Making a Case for Deep Cover Mortgage Insurance

Spring 2016 Volume 22 Number 1 www.iijsf.com Making a Case for Deep Cover Mortgage Insurance STEVE MACKEY AND TED DURANT The Voices of Influence iijournals.com

Making a Case for Deep Cover Mortgage Insurance STEVE MACKEY AND TED DURANT STEVE MACKEY is an executive vice president and chief risk officer at MGIC Investment Corporation in Milwaukee, WI. steve_mackey@mgic.com TED DURANT is a vice president and a risk officer at MGIC Investment Corporation in Milwaukee, WI. ted_durant@mgic.com The current state of the U.S. housing finance system has evolved out of the financial crisis and was formed when the government-sponsored enterprises (GSEs), Fannie Mae and Freddie Mac, were placed into conservatorship by the Federal Housing Finance Authority (FHFA) in 2008. Over eight years and approximately $12 trillion in mortgage originations later, there has been a host of changes affecting housing finance, from HAMP and HARP to the implementation of rules under the Dodd Frank Act. Yet, the fundamental structure of housing finance has not been addressed in any meaningful manner, which has resulted in the U.S. taxpayer providing the financial backstop for the vast majority of the nearly $10 trillion in outstanding mortgage credit risk. In 2008, prior to conservatorship of the GSEs, the federal government provided explicit support for about 6% of mortgage debt outstanding (MDO), 1 the GSEs held about 41%, and the remaining 53% was split evenly between depositories and other private sector holders. Today, the federal government directly supports 62% of MDO, depositories hold 25%, and the remaining 13% is held by other private sector investors. Whether you call that an increase of 10 times (6% to 62%) or an increase of 33% (47% to 62%), it is still a very large increase and represents a significant risk exposure for the taxpayer. Quantifying this risk exposure is a tough problem to solve without having the ability to model the loan portfolios of both the government agencies and the GSEs. As well, there is not nearly enough information disclosed publicly to accomplish this task. We can look to losses that were realized during the financial crisis as a ballpark estimate of this risk exposure. Based on analysis by Mark Zandi at Moody s Analytics, 2 total losses recognized between 2006 and 2013 as a percentage of MDO at year-end 2007 were 9.1% for all loan types and 4.3% for loans backed by the GSEs and government agencies. Given the current amount of MDO and the 62% backstopped by the taxpayer, the implied risk exposure to the taxpayer is between $267 billion and $564 billion. In recognition of the significant risk exposure to the taxpayer, FHFA has mandated that the GSEs establish risk sharing programs that enable the GSEs to sell mortgage credit risk to sources of private capital. To date, both have developed capital market and reinsurance executions to satisfy the regulatory mandate of FHFA. So far, these new programs have been successful in selling portions of the credit risk the GSEs hold. The advent of these risk sharing programs has led the GSEs and FHFA from an acquire and hold framework to an acquire, warehouse, and distribute framework. Although these programs start to reduce the risk exposure MAKING A CASE FOR DEEP COVER MORTGAGE INSURANCE SPRING 2016

