Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 849342 | 64 pages | https://doi.org/10.1155/2011/849342

Profit and Risk under Subprime Mortgage Securitization

Academic Editor: Binggen Zhang
Received07 Mar 2011
Accepted29 Apr 2011
Published27 Jul 2011

Abstract

We investigate the securitization of subprime residential mortgage loans into structured products such as subprime residential mortgage-backed securities (RMBSs) and collateralized debt obligations (CDOs). Our deliberations focus on profit and risk in a discrete-time framework as they are related to RMBSs and RMBS CDOs. In this regard, profit is known to be an important indicator of financial health. With regard to risk, we discuss credit (including counterparty and default), market (including interest rate, price, and liquidity), operational (including house appraisal, valuation, and compensation), tranching (including maturity mismatch and synthetic) and systemic (including maturity transformation) risks. Also, we consider certain aspects of Basel regulation when securitization is taken into account. The main hypothesis of this paper is that the SMC was mainly caused by the intricacy and design of subprime mortgage securitization that led to information (asymmetry, contagion, inefficiency, and loss) problems, valuation opaqueness and ineffective risk mitigation. The aforementioned hypothesis is verified in a theoretical- and numerical-quantitative context and is illustrated via several examples.

1. Introduction

The mid-2007 to 2009 subprime mortgage crisis (SMC) was preceded by a decade of low interest rates that spurred significant increases in both the financing of residential mortgage loans—hereafter, simply called mortgages—and house prices. This environment encouraged investors (including investment banks) to pursue instruments that offer yield enhancement. In this regard, subprime mortgages generally offer higher yields than standard mortgages and consequently have been in demand for securitization. In essence, securitization offers the opportunity to transform below investment grade assets (the investment or reference portfolio) into AAA and investment grade liabilities. The demand for increasingly intricate structured mortgage products (SMPs) such as residential mortgage-backed securities (RMBSs) and collateralized debt obligations (CDOs) which embed leverage within their structureexposed investing banks—hereafter, called investors—to an elevated risk of default. In the light of relatively low interest rates, rising house prices and investment grade credit ratings (usually AAA) given by the credit rating agencies (CRAs), this risk was not considered to be excessive. A surety wrap—insurance purchased from a monoline insurer—may also be used to ensure such credit ratings.

The process of subprime mortgage securitization is briefly explained below. The first step is where mortgagors—many first-time buyers—or individuals wanting to refinance seeked to exploit the seeming advantages offered by subprime mortgages. Next, mortgage brokers entered the lucrative subprime market with mortgagors being charged high fees. Thirdly, originators offering mortgages solicited funding that was often provided by Wall Street money. After extending mortgages, these originators quickly sold them to dealer (investment) banks and associated special-purpose vehicles (SPVs) for more profits. In this way, originators outsourced credit risk while relying on income from securitization to fund new mortgages. The fourth step involved Wall Street dealer banks pooling risky mortgages that did not meet the standards of the government-sponsored enterprises (GSEs) such as Fannie Mae and Freddie Mac and sold them as “private label,” nonagency securities. This is important because the structure of securitization will have special features reflecting the design of the reference mortgage portfolios. Fifthly, CRAs assisted dealer banks trading structured mortgage products (SMPs), so that these banks received the best possible bond ratings, earned exorbitant fees, and made SMPs attractive to investors including money market, mutual and pension funds. However, during the SMC, defaults on reference mortgage portfolios increased, and the appetite for SMPs decreased. The market for these securities came to a standstill. Originators no longer had access to funds raised from pooled mortgages. The wholesale lending market shrunk. Intra- and interday markets became volatile. In the sixth step, the SMPs were sold to investors worldwide thus distributing the risk.

The main hypothesis of this paper is that the SMC was mainly caused by the intricacy and design of subprime structures as well as mortgage origination and securitization that led to information (asymmetry, contagion, inefficiency, and loss) problems, valuation opaqueness and ineffective risk mitigation. More specifically, information was lost due to intricacy resulting from an inability to look through the chain of mortgages and SMPs—reference mortgage portfolios and RMBSs, ABS CDOs, structured investment vehicles (SIVs), etc. This situation was exacerbated by a lack of understanding of the uniqueness of subprime securities and their structural design. It is our opinion that the interlinked security designs that were necessary to make the subprime market operate resulted in information loss among investors as the chain of SMPs stretched longer and longer. Also, asymmetric information arose because investors could not penetrate the SMP portfolio far enough to make a determination of the risk exposure to the financial sector. An additional problem involves information contagion that played a crucial role in shaping defensive retrenchment in interbank as well as mortgage markets during the SMC. As far as valuation problems are concerned, in this contribution, problems with SMPs result from the dependence of valuation on house prices and its independence from the performance of the reference mortgage portfolios. Also, issues related to mortgage and investor valuation are considered. With regard to the latter, we identify a chain of valuations that starts with the valuation of mortgages and SMPs then proceeds to cash flow, profit, and capital valuation and ends up with the valuation of the investors themselves. Finally, we claim that the SMC primarily resulted from mortgage agents' appetite for rapid growth and search for high yields—both of which were very often pursued at the expense of risk mitigation practices. The subprime structure described above is unique to the SMC and will be elaborated upon in the sequel.

