Government-Sponsored Mortgage Securitization and Financial Crises

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) Government-Sponsored Mortgage Securitization and Financial Crises Wayne Passmore and Roger Sparks 2024-002 Please cite this paper as: Passmore, Wayne, and Roger Sparks (2024). “Government-Sponsored Mortgage Securitization and Financial Crises,” Finance and Economics Discussion Series 2024-002. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2024.002. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Government-Sponsored Mortgage Securitization and Financial Crises By 1 Wayne Passmore and Roger Sparks This paper analyzes a model of the mortgage market, considering scenarios with and without government-sponsored mortgage securitization. Conventional wisdom says that securitization, by fostering diversification and creating a “safe” asset in the form of mortgage-backed security (MBS), will reduce risk and enhance liquidity, thereby mitigating financial crises. We construct a strategicgame framework to model the interaction between the securitizer and banks. In this framework, the securitizer initiates the process by setting the MBS contract terms, which includes the guaranteed rate and the criterion that qualifies a mortgage for securitization. The bank then selects which qualifying mortgages to exchange for the MBS. Our investigation leads to a key result: government-sponsored securitization, somewhat counterintuitively, is more likely to exacerbate the severity and frequency of financial crises. 1 Wayne Passmore is a Senior Advisor at the Board of Governors of the Federal Reserve System. Roger Sparks is a Professor of Economics at Northeastern University. Corresponding author: Roger Sparks, r.sparks@northeastern.edu. We thank Colleen Faherty for her excellent research assistance. The views and opinions expressed here are the authors’ own. They are not necessarily those of the Board of Governors of the Federal Reserve System, its members or its staff. 1

1. Introduction This paper analyzes the impact of government-sponsored mortgage securitization on contemporary financial crises, exploring whether it acts as a mitigating or aggravating factor. The securitization process is often perceived as a means of increasing access to home ownership while reducing financial risks to both mortgage originators and investors who buy mortgage-backed securities. Conventional wisdom suggests that securitization creates a safe asset, the mortgagebacked security, which reduces risks by fostering diversification and enhancing the liquidity of otherwise illiquid assets (such as mortgages). Consequently, it is believed to dampen financial crises and lowe r their frequency of occurrence (Kara A, Ozkan A, Altunbas Y, 2016 and Deku SY, Kara A., 2017). This uncomplicated story of safe assets providing stability and liquidity during periods of economic volatility ignores some key elements in the production process that quasi-government entities use to create those ‘safe’ assets. In creating a mortgage-backed security (MBS), the input suppliers, mortgage originators, select which qualifying mortgages to exchange for the MBS. The originators, therefore, have an opportunity and an incentive to retain in their own portfolios the mortgages with lower default risk, adversely selecting mortgages with higher default risks as inputs into the ‘safe’ asset production process, potentially undermining the goal of creating a safe asset. However, in anticipation of the originator’s decision, the securitizer strategically sets the MBS contract terms, aiming to influence the originators’ choices. This process of creating these government guarantees involves strategic interactions that can result in instability, as evidenced by the safe asset guarantees of Fannie Mae and Freddie Mac, and the Federal Home Loan banks 2 (FHLBs). Their production of quasi-sovereign mortgage-backed securities and quasi-sovereign 2 For evidence that the expansion of mortgage credit accessibility contributed to greater mortgage securitization and more defaults in 2007, see Mian, A., and Sufi, A. (2009). 2

debt provided liquidity advantages to the mortgage market, but their corporate failures aggravated the 2008 financial crisis. we show government-sponsored securitizBayti aonn ahlaysz itnhge ap omteondtieall otof tahmep sleifcyu rthitei ztahtrieoant process, of financial crises. Our model emphasizes two critical factors: the first-mover advantage of originators, who possess the ability to select which mortgages to retain in their portfolio, and the behavior of originators, who determine the guaranteed rate and credit quality standard in the MBS contract. These two elements undermine the ability of securitization, with its liquidity and diversification advantages, to lower investment risks and thereby mitigate financial crises. The adverse consequences of securitization are a result of originators opting out of holding mortgages from borrowers with moderate and/or high default risks that meet the qualifying criteria. Instead, they choose to transfer these mortgages to the securitizer while retaining lowerrisk mortgages in their own portfolios. This adverse selection problem is aggravated when the prevailing demand for mortgages is low enough so that originators choose not to hold any of the higher-risk mortgages. In this case, securitization serves to decrease the equilibrium mortgage rate, lower the guaranteed MBS rate, and expand accessibility to mortgages, while also shifting the burden of default risk from the originator to the securitizer. A key consequence is that both parties experience diminished profit margins, rendering them susceptible to losses stemming from low or negative rates of home price appreciation. In the context of our model, we find that government subsidies are more likely to be successful in lowering mortgage rates and enhancing access to mortgages if they are directed towards mortgage originators rather than securitizers. This consideration hinges on the level of mortgage demand. If demand is robust, leading banks to retain mortgages with marginal nodefault probabilities, the efficacy of the subsidy depends on it being directed towards banks. On 3

the other hand, if demand is weak, leading to securitization of the marginal mortgage, directing 2th. eT shueb Ssuidpyp tloyw aanrdd sD eeimthaern dba fnokrs S oarf es eAcsusreittisz ers can yield the intended effects. 3 We develop two versions of a supply and demand model for safe assets. The baseline model has three categories of actors and two dates when consumption takes place. There are two types of households: Wealthy households inherit homes and put their savings at time 0 into bank equity or money in the form of insured deposits held by banks. Less-wealthy households can apply for mortgages to buy housing, and if they do not obtain a mortgage, they rent housing. To keep our analysis focused, we set aside the issue of down payments and assume that successful mortgage applicants borrow the full house price. Each household that obtains a mortgage has a positive probability of defaulting on the mortgage, with the probability of default being drawn from a common probability density function. The third actor is the bank, which originates mortgages, invests in Treasury Bills, and holds deposits. The extended model of section 4 introduces a fourth actor, a Government Sponsored Enterprise (GSE), that buys bundles of mortgages from banks by issuing mortgage-backed securities and raises equity investments from the wealthy. The baseline model considers insured deposits and Treasury bills as the safe assets, whereas the extended model includes mortgage-backed securities as an additional ‘safe’ asset. The tangible assets considered are housing, bank equity, and GSE equity. The actors make a series of sequential decisions. The first decision is made by the GSE, which sets two parameters for a contract offer to banks, the guaranteed interest rate offered on the MBS and the credit standard 3 The models created in this paper present a substantial advancement compared to previous research (Heuson, Passmore, and Sparks, 2001, Credit Scoring and Mortgage Securitization: Implications for Mortgage Rates and Credit Availability). Here, we include more assets (bank equity, GSE equity, and money), a more complete analysis of the securitizer’s optimization problem, intertemporal utility maximization by households, analyses of the profits and balance sheets for the bank and securitizer, differentiation of households by wealth, parameterization of house price appreciation, and simulation results. 4

that mortgages must meet to qualify for GSE securitization. Next, households make decisions regarding their wealth allocation. They determine the amount of wealth to consume at time 0, the portion to save, and whether they should apply for a mortgage. Wealthy households also decide how much of their wealth to invest in GSE equity. Subsequently, at time 1, profit-maximizing banks decide which mortgage applications to reject, which to accept and securitize using the GSE contract, and which ones to accept and retain in their own portfolios. At time 1, households either pay off their mortgages or default, and all parties involved receive their respective payoffs. The models have two types of uncertainty. The first type pertains to low-wealth households and their probability of not defaulting on a mortgage. This probability is represented by 𝑞𝑞𝑗𝑗 ∈ [0,1], with . Both the probabilities and the probability density function of probabilities are common knowledge at time 0. The securitizer accepts mortgages for securitization only if they meet its underwriting standard as specified in the GSE contract. The second type of uncertainty revolves around the rate of home price appreciation between time 0 and time 1. This rate, denoted by the parameter 𝛿𝛿 ∈ [−1,𝑣𝑣 ],𝑣𝑣 > 0, applies to all homes purchased at time 0, and is revealed to all agen3t.s Tath teim Bea s1e. line Model, with no GSE To isolate the effects of having a GSE, we first analyze a baseline model in which a GSE is not present. In this model, each household maximizes its expected utility over time by choosing at time 0 how much of its exogenous wealth to allocate to immediate consumption and the portion to be saved for future consumption at time 1. Wealthy households posses inherited homes and have the option to invest their unspent wealth at time 0 in bank equity, which has a positive expected rate of return, or keep it as money (i.e., insured demand deposits) yielding a zero return. In contrast, some low-wealth households resort to borrowing in order to purchase a house at time 0, incurring an obligation to repay the loan with interest at time 1. Other low-wealth households 5