to the taxpayer, a robust framework to measure and control the amount of risk that is prudent to take in the warehouse stage of the pipeline has not been developed. In addition, it is uncertain how market liquidity needed to support these risk sharing programs will hold up in times of market stress, but if history is any type of guide it is probable that liquidity for this type of risk will be volatile and at times may even be non-existent. Because reengineering the U.S. housing finance system appears to be a low priority in Washington, we need to find a way to live with the current system while reducing the risk exposure that taxpayers bear. We believe that a very effective way to do this is already in place and can be prudently expanded to bring additional private capital into housing finance. By expanding the use of mortgage insurance (MI), both in terms of the initial loan-to-value ranges covered by the insurance and the depth of coverage for each loan, the majority of the credit risk can be taken by private capital before a loan even gets to the GSEs. This deep cover MI is an additional form of risk sharing and is meant not to replace but to complement the existing programs and to provide enhanced liquidity through the cycle. Furthermore, if deep cover MI were combined with the back-end risk sharing programs already in use, the risk exposure taken by the GSEs and backstopped by the taxpayer could be limited to low probability, extreme tail risk levels. Although mortgage insurance is subject to counterparty risks, there are several factors that indicate these risks should not be sufficient to deter greater reliance on mortgage insurance to de-risk the GSEs. First, despite the limited role played by MI in the past, the industry absorbed approximately $50 billion in losses, substantially reducing the amount required by the Treasury to keep the GSEs solvent. Second, the majority of MI providers raised capital during the crisis, demonstrating the ability and willingness to add more capital to cover excessive losses and to continue insuring new loans. Third, mortgage insurers have worked with their state regulators, FHFA, and the GSEs to revise master policy terms and increase capital requirements. These improvements, coupled with a dramatically less risky mortgage market, have already reduced counterparty risk to a very low level, and additional changes to contract terms using standard insurance techniques can further reduce uncertainty. In the remainder of this article, we will review key considerations in structuring and pricing deep cover MI as well as a framework for pricing mortgage credit risk, deep cover MI, and the residual risk to the GSEs. We will also quantify the incremental counterparty credit risk associated with deep cover MI. In general, the key features of deep cover MI would include the following: A stated level of coverage for each loan (e.g., 40%) or coverage down to a specific loan-to-value ratio (LTV) (e.g., 50% LTV). Coverage that is supplemental to any MI provided for loans with greater than 80% LTV. Applicable to loans with initial LTV greater than 60%. Paid for by the lender and thus not subject to the termination requirements of the Home Ownership and Equity Protection Act (HOEPA). Structured with lenders and the GSEs in a manner that will ensure eligibility for the TBA (to be announced) market. We believe that deep cover MI is most appropriately structured and priced at the loan level for a number of reasons. Operationally it is much simpler to assign coverage to each individual loan than to track the losses and loss limits that are typically associated with a pool of loans. Loan-level coverage follows the loan over its life regardless of who holds the servicing and does not require that the pool be kept intact. And, as will be discussed in more detail, a mortgage has a significant amount of idiosyncratic credit risk that is most appropriately priced at the loan level. Deep cover MI at the loan level could be combined with aggregate loss limits to provide an absolute limit to the risk exposure assumed by the MI or structured with other features although, additional features reduce operational simplicity and ease of execution and servicing. The credit risk inherent in a mortgage is a function of the likelihood of borrower default, the loss given default, and the interrelationship between the two. In simple terms, mortgage default risk is conditional on two occurrences: a change to the borrower s ability to pay and a decline in the value of the property below the outstanding loan amount. As clearly evidenced in the financial crisis, a significant decline in property value may create enough incentive for borrowers to exercise their default option and thus may be both a necessary and sufficient condition for default. Standard portfolio theory says that idiosyncratic risk is not priced as SPRING 2016 THE JOURNAL OF STRUCTURED FINANCE