1.1. Literature Review

The discussions above and subsequently in Sections 2, 3, 4, and 5 are supported by various strands of existing literature.

The paper [1] examines the different factors that have contributed to the SMC (see, also, [2, 3]). The topics that these papers have in common with our contribution are related to yield enhancement, investment management, agency problems, lax underwriting standards, CRA incentive problems, ineffective risk mitigation, market opaqueness, extant valuation model limitations, and structured product intricacy (see Sections 2 and 3 as well as [4] for more details). Furthermore, our paper discusses the aforementioned issues and offers recommendations to help avoid future crises as in [5, 6].

In [7], light is shed on subprime mortgagors, mortgage design, and their historical performance. Their discussions involve predatory borrowing and lending that are illustrated via real-life examples. The working paper [8] firstly quantifies how different determinants contributed to high delinquency and foreclosure rates for vintage 2006 mortgages—compare with examples in Sections 5.2 and 5.3. More specifically, they analyze mortgage quality as the performance of mortgages adjusted for differences in mortgagor characteristics (such as credit score, level of indebtedness, and ability to provide documentation), mortgage characteristics (such as product type, amortization term, mortgage amount, and interest rate), and subsequent house appreciation (see, also, [3, 4]). Their analysis suggests that different mortgage-level characteristics as well as low house price appreciation were quantitatively too small to explain the bad performance of 2006 mortgages (compare with Table 1 in Section 3). Secondly, they observed a deterioration in lending standards with a commensurate downward trend in mortgage quality and a decrease in the subprime-prime mortgage rate spread during the 2001–2006 period (refer, e.g., Section 5.3). Thirdly, Demyanyk and Van Hemert show that mortgage quality deterioration could have been detected before the SMC (we consider “before the SMC” to be the period prior to July 2007 and “during the SMC” to be the period between July 2007 and December 2009). “After the SMC” is the period subsequent to December 2009. (see, also, [5, 6]).


Global CDO issuance ($millions)
Total issuance Structured finance Cash flow and hybrid Synthetic funded Arbitrage Balance sheet

Q1:04 24 982.5 NA 18 807.8 6 174.7 23 157.5 1 825.0
Q2:04 42 864.6 NA 25 786.7 17 074.9 39 715.5 3 146.1
Q3:04 42 864.6 NA 36 106.9 5 329.7 38 207.7 3 878.8
Q4:04 47 487.8 NA 38 829.9 8 657.9 45 917.8 1 569.9
2004 Tot. 157 418.5 NA 119 531.3 37 237.2 146 998.5 10 419.8
% of Tot. 75.9% 23.7% 93.4% 6.6%

Q1:05 49 610.2 28 171.1 40 843.9 8 766.3 43 758.8 5 851.4
Q2:05 71 450.5 46 720.3 49 524.6 21 695.9 62 050.5 9 400.0
Q3:05 52 007.2 34 517.5 44 253.1 7 754.1 49 636.7 2 370.5
Q4:05 98 735.4 67 224.2 71 604.3 26 741.1 71 957.6 26 777.8
2005 Tot. 271 803.3 176 639.1 206 225.9 64 957.4 227 403.6 44 399.7
% of Tot. 65.0% 75.9% 23.9% 83.7% 16.3%

Q1:06 108 012.7 66 220.2 83 790.1 24 222.6 101 153.6 6 859.1
Q2:06 124 977.9 65 019.6 97 260.3 24 808.4 102 564.6 22 413.3
Q3:06 138 628.7 89 190.2 102 167.4 14 703.8 125 945.2 12 683.5
Q4:06 180 090.3 93 663.2 131 525.1 25 307.9 142 534.3 37 556.0
2006 Tot. 551 709.6 314 093.2 414 742.9 89 042.7 472 197.7 79 511.9
% of Tot. 56.9% 75.2% 16.1% 85.6% 14.4%

