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Article Excerpt
We study the valuation of mortgage-backed securities when borrowers may have to refinance at premium rates because of their credit. The optimal refinancing strategy often results in prepayment being delayed significantly relative to traditional models. Furthermore, mortgage values can exceed par by much more than the cost of refinancing. Applying the model to an extensive sample of mortgage-backed security prices, we find that the implied credit spreads that match these prices closely parallel borrowers' actual spreads at the origination of the mortgage. These results suggest that models that incorporate borrower credit into the analysis may provide a promising alternative to the reduced-form prepayment models widely used in practice.
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Since its inception in the 1970s, the mortgage-backed security market has experienced dramatic growth in the United States. As of June 30, 2002, the total notional amount of agency mortgage-backed securities and collateralized mortgage obligations outstanding was more than $3.9 trillion. This means that the size of these markets now exceeds the $3.5 trillion notional amount of publicly traded U.S. Treasury debt.
Despite the importance of these markets, however, the goal of developing a fundamental theory of mortgage valuation represents an ongoing challenge to researchers. The key element that has proven difficult to explain within a rational model is how mortgage borrowers choose to refinance their loans. Influential early work by Dunn and McConnell (1981a,b), Brennan and Schwartz (1985) and others applies contingent claims techniques to the problem by modeling prepayment as an endogenous decision made by the borrower in minimizing the present value of his current mortgage. More recently, Dunn and Spatt (1986) and Stanton and Wallace (1998) extend this classical approach in an important way by modeling the prepayment decision as the result of the borrower minimizing his lifetime mortgage costs. As discussed by Schwartz and Torous (1989, 1992, 1993), however, actual prepayment behavior appears very suboptimal relative to the optimal behavior implied by these models. Furthermore, these models all have the property that mortgage-backed security prices cannot exceed par plus the number of points paid to refinance the loan. As demonstrated by Stanton (1995), Boudoukh et al. (1997) and others, this upper bound is nearly always violated in practice.
A common feature throughout the earlier literature is the assumption that the rate at which a borrower can refinance his loan is independent of his financial status. In actuality, however, a borrower who does not satisfy the strictest underwriting guidelines may have to refinance at a higher rate than other more-qualified borrowers. This clearly would reduce his incentives for prepaying his current mortgage and affect his optimal refinancing strategy. Thus, even though principal and interest on an agency mortgage-backed security are guaranteed, the actual timing of these cash flows could be affected by the credit of the borrower, which in turn would be reflected in the value of the security.
In this article, we study the valuation of mortgage-backed securities when the borrower may only be able to refinance at a premium rate over the par mortgage rate because of his credit. Following Dunn and Spatt (1986) and Stanton and Wallace (1998), we solve for the optimal refinancing strategy of a borrower whose objective is to minimize his lifetime mortgage costs. Our approach also takes into account the effects of mortgage refinancing transaction costs and the probability of prepaying for exogenous reasons. Because the nature of the problem requires considering the effects of decisions going far beyond the current mortgage, it is important to use a modeling framework that reflects the actual intertemporal behavior of the term structure. In light of this, we develop the model within a realistic multifactor term-structure setting that matches both the current term structure and the values of fixed-income options. The use of a multifactor model is also consistent with Boudoukh et al. (1997) and Downing, Stanton and Wallace (2002) who find evidence that mortgage-backed security prices are driven by multiple term-structure factors. Because the recursive refinancing problem is essentially an American option valuation problem, we use the least-squares simulation method of Longstaff and Schwartz (2001) to solve for the optimal recursive strategy and value mortgage-backed securities in this multifactor framework.
We show that the optimal refinancing strategy implies prepayment behavior very different from that given by classical rational prepayment models. For example, the mean time to prepayment of the current mortgage can be several years later under the optimal recursive strategy when the borrower is not able to refinance at the par mortgage rate. Furthermore, a borrower attempting to minimize lifetime mortgage costs has a strong incentive to minimize the number of times he refinances his mortgage because of the associated transaction costs. Thus, a borrower who believes that mortgage rates may be going down further in the near future may choose to delay refinancing his loan. This is particularly true when there is a significant possibility that the borrower may need to pay off the loan for exogenous reasons in the near future anyway. Because the first prepayment may occur much later when the optimal recursive strategy is followed, the value of a mortgage-backed security can be significantly higher than par plus the costs of refinancing.