either do not apply or fail to meet the criteria for obtaining a mortgage. Consequently, they opt to rent housing instead. We employ the following notation throughout the analysis: and are consumption levels at times 0 and 1, respectively, with the addition of superscripts occasionally used to differentiate between high-wealth and low-wealth households. The variable (with appropriate subscripts) denotes the money holdings of households, set aside at time 0 to support consumption at time 1. Exogenous to the model are the initial levels of wealth of low and high-wealth households, respectively, and the initial house price, , at time 0. We assume that the house price changes to at time 1, where is a random variable drawn from the common pdf . We assume that every household, regardless of their wealth, shares the same intertemporal utility function that depends on consumption at times 0 and 1: , with potentially being a random variable. To ensure consistency, we normalize the utility function for both types of households so that residing in a home yields a multiplicative 1 in the utility function. We begin by analyzing the utility-maximizing decisions of a representative high-wealth household. For the sake of simplicity in notation, we omit the subscript/superscript for household-type for all variables, except wealth. Thus, we can express the problem of maximizing expected utility for the wealthy household as: where is the expectations operator, represents funds invested in bank equity, denotes the expected rate of return on bank equity (with the limited liability of 6

shareholders setting a lower bound of ), and is the level of money holding (in insured deposits, a safe asset) necessary to support survival consumption at time 1. Assuming the expected return for investing in bank equity is positive, we derive in Appendix A the wealthy household’s optimal choices for holding money and investing in bank equity: (1) (2) Now consider the problem facing low-wealth households, which we assume lack the option of investing in bank equity. We later show that these households get segmented into two distinct groups, those who secure mortgages and those who do not, a differentiation based on a specific threshold value for the probability of not defaulting on a mortgage. We use the subscript to denote householdRs that fail to obtain a mortgage. At time 0, these households are obliged to pay rent, denoted as ,for their housing. Hence, their utility maximization problem is: Assuming , we get a straightforward solution to Problem II: . (3) Households that obtain mortgages choose money holdings, denoted below with subscript 4 , to solve: 4 For notational ease we suppress the subscript on the subscript so that . 7

where is the expected rate of home price appreciation from time 0 to 1, and is the mortgage interest rate between those times. If the household pays off its mortgage at time 1, it has a cash outflow of but gains ownership of a house expected to have value . Hence, the household anticipates its net wealth will change by if it pays off its mortgage. In the event of default, the household does not pay the interest and principal owed on the mortgage, nor does it own the home. We assume, however, there are legal and other costs associated with defaulting, represented by , where is a known constant. The condition guarantees the borrower is made worse off from mortgage default compared to the avoided cost of paying rent. Finally, the constraint guarantees that in the event of default, the household has time 1 consumption of at least . Putting these elements together, we represent households obtaining mortgages as solving the Lagrangean: . In Problem III, a mortgage borrower’s expected utility is given by the product of its consumption at time 0 and its expected value of consumption at time 1, which is the sum of its money holdings and two terms: the probability of mortgage repayment times the expected wealth gain if the mortgage gets paid off plus the probability of default times the cost of default. The last term is the product of the Lagrange multiplier γ and the minimum consumption constraint at time 8

1. As shown in Appendix C, mortgage borrowers, with a continuum of no default probabilities, choose aggregate money holdings: (4) where . Equation (4) shows the total money holdings for all mortgage borrowers. Those with no-default probabilities below the threshold value hold the minimum amount of money to guarantee consumption at time 1, while other borrowers hold money balances that vary inversely with their no-default probabilities. We assume banks are risk neutral and make loans from their holdings of demand deposits and equity. Banks lend to home-buying households at the mortgage rate and to the government at the T-bill rate . They also hold household money balances that pay zero interest. When a household applies for a mortgage, the bank observes the applicant’s no-default probability and offers a mortgage to any applicant who incrementally adds to the lender’s expected profit. Thus, successful mortgage applicants must satisfy , (5) where is the rate paid to the lender if the borrower does not default, is the expected rate of return to the bank in the event of default, with being the bank’s cost of foreclosure per dollar of mortgage. The parameter is the return on the alternative risk-free investment, which we assume is the T-bill. Writing (5) as an equality and solving for defines the lowest mortgage rate the lender is willing to accept, i.e., the inverse supply function for mortgages with : 9

. (6) From (6) we see that is increasing in and but decreasing in and . Notice also that for any household with a no-default probability , banks are willing to charge a mortgage rate as low as the T-bill rate. Consider next the household’s decision of whether to apply for a mortgage. For a household to prefer a mortgage to renting, the expected utility from obtaining the mortgage must be at least as great as the expected utility from renting. That is, the maximized value of the solution to Problem III must be greater than or equal to the maximized value in Problem II. So, a household with no default probability will want a mortgage if and only if , (7) which we show in Appendix B is equivalent to: . (8) The weak inequality (8) says the expected net benefit from obtaining a mortgage is non negative. This net benefit consists of the probability of no default times the net expected return to the nondefaulting borrower minus the probability of defaulting times the cost of default plus the avoided rental cost. Now we set (8) as an equality, assuming , and find . (9) 10

Equation (9) is the inverse-demand function showing that the highest mortgage rate a household 5 is willing to pay is decreasing in and but increasing in , , and . Assuming there are many price-taking lenders, we equate (6) to (9) and solve for the market equilibrium no-default probability, , for the marginal borrower obtaining a mortgage . (10) Substituting (10) into (6) or (9), we find the market equilibrium mortgage rate: . (11) Figure 1 illustrates. Several comparative statics of the baseline model are relevant to our investigation. If the expected rate of home price appreciation declines, the inverse demand function shifts downward 5 Again, we presume the household does not benefit from certain mortgage default compared to paying rent, i.e., . For a household with , (8) implies . 11

while the inverse supply function shifts upward, with the former shift being larger, causing the equilibrium mortgage rate to fall and the no-default probability cutoff to rise. Thus, mortgages become less expensive but more difficult to obtain. Other comparative static results are similarly consistent with what we expect to see in the real world. A decrease in the bank’s cost of foreclosure causes the inverse supply function to shift down, lowering the equilibrium mortgage rate and marginal no-default probability. Finally, an increase in the T-bill rate causes the inverse supply function to shift upwards, raising the equilibrium mortgage rate and the no-default cutoff. Turning our attention to a representative bank in the baseline model, we assume the bank is willing to hold all deposits, which along with bank equity, help fund two types of loans: mortgages to households and loans to the federal government in the form of T-Bill purchases, which have a safe rate of return . We write the bank balance sheet in the baseline model as consisting of two assets and two liabBilaitnieks B. alance Sheet Assets Liabilities Mortgages and rental properties Demand Deposits (money needed by households at time 1) T-Bills Equity (to cover losses from mortgage defaults) Consider bank liabilities in the model at time 0. Low-wealth households with both desire and qualify for a mortgage, while the other low-wealth households with end up renting a dwelling. Therefore, the proportion of low-wealth households that obtain mortgages is and the proportion that rent is . Since the total number of low-wealth households is , the total value of demand deposits is , (12) 12

where is the number of high-wealth households, and is given by (4). From (2), total bank equity is . On the asset side, the total value of mortgages at time 0 is , while the value of rental income is . Consequently, the value of T-bills is . (13) We now assess the possibility of a financial threat triggered by a decrease in home price that results in negative returns to banks. At time 1, the bank makes a profit on each defaulted mortgage of , which becomes negative if the realized value of falls below the foreclosure cost parameter . On the other hand, for a mortgage that is paid off, the bank realizes a profit of . We can write the expected value of for mortgage-qualifying households as . Then from the LHS of (5), we can write the bank’s expected rate of return on mortgages as . (14) The proportion of all mortgages that do not default is , the average no-default probability of successful borrowers, while the proportion that default is . Given that is the cutoff nodefault probability for obtaining a mortgage, it follows that the proportion of low-wealth households that obtain mortgages and do not default is given by while the proportion of those households that get mortgages and default is . From their mortgage investments, lenders realize negative time 1 profits if . (15) 13