it can be reduced to negligible levels through hedging or diversification, and therefore only systematic risks should be priced. This may be true where a portfolio can be diversified across a range of asset classes in the economy or where hedges can be constructed through a replicating portfolio. For an entity that is limited to only investing in mortgage credit, like the GSEs or a mortgage insurer, however, we believe that the idiosyncratic risk of a mortgage cannot be reduced to negligible levels and must be priced. The default option in a mortgage is highly dependent on borrower behavioral response to the key drivers of risk, such as changes in home prices, income, employment, and personal events such as divorce. The behavioral response to many of these drivers is unique to each borrower, or set of borrowers, and cannot be hedged or diversified away to a negligible level across a portfolio consisting of only mortgages. Therefore, we believe that loan-level pricing at origination is the most appropriate method to price both the idiosyncratic and systematic risks. The following analysis reveals a wide range of prices for credit risk across credit score and LTV cohorts, demonstrating the significant variation of risk across loans. Theoretically, pricing may be practical at a pool level for homogenous loans. In reality, the range of identifiable risk factors in mortgage lending makes it impractical to create pools of any meaningful size of truly homogenous loans. Those risk factors that are readily identifiable at the loan level, then, are best priced at the loan level at origination. QUANTIFYING MORTGAGE CREDIT RISK We present two methodologies to quantify and price the credit risk associated with a mortgage and to evaluate the feasibility of deep cover MI, structured as 40% total coverage on all loans, regardless of initial LTV. These methodologies are described by Davidson and Levin [2014] in Mortgage Valuation Models, and the analysis we are presenting was performed by Andrew Davidson & Co. (AD&Co) and commissioned by MGIC. Both methodologies make use of industry standard prepayment, default, severity, interest rate, and home price models developed by AD&Co, as well as their modeling platform. These models, like all mortgage models, are abstractions of reality and are tools used to inform judgment. The use of models in understanding borrower behavior and mortgage performance introduces model risk that has to be taken into consideration in evaluating analytical results. We believe that by using an industry standard modeling platform for the deep cover analysis that the level of model risk is limited to an acceptable level. The first is a capital charge methodology (CC) and is fundamentally a present value of lifetime future cash flows approach. The cash flows are generated using the AD&Co analytical framework and their three-group Vasicek model of loss distribution. The key features of the CC approach are solving for the economic capital requirement and for the premium required to generate a target return on capital (ROC). The second is Davidson and Levin s risk-neutralization (RN) approach to shifting, or calibrating, the loss distribution to find the best fit across observed market prices of the credit risk transfer securities, CAS and STACR, issued by the GSEs. Capital Charge Method The CC methodology utilizes the expected shortfall (ES) approach to quantify the tail risk associated with a mortgage in order to determine the amount of economic capital required. ES represents the probability-weighted expected loss beyond a specified confidence level of the loss distribution. The required ROC represents a return target and is a user-specified input. In our analysis we looked at results for a range of ES confidence levels and ROC assumptions. In this article, we present the results for an ES confidence level of 97% and a ROC of 16%. The CC approach of Davidson and Levin (see the Appendix) solves for both capital c and annual premium p simultaneously. To simplify the calculation, the annual premium is transformed to a single upfront premium P using an IO (interest-only) multiple. This allows for a closed-form solution to the equation. Both c and P are held in a reserve account and released over time, proportional to the reduction in risk. The assumption of capital and premiums being held in a reserve account is an important one that enters into the consideration of counterparty risk and ensures that adequate capital is maintained to support the transaction until maturity. The single premium P is the value of the credit risk given economic capital at the specified ES and the required ROC. This represents a model estimate for the fair value of the credit risk at which an arm s-length transaction would take place. In the results that follow, we present the total value of the credit risk and how this would be allocated between the deep cover MI and the MAKING A CASE FOR DEEP COVER MORTGAGE INSURANCE SPRING 2016

residual risk (RR) of the GSEs for a cohort of 56 theoretical loans with initial LTV ranging from 65% to 97% and FICO credit scores ranging from 600 to 770. Home prices for these loans are modeled using a national-level index. The results for deep cover MI include the total benefit of MI for any loans >80% LTV that would have primary coverage. The results presented in Exhibit 1 show that the total value of the credit risk on this set of loans ranges between 0.42% for a low-risk loan of 65 LTV and 770 FICO to 12.85% for a high risk loan of 97 LTV and 600 FICO. The value of the deep cover MI share of the credit risk is presented in Exhibit 2 and ranges between 0.33% and 8.05% for these same loans. The value of the E XHIBIT 1 Total Value of Credit Risk, CC Method at 97% ES/16% ROC (theoretical loans) GSE share of the credit risk is presented in Exhibit 3 and ranges between 0.08% and 4.79%. These results demonstrate that deep cover MI would absorb between 63% and 78% of the total credit risk; thereby, reducing the residual risk held by the GSEs to between 37% and 22%;. See Exhibit 4. The economic capital required to support this amount of credit risk (at 97% ES and 16% ROC) and how it would be allocated between the deep cover MI and RR of the GSEs is presented in Exhibits 5, 6, and 7. The economic capital is commensurate with the relative risk presented by the loan and the risk sharing across the holders of this risk. The annual premium charge for the credit risk borne by deep cover MI and RR of the GSEs is presented in Exhibits 8 and 9. E XHIBIT 3 Residual Value of Credit Risk, CC Method at 97% ES/16% ROC (theoretical loans) E XHIBIT 2 Deep Cover MI Value of Credit Risk, CC Method at 97% ES/16% ROC (theoretical loans) E XHIBIT 4 Ratio of Deep Cover MI to Total Value of Credit Risk at 97% ES/16% ROC (theoretical loans) SPRING 2016 THE JOURNAL OF STRUCTURED FINANCE

