the upfront with financing case, defaulters pay the least amount of premiums and prepayers pay the same amount of premiums as borrowers holding the mortgages to maturity. The earlier a borrower defaults, the less the total premium payment is. Comparison of Alternative Premium Structures

The comparison of the present value of ex-post premium payments is based upon the premium schedules obtained from a set of default and prepayment probabilities and a fixed gross margin. We separately compare payments for defaulters, prepayers and borrowers holding mortgages to maturity. We use superscripts AN UF, RF, FG to represent Cases 14, respectively. For example, DF,UF represents the ex-post payment for time t defaulters under the upfront premium structure (Case 2). The definitions and abbreviated names for these four cases are summarized as below:

Borrowers Holding Mortgages to Maturity. For borrowers holding mortgages to maturity, the payment in the Upfront case is lower than that in the Annual case because in the Annual case both defaulters and prepayers are paying less than borrowers holding the mortgages to term. In comparison with the Upfront case, under the Annual case borrowers holding the mortgages to maturity, as well as the late defaulters and prepayers, must pay higher premiums to make up for the lower premiums paid by the early defaulters and prepayers. The premium payment in the Refund case is greater than that in the Upfront case because the Refund case prepayers are refunded for the unused premiums while the Upfront case prepayers are not. The premium payments made by early defaulters are lower in the Annual case than those in the Refund case, implying that borrowers holding mortgages to maturity will pay more in the Annual case than under the Refund case. The relationship of the accumulated payments for borrowers holding mortgages to maturity is:

The payments for defaulters in the Refund case are higher than those in the Upfront case, i.e., DF,RF > DF,UF. The difference between the payment made by a defaulter in the Upfront case and that in the Refund case is positively correlated with prepayment rates. A defaulter's premium payments in the Refund case reduce to those in the Upfront case when the rates of prepayment are zero. Under normal circumstances when the prepayment rates are not expected to be close to zero, a defaulter's payment in the Financing case is always below that in the Annual case. However, when prepayment rates are expected to be very low, the relative size of the premium payment between the Annual case and the Financing case depends upon the time of default. Early defaulters pay lower premium in the Financing case than in the Annual case but the late defaulters pay higher premium in the Financing case than in the Annual case. Prepayers. Borrower premium payments in the Refund case (PP,RF) are lower than those in the Annual case (pp,AN) if the prepayment occurs in the early and late policy years, but may become larger than those in the Annual case if the prepayment occurs during the periods of high default probability. PPt' is also lower than DF,RF and approaches it as t approaches T . The prepayers always pay more in the Financing case than in the Upfront case because insurers lose the unamortized premium balance in the Financing case once default occurs. The premium payments in the Annual case and the Refund case are lower than those in the Upfront case and the Financing case for early prepayers but higher for late prepayers.

Numerical Results

To illustrate the above framework and compare the alternative premium programs, we constructed numerical examples. The underlying conditional prepayment and default rates used to determine the fair insurance premium rates for the four programs are reported in Price Waterhouse LLP (1997). The actual FHA default and prepayment experience is computed for policy years 1 to 22 and the remaining years are estimated with grouped logit models. The rates are estimated by taking a simple average across all thirty-year fixed-rate mortgages for each policy year. The resulting default and prepayment rates are displayed in Exhibit 1.

Given the conditional prepayment and default rates, we calculated the premium schedule for each structure case assuming zero net present values for the insurers and a claim loss rate of 40% of the remaining principal balance. Expressed as a percentage of the initial loan amount, the premium rates are 0.55% per annum for the Annual case, 3.36% for the Upfront case, 4.04% for the Refund case, and 3.76% for the Financing case.

Borrower's Perspective: Comparison of Alternative Premium Paid by Premium Structures Exhibits 2 and 3 display the present value of the ex-post premium payments for prepayers and defaulters, respectively, by year of termination. The dramatic difference in total premiums paid, depending upon year of termination and premium program, is apparent in these exhibits. These differences in total premiums paid provide incentives for borrowers to select the most financially advantageous program, i.e., the program with lowest total cost. Which program is most attractive to a prepayer depends upon the year in which termination occurs.