But negative mortgage profit will not cause total bank profits to become negative as long as the bank has sufficient positive cash inflows from other sources to cover deposit liabilities. In our model, the other inflows are from Treasuries and invested rental income. Therefore, the bank only realizes negative profits at time 1 if the sum of the net returns from the purchase of T-bills, the appreciated 6 value rental housing, and mortgage investments is less than total demand deposits, : (16) where is the value of rental income received at time 0 and immediately invested in 7 Treasuries, and is the appreciation in rental housing. Isolating in (16), we can write the critical range for home price appreciation that yields negative bank profit as (17) If falls below the RHS of (17), banks face losses due to a low, and possibly negative, rate of home price appreciation for both repossesed homes after mortgage default and rented-out homes. As indicated in (17), the critical value of , and therefore the probability of bank nonviability, increases with the cost of foreclosure, but decreases with the values of Treasuries and 8 the mortgage rate. 6 We assume so that wealthy households prefer investing in bank equity to investing in Treasury securities. 7 Funding of mortgages in the model is assured since the demand for mortgages is less than or equal to the sum of demand deposits and bank equity, i.e., , which follows from the balance sheet equation with . 8 In the event that (17) holds, the bank’s shareholders will incur losses unless the government intervenes to shift the loss burden to taxpayers as seen in programs such as the TARP program during the 2008 financial crisis. Alternatively, the government could opt to assist mortgage borrowers facing a low rate of home price appreciation that threatens shareholder returns. An example is the HAMP Program (2009-2016) in which homeowners at risk of foreclosure were allowed to make reduced monthly mortgage payments that were more affordable. 14

Government-sponsored mortgage securitization is frequently heralded as a proactive strategy for reducing the likelihood of bank insolvency caused by adverse changes in home prices. Through securitization, the risks of mortgage defaults are shifted from banks to shareholders in entities like Fannie Mae and Freddie Mac. These shareholders take on the default risk and receive a guarantee fee in return. In the next section, we delve into the likely impact of securitization on the likli4h.o oTdh ean Edx mteangdneidtu Mdeo odfe nl ewgiatthiv ae GreStEu rns for bank shareholders and investors in GSE equity. We now extend the baseline model to include a government-sponsored securitizer (GSE) that aims to encourage home loans by relieving banks of some default risk while also providing investors with a relatively safe investment, GSE equity. To further these goals and its own profitability, the securitizer has two parameter values to choose: the minimum credit standard (i.e., no-default probability) that qualifies a mortgage for securitization and the level of a guaranteed rate of return to the bank that sells the mortgage to the securitizer in exchange for a mortgagebacked security (MBS), which carries a liquidity premium in addition to the guaranteed rate. To balance model simplicity and realism, we assume that only wealthy households purchase GSE equity, which offers a positive expected rate of return. Wealthy households prefer holding GSE equity over money because, unlike money, it offers a positive expected rate of return along with the implicit backing of the government, which ensures that any downside risk is borne by taxpayers. As in the baseline model, wealthy households can also invest in bank equity and must satisfy a minimum consumption constraint at time 1. The behavior of low-wealth households remains unchanged from the baseline model. We now modify Problem 1 by adding GSE equity as an investment alternative for wealthy households. We denote the level of this investment by and assume investors believe the 15

implicit government backing guarantees a positive rate of return, . We can, therefore, write the expected utility maximization problem for the wealthy household as: is the amount of cash (from the sale of the MBS) necessary to support survival 9 consumption at time 1. Appendix D solves Problem IV and finds: (18) and , (19) under the assumption that an investment in bank equity is perceived to have a greater expected return than the return on GSE equity, i.e., . Equation (18) reveals an intriguing relationship wherein investment in securitizer equity declines as its rate of return increases. This somewhat puzzling result stems from the role of GSE equity investment in the model as a guarantee, ensuring a minimum consumption level at time 1. Consequently, the higher the rate of return (denoted as ), the less investment ( ) is required at time 0. Equation (19) shows that the size of the wealthy household’s investment in bank equity increases with the expected rates of return on bank equity and GSE equity. Furthermore, we note that the household invests less in bank equity when GSE equity serves as an available option, as shown by comparing (19) and (2). We now analyze the securitizer’s behavior as a sequential game played in conjunction with a representative bank. At time 0, the securitizer sets the two contractual terms for swapping a 9 Our modeling effort abstracts away from interest-rate risk and focuses instead on default risk. 16

mortgage-backed security (MBS) for a mortgage, while knowing the probability distributions for no-default probabilities and for home-price appreciation but not their realized values. During this initial phase, the securitizer establishes a guaranteed interest rate offered on the MBS and a minimum credit standard (i.e., no-default probability) that every mortgage must meet to qualify for securitization. Subsequently, if the bank opts to hold the MBS rather than the mortgage itself, it receives an exogenous liquidity premium, . This additional compensation factor is independent of the securitizer’s decisions and reflects the specific liquidity advantage associated with holding a MSB. Once the securitizer has set the MBS contractual terms, the bank begins receiving mortgage loan applications. These applications reveal the applicants’ loan default probabilities, which are also observed by the securitizer. The bank uses the information on default probabilities and MBS contractual terms to decide which households’ mortgage applications will be rejected and which will be offered mortgages. The bank also decides whether to keep these mortgages in portfoilo or immediately swap them for a mortgage-backed security. At time 1, borrowers who have been granted mortgages either default on their mortgages, and obtain a payoff equal to their money holdings minus the costs of default , or pay off their mortgage in full, getting a payoff of , their money holdings plus the difference between the realized house appreciation rate and the mortgage rate multiplied by the original house price. The sequential game is depicted in Figure 2. The securitizer first chooses and . Then, both the bank and securitizer observe the applicants’ default probabilities. For mortgage applications that meet both the bank’s credit cutoff and the securitizer’s conforming requirement , the bank decides either to hold the mortgage in its own portfolio or to sell the 17

mortgage to the securitizer. For applications that do not meet the conforming standard, the bank decides whether to offer a mortgage and keep it in portfolio (for ) or reject the application (for ). Finally at time 1, borrowers either default or do not default, and all parties 10 (securitizer, bank, and households) receive their payoffs. 10 On a mortgage kept in the bank’s portfolio, the time 1 payoffs to the securitizer, bank, and low-wealth household with the mortgage are, respectively, in the event of no default and if default occurs. For securitized mortgages, these payoffs are for those who do not default and for those who default. For households denied mortgages, the returns are , as the bank invests its rental funds in T-bills and the household pays rent. Not shown in Figure 2 are the payoffs to high-wealth households, which can be expressed as where is the realized rate of return on bank equity, which varies directly with . 18

We now shift our attention towards events and actions that could serve as precursors to a financial crisis in the extended model. As in the baseline model, we assume households apply for mortgages if the expected utility from home ownership outweighs that of renting. Consequently, a household applies for a mortgage if inequality (8) is satisfied, which can be rearranged to show the threshold or cutoff no-default probability at which obtaining a mortgage becomes advantageous to the household: . (20) Households with no-default probabilities less than do not apply for mortgages. A conforming loan, which qualifies for securitization, must have a no-default probability at least as high as the threshold set by the securitizer. Thus, the proportion of households that qualify for securitization is for , which is decreasing in . We now examine more closely the bank’s decision either to accept or reject a mortgage application. As noted earlier, if a particular application is accepted, then the bank either keeps the mortgage in its portfolio or trades it for a mortgage-backed security, if it is conforming. To induce a bank to originate and securitize a mortgage, the return generated from securitization must be at least as great as the alternative return the bank could earn by holding the mortgage in its portfolio. The total return to the bank from securitizing a mortgage is , where is the liquidity value to the bank from holding a mortgage-backed security, as opposed to the mortgage itself. The bank also considers the borrower’s credit quality. To hold a mortgage in its own portfolio, the bank has a minimum requirement for the probability of no default, which is derived from (5). The bank will refuse to hold any mortgage that fails to satisfy (5), implying a probability of default in the interval: 19

. (21) With the possibility of securitization and assuming the bank’s return to securitization exceeds its alternative return, i.e., , the bank only rejects a mortgage application if it fails to meet both its own credit standard and that of the securitizer, i.e., and . If either of the next two conditions are met, the bank will offer a mortgage and hold it in its own portfolio. The first condition states that the expected return is higher from holding the asset in portfolio rather than securitizing it: , which implies the no-default probability satisfies . (22) The next condition stipulates that the no-default probability of the mortgage application satisfies the bank’s minimum requirement but falls below the qualifying standard for securitization: . (23) The bank offers and securitizes a mortgage if it qualifies for securitization and the bank finds the swap more profitable than holding the mortgage in its own portfolio: , (24) where the securitizer’s choice of remains to be determined. Figure (3) partitions the probability density function of no-default probabilities into regions indicating mortgage outcomes ranging from no application, to being selected for the bank’s portfolio, to being swapped for a mortgagebacked security. 20