E XHIBIT 5 Total Economic Capital, CC Method at 97% ES/16% ROC (theoretical loans) E XHIBIT 8 Deep Cover Annual Risk Premium, CC Method at 97% ES/16% ROC (theoretical loans) E XHIBIT 6 Deep Cover MI Capital, CC Method at 97% ES/16% ROC (theoretical loans) E XHIBIT 9 Residual Risk Annual Risk Premium, CC Method at 97% ES/16% ROC (theoretical loans) Risk-Neutral Valuation E XHIBIT 7 Residual Risk Capital, CC Method at 97% ES/16% ROC (theoretical loans) The RN approach has been calibrated to the population of existing GSE credit risk sharing transactions (CRT) in the market place through roughly June 2015. The RN approach incorporates the market price of risk implied by the bond prices of the CRT structures beyond the credit and prepayment risk ref lected in the modelbased cash flows generated by a Monte Carlo simulation (i.e., credit option-adjusted spread, or OAS). Levin and Davidson first published their credit option-adjusted spread (croas) in 2008. As with traditional OAS, the goal is to neutralize known risk factors to calculate a residual spread that reflects technical (e.g., liquidity and financing) risks not captured in the modeled cash flows. CRT bond returns are driven primarily by interest rate and home price movements, which translate MAKING A CASE FOR DEEP COVER MORTGAGE INSURANCE SPRING 2016

into prepayments and defaults. AD&Co models simulate future paths of rates and home prices and estimate cash flows to the bond holders along each path. They then shift the probability distribution of the cash f lows sufficiently to produce approximately equal croas spreads among tranches of existing STACR deals. An example of the RN approach is presented in Exhibit 10 and is based on the loans underlying Freddie Mac s STACR 2015-DNA2 structure. This example is based on the three-group Vasicek approach and the analysis performed by AD&Co. The 20 scenarios can be thought of simplistically as quartiles, from left to right, as benign, base case, stress, and severe stress. The best fit of the pure modelbased cash flows has a median default rate that is close to scenario 7 (solid line). The risk-neutralization process results in a shift in the loss distribution so that the median default rate is closer to scenario 12 (dotted line), which is considered a stress scenario. This indicates that the market is pricing in a greater expectation of loss than is reflected in the purely model-based cash flows. The columns in the chart reflect the losses being absorbed by the various bonds in the structure as cumulative defaults increase. By comparing the CC and RN approaches for deep cover MI, we are able to identify the combination of ROC and ES that most closely relates to the market price of risk implied in the calibrated RN results and identify the point at which an investor in credit risk would be indifferent between deep cover MI and a CRT bond, like STACR or CAS. The results from applying the RN method to value the same set of theoretical loans presented here for the CC method yield the following. The results, shown in Exhibit 11, demonstrate that the total RN value of the credit risk on this set of loans ranges between 0.33% for a low-risk loan of 65 LTV/770 FICO to 9.18% for a high risk loan of 97 LTV/600 FICO. The value of the deep cover MI share of the credit risk is presented in Exhibit 12 and ranges between 0.26% and 6.12% for these same loans. The value of the GSE share of the credit risk is presented in Exhibit 13 and ranges between 0.07% and 3.06%. These results demonstrate that deep cover MI would absorb between 67% and 79% of the total credit risk, thereby reducing the residual risk held by the GSEs to between 33% and 21%, as seen in Exhibit 14. E XHIBIT 10 Risk-Neutral Valuation Using Scenario Grid Note: Solid line = base case default rate; dashed line = calibrated default rate. Source: Andrew Davidson & Co. SPRING 2016 THE JOURNAL OF STRUCTURED FINANCE