Q1:07 186 467.6 101 074.9 140 319.1 27 426.2 156 792.0 29 675.6
Q2:07 175 939.4 98 744.1 135 021.4 8 403.0 153 385.4 22 554.0
Q3:07 93 063.6 40 136.8 56 053.3 5 198.9 86 331.4 6 732.2
Q4:07 47 508.2 23 500.1 31 257.9 5 202.3 39 593.7 7 914.5
2007 Tot. 502 978.8 263 455.9 362 651.7 46 230.4 436 102.5 66 8769.3
% of Tot. 52.4% 72.1% 9.1% 86.8% 13.3%

Q1:08 12 846.4 12 771.0 75.4 18 607.1 1 294.6
Q2:08 16 924.9 15 809.7 1 115.2 15 431.1 6 561.4
Q3:08 11 875.0 11 875.0 10 078.4 4 255.0
Q4:08 3 290.1 3 140.1 150.0 3 821.4 1 837.8
2008 Tot. 44 936.4 43 595.8 1 340.6 47 938.0 13 948.8
% of Tot. 32.4% 91.2.1% 1.6% 89.4% 10.6%

Q1:09 296.3 196.8 99.5 658.7 99.5
Q2:09 1 345.5 1 345.5 1 886.4
Q3:09 442.9 337.6 105.3 208.7 363.5
Q4:09 730.5 681.0 49.5 689.5 429.7
2009 Tot. 2 815.2 2 560.9 254.3 3 443.3 892.7
% of Tot. 40.4% 91.2.1% 1.6% 89.4% 10.6%

Q1:10 2 420.8 2 378.5 42.3 2 420.7
Q2:10 1 655.8 1 655.8 598.1 1 378.9
Q3:10 2 002.7 2 002.7 2002.7
Q4:10
2010 Tot. 6 079.3 6 037.0 42.3 2 600.8 3 799.6
% of Tot. 44.1% 91.2.1% 1.6% 89.4% 10.6%

The literature about mortgage securitization and the SMC is growing with our contribution, for instance, having close connections with [7] where the key structural features of a typical mortgage securitization is presented (compare with Figure 1 in Section 1.2.4). Also, that paper demonstrates how CRAs assign credit ratings to asset-backed securities (ABSs) and how these agencies monitor the performance of reference mortgage portfolios (see Sections 2.1, 2.2, and 2.3). Furthermore, that paper discusses RMBS and CDO architecture and is related to [9] that illustrates how misapplied bond ratings caused RMBSs and ABS CDO market disruptions (see Sections 3.2, 3.3, and 3.4). In [8], it is shown that the subprime mortgage market deteriorated considerably subsequent to 2007 (see, also, [4]). We believe that mortgage standards became slack because securitization gave rise to moral hazard, since each link in the securitization chain made a profit while transferring associated credit risk to the next link (see, e.g., [10]). At the same time, some financial institutions retained significant amounts of the mortgages they originated, thereby retaining credit risk and so were less guilty of moral hazard (see, e.g., [11]). The increased distance between originators and the ultimate bearers of risk potentially reduced originators' incentives to screen and monitor mortgages (see [12] for a preSMC description). As claimed in the present paper, the SMC and its impact on mortgage prices were magnified by the sale of SMPs. The enhanced intricacy of markets related to these products also reduces investor's ability to value them correctly where the value depends on the correlation structure of default events (see, e.g., [3, 11, 13]). Reference [14] considers parameter uncertainty and the credit risk associated with ABS CDOs (see, also, [46]). In [15] it is claimed that ABS CDOs opened up a whole new category of work for monoline insurers who insured the senior tranches of SMPs as part of the credit enhancement (CE) process (see, also, [4]). The working paper [1] asserts that, since the end of 2007, monoline insurers have been struggling to keep their AAA rating (compare with Figure 1 in Section 1.2.4). By the end of 2009, only MBIA and Ambac as well as a few others less exposed to mortgages such as financial security assurance (FSA) and assured guaranty, have been able to inject enough new capital to keep their AAA credit rating. In our paper, the effect of monoline insurance is tracked via the term 𝑐𝑖Σ (see (2.1) for an example).