Inequality (22) reflects the bank’s ability to cherry pick borrowers with high no-default probabilities by using its first-mover advantage in choosing which mortgages to hold versus securitize, while (23) captures its ability to offer and keep in its portfolio mortgages the securitizer deems lemons, not worthy of securitization. Later in our analysis, we demonstrate that in a particular type of market equilibrium, collapses and becomes equal to , implying that mortgages with the lowest no-default probabilities are securitized. We now investigate how the securitizer sets the MBS contract terms, which has important effects on the bank’s subsequent mortgage-portfolio decisions and the payoffs to both the bank and securitizer. We assume, while taking as given , the securitizer chooses and to maximize its expected profit: . (25) Assuming positive solutions for and , we compute the following first-order conditions: (26) 21

and (27) By prior assumptions, , which implies from (26) that . (28) Recalling (22), we see that , which later we show is positive. Equation (28) shows the securitizer would earn zero profit by securitizing the marginal qualifying mortgage, a standard profit-maximization result. Together (26) and (27) imply (29) which geometrically says the area of a rectangle with height and width equals the a b area under the pdf between and , implying in Figure 4 that areas and are equal. 22

Equation (29) has an appealing economic interpretation. The LHS is the securitizer’s marginal benefit from raising the guaranteed rate . A marginal increase in increases the proportion of low-wealth households with securitized mortgages at the upper end of the securitized group at rate . The marginal increase in also improves the credit quality interval of the securitized group by raising both the upper and lower-bound probabilities, and , by the same magnitude. So, the product can be interpreted as the change in the securitized proportion times the gain in credit quality, the marginal benefit of raising . The RHS of (29) measures the marginal cost of raising as the increased proportion of all low-wealth households that end up with securitized mortgages (costing the securitizer ). This proportion is increasing at the optimal solution since must be true for (29) to hold. In Appendix E, we derive second-order conditions for the securitizer’s problem. To derive implications regarding the impact of securitization on the liklihood of financial crises, it is essential to obtain an explicit solution for the securitizer’s choice of . Furthermore, obtaining this solution requires a specific form for the probability density function of no-default 11 probabilities, . The beta distribution, often referred to as the ‘probability distribution for probabilities’, provides us with a suitable flexible mathematical form. To estimate the shape parameters, and , of the Beta distribition for , we use quarterly U.S. data from the period 11 The beta distribution is with , support , and where denotes the gamma distribution such that . 23

12 spanning 2013.2 to 2020.1 on the probability of default on consumer mortgages. By fitting the Beta distribution to the observed default data, we estimate parameter values and . These values reflect the characteristics and behavior of mortgage defaults during the 13 given time frame. Plugging these parameter values into the Beta distribution yields 14 that has the following graph, which is unsurprisingly left skewed. For the Beta distribution, the RHS of (29) becomes , while the LHS becomes . In Appendix G, we demonstrate that the securitizer’s profit-maximizing MBS rate varies directly with the mortgage rate, a result that is well supported 12 Source: https://www.federalreserve.gov/releases/efa/efa-project-mortgage-and-consumer-loans-byprobability-of-default.htm. The probability of default is defined as the probability of being 90+ days past due in the previous two-year period. The historical data only go back to 2013 q2, and choosing an end date of 2 1 14 3 0 S2o0u qrc1e :a hvtotpidss:/ /thhoem peepraiogde. doifv tmhes. uCioovwida. 1ed9u p/~amndbeomgniacr. / applets/beta.html. For derivations see the spreadsheet in Appendix F. 24

by empirical data and aligns with economic intuition. In our model, this relationship plays a crucial role when securitization lowers both the equilibrium mortgage rate and MBS rate, thereby shrinking bank profit margins on both securitized and retained mortgages. When borrowers enjoy lower mortgage rates from banks, the securitizer seizes the opportunity to offer banks a reduced MBS rate. Doing so improves the securitizer’s profit margin while maintaining bank incentives to securitize those borrowers with no default rates in the gap between and . Therefore, when securitization successfully lowers the equilibrium mortgage rate, it leads to enhanced access to securitization. Equation (22) shows that the bank holds mortgages with no-default probabilities above , which is decreasing in the mortgage rate for a fixed MBS rate: (30) From (22) and (28), we know that the gap is decreasing in after adjustments in are taken into account. Any change in leads to a direct change in and by the same magnitude. Solving (6) for yields the minimum no-default probability that causes the bank to be willing to hold rather than reject a mortgage: . For the bank to favor securitization over investing in T-bills it must be the case that the return from securitization is greater than or equal to the return on T-bills: . Substituting (28), which shows how the securitizer’s MBS rate varies with the mortgage rate, into this inequality, we find , (31) which is the minimum no-default probability for mortgages that banks will accept when they intend to securitize the mortgage. The difference between these two probabilities is 25

, which is increasing in , and but decreasing in and . The GSE’s willingness to securitize a mortgage depends on the borrower’s no-default probability and the mortgage rate, as shown in Appendix G, which also demonstrates that increases with . For a mortgage to be securitized, both the bank and GSE must be willing to exchange the mortgage (with its risk of default) for the GSE’s guaranteed payment of rate to the bank. At any mortgage rate , the willingness of both parties to securitize the mortgage is given by the right envelope of the and functions, shown in bold as the “short side” of the market in Figure 5. The bold segments of and thus show the lowest probability of no default on a securitized mortgage as a function of the mortgage rate. All mortgages plotting to the right of the bold segments are acceptable for securitization. We derive the market inverse supply function as the set of lowest mortgage rates for which a mortgage is offered and held either by the bank or securitizer. For any , the lowest rate is given 26

by the function. In that interval of no-default probabilities, the securitizer is willing and able to hold mortgages at a lower mortgage rate than are banks. In addition, banks are willing to securitize these mortgages at the MBS rate rather than reject them. On the other hand, for , the lowest mortgage rate is given implicitly by . In the interval , the securitizer and bank do not find a mutually agreeable combination for securitizing the marginal mortgage. Essentially, the securitizer recoils at the idea of holding mortgages with no-default probabilities below . Recall that and solve (31) for to obtain . (32) Thus, the inverse supply function is for and for as shown in Figure 6. The discontinuity at shows that securitization lowers the marginal cost of supplying mortgages to borrowers with good credit risk in the interval but has no effect on the supply of mortgages to borrowers with poor credit risk . The supply function has a discontinuity at where it jumps downwards by . To summarize, the market inverse supply in the extended model is for and for , as shown in Figure 6. 27

The inverse demand for mortgages remains unchanged between the extended and baseline models. In Figure 7, we graph both the demand and supply functions for the extended model and show two possible equilibria, corresponding to ‘high’ versus ‘low’ demand for mortgages. In situations where mortgage demand is sufficiently high, the market equilibrium is characterized by the bank holding the marginal mortgage. Securitization, in this context, does not affect the equilibrium, yielding the same mortgage rate and marginal no-default probability as in the baseline model. However, when mortgage demand is low enough, the securitizer holds the marginal mortgage in equilibrium and, with its liquidity premium, is able to depress the 15 equilibrium mortgage rate and marginal no-default probability. An interesting empirical question, not addressed herein, pertains to the likelihood of either of these two equilibria, especially considering that their probability relies on the value of , determining the position of the discrete jump, which in turn depends on the values of and according to (28). 15 A third type of market equilibrium emerges if the function crosses through the discontinuous jump in the inverse supply function. In such instances, the equilibrium is a pair , with securitization contributing to a reduction in the equilibrium mortgage rate and marginal no-default probability, as in the low-demand case. 28

In the scenario of high mortgage demand, the equilibrium is characterized by , where the relevant portion of the demand function is the same as in baseline model, and generates the same equilibrium marginal no-default probability and mortgage rate, as shown in (10) and (11). This result is attributed to the fact that the securitizer is only willing to securitize mortgages with no-default rates at or above , which exceeds the market equilibrium cutoff with high demand. With high mortgage demand, households exhibit a willingness to pay higher mortgage rates, prompting banks to retain those with low no-default probabilities in their portfolios. It is these marginal mortgages that dictate the equilibrium mortgage interest rate, . 29

The low mortgage demand equilibrium in the extended model arises from equating the inverse demand and supply at , which implies (33) Substituting (33) back into , we find the low-demand equilibrium mortgage rate with the GSE: (34) Comparing (34) to the equilibrium mortgage rate given by (11) for the baseline model, we find the two equations differ only in that the liquidity premium (with a negative sign) appears in the lowdemand equation. This analysis shows that for given values of , securitization lowers the equilibium mortgage rate if demand is “low” (when the securitizer holds the marginal mortgage) but has no effect if demand is “high.” Another interesting implication of our model pertains to a policy aimed at reducing mortgage rates and expanding accessibility to home ownership. As depicted in Figure 7, an effective approach would involve a government subsidy directed toward banks (achieved by reducing the bank’s foreclosure cost of foreclosure ). By doing so, the northwestern and 16 southeastern sections of the inverse supply function would shift downwards, achieving the desired result regardless of whether demand is high or low. Conversely, providing a subsidy to the securitizer that increases the liquidity premium only shifts downward the southeastern section of the supply function, and therefore is only effective if demand is low. 16 See (32). 30