E XHIBIT 11 Total Value of Credit Risk, RN Method (theoretical loans) E XHIBIT 13 Residual Value of Credit Risk, RN Method (theoretical loans) E XHIBIT 12 Deep Cover MI Value of Credit Risk, RN Method (theoretical loans) E XHIBIT 14 Ratio of Deep Cover MI to Total Value of Credit Risk, RN Method (theoretical loans) The results on these theoretical loans reflect a fairly significant difference between the RN and CC methods. As indicated earlier, in the CC method, we have used an ES of 97% and ROC of 16% to price these theoretical loans, and it has resulted in a significantly higher value than the RN results. In fact, to approximate the RN results on these theoretical loans, we need reduce the ES to 95% and the ROC to 10%, which reduces both the amount of capital deployed and the return on that capital. Why do the valuations of the CC and RN approach for the theoretical loans differ so significantly? Based on the analysis that we have worked on with AD&Co, it is largely driven by the geographic assumption used for the theoretical loans. Because these loans are using a national home price index (HPI), the home price volatility is significantly lower than a portfolio of loans, like STACR, that includes a geographically diverse group of loans. As noted in Davidson and Levin, this is due to the nonlinearity of default option exercise and that while home price changes across regions can be averaged out and offset each other, the actual loan losses cannot be averaged out and will not offset. In other words, average volatility of loans across a large number of local HPIs is always greater than the volatility of the average national HPI. The RN method is calibrated to actual CRT trans- MAKING A CASE FOR DEEP COVER MORTGAGE INSURANCE SPRING 2016

actions that are geographically diverse portfolios of loans that have HPI volatility significantly greater than a national HPI. By using this calibration to price these theoretical loans, which are based on a national HPI, we are overstating the volatility, and thus, the RN approach is mispricing these theoretical loans. This is analogous to mispricing an option by overstating the implied volatility when pricing a swaption or using an overstated OAS to value an MBS. This is reflected in the lower capital and ROC required to be used in the CC approach to achieve similar values for the credit risk. E XHIBIT 16 Value of Credit Risk Comparison, CC vs. RN Methods (STACR 2015- DNA2 portfolio) ANALYSIS OF STACR 2015-DNA2 Now that we have explored the CC and RN methods on a set of theoretical loans, let us apply these methods to actual collateral underlying a transaction in the capital markets. The collateral underlying STACR 2015-DNA2 (DNA2) consists of approximately 142,000 geographically diverse loans. Unlike the theoretical loans discussed previously, the RN method is properly calibrated for this portfolio of loans. Freddie Mac brought this structure to market in June 2015 and losses are based on actual severities. The LTV and FICO stratifications of this pool of loans are presented in Exhibit 15. This pool did not include any delinquent loans, and unlike our theoretical loan example, LTVs ranged from 65% to 80%. The portfolio level results are presented in Exhibit 16 and show that the total value of 1.75% for the CC method at 97% ES and 16% ROC is reasonably close to the RN valuation of 1.84%; a difference of less than 5%. The deep cover MI takes 66% of the total risk leaving the RR of the GSEs at 34%. The total amount of capital required to be held against this portfolio is 1.57%, with deep cover MI providing 1.02% and the RR position of the GSEs providing 0.55%. The total annual premium charged for the portfolio is 0.29% with the deep cover MI share at 0.19% and the RR position share of the GSE at 0.10%. Cohort level results for a range of LTV and FICO combinations are presented in Exhibit 17. These results display some variation across the range of LTV and FICO cohorts where RN may exceed CC total value, and it is reversed in other cohorts. Most all of the cohorts are reasonably close in value. So what does the comparison of the CC and RN methods indicate? These results indicate that based on market prices at the time of the 2015 DNA-2 transaction, an investor in credit risk would be roughly indifferent between the cost of the DNA-2 structure and the E XHIBIT 15 STACR 2015-DNA2 Loan Distribution by LTV and Credit Score SPRING 2016 THE JOURNAL OF STRUCTURED FINANCE