To explore the valuation implications of the model, we collect monthly prices for mortgage-backed securities with coupon rates ranging from 5.50% to 9.50% for the 1992-2002 period. Calibrating the model to the swap curve and the prices of interest rate caps and swaptions for the corresponding months, we solve for the implied mortgage turnover rate and credit spreads that match the historical data. The mean mortgage turnover rate implied from the data is 5.59%, which closely approximates the actual historical turnover rate of 6.04%. Similarly, the implied credit spreads needed to match the cross-section of mortgage-backed security prices in the sample range from zero to about 150 basis points. These implied credit spreads closely parallel actual credit spreads observed in the market at the time the mortgages are originated and are also consistent with mortgage spreads in the rapidly growing subprime mortgage market. These results strongly suggest that mortgage-backed security prices can be reconciled within a rational model by taking into account the credit of the borrower. We also examine the out-of-sample pricing performance of the model by calibrating it using ex ante data and then comparing implied model prices to actual market mortgage-backed security prices. We find that the model is able to capture more than 97% of the variation in the data over the past decade and provides unbiased estimates of overall mortgage-backed security prices.
Although providing a promising start, there are clearly many possible extensions and refinements of this analysis which would be worthwhile to pursue in future work. For example, borrower heterogeneity could be introduced into the model. As in Stanton (1995), this would allow us to study the "burnout" phenomenon in mortgage markets as low-credit-risk borrowers exit a mortgage pool and leave higher-credit-risk borrowers in the pool over time. Although we focus on pricing in this article, the analysis could also be applied to model prepayment behavior itself. Note, however, that simply matching current prepayment rates for mortgages is a less ambitious undertaking than matching prices. This is because mortgage prices are determined not only by current mortgage prepayment behavior, but also by future prepayment behavior under all possible future states of the term structure. Finally, we could extend the analysis to allow borrowers to discount their cash flows along a different discount curve than those used by the market to value mortgage-backed securities. Numerical examples, however, suggest that this possible extension has only a minor effect on the results.
The remainder of this article is organized as follows. The next section describes the mortgage-backed security market. The third section presents the optimal recursive refinancing model. The fourth section provides numerical examples. The fifth section describes the data used in the study. The sixth section provides a preliminary regression analysis of the properties of mortgage-backed security prices. The seventh section examines the implications of the model for mortgage-backed security valuation. The eighth section examines the out-of-sample performance of the recursive model in valuing mortgage-backed securities. The final section summarizes the results and makes concluding remarks.
Mortgage-Backed Securities
A mortgage-backed security is a claim to the cash flows generated by a specific pool of mortgages. Most mortgage-backed securities are issued by one of the three government-sponsored enterprises or agencies known as Ginnie Mae (GNMA), Freddie Mac (FHLMC) and Fannie Mae (FNMA), although there is a growing trend toward mortgage-backed securities being issued directly by large mortgage lenders. Since their inception in the 1970s, mortgage-backed securities have become very popular as an investment vehicle among individual and institutional fixed-income investors. Key reasons for this popularity are that mortgage-backed securities offer attractive yields, have little or no credit risk and trade in a liquid secondary market.
To illustrate the mechanics of how mortgage-backed securities work, let us consider the GNMA I mortgage-backed security program as a specific example. Under this program, the creation of a mortgage-backed security begins with a GNMA-approved mortgage lender or issuer originating a pool of Federal Housing Administration (FHA) insured or Veterans Administration (VA) guaranteed mortgage loans secured by single-family homes. To be eligible for the GNMA I program, these mortgage loans must have the same mortgage rate and meet specific documentation and origination guidelines. As an example, assume that the pool of mortgages consists of recently originated 30-year loans with a fixed coupon rate of 7.00%. Securitized by these loans, the issuer then structures a mortgage-backed security with a coupon rate equal to 6.50% and a total notional amount equal to the aggregate principal amount of the underlying mortgage pool ($1 million minimum pool size). The issuer then sells the mortgage-backed security to investors either directly or through a network of Wall Street dealers.