Shifting our focus to empirical matters, we now take parameter values based on U.S. 17 economic data and insert them into the first-order conditions for high and low demand, yielding equations with as the only unknown. We develop simulations that offer broad insights into causal relationships, aiming to illuminate overarching patterns rather than furnish precise predictions applicable to the real world. Since is no straightforward closed-form solution for , we solve for it iteratively and find 18 for high demand and . Next, we insert the same estimated parameter values into the low-demand equilibrium and solve iteratively for , showing the securitizer’s guartanteed rate is lower in the low-demand case as compared to the high-demand 19 case. Also, the equilibrium mortgage rate is below the rate determined in the baseline model. Furthermore, the equilibrium marginal no-default probability is , whch is less than the correponding value of in the high-demand case. Thus, in the lowdemand scenario, the presence of the securitizer lowers the equilibrium mortgage rate and makes mortgages accessible to households with lower credit worthiness. Graphically, this is shown by a comparison of points A and B in Figure 7. At this juncture, the model’s results align with the prevailing consensus that government- 20 sponored mortgage securitization can increase the affordability and accessiblity of mortgages. Nevertheless, our primary objective remains focused on examining the risk of financial crisis 17 See simulation spreadsheet, sheet 1. These simulations offer broad insights into causal relationships, aiming to illuminate overarching patterns rather than furnish precise predictions applicable to the real world. 18 Ibid, sheet 3. See Ambrose and Warga (2004) on the estimate of 25 basis points for . 19 Ibid, sheet 2. 31

stemming from the impact of house price appreciation on the the profits of both the securitizer and the bank. The securitizer’s profit at time 1, after the realization of , is given by , (35) where is the average no-default probability for the mortgages that are securitized. If , (36) then the securitizer’s profits are negative. Next, we write this inequality using the beta distribution and the model parameter values specified in the spreadsheet. For the high-demand 21 case, we compute . Then, we substitute into (36), to find , which means that in the high-demand case, a 3.142 percent drop in the price of housing is required to cause the 22 securitizer’s profits to become negative. Next, we compute for the low-demand case. Substitute into (36) to find , showing that when demand is low, the securitizer’s profits are negative for any rate of house price appreciation below 1.65 percent. Putting these results together, we see that the securitizer’s profits are subject to greater downside risk when mortgage demand is low, when the securitizer’s presence brings down the equilibrium mortgage rate, as opposed to when demand is high. 21 The computation is shown in simulation spreadsheet, sheet 4. 22 Ibid. 32

We now focus on the bank’s profitability. The bank earns profit from three segments of the mortgage market. By holding the mortgages of households with low, but profitable, no default probabilities in the interval , the bank earns in the high-demand state: (37) Where is the average no-default probability for the profitable but non-qualifying borrowers in the high-demand case. In the low-demand case, the bank does not hold any mortgages with default probabilities below , the cherry-picking rate, so that . From securitized mortgages the bank earns in each state: (38) which is always positive. Finally, the bank has profits from qualifying mortgages that are cherry picked and kept in the bank’s portfolio: , (39) where is the average no default probability for cherry-picked mortgages kept by the bank. In each state, bank profits become negative in period 2, compelling the bank to draw on its capital buffer, if . (40) In the baseline model, negative bank profits arise if (17) holds. Solutions for 23 are shown in the following table, and given these values and the levels of the model’s exogenous variables, we compute the critical house appreciation rate in the baseline case as . 23 See simulation spreadsheet, sheet 1. 33

Any change house price that is more negative than -45.6 percent causes bank profits to become Enndeoggaetnivoeu. s variable Baseline model Extended-high Extended-low Equilibrium mortgage rate Equilibrium no-default rate Qualifying no-default rate NA Same as bove Cherry pick no-default rate NA Guaranteed rate NA Average no-default probability for households with mortgages: lower bank partition, upper bank partition, securitizer partition Minimum no-default NA probability for bank accepting mortgage application Bank: Critical home price appreciation rate For the extended model (with the GSE), we substitute (37), (38), and (39) into (40) and isolate to find that negative bank profit arises in the case of high demand if . (41) 34

In the low-demand state, the critical house appreciation rate is . (42) A key question is whether the RHS of (41) and/or (42) is larger or smaller than the RHS of (17). This comparison will indicate whether a banking crisis is more or less likely with GSE securitization of mortgages. We address this question by running a simulation of the model and finding the critical rate of home price appreciation that causes negative bank profits with 24 securitization is -5.73 percent with high demand and +30.28 percent with low demand. Thus, in our modeling exercise, securitization increases the likelihood of financial crisis in the form of bank runs whether demand is high or low, but the threat is much greater when demand is low. Securitization poses a risk for banks, even in times of robust mortgage demand, as it grants securitizers the ability to wield first-mover market influence when establishing the guaranteed interest rate and qualifying standard. Furthermore, the securitizer, is exposed to potential losses if home price appreciation is more negative than -3.142 percent in the high-demand state and less t5h. aCno 1n.c6l5u spieornc ent in the low-demand state. In our model, a systemic financial threat emerges when the change in home prices results in banking assets generating returns that are insufficient to cover deposit liabilities. Our analysis demonstrates the potential impact of government-sponsored securitization in amplifying these threats. It is important to note that the subsequent progression of such threats into full-blown bank failures and other markers of a financial crisis is contingent upon factors outside this paper’s scope. 24 See simulation spreadsheet sheet 5. 35

Our model is based on several key observations about the U.S. economy. Firstly, the process of mortgage securitization serves to transfer credit risk from banks to government-sponsored enterprises in a setting whereby the securitizer, anticipating bank responses, sets the terms of the MBS contract. Secondly, unstable fluctuations in housing prices have proven to be a major catalyst for financial crises. Lastly, declines in asset prices often trigger bank runs. We have identified both positive and negative effects stemming from mortgage securitization. On the one hand, securitization helps to diversify default risk and, in certain cases, leads to lower mortgage rates and increased loan accessibility for borrowers. However, these benefits come with the downsides of shifting risk to government-sponsored securitizers and, subsequently, to the public, as well as putting banks at risk of being unable to cover their deposit liabilities. Contrary to the belief that securitization safeguards banks, our analysis reveals that it actually exposes them to greater downside risk. In the low-demand state, when securitization lowers the mortgage rate, the likelihood of a bank run becomes substantially greater than in the baseline model without securitization. This effect arises because securitization lowers the equilibrium mortgage rate while leaving virtually unchanged the average no default 25 probability of the mortgages held by the bank, thereby cutting into bank profit margins. Interestingly, our model also demonstrates that the securitizer is exposed to m o re risk preciseWly ew choennc eitdse pthoalitc iine sp prarcotivcea le rffeeaclittiyv,e t hine epxopteanncdyi nogf tmheosret gthargeea tasc tcoe sinsdibuicleit ya. financial crisis is diminished when robust capital requirements and stringent liquidity standards are in place, ensuring banks have sufficient buffers to absorb losses. Within this context, the outcomes yielded 25 As shown in the table, . 36

by our model lend support to the idea that macroprudential policies, such as capital and liquidity requirements, would be stabilizing. An alternative policy approach involves mitigating the potential adverse impact on bank profits. One avenue for achieving this is through reforming the process for determining the MBS contract. A potential transformation entails promoting competition in the securitization market by fostering many independent MBS suppliers while also reducing banks’ discretion in selecting the mortgages for securitization. 37

Refere nces Ambrose, B.W., and Warga, A.D. (2004) Measuring Potential Gains from Mortgage Pooling in Agency Securitization Programs. Journal of Money, Credit, and Banking, 36(1), 1-22. Ashcraft, Adam B. and Schuermann, Til, Understanding the Securitization of Subprime Mortgage Credit (March 2008). Wharton Financial Institutions Center Working Paper No. 07-43, FRB of New York Staff Report, No. 318, Available at SSRN: https://ssrn.com/abstract=1071189 or http://dx.doi.org/10.2139/ssrn.1071189 Basel Committee on Banking Supervision (2013) “Basel III: The Liquidity Coverage Ratio and liquidity risk monitoring tools,” January Bernanke, Ben S., “The Global Savings Glut and the U.S. Current Account Deficit,” 2005, Sandridge Lecture, th Virginia Association of Economics, March 10 . Caballero, Ricardo J., Emmanuel Farhi, and Pierre-Olivier Gourinchas (2017), “The Safe Assets Shortage Conundrum, "Journal of Economic Perspectives, Vol. 31, No. 3, pp. 29-46, SummSeerc.u ritization: Deku SPYa, sKta, Prare Ase. Entff aenctds Fouf tSuerceu. rPiatilzgartaivoen Mona cbmainllkasn a Sntdu dthiees fiinn aBnacnikailn sgy satnedm F iinn:a ncial Institutions . 2017: 93–111. Palgrave Macmillan; 10.1007/978-3-319-60128-1. Gourinchas, Pierre-Olivier and Olivier Jeanne (2012), “Global Safe Assets,” BIS Working Paper No, 399, December. Golec, Pascal and Enrico Perotti (2017), “Safe assets: a review,” ECB Working Paper Series No. 2035. March. Gorton, Gary B . (2016), “The History and Economics of Safe Assets,” NBER Working Paper 22210, April He, Zhiguo, Arvind Krishnamurthy, and Konstantin Milbradt (2016). “What Makes US Government Bonds Safe Assets? American Economic Review: Papers and Proceedings, pp.519-523. 38