The relationship between total premiums paid under the different premium programs and by termination year is based largely upon the extent to which either of two types of subsidization is implicit in the programs: (1) subsidization across termination years, and (2) subsidization of defaulters by non-defaulters. To the extent premiums are charged each year equal to the present value of the expected future loss in the respective year, no subsidy across termination years will exist. However, for a given book of business, i.e., mortgage loans originated during the same year, this program would require an annual premium that varies by year. In addition, the premium pattern would be different for each book of business due to differences in expected loss patterns by book. For simplicity, most annual premium programs charge a constant premium rate that results in premiums being less than future expected losses during the early years of the policy and premiums being greater than future expected losses during later years of the policy. Thus, borrowers who do not prepay or prepay later in the life of the loan subsidize borrowers who prepay early in the life of their loan.

Even if an actuarially sound premium is charged that matches the timing of the premium with the pattern of expected future losses, non-defaulters will subsidize defaulters (that is the nature of insurance). However, the structure of some premium programs result in defaulters not even paying an actuarially sound total premium, while non-defaulters pay a greater than actuarially sound total premium amount. In such cases, non-defaulters are "extra-subsidizing" defaulters.

It is possible to generally categorize premium structures as being termination year subsidizers or defaulter extra-subsidizers. However, the same premium program may not have the same subsidization implications for both prepayers and defaulters. For example, for prepayers, the Upfront case and the Financing case are heavy termination year subsidizer programs. In both cases, the present value of the total premium paid is invariant with regard to termination year, so early prepayers pay a premium that is larger than the present value of expected future losses and late prepayers pay a total premium that is lower than future expected losses. The Annual cases and the Refund case have less implicit termination year subsidization because the present value of total premiums paid varies by termination year. Given an actuarially sound refund schedule, the Refund case would match net premiums paid to the present value of expected future losses, and thus would have no implicit termination year subsidization.

Referring to Exhibit 2, borrowers prepaying in year 6 or earlier financially prefer the Annual case or the Refund case to either the Upfront premium case or the Financing case, while borrowers prepaying after six years have the opposite preference. This preference pattern results from the greater amount of termination year subsidization inherent in the two upfront premium without refund cases (the Upfront case and the Financing case) relative to the Annual case and the Refund case. The least preferred premium program for late prepayers (after year 11) is the Annual case.

While both the Upfront case and the Financing case have identical degrees of termination year subsidization, the total premium cost to the Financing case prepayers is greater than that to the Upfront case prepayers because the Financing case has a greater amount of extra defaulter subsidization than the Upfront case. In the Upfront case both prepayers and defaulters pay the same upfront premium, none of which is refunded no matter what the timing or reason for termination. In contrast, under the Financing case, the premium is included in the loan principal amount and amortized over the life of the loan. Early prepayers must pay the unpaid principal balance and the unpaid premium.

However, when a borrower defaults he defaults both on the unpaid principal balance and on the unpaid premium. Thus, the present value of the amortized premium equals the amount of the upfront premium (as long as the discount rate equals the mortgage interest rate) for prepayers, but is less than the amount of the upfront premium for defaulters.

Therefore, non-defaulters must pay a higher premium to make up for the nonpayment by defaulters, resulting in extra defaulter subsidization.

After year 11, total premiums in the Refund case become increasingly lower than those in the Annual case. This occurs because the Refund case has significantly less extra defaulter subsidization than the Annual case does after year 11. While both the Annual case and the Refund case match expected future losses fairly closely through year 11, at that point in time defaults decline markedly. This decline in future defaults is reflected in the refund schedule of the Refund case, but is not reflected in the Annual case which continues to remain unchanged. Under the Annual case, defaulters avoid paying annual premiums for the remainder of the loan period. However, since defaulters lose their refund under the Refund case, they effectively still pay the premium that covers the remainder of the loan term. Thus, the Refund case has a reduced extra defaulter subsidization.

Exhibit 3 displays similar information for defaulters. The Financing case is the most preferred for all but the latest defaulters. This general preference for the Financing case is due to the heavy implicit extra defaulter subsidization. The Upfront case and the Refund case have substantial implicit termination year subsidization for defaulters, while the Annual case and the Financing case have less. Thus, for both prepayers and defaulters, upfront premiums have substantial termination year subsidization. While for prepayers, the Financing case has substantial termination year subsidization, for defaulters this case has much less termination year subsidization. This occurs because, since defaulters default on the unpaid amortized premium, the later they default the more total premium they pay. Non-defaulters pay the same total present value premium regardless of when they prepay. In contrast, the Refund case has substantial termination year subsidization for defaulters since they give up their refund upon default and therefore pay the same total premium regardless of when they default. The Refund case has the least termination year subsidization for prepayers.