Before the SMC, risk management and control put excessive confidence in credit ratings provided by CRAs and failed to provide their own analysis of credit risks in the underlying securities (see, e.g., [16]). The paper [17] investigates the anatomy of the SMC that involves mortgages and their securitization with operational risk as the main issue. At almost every stage in the subprime process—from mortgage origination to securitization—operational risk was insiduously present but not always acknowledged or understood. For instance, when originators originated mortgages, they were outsourcing their credit risk to investors, but what they were left with evolved into something much larger—significant operational and reputational risk (see, e.g., [17]). Before the SMC, the quantity of mortgages originated was more important than their quality while an increased number of mortgages were originated that contained resets. The underwriting of new mortgages embeds credit and operational risk. House prices started to decline and default rates increased dramatically. Also, credit risk was outsourced via mortgage securitization which, in turn, funded new mortgage originations. Securitization of mortgages involves operational, tranching, and liquidity risk. During the SMC, the value of these securities decreased as default rates increased dramatically. The RMBS market froze and returns from these securities were cut off with mortgages no longer being funded. Financial markets became unstable with a commensurate increase in market risk which led to a collapse of the whole financial system (compare with Sections 2.1 and 3.2). The paper [16] discusses several aspects of systemic risk. Firstly, there was excessive maturity transformation through conduits and SIVs—this ended in August 2007. The overhang of SIV ABSs subsequently put additional downward pressure on securities prices. Secondly, as the financial system adjusted to mortgage delinquencies and defaults and to maturity transformation dysfunction, the interplay of market malfunctioning or even breakdown, fair value accounting and the insufficiency of equity capital at financial institutions, and, finally, systemic effects of prudential regulation created a detrimental downward spiral in the overall banking system. Also, [16] argues that these developments have not only been caused by identifiably faulty decisions, but also by flaws in financial system architecture. We agree with this paper that regulatory reform must go beyond considerations of individual incentives and supervision and pay attention to issues of systemic interdependence and transparency. The aforementioned paper also discusses credit, market, and tranching (including maturity mismatch) risks. Furthermore, [4, 18] provides further information about subprime risks such as credit (including counterparty and default), market (including interest rate, price, and liquidity), operational (including house appraisal, valuation, and compensation), tranching (including maturity mismatch and synthetic) and systemic (including maturity transformation) risks (see, the discussion in Section 2.1.1).

Our hypothesis involves the intricacy and design of mortgage origination, securitization and systemic agents as well as information (loss, asymmetry and contagion) problems, valuation opaqueness and ineffective risk mitigation. In this regard, [19] investigates the effects of agency and information asymmetry issues embedded in structural form credit models on bank credit risk evaluation, using American bank data from 2001 to 2005. Findings show that both the agency problem and information asymmetry significantly cause deviations in the credit risk evaluation of structural form models from agency credit ratings (see, also, [3, 16]). Additionally, the aforementioned papers involve both the effects of information asymmetry and debt-equity agency positively relate to the deviation while that of management-equity agency relates to it negatively. The paper [20] is specifically focussed on the issue of counterparty risk and claim that the effects on counterparties in the SMC are remarkably small (see, e.g., Section 2.1.1).

1.2. Preliminaries about Subprime Mortgages and Their Securitization

In this subsection, we provide preliminaries about mortgages and risks as well as a subprime mortgage model that describes the main subprime agents. All events take place in either period 𝑡 or 𝑡+1.

1.2.1. Preliminaries about Subprime Mortgages

Subprime mortgages are financial innovations that aim to provide house ownership to riskier mortgagors. A design feature of these mortgages is that over short periods, mortgagors are able to finance and refinance their houses based on gains from house price appreciation (see [4] for more details). House appraisals were often inflated with originators having too much influence on appraisal companies. No-income-verification, mortgages led to increased cases of fraud and contain resets. Before, during and after the SMC, mortgage brokers were compensated on volume rather than mortgage quality. This increased volume led to a poor credit culture. Before the SMC, house values started to decline. Mortgagors were unable to meet mortgage terms when they reset resulting in increased defaults.