Heuson, Andrea J. and Passmore, Stuart Wayne and Sparks, Roger W., Credit Scoring and Mortgage Securitization: Implications for Mortgage Rates and Credit Availability. https://ssrn.com/abstract=302132 Kacperczyk, Marcin, Christophe Perignon, and Guillaume Vuillemey (2017), “The Private Production of safe Assets,” September 24. Review of Kara AB, Oehzakvaino rAa, lA Flitnuannbcaes Y. Securitisation and banking risk: what do we know so far? . 2016;8(1): 2–16. 10.1108/RBF-07-2014-0039. Journal Mayer, oCfh Ercisotnoopmheicr ,P Kearsrpeenc Ptievnesce, and Shane M. Sherlund (2009). “The Rise in Mortgage Defaults,” , 23(1), 27–50. Mian, A., and Sufi, A. (2009). “The Consequences of Mortgage Credit Expansion: Evidence from the US Mortgage Default Crisis,” Quarterly Journal of Economics, 124(4), 1449-1496. Nadauld, Taylor D. and ShJaonuer nMa.l Sohf eFrinluanndc i(a2l 0E1c3o)n. o“mThices Impact of Securitization on the Expansion of Subprime Credit,” , 107(2), 454–476. Passmore, Wayne and Alexander H. von Hafften (2017). “Are Basel’s Capital Surcharges for Global Systemically Important Banks Too Small?” FEDS 2017-21. Passmore, Wayne and Alexander H. von Hafften (forthcoming). “GSE Guarantees, Financial Stability, and Home Equity Accumulation,” Federal Reserve Bank of New York Economic Policy Review. 39

Appendix A Substituting the two equality constraints into the objective function and writing the objective function and the remaining constraint with 𝛹𝛹 as the Lagrangean function, we have: m𝑀𝑀,a𝐸𝐸𝐵𝐵x𝛹𝛹 = [𝑊𝑊𝐻𝐻 −𝐸𝐸𝐵𝐵 −𝑀𝑀][𝑀𝑀+(1+𝑟𝑟‾𝐵𝐵)𝐸𝐸𝐵𝐵]+𝜆𝜆[𝑀𝑀−𝑀𝑀‾] (A.1) The Karush-Kuhn-Tucker conditions for maximizing (A.1) are: ∂𝛹𝛹 = −[𝑀𝑀+(1+𝑟𝑟‾)𝐸𝐸𝐵𝐵]+[𝑊𝑊𝐻𝐻 −𝐸𝐸𝐵𝐵 −𝑀𝑀]+𝜆𝜆 ≤ 0 (A.2) ∂𝑀𝑀 ∂𝛹𝛹 = 0 (A.3) ∂𝑀𝑀 ∂𝛹𝛹 = −[𝑀𝑀+(1+𝑟𝑟‾)𝐸𝐸𝐵𝐵]+(1+𝑟𝑟‾)[𝑊𝑊𝐻𝐻 −𝐸𝐸𝐵𝐵 −𝑀𝑀] ≤ 0 (A.4) ∂𝐸𝐸𝐵𝐵 ∂𝛹𝛹 𝐸𝐸𝐵𝐵 = 0 (A.5) ∂𝐸𝐸𝐵𝐵 ∂𝛹𝛹 = 𝑀𝑀−𝑀𝑀‾ ≥ 0 (A.6) ∂𝜆𝜆 ∂𝛹𝛹 𝜆𝜆 = 0 (A.7) ∂𝜆𝜆 We assume that the expected return for investing in bank equity is greater than the zero return on holding money, i.e., 𝑟𝑟‾𝐵𝐵 > 0. The requirement 𝑀𝑀 ≥ 𝑀𝑀‾ > 0 implies from (A.3) that (A.2) holds with equality so that {𝑊𝑊𝐻𝐻 = 2[𝐸𝐸𝐵𝐵 +𝑀𝑀]+𝑟𝑟‾𝐵𝐵𝐸𝐸𝐵𝐵 −𝜆𝜆, which substituted into 𝑟𝑟‾𝐵𝐵𝑀𝑀 (A.4) implies 𝜆𝜆 ≥ (1+𝑟𝑟‾)+𝑟𝑟‾𝐵𝐵𝐸𝐸𝐵𝐵 > 0. . 0 implies that (A.6) holds with equality, i.e., 𝑀𝑀 = 𝑀𝑀‾. Assuming the wealthy household has initial wealth that exceeds the survival constraint, 𝑊𝑊𝐻𝐻 > 𝑀𝑀‾, it will invest a positive amount in bank equity, i.e., 𝐸𝐸𝐵𝐵 > 0, and then (A.4) holds with equality. Consequently, and using the fact that 𝑀𝑀 = 𝑀𝑀‾, we find from (A.4): 𝑊𝑊𝐻𝐻 (2+𝑟𝑟‾𝐵𝐵)𝑀𝑀‾ 𝐸𝐸𝐵𝐵 = − . (A.8) 2 2(1+𝑟𝑟‾𝐵𝐵) 40

That is, a wealthy household invests in bank equity half its wealth minus a proportion (2+𝑟𝑟‾𝐵𝐵) 2(1+𝑟𝑟‾𝐵𝐵) of the money holdings needed to ensure survival at time 1. Not surprisingly,the size of the household?s utility-maximizing investment in bank equity is increasing in the expected rate of return on bank equity. If we plug 𝑀𝑀 = 𝑀𝑀‾ and (A.8) back into the consumption levels defined in Problem I, we find that time 0 consumption is 1 𝑟𝑟‾𝐵𝐵𝑀𝑀‾ 𝐶𝐶1 = �𝑊𝑊𝐻𝐻 − �, (A.9) 2 (1+𝑟𝑟‾𝐵𝐵) and expected consumption at time 2 is 1 𝐸𝐸(𝐶𝐶2) = [𝑊𝑊𝐻𝐻 +𝑟𝑟‾𝐵𝐵(𝑊𝑊𝐻𝐻 −𝑀𝑀‾)], (A.10) 2 Under the assumptions that 𝑟𝑟‾𝐵𝐵 > 0 and 𝑊𝑊𝐻𝐻 > 𝑀𝑀‾, it is clear these households make optimizing choices leading to the expectation that their consumption level will be higher at time 1 than at time 0. This behavior reflects a standard tradeoff between consumption and investment. If a household starts from an allocation of equal expected consumption levels at each time, it realizes it will gain total utility if it sacrifices a unit of time 0 consumption by investing more in bank capital, with its positive expected return, thereby generating more consumption at time 1 than was sacrificed at time 0. If the expected rate of return on bank equity were negative, then wealthy households would switch out of bank equity to hold only money balances. Specifically, 𝑟𝑟‾𝐵𝐵 > 0 causes the weak inequality in 𝑊𝑊𝐻𝐻 to imply a strong inequality in (A.4), which in (A.5) implies 𝐸𝐸𝐵𝐵 = 0 and 𝑀𝑀 = 2 . That is, wealth not consumed in period 1 is plunged entirely into money holdings; the wealthy hold no bank equity. 41

Appendix B Substituting (3) and (4) into (7) and letting 𝑋𝑋 ≡ −𝑞𝑞𝑗𝑗�𝛿𝛿‾−𝑖𝑖�𝑃𝑃+�1−𝑞𝑞𝑗𝑗�𝑘𝑘𝑃𝑃, we obtain (𝑊𝑊𝐿𝐿 −𝑋𝑋) (𝑊𝑊𝐿𝐿 −𝑋𝑋) (𝑊𝑊𝐿𝐿 +𝑅𝑅) (𝑊𝑊𝐿𝐿 +𝑅𝑅) �𝑊𝑊𝐿𝐿 − �� +𝑋𝑋� ≥ �𝑊𝑊𝐿𝐿 − �� −𝑅𝑅� 2 2 2 2 which becomes 2 2 (𝑊𝑊𝐿𝐿 +𝑋𝑋) (𝑊𝑊𝐿𝐿 −𝑅𝑅) � � ≥ � � . 2 2 Then substituting in the definition of 𝑋𝑋, we get 2 𝑊𝑊𝐿𝐿 𝑞𝑞𝑗𝑗�𝛿𝛿‾−𝑖𝑖�𝑃𝑃−�1−𝑞𝑞𝑗𝑗�𝑘𝑘𝑃𝑃 𝑊𝑊𝐿𝐿 𝑅𝑅 � + � ≥ � − �. (B.1) 2 2 2 2 Take the positive square roots of (B.1) to get the consumption levels at each time for owning versus renting and then rearrange and simplify to obtain: 𝑞𝑞𝑗𝑗�𝛿𝛿‾−𝑖𝑖�𝑃𝑃−�1−𝑞𝑞𝑗𝑗�𝑘𝑘𝑃𝑃+𝑅𝑅 ≥ 0 (B.2) 42