In summary, early prepayers (through year 7) will tend to choose either the Annual case or the Refund case, while later prepayers will prefer the Upfront case. All but the latest defaulters (after year 22) will tend to choose the Financing case. The latest defaulters, like prepayers, will choose the Upfront case.

Insurer's Perspective: Alternative Premium Structure and Adverse Selection

The financial effect of the different premium structures on insurers will depend on the revenues generated relative to the claim costs incurred. While the claim costs are independent of the premium structure (absent ancillary or feedback effects), the revenues depend upon the combination of the prepayment and default patterns, and the premium structure. To evaluate the financial effect, we construct a set of nine types of prepayment/default patterns based upon the FHA portfolio. The nine types comprise combinations of low, medium and high rates for both prepayments and defaults. Medium rates are FHA average conditional prepayment and default rates. High rates are 150% of medium, and low rates are 50% of medium. The same mix of borrowers is used for each premium program. Premium levels are calculated such that total premium payments generate zero net (of claim costs) present value under the average FHA prepayment and default rates (the medium prepayment/medium default case). Exhibit 4 presents revenues (expressed as a percent of original principal balance) for each of the termination combinations and for each premium program, as well as claim costs for each termination combination. The means and standard deviations are also presented. Exhibit 5 presents the net present value of the different premium programs by termination combination. Because premiums were set based upon average defaults and prepayments, the net present value of the medium default/medium prepayment (M Def M Prep) case is zero for all premium programs. If borrowers are evenly distributed among the nine termination rate categories, the nonlinearity of revenues with regard to the termination profiles results in average net present value across all termination combinations different from zero. In other words, pricing based upon the mean only, not taking into account the variance, results in biased pricing.

In fact, as illustrated in Exhibits 2 and 3, borrowers have incentives to select the premium program most financially beneficial to them. If this self-selection is not taken into account in pricing the programs, insurers can experience serious financial consequences. To take this self-selection into account, we assign a distribution of borrowers characterized by combinations of high, medium and low default and prepayment rates for each premium program, based upon the incentives embedded in the total present value premium payments. We assume that the marginal distribution of borrowers in terms of both the default and the prepayment rate categories is one half, one third and one sixth. That is, depending upon the premium program, one half of the borrowers are assumed to fall in the most likely default (or prepayment) rate category (which could be high, medium, or low), one third in the next likely category, and the rest in the least likely category. We further assume that the distribution of borrowers in terms of default rates and in terms of prepayment rates is independent. Thus, the borrower percentage in each of the nine combined termination rate categories is the product of the two borrower percentages in the corresponding default rate and prepayment rate categories. For the Annual case, the most likely borrowers are those with high default and high prepayment rates. The borrower distribution in terms of default rates is one half of the borrowers with high default rates (1.5 times FHA conditional rates), one third with medium default rates (FHA conditional rates), and one sixth with low default rates (one half of FHA rates). The Annual case borrowers' distribution in terms of prepayment rates is one half with high prepayment rates, one third with medium prepayment rates, and one sixth with low prepayment rates. Therefore, among all borrowers in the insurance pool, 25% (=1/2x 12) have high default and high prepayment rates, 16.67% (=1/2xl/3) have high default and medium prepayment rates, 8.33% (= 1/2x 1/6) have high default and low prepayment rates, 2.78% (= 1/6x 1/6) have low default and low prepayment rates, and so on. The borrowers' distribution for the other three insurance premium cases can be derived similarly given borrowers' self-selection behavior. For the Upfront case, the most likely borrowers are those with low default and low prepayment rates; for the Refund case, those with low default and high prepayment rates; for the Financing case, those with high default and low prepayment rates. The borrower distributions in the nine combinations of default and prepayment rates are presented in Exhibit 6. Based upon the borrower distribution as shown in Exhibit 6, we can calculate the fair premiums for each program that take borrowers' self-selection behavior into account. We distinguish two types of premium pricing: deterministic pricing and behavioral pricing.