E XHIBIT 17 Comparison of CC vs. RN Results by Cohort (STACR 2015-DNA2) deep cover MI priced at 97% ES and 16% ROC. In this simplified example (no expenses or taxes for either alternative), the results indicate that a mortgage insurance provider could 1) hold a prudent level of economic capital against the credit risk and 2) earn a reasonable level of return on that economic capital. QUANTIFYING COUNTERPARTY CREDIT RISK As the guarantor of the timely payment of interest and principal on a mortgage-backed security, the GSEs take on counterparty risk from entities that provide insurance or other types of financial guarantees. Counterparty risk presents itself as a combination of concerns over willingness and ability to pay contractual obligations. Willingness-to-pay issues, while important, are best addressed through contractual terms and business practices, which are outside the scope of this article. Ability to pay depends on the capitalization of the counterparty and the ability of the entity to continue to pay the contractual amounts due during severe stress scenarios. The required amount of capital is an important component of the capital and pricing models presented in this article, and that analysis provides a framework for valuing the incremental counterparty risk associated with deep cover MI. For a specified confidence level, the expected shortfall approach generates a capital level that, in combination with the premiums received, ensures that the guarantor is able to pay all of the expected claims in the tail portion of the loss distribution. This is in contrast to value-at risk (VAR), or threshold loss, which is a point estimate of the losses at a specified level on the distribution. In simple terms, ES is the expected value of the tail area of the distribution, and VAR is a single point estimate on the distribution. Given the specification of the loss distribution and the relationship between ES and VAR, we can estimate the maximum losses that can be absorbed by the guarantor and the probability-weighted average of losses MAKING A CASE FOR DEEP COVER MORTGAGE INSURANCE SPRING 2016

in the extreme tail beyond that probability level. We refer to this as excess expected shortfall (EES). An example of EES is presented in Exhibit 18 for a theoretical loan with an 80% LTV and 750 FICO. At the 97% confidence level, the VAR is 1.55% and the ES (the expected loss conditional on being beyond the 97% point) is 2.06%. The ES of 2.06% is the amount of capital and premium that must be held against this loan in the CC valuation method. Cumulative losses of 2.06% occur at a point on the distribution that is roughly the 99th percentile VaR in this example, also noted on the x-axis. The EES is the probability-weighted expected losses in the tail of the distribution beyond the 2.06% point and has a value of 1.2 basis points (bps). Thus, the incremental counterparty credit risk to the GSEs from deep cover MI on this loan is only 1.2 bps. To put this in context, deep cover MI on this theoretical loan would reduce the risk to the GSEs by 130 bps, while the GSEs would be assuming an incremental 1.2 bps in counterparty risk from the transaction. On a net basis, that is a huge reduction in risk to the taxpayers. An important assumption behind this measurement of counterparty risk is that the premiums and capital are captured within the system. In other words, there is no opportunity for capital to be used elsewhere, either extracted by the owners of the guarantor or used to pay other claimants. In addition, mismatched timing between losses and premium may create a situation where the present value of future premium is sufficient, but there are insufficient liquid assets to pay claims when due. There are straightforward structural and contractual solutions, however, that can address these potential issues in an actual transaction. FINAL THOUGHTS In this article and the supporting analysis, we have demonstrated a sound analytical framework for evaluating and pricing mortgage credit risk and sharing that risk through a deep cover MI structure. Although this analysis includes simplifying assumptions, it clearly indicates that deep cover MI can be an economically viable and prudently capitalized complement to the E XHIBIT 18 Illustration of Expected Shortfall (ES) and Excess Expected Shortfall (EES) SPRING 2016 THE JOURNAL OF STRUCTURED FINANCE