Appendix C (Problem III) The Lagrangean objective is: m𝑀𝑀a𝑞𝑞x𝛺𝛺 = �𝑊𝑊𝐿𝐿 −𝑀𝑀𝑞𝑞��𝑀𝑀𝑞𝑞 +𝑞𝑞�𝛿𝛿‾−𝑖𝑖�𝑃𝑃+(1−𝑞𝑞)(−𝑘𝑘𝑃𝑃)�+𝛾𝛾�𝑀𝑀𝑞𝑞 −𝑘𝑘𝑃𝑃−𝑀𝑀‾� (C.1) The Karush-Kuhn-Tucker conditions for maximizing (C.1) are: ∂𝛺𝛺 = −�𝑀𝑀𝑞𝑞 +𝑞𝑞�𝛿𝛿‾−𝑖𝑖�𝑃𝑃+(1−𝑞𝑞)(−𝑘𝑘𝑃𝑃)�+�𝑊𝑊𝐿𝐿 −𝑀𝑀𝑞𝑞�+𝛾𝛾 ≤ 0 (C.2) ∂𝑀𝑀𝑞𝑞 ∂𝛺𝛺 𝑀𝑀𝑞𝑞 = 0 (C.3) ∂𝑀𝑀𝑞𝑞 ∂𝛺𝛺 = 𝑀𝑀𝑞𝑞 −𝑘𝑘𝑃𝑃−𝑀𝑀‾ ≤ 0 (C.4) ∂𝛾𝛾 ∂𝛺𝛺 𝛾𝛾 = 0 (C.5) ∂𝛾𝛾 Since 𝑀𝑀𝑞𝑞 > 𝑀𝑀‾ > 0,(𝐶𝐶.3) implies that (𝐶𝐶.2) holds with equality, which solved yields 𝑊𝑊𝐿𝐿 +(1−𝑞𝑞)𝑘𝑘𝑃𝑃−𝑞𝑞�𝛿𝛿‾−𝑖𝑖�𝑃𝑃+𝛾𝛾 𝑀𝑀𝑞𝑞 = . (C.6) 2 Substitute (C.6) into (C.4) to find 𝛾𝛾 ≥ 2𝑀𝑀‾ +𝑘𝑘𝑃𝑃−𝑊𝑊𝐿𝐿 +𝑞𝑞�𝑘𝑘 +𝛿𝛿‾−𝑖𝑖�𝑃𝑃. (C.7) Next, solve (C.7) for the value of q that makes 𝛾𝛾 = 0: 𝑊𝑊𝐿𝐿 −2𝑀𝑀‾ −𝑘𝑘𝑃𝑃 𝑞𝑞� = (C.8) �𝑘𝑘+𝛿𝛿‾−𝑖𝑖�𝑃𝑃 All households with 𝑞𝑞 > 𝑞𝑞� have 𝛾𝛾 > 0, which implies by (C.5) that (C.4) holds with equality. It follows that mortgage borrowers hold money in amounts: 43

𝑀𝑀𝑞𝑞 = 𝑘𝑘𝑃𝑃+𝑀𝑀‾ for 𝑞𝑞 > 𝑞𝑞� (C.9) 𝑊𝑊𝐿𝐿 +(1−𝑞𝑞)𝑘𝑘𝑃𝑃−𝑞𝑞�𝛿𝛿‾−𝑖𝑖�𝑃𝑃 𝑀𝑀𝑞𝑞 = for 𝑞𝑞 ≤ 𝑞𝑞�. (C.10) 2 Total money holdings for these two groups are therefore: 1 (1−𝑞𝑞�𝐿𝐿)(𝑘𝑘𝑃𝑃+𝑀𝑀‾)+𝑞𝑞�𝐿𝐿� 𝑓𝑓(𝑞𝑞)𝑞𝑞𝑞𝑞𝑞𝑞. (C.11) 𝑞𝑞� 44

Appendix D (Problem IV) Substituting the two equality constraints into the objective function, while using 𝛷𝛷 as the Lagrangean function and 𝜆𝜆 as the Lagrange multiplier, we have: 𝐸𝐸m𝐺𝐺,a𝐸𝐸x𝐵𝐵𝛷𝛷 = [𝑊𝑊𝐻𝐻 −𝐸𝐸𝐵𝐵 −𝐸𝐸𝐺𝐺][(1+𝑟𝑟𝐺𝐺)𝐸𝐸𝐺𝐺 +(1+𝑟𝑟‾𝐵𝐵)𝐸𝐸𝐵𝐵]+𝜆𝜆[(1+𝑟𝑟𝐺𝐺)𝐸𝐸𝐺𝐺 −𝑀𝑀‾] (D.1) The Karush-Kuhn-Tucker conditions for maximizing (D.1) are: ∂𝛷𝛷 = −[(1+𝑟𝑟𝐺𝐺)𝐸𝐸𝐺𝐺 +(1+𝑟𝑟‾𝐵𝐵)𝐸𝐸𝐵𝐵]+(1+𝑟𝑟𝐺𝐺)[𝑊𝑊𝐻𝐻 −𝐸𝐸𝐵𝐵 −𝐸𝐸𝐺𝐺]+𝜆𝜆(1+𝑟𝑟𝐺𝐺) ≤ 0 (D.2) ∂𝐸𝐸𝐺𝐺 ∂𝛷𝛷 𝐸𝐸𝐺𝐺 = 0 (D.3) ∂𝐸𝐸𝐺𝐺 ∂𝛷𝛷 = −[(1+𝑟𝑟𝐺𝐺)𝐸𝐸𝐺𝐺 +(1+𝑟𝑟‾𝐵𝐵)𝐸𝐸𝐵𝐵]+(1+𝑟𝑟‾𝐵𝐵)[𝑊𝑊𝐻𝐻 −𝐸𝐸𝐵𝐵 −𝐸𝐸𝐺𝐺] ≤ 0 (D.4) ∂𝐸𝐸𝐵𝐵 ∂𝛷𝛷 𝐸𝐸𝐵𝐵 = 0 (D.5) ∂𝐸𝐸𝐵𝐵 ∂𝛷𝛷 = (1+𝑟𝑟𝐺𝐺)𝐸𝐸𝐺𝐺 −𝑀𝑀‾ ≥ 0 (D.6) ∂𝜆𝜆 ∂𝛷𝛷 𝜆𝜆 = 0 (D.7) ∂𝜆𝜆 In the following, we assume the expected return for investing in bank equity is greater than the positive return on securitizer equity, i.e., 𝑟𝑟‾𝐵𝐵 > 𝑟𝑟𝐺𝐺 > 0. Constraint (D.6) implies that (1+𝑟𝑟𝐺𝐺)𝐸𝐸𝐺𝐺 ≥ 𝑀𝑀‾ > 0, which in turn implies 𝐸𝐸𝐺𝐺 > 0, which in turn implies (D.2) holds (2+𝑟𝑟𝐺𝐺+𝑟𝑟‾𝐵𝐵)𝐸𝐸𝐵𝐵 with equality, which further implies 𝑊𝑊𝐻𝐻 = (1+𝑟𝑟𝐺𝐺) +2𝐸𝐸𝐺𝐺 −𝜆𝜆. Substituting this expression into (D.4), we find that 𝜆𝜆 > 0, which implies (D.6) holds with equality; i.e., 𝐸𝐸𝐺𝐺 = 𝑀𝑀‾ (1+𝑟𝑟𝐺𝐺). 45

Assuming the wealthy household has initial wealth that exceeds the survival constraint, 𝑊𝑊𝐻𝐻 > 𝑀𝑀‾, it will invest a positive amount in bank equity, i.e., 𝐸𝐸𝐵𝐵 > 0. It follows that (D.4) 𝑊𝑊𝐻𝐻 (2+𝑟𝑟𝐺𝐺+𝑟𝑟‾𝐵𝐵)𝑀𝑀‾ holds with equality and 𝐸𝐸𝐵𝐵 = 2 −2(1+𝑟𝑟𝐺𝐺)(1+𝑟𝑟‾𝐵𝐵). 46