A traditional mortgage model for profit with mortgages at face value is built by considering the difference between cash inflow and outflow in [4] (compare with [21]). For this profit, in period 𝑡, cash inflow is constituted by returns on risky marketable securities, 𝑟𝐵𝑡𝐵𝑡, mortgages, 𝑟𝑀𝑡𝑀𝑡 and Treasuries, 𝑟𝚃𝑡𝚃𝑡. Furthermore, we denote the recovery amount, mortgage insurance payments per loss and present value of future profits from additional mortgages based on current mortgages by 𝑅𝑡, 𝐶(𝑆(𝒞𝑡)), and Π𝑝𝑡, respectively. Also, we consider the cost of funds for 𝑀, 𝑐𝑀𝜔𝑀𝑡, face value of mortgages in default, 𝑟𝑆𝑀𝑡, recovery value of mortgages in default, 𝑟𝑅𝑀𝑡, mortgage insurance premium, 𝑝𝑖(𝒞𝑡)𝑀𝑡, the all-in cost of holding risky marketable securities, 𝑐𝐵𝑡𝐵𝑡, interest paid to depositors, 𝑟𝐷𝑡𝐷𝑡, cost of taking deposits, 𝑐𝐷𝐷𝑡, interest paid to investors, 𝑟𝙱𝑡𝙱𝑡, the cost of borrowing, 𝑐𝙱𝙱𝑡, provisions against deposit withdrawals, 𝑃𝚃(𝚃𝑡), and the value of mortgage losses, 𝑆(𝒞𝑡), to collectively comprise cash outflow. Here 𝑟𝐷 and 𝑐𝐷 are the deposit rate and marginal cost of deposits, respectively, while 𝑟𝙱 and 𝑐𝙱 are the borrower rate and marginal cost of borrowing, respectively. In this case, we have that a traditional model for profit with defaulting, refinancing, and fully amortizing mortgages at face value may be expressed asΠ𝑡=𝑟𝑀𝑡𝑐𝑡𝑀𝜔𝑝𝑖𝑡+𝑐𝑝𝑡𝑟𝑓𝑡1𝑟𝑅𝑡𝑟𝑆𝑡𝑀𝑡𝐄𝑆𝒞+𝐶𝑡+𝑟𝐵𝑡𝑐𝐵𝑡𝐵𝑡+𝑟𝚃𝑡𝚃𝑡𝑃𝚃𝚃𝑡𝑟𝐷𝑡+𝑐𝐷𝑡𝐷𝑡𝑟𝙱𝑡+𝑐𝙱𝙱𝑡+Π𝑝𝑡.(1.1) From [4], the originator's balance sheet with mortgages at face value may be represented as 𝑀𝑡+𝐵𝑡+𝚃𝑡=(1𝛾)𝐷𝑡+𝙱𝑡+𝐾𝑡.(1.2) Also, the originator's total capital constraint for mortgages at face value is given by𝐾𝑡=𝑛𝑡𝐸𝑡1+𝑂𝑡𝜔𝒞𝜌𝑡𝑀𝑡+𝜔𝐵𝐵𝑡+12.5𝑓𝑀(𝑚𝑉𝑎𝑅+𝙾),(1.3) where 𝜔(𝒞𝑡) and 𝜔𝐵 are the risk weights related to 𝑀 and 𝐵, respectively, while 𝜌—Basel II pegs 𝜌 at approximately 0.08—is the Basel capital regulation (by Basel capital regulation, we mean the regulatory capital framework set out by Basel II and beyond) ratio