existing CRTs and would significantly reduce the risk exposure of the GSEs and the taxpayers. Furthermore, the recent market volatility, the associated widening of spreads, and the reduction in liquidity for CRTs underscores the need for the GSEs to develop additional risk sharing executions, especially with counterparties that have demonstrated capability to provide liquidity throughout the cycle. By adding a deep cover MI execution to their suite of risk management transactions, the GSEs are only strengthening their abilities to effectively reduce the risk they present to the taxpayers. A PPENDIX DAVIDSON AND LEVIN CAPITAL CHARGE METHOD EXPLANATION The purpose of this method is to derive the premium and capital requirement for bearing credit risk. The method assumes that there is no valuation benchmark or price of credit risk observable in the capital markets. It presents an analysis from the point of view of an MI company, GSE, or other bearer of credit risk. The method is published in Chapters 4 and 19 of Mortgage Valuation Models by Davidson and Levin [2014]. Single-Period Model In order to understand the approach, the analysis starts from a single-period model. The MI faces expected loss (μ) and maximal loss (given confidence level) called expected shortfall (l ES ). At the beginning of the period, the MI collects premium (p) and allocates capital (c) that is kept in a risk-free account earning r. Those two unknowns are found by solving two equations: 1. Expected return on equity (ROE) is equal to the desired target rate R. 2. The account is large enough to cover expected shortfall with probabilistic confidence 1 α l ES 1 1 = α l ( F ) df 1 α where l(f) denotes loss as a function of cumulative probability F. Collecting all cash flow terms and solving for p and c, we will have R r p(1 + r) =μ+ ( LES μ) 1+ R LES μ c = 1+ R The first equation presents premium p in a form of expected loss plus capital charge. The capital-charge term grows with R, thereby increasing the premium but decreasing capital. If we allow R to become infinite, economic capital vanishes, and the premium becomes equal to the expected shortfall all by itself. On the other end, if R = r, the premium is going to cover only expected loss μ, whereas the capital will cover all unexpected losses. Unlike the value of R, the expected-shortfall confidence increases both premium and capital. Multi-Period Model The single-period model has an expositional value but can t be used to determine the capital and premium of realistic MI. First, expected shortfall cannot be set on an annual basis. Second, capital needs to be released over time. In addition, premium becomes a stream that can be sold to the market as an IO. None of those issues exist in the single-premium model. In order to simplify the problem and solve it in a closed-end form, we make several assumptions. 1. GSE/MI sells guarantee as IO at price P upfront; this will minimize capital. 2. Capital and P get placed in a risk-free account and released with balance amortization. 3. Expected lifetime shortfall is used given confidence. 4. No recapitalization is allowed. The solution shown in the book is P = L(R) + L ES (r)(r r)iom(r) c = L ES (r) P p = P/IOM(r) where L(R) is total losses discounted at R, L ES (r) is expected shortfall discounted at r, p is the annual premium, IOM is IO multiple discounted at r. The main properties of the multi-period model are the same as for the single-period model, including the structure of the solution. ENDNOTES 1 Derived from Mortgage Debt Outstanding information published by the U.S. Federal Reserve Board. 2 Data from Zandi [2013] were updated by the authors as of December 2014. MAKING A CASE FOR DEEP COVER MORTGAGE INSURANCE SPRING 2016

REFERENCES Davidson, A., and A. Levin. Mortgage Valuation Models. New York: Oxford University Press, 2014. Levin, A., and A. Davidson. The Concept of Credit OAS in Valuation of MBS. The Journal of Portfolio Management, Spring 2008, pp. 41-55. Zandi, M. Essential Elements of Housing Finance Reform. Testimony before the Senate Banking Committee, Washington, DC, 2013. To order reprints of this article, please contact Dewey Palmieri at dpalmieri@iijournals.com or 212-224-3675. SPRING 2016 THE JOURNAL OF STRUCTURED FINANCE