The coupon rate on the mortgage-backed security is lower than the mortgage rate because the issuer receives a 50-basis-point fee for servicing the mortgages and passing through interest payments and scheduled and unscheduled principal to investors. In the case of delinquent loans, the mortgage-backed security issuer has the responsibility to advance funds from his own account to make the scheduled payments. Out of the 50-basis-point fee received by the issuer, however, the issuer must also pay GNMA a six-basis-point guarantee fee. For this fee, GNMA guarantees the timely payment of principal and interest on the mortgage-backed security to the investors. Thus, because of the GNMA guarantee, investors do not face the risk of losses arising from delinquent payments or from foreclosures or bankruptcies among the mortgages in the underlying mortgage pool. The GNMA guarantee is backed by the full faith and credit of the United States. Mortgage-backed securities issued by FHLMC or FNMA are guaranteed by the respective agencies and bear credit risk similar to their debt, which is rated AAA (or better). GNMA I mortgage-backed securities generally exist only in book-entry form and are issued in minimum denominations of $25,000.
To simplify the pass-through of interest and principal from borrowers to the mort- gage-backed security investors, monthly payments are made on a pre-specified schedule. In particular, the interest and principal collected by the mortgage servicer during a month is paid out to the investors 15 days after the end of the month in which the interest accrues. For other mortgage-backed security programs, the delay can be between 15 and 45 days. Because of this, there can be a slight timing mismatch between the cash flows on the underlying pool of mortgages and those on the mortgage-backed security.
If borrowers did not have the option to prepay their loans, the value of a mortgage-backed security would simply be the value of a fixed 30-year annuity. Because of the prepayment option, however, principal is returned at varying times. Mortgage-backed security valuation is complicated because the timing of these prepayments may be determined by both random and strategic factors. For example, a borrower may prepay his mortgage for exogenous reasons such as moving even though market rates may be higher than his current mortgage rate. Alternatively, a borrower may strategically decide to refinance his mortgage when market rates are below his current mortgage rate.
It is important to observe, however, that the decision to refinance a mortgage does not depend solely on the relation between the borrower's current mortgage rate and the prevailing market rate. To prepay his existing mortgage and refinance the loan, the borrower needs to qualify for a new mortgage loan. To do this, the borrower needs to meet a number of income, credit, documentation, employment, loan-to-value and debt-to-income standards. If the borrower's personal financial situation is such that he or she is marginal in meeting some of these criteria, the borrower may have fewer choices of lenders available and may only be able to refinance at a premium rate (if at all). To illustrate how the borrower's financial situation may affect the rate at which he or she can refinance, note that a VA borrower who refinances a mortgage but puts down less than 5% of the value of the home must pay a three-point VA funding fee in addition to the usual origination costs paid to a mortgage lender. By making a larger down payment, however, the VA borrower may be able to reduce this fee to 1.25 points. A similar sliding schedule of mortgage insurance fees applies to standard FHA mortgages. Thus, a mortgage borrower who is cash-constrained or whose home has declined in value relative to the mortgage balance faces significantly higher costs to refinance. Because these costs are typically absorbed into the refinanced mortgage, they serve to increase the effective rate at which the borrower is able to refinance. The GNMA I program allows premium mortgages with rates as high as 150 basis points above the prevailing GNMA mortgage rate to be pooled into a mortgage-backed security.
Furthermore, FHA or VA borrowers who have recently gone through bankruptcy, a period of unemployment or other credit problems, or who have judgments against them are unable or unwilling to document their income or in other ways fail to meet standard underwriting requirements, may not be able to refinance into new FHA or VA loans. In some cases, these nonconforming borrowers may be able to find conventional lenders with less stringent criteria who would be willing to refinance a loan, albeit at a premium rate. If a borrower's credit is sufficiently bad that he or she cannot refinance with a conventional lender at all, the borrower may then need to go to the subprime mortgage market. This is a rapidly growing sector of the mortgage market that specializes in making loans to borrowers with impaired credit. A quick search of the Web reveals a vast array of subprime mortgage products available to borrowers (often with names such as "bad credit mortgage loans"). Laderman (2001) reports that subprime mortgage originations as a share of total mortgage originations grew from 5% percent in 1994 to 13.4% in 2000. Laderman also shows that subprime mortgage rates averaged roughly 370 basis points higher than prime mortgage rates during the 1994-2000 period and were about 300 basis points at the end of 2001. These considerations make clear that the financial situation...
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