In the deterministic pricing, the zero net present value premiums are calculated assuming borrowers default and prepayment patterns follow FHA experience; in the behavioral pricing, the premiums are determined assuming borrowers self-select into the premium programs that are beneficial to them and the borrower distribution follows those reported in Exhibit 6. We also distinguish two types of borrower choice outcomes: static outcome and behavioral outcome. The static outcome is the scenario that borrowers are uniformly distributed among the nine combined termination rate categories, while the behavioral outcome is the one that borrowers self-select so that their distribution is the same as those reported in Exhibit 6. In Exhibit 7, we present the premium rates and the net present values for the four sets of insurer pricing / borrower selection scenarios: (1) deterministic pricing / static outcome, (2) deterministic pricing / behavioral outcome, (3) behavioral pricing / static outcome, and (4) behavioral pricing / behavioral outcome.

If pricing does not take self-selection into account, and estimation of the resulting net present values ignores self-selection as well, the results indicate that premiums just about cover claim costs for each program, with the Upfront program and the Financing program being slightly negative and the other two programs being slightly positive (see the top bank of numbers). If the effect of deterministic pricing is measured properly, taking into account that borrowers will self-select, it becomes apparent that some programs will be significant losers (have negative net present value) and others may be significant gainers (second from top bank of numbers). As expected, the two programs preferred by the majority of defaulters (the Annual program and the Financing program) are the programs at most financial risk due to adverse selection, having -0.48 and -0.88 net present values. Of course, if prices are set taking into account the self-selection of borrowers, and the measurement of the results also takes self-selection into account, then each program's prices exactly cover total claims cost, and the mean net present value is zero for all programs.

Exhibits 8 and 9 display the effect of incorporating self-selection in pricing on the total present value of premium payments by non-defaulters and defaulters by termination year (the equivalent of Exhibits 2 and 3). As expected, the profile of the total premium paid curves is not changed by the premium adjustment, since the adjustment is essentially a level adjustment and not a time pattern adjustment. For both non-defaulters and defaulters, the total premium curves for the programs preferred by defaulters (the Annual case and the Financing case) shift upward, reflecting the higher premium that is charged for these programs when self-selection is taken into account. Similarly, the curves for the programs preferred by non-defaulters shift downwards relative to pricing that does not take self-selection into account.

Conclusions

In this study, we develop a framework for determining the premium structure of mortgage insurance. Mortgage insurance has the unique features of being a contract with multi-period coverage, finite life, decreasing risk over time, and high systematic (catastrophic) risk, and is a mandatory contract to cover the risk to the lender (instead of the borrower who pays the premium). Because of these unique features, the existing insurance theories and premium structures in other types of insurance may not be applicable. The framework we developed allows the insurer to calculate the premium amount needed to cover the expected claim cost and earn economic profit. The rationale of different premium structures is analyzed as a mechanism to address the problem of adverse selection caused by the information asymmetry between the lender and borrower regarding the borrower's tenure plan and level of default risk. With the combination of upfront premiums, annual premiums and premium refunds, it is possible to reduce the degree of cross-subsidy from stable borrowers to mobile borrowers and from low-risk borrowers to high-risk borrowers. However, upfront premiums tend to increase homebuyer downpayment burdens and decrease housing affordability. For most mortgage insurers with a social mission, the amount of upfront premiums to charge would depend on the trade-off between economic and social benefits.

We find that an important determinant of the effect of premium structures on borrowers and insurers is the extent to which either of the two forms of subsidization is inherent in the structure: (1) termination year subsidization and (2) extra defaulter subsidization. Interestingly, the same premium structure may have very different subsidization implication for prepayers and defaulters.

The content and views expressed in this work are those of the authors, and do not reflect the position or views of Price Waterhouse LLP or its clients. Questions on the content should be directed to the authors.

References

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Barry Dennis* Chionglong Kuo* Tyler T. Yang*

+1997 Real Estate Finance award, sponsored by the Fannie Mae Foundation.

*Price Waterhouse LLP, 1616 N. Fort Myer Drive, Arlington, Virginia 22209.

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