of regulatory capital to risk weighted assets. Furthermore, for the function𝐽𝑡=Π𝑡+𝑙𝑡𝐾𝑡𝜔𝒞𝜌𝑡𝑀𝑡+𝜔𝐵𝐵𝑡+12.5𝑓𝑀(𝑚𝑉𝑎𝑅+𝙾)𝑐𝑡𝑑𝑤𝐾𝑡+1+𝐄𝑡𝛿𝑡,1𝑉𝐾𝑡+1,𝑥𝑡+1,(1.4) the optimal originator valuation problem is to maximize its value by choosing 𝑟𝑀, 𝐷, 𝚃, and 𝐾, for𝑉𝐾𝑡,𝑥𝑡=max𝑟𝑀𝑡,𝐷𝑡,𝚃𝑡𝐽𝑡,(1.5) subject to mortgage, cash flow, balance sheet, and financing constraints given by𝑀𝑡=𝑚0𝑚1𝑟𝑀𝑡+𝑚2𝒞𝑡+𝜎𝑀𝑡,(1.6) equations (1.1), (1.2), and𝐾𝑡+1=𝑛𝑡𝑑𝑡+𝐸𝑡+1+𝑟𝑂𝑡𝑂𝑡Π𝑡+Δ𝐹𝑡,(1.7) respectively. In the value function, 𝑙𝑡 is the Lagrange multiplier for the capital constraint, 𝑐𝑡𝑑𝑤 is the deadweight cost of capital, and 𝛿𝑡,1 is a stochastic discount factor. In the profit function, 𝑐Λ𝜔 is the constant marginal cost of mortgages (including the cost of monitoring and screening). In each period, banks invest in fixed assets (including buildings and equipment) which we denote by 𝐹𝑡. The originator is assumed to maintain these assets throughout its existence, so that it must only cover the costs related to the depreciation of fixed assets, Δ𝐹𝑡. These activities are financed through retaining earnings and eliciting additional debt and equity, 𝐸𝑡, so thatΔ𝐹𝑡=𝐸𝑟𝑡+𝑛𝑡+1𝑛𝑡𝐸𝑡+𝑂𝑡+1.(1.8) Suppose that 𝐽 and 𝑉 are given by (1.4) and (1.5), respectively. When the capital constraint given by (1.3) holds (i.e., 𝑙𝑡>0), a solution to the originator's optimal valuation problem yields an optimal 𝑀 and 𝑟𝑀 of the form𝑀𝑡=𝐾𝑡𝒞𝜌𝜔𝑡𝜔𝐵𝐵𝑡+12.5𝑓𝑀(𝑚𝑉𝑎𝑅+𝙾)𝜔𝒞𝑡,𝑟(1.9)𝑡𝑀=1𝑚1𝑚0+𝑚2𝒞𝑡+𝜎𝑀𝑡𝑀𝑡,(1.10) respectively. In this case, the originator's corresponding optimal deposits, provisions for deposit withdrawals, and profits are given by𝐷𝑡=11𝛾𝐷+𝐷𝑟𝑝𝑡𝑟𝚃𝑡+𝑟𝐵𝑡𝑐𝐵𝑡+𝑟𝙱𝑡+𝑐𝙱𝑡1𝑟1𝛾𝐷𝑡+𝑐𝐷𝑡+𝐾𝑡𝒞𝜌𝜔𝑡𝜔𝐵𝐵𝑡+12.5𝑓𝑀(𝑚𝑉𝑎𝑅+𝙾)𝜔𝒞𝑡+𝐵𝑡𝐾𝑡𝙱𝑡,𝚃(1.11)𝑡=𝐷+𝐷𝑟𝑝𝑡𝑟𝚃𝑡+𝑟𝐵𝑡𝑐𝐵𝑡+𝑟𝙱𝑡+𝑐𝙱𝑡1𝑟1𝛾𝐷𝑡+𝑐𝐷𝑡,Π(1.12)𝑡=𝐾𝑡𝒞𝜌𝜔𝑡𝜔𝐵𝐵𝑡+12.5𝑓𝑀(𝑚𝑉𝑎𝑅+𝙾)𝜔𝒞𝑡×1𝑚1𝑚0𝐾𝑡𝒞𝜌𝜔𝑡+𝜔𝐵𝐵𝑡+12.5𝑓𝑀(𝑚𝑉𝑎𝑅+𝙾)𝜔𝒞𝑡+𝑚2𝒞𝑡+𝜎𝑀𝑡𝑐𝑡𝑀𝜔𝑐𝑝𝑡𝑟𝑓𝑡+𝑝𝑖𝑡+1𝑟𝑅𝑡𝑟𝑆𝑡+𝑟𝐷𝑡+𝑐𝐷𝑡1𝑟(1𝛾)𝐷𝑡+𝑐𝐷𝑡1(𝐵1𝛾)𝑡𝐾𝑡𝙱𝑡+𝐷+𝐷𝑟𝑝𝑡𝑟𝚃𝑡+𝑟𝐵𝑡𝑐𝐵𝑡+𝑟𝙱𝑡+𝑐𝙱𝑡1𝑟1𝛾𝐷𝑡+𝑐𝐷𝑡𝑟𝚃𝑡𝑟𝐷𝑡+𝑐𝐷𝑡1+𝑟(1𝛾)𝐵𝑡𝑐𝐵𝑡𝐵𝑡𝑟𝙱𝑡+𝑐𝙱𝑡𝙱𝑡𝐄𝑆𝒞+𝐶𝑡𝑃𝚃𝚃𝑡+Π𝑝𝑡,(1.13) respectively.