Appendix E (second-order conditions) (i) Derivation of 𝑓𝑓′ < 0. To satisfy the second-order conditions for a maximum to the securitizer’s problem, the quadratic associated with the Hessian matrix 2 2 ∂ 𝜋𝜋𝑠𝑠 ∂ 𝜋𝜋𝑠𝑠 ⎡ 2 ⎤ ⎢ ∂𝑞𝑞� ∂𝑞𝑞�∂𝑟𝑟𝑠𝑠 ⎥ 𝐻𝐻 = 2 2 ⎢ ∂ 𝜋𝜋𝑠𝑠 ∂ 𝜋𝜋𝑠𝑠 ⎥ ⎢ 2 ⎥ ⎣∂𝑟𝑟𝑠𝑠∂𝑞𝑞� ∂𝑟𝑟𝑠𝑠 ⎦ must be negative definite. Taking partial derivative of (27) and (28), we display two of the second order conditions: 2 ∂ 𝜋𝜋𝑠𝑠 ′�𝑞𝑞��𝑖𝑖+𝑐𝑐−𝑘𝑘−𝛿𝛿‾�+𝛿𝛿‾+𝑘𝑘−𝑐𝑐−𝑟𝑟𝑠𝑠� 2 = −𝑃𝑃𝐿𝐿𝑓𝑓 −𝑃𝑃𝐿𝐿𝑓𝑓(𝑞𝑞�)�𝑖𝑖 +𝑐𝑐−𝑘𝑘 −𝛿𝛿‾� < 0 (E.1) ∂𝑞𝑞� and 2 ∂ 𝜋𝜋𝑠𝑠 𝑃𝑃𝐿𝐿𝑓𝑓′(𝑞𝑞′) 2 = 2�𝑞𝑞′�𝑖𝑖 +𝑐𝑐 −𝑘𝑘 −𝛿𝛿‾�+𝛿𝛿‾+𝑘𝑘−𝑐𝑐 −𝑟𝑟𝑠𝑠)� < 0. (E.2) ∂𝑟𝑟𝑠𝑠 �𝑖𝑖 +𝑐𝑐 −𝑘𝑘−𝛿𝛿‾� Given the term in the square brackets of (E.1) is zero by (27) while 𝑞𝑞′ > 𝑞𝑞�, 𝑃𝑃𝐿𝐿 > 0 and 𝑖𝑖+ 𝑐𝑐 −𝑘𝑘−𝛿𝛿‾ > 0, then the term in square brackets of (E.2) must be positive, which implies ′ 𝑓𝑓′(𝑞𝑞 ) < 0 (E.3) (ii) Another second-order condition is derived by substituting (23) and (29) into (30): 𝑟𝑟+𝜏𝜏+𝑐𝑐 −𝑘𝑘 −𝛿𝛿‾ 𝜏𝜏 𝑟𝑟𝑠𝑠 +𝜏𝜏+𝑐𝑐 −𝑘𝑘 −𝛿𝛿‾ 𝑟𝑟𝑠𝑠 +𝑐𝑐 −𝑘𝑘−𝛿𝛿‾ 𝑓𝑓� �� � = 𝐹𝐹� �−𝐹𝐹� �, 𝑖𝑖 +𝑐𝑐 −𝑘𝑘 −𝛿𝛿‾ 𝑖𝑖 +𝑐𝑐 −𝑘𝑘 −𝛿𝛿‾ 𝑖𝑖 +𝑐𝑐 −𝑘𝑘 −𝛿𝛿‾ 𝑖𝑖 +𝑐𝑐 −𝑘𝑘−𝛿𝛿‾ 47

which is an equation with one endogenous variable 𝑟𝑟𝑠𝑠. Substituting (29) into (28) and differentiating the resulting expression with respect to 𝑟𝑟𝑠𝑠, we obtain the second-order condition that 𝜏𝜏 0 𝑓𝑓′� �−𝑓𝑓+𝑓𝑓 < 0 (E.4) 𝑖𝑖 +𝑐𝑐 −𝑘𝑘−𝛿𝛿‾ 𝑟𝑟𝑠𝑠+𝜏𝜏+𝑐𝑐−𝑘𝑘−𝛿𝛿‾ where 𝑓𝑓′ denotes the first derivative of 𝑓𝑓( ) evaulated at 𝑖𝑖+𝑐𝑐−𝑘𝑘−𝛿𝛿‾ , 𝑓𝑓 is the value of the 𝑟𝑟𝑠𝑠+𝜏𝜏+𝑐𝑐−𝑘𝑘−𝛿𝛿‾ 0 𝑟𝑟𝑠𝑠+𝑐𝑐−𝑘𝑘−𝛿𝛿‾ pdf evaluated at 𝑖𝑖+𝑐𝑐−𝑘𝑘−𝛿𝛿‾ , and 𝑓𝑓 is the pdf evaluated at 𝑖𝑖+𝑐𝑐−𝑘𝑘−𝛿𝛿‾ . Thus, (E.3) and (E.4) are second-order conditions for the securitizer’s choice of 𝑟𝑟𝑠𝑠 to maximize profits. 48

Appendix F (showing the guaranteed rate and conforming cut off vary directly with the mortgage rate) Inserting the estimated beta distribution and values for 𝑞𝑞′ and 𝑞𝑞� into (29), we obtain: 97.174 1.768 1 𝑟𝑟𝑠𝑠+𝜏𝜏+𝑐𝑐−𝑘𝑘−𝛿𝛿‾ 𝑖𝑖−𝑟𝑟𝑠𝑠−𝜏𝜏 𝜏𝜏 𝑟𝑟𝑠𝑠+𝜏𝜏+𝑐𝑐−𝑘𝑘−𝛿𝛿‾ 𝑟𝑟𝑠𝑠+𝑐𝑐−𝑘𝑘−𝛿𝛿‾ � � � � � �=𝐹𝐹� �−𝐹𝐹� �.(F.1) 𝛽𝛽(98.174,2.768) 𝑖𝑖+𝑐𝑐−𝑘𝑘−𝛿𝛿‾ 𝑖𝑖+𝑐𝑐−𝑘𝑘−𝛿𝛿‾ 𝑖𝑖+𝑐𝑐−𝑘𝑘−𝛿𝛿‾ 𝑖𝑖+𝑐𝑐−𝑘𝑘−𝛿𝛿‾ 𝑖𝑖+𝑐𝑐−𝑘𝑘−𝛿𝛿‾ To investigate how the securitizer’s choice of 𝑟𝑟𝑠𝑠 responds to changes in the mortgage rate 𝑖𝑖, we take the total differential of (29) with respect to 𝑟𝑟𝑠𝑠 and 𝑖𝑖 and find ′ 2 𝜏𝜏 0 �𝑟𝑟𝑠𝑠+𝑐𝑐−𝑘𝑘−𝛿𝛿‾� ′ 𝜏𝜏 0 𝑓𝑓 𝜏𝜏 �𝑓𝑓′ +𝑓𝑓 −𝑓𝑓�𝑞𝑞𝑟𝑟𝑠𝑠−� �𝑓𝑓 +𝑓𝑓 −𝑓𝑓�+ 2�𝑞𝑞𝑖𝑖=0.(F.2) �𝑖𝑖+𝑐𝑐−𝑘𝑘−𝛿𝛿‾� �𝑖𝑖+𝑐𝑐−𝑘𝑘−𝛿𝛿‾� �𝑖𝑖+𝑐𝑐−𝑘𝑘−𝛿𝛿‾� �𝑖𝑖+𝑐𝑐−𝑘𝑘−𝛿𝛿‾� Since the term inside the two square brackets above is negative, while the terms in parentheses are positive and 𝑓𝑓′ is negative, it follows that ∂𝑟𝑟𝑠𝑠 > 0. ∂𝑖𝑖 That is, the securitizer’s profit-maximizing MBS rate varies directly with the mortgage rate, a result that is well supported by empirical data and aligns with economic intuition. Turning to the securitizer’s choice of the conforming no-default probability, we use (F.1) and (F.2) while differentiating (28) to find: 2 ∂𝑞𝑞� 𝑓𝑓′𝜏𝜏 1 = 3� 𝑓𝑓′𝜏𝜏 0� > 0, ∂𝑖𝑖 �𝑖𝑖 +𝑐𝑐 −𝑘𝑘 −𝛿𝛿‾� �𝑖𝑖+𝑐𝑐−𝑘𝑘−𝛿𝛿‾�−𝑓𝑓+𝑓𝑓 indicating that the securitizer lowers the qualifying credit standard in response to decreases in the mortgage rate. 49