1.2.2. Preliminaries about Residential Mortgage-Backed Securities (RMBSs)

In this subsection, we discuss the main design features of subprime RMBSs. In particular, we provide a description of SPVs, cost of mortgages, default, collateral, adverse selection, and residual value (see [4] for more details). Further discussions of these features are included in Section 2.

In terms of the organization of the SPV, there are various states that can be associated with corporate forms such as a trust (denoted by 𝙴1), a limited liability corporation (LLC; denoted by 𝙴2), LLP (𝙴3) or a C-corporation (𝙴4). Our interest is mainly in 𝙴1 trusts and 𝙴2 LLCs that have their own unique tax benefits and challenges as well as degree of mortgage protection and legal limited liability. For our purposes, the optimal state during the lifetime of 𝙴𝑖, 𝑖{1,2} is denoted by 𝙴, such that the deviation from 𝙴 is given by||𝙴𝑖𝑡𝙴||.(1.14) These deviations measure the loss and opportunity costs arising from the use of suboptimal corporate structures such as SPVs. Usually such loss is in the form of increased legal fees, losses due to low limited liability, and the value of additional time spent in dispute resolution. In this case, the formula for 𝙴 is given by𝙴𝑡||𝙴=max𝑖𝑡𝙴||,0.(1.15) Banks may monitor mortgagor activities to see whether they are complying with the restrictive agreements and enforce the agreements if they are not by making sure that mortgagors are not taking on risks at their expense. Securitization dissociates the quality of the original mortgage portfolio from the quality of the cash flows from the reference mortgage portfolio, 𝑓Σ𝑀, to investors, where 𝑓Σ is the fraction of the face value of mortgages, 𝑀, that is securitized. Whether on-balance sheet or in the market, the weighted average of the cost of mortgages summarizes the cost of various funding solutions. It is the weighted average of cost of equity and debt on the originator's balance sheet and the weighted average of the costs of securitizing various mortgages. In both cases, we use the familiar weighted average cost of capital. As a consequence, cost of funds via securitization, 𝑐𝑀Σ𝜔 (includes monitoring and transaction costs for 𝑓Σ𝑀 denoted by 𝑐𝑚Σ and 𝑐𝑡Σ, respectively, as well as the cost of funds in the market through securitizing mortgages), does not have to coincide with the originator's cost of mortgages for 𝑀, 𝑐𝑀𝜔 (includes monitoring and transaction costs for 𝑀 denoted by 𝑐𝑚 and 𝑐𝑡, resp.). We note that 𝑐𝑡 may include overhead, fixed costs, and variable costs per transaction expressed as a percentage of 𝑀. This suggests that if𝑐𝑀Σ𝜔<𝑐𝑀𝜔,(1.16) then the securitization economics is favorable and conversely.

In the sequel, the notation 𝑟𝑡𝑆Σ represents the default rate on securitized mortgages, 𝑓Σ𝑡 denotes the fraction of 𝑀 that is securitized, while 𝑓Σ𝑡 denotes the fraction of the originator's reference mortgage portfolio realized as new mortgages in securitization as a result of, for instance, equity extraction via refinancing. In the sequel, collateral is constituted by the reference mortgage portfolio that is securitized and relates to the underlying cash flows. Such flow and credit characteristics of the collateral will determine the performance of the securities and drive the structuring process. Although a wide variety of assets may serve as collateral for securitization, mortgages are the most widely used form of collateral. In the case of a mortgage secured by collateral, if the mortgagor fails to make required payments, the originator has the right to seize and sell the collateral to recover the defaulted amount.

RMBSs mainly use one or both of the sen/sub-shifting of interest structure, sometimes called the 6-pack structure (with 3 mezz and 3 sub-RMBS bonds junior to the AAA bonds), or an XS/OC structure (see, e.g., [3]). Here, XS and OC denote excess spread and overcollaterization, respectively. Like sen/sub deals, XS is used to increase OC, by accelerating payments on sen RMBS bonds via sequential amortization—a process known as turboing. An OC target, 𝑂𝑐, is a fraction of the original mortgage par, 𝑀, and is designed to be in the second loss position against collateral losses with the interest-only (IO) strip being first. Typically, the initial OC amount, 𝑂𝑐𝑖, is less than 100% of 𝑂𝑐 and it is then increased over time via the XS until 𝑂𝑐 is reached. When this happens, the OC is said to be fully funded and nett interest margin securities can begin to receive cash flows from the RMBS bond deal. Once 𝑂𝑐 has been reached, and subject to certain performance tests, XS can be released for other purposes, including payment to residual (residual value is the payout received by the RMBS bond holder—in our case the investor—when bonds have been paid off and cash flows from the reference mortgage portfolio (collateral) are still being generated. Residual value also arises when the proceeds amount from the sale of this reference portfolio as whole mortgages is greater than the amount needed to pay outstanding bonds.) bond holders. In this contribution, we assume that the investor is also a residual bond holder. For our purposes, the symbol 𝑟𝑟𝑡, represents the average residual rate in a period 𝑡 securitization. It is defined as the difference between the average interest rate paid by mortgagors, 𝑟𝑀, and the present value of interest paid on securitized mortgages, 𝑟𝑝Σ, so that𝑟𝑟=𝑟𝑀𝑟𝑝Σ.(1.17)Adverse selection is the problem created by asymmetric information in originator's mortgage originations. It occurs because high-risk (e.g., subprime) mortgagors that are most likely to default on their mortgages, usually apply for them. In other words, subprime mortgages are extended to mortgagors who are most likely to produce an adverse outcome. We denote the value of the adverse selection problem by 𝑉𝑎. For the sake of argument, we set𝑉𝑎𝑡=𝑎𝑓Σ𝑡𝑀𝑡,(1.18) where 𝑉𝑎 is a fraction 𝑎, of the face value of mortgages in period 𝑡.

1.2.3. Preliminaries about Collateralized Debt Obligations (CDOs)

RMBS CDOs are sliced into tranches of differing risk-return profiles. SIVs assist hedge funds and banks to pool a number of single RMBS tranches to create one CDO. As with RMBSs, risk associated with CDOs is shifted from sen to subtranches. The funds generated by the sale of CDOs enable CDO issuers to continue to underwrite the securitization of subprime mortgages or continue to purchase RMBSs. Before the SMC, major depository banks around the world used financial innovations such as SIVs to circumvent capital ratio regulations. This type of activity resulted in the failure of Northern Rock, which was nationalized at an estimated cost of $150 billion.

Certain features of RMBS CDOs make their design more intricate (compare with Question 3). For instance, many such CDOs are managed by managers that are to a limited extent allowed to buy and sell RMBS bonds over a given period of time. The reason for this is that CDOs amortize with a longer maturity able to be achieved by reinvestment. In particular, managers are able to use cash that is paid to the CDO from amortization for reinvestment. Under the conditions outlined in Section 1.2, they can sell bonds in the portfolio and buy other bonds with restrictions on the portfolio that must be maintained. CDO managers typically owned all or part of the CDO equity, so they would benefit from higher yielding assets for a given liability structure. In short, CDOs are managed funds with term financing and some constraints on the manager in terms of trading and the portfolio composition. Further discussion of RMBS CDOs is provided in Section 3.

Table 1 below elucidates CDO issuance with Column 1 showing total issuance of CDOs while the next column presents total issuance of RMBS CDOs. This table suggests that CDO issuance has been significant both before and after the SMC with the majority being CDOs with structured notes as collateral. In addition, Table 1 suggests that the motivation for CDO issuance has primarily been arbitrage.

From Table 1, we note that issuance of RMBS CDOs roughly tripled over the period 2005–07 and RMBS CDO portfolios became increasingly concentrated in subprime RMBSs. In this regard, by 2005, spreads on subprime BBB tranches seemed to be wider than other structure mortgage products with the same rating, creating an incentive to arbitrage the ratings between subprime RMBS and CDO tranches ratings. Subprime RMBSs increasingly dominated CDO portfolios, suggesting that the pricing of risk was inconsistent with the ratings. Also, concerning the higher-rated tranches, CDOs may have been motivated to buy large amounts of structured mortgage products, because their AAA tranches would input profitable negative basis trades (According to [3], in a negative basis trade, a bank buys the AAA-rated CDO tranche while simultaneously purchasing protection on the tranche under a physically settled CDS. From the bank's viewpoint, this is the simultaneous purchase and sale of a CDO, which meant that the bank lender could book the nett present value (NPV) of the excess yield on the CDO tranche over the protection payment on the CDS. If the CDS spread is less than the bond spread, the basis is negative. An example of this is given below. Suppose the bank borrows at LIBOR + 5 and buys an AAA-rated CDO tranche which pays LIBOR + 30. Simultaneously, the investor buys protection for 15 bps (basis points). So the investor makes 25 bps over LIBOR nett on the asset, and they have 15 bps in costs for protection, for a 10 bps profit. Note that a negative basis trade swaps the risk of the AAA tranche to a CDS protection writer. Now, the subprime-related risk has been separated from the cash host. Consequently, even if we were able to locate the AAA CDO tranches, this would not be the same as finding out the location of the risk. Refernce [3] suggests that nobody knows the extent of negative basis trades.) As a consequence, the willingness of CDOs to purchase subprime RMBS bonds increased. In the period 2008-2009, during the height of the SMC, there was a dramatic decrease in CDO issuance. During Q1:10 there was a marked increase in RMBS CDO issuance by comparison with Q3:09 and Q4:09 indicating an improvement in the CDO market.

Table 2 shows estimates of the typical collateral composition of sen and mezz RMBS CDOs before the SMC. It is clear that subprime and other RMBS tranches make up a sizeable percentage of both these tranche types.


Typical collateral composition of RMBS CDOs (%)
High-grade RMBS CDOs Mezzanine RMBS CDOs

Subprime RMBS tranches 50% 77%
Other RMBS tranches 25 12
CDO tranches 19 6
Other 6 5

Table 3 below demonstrates that increased volumes of origination in the mortgage market led to an increase in subprime RMBSs as well as CDO issuance.


Subprime-related CDO volumes
VintageMezz RMBS CDOs High srade RMBS CDOs All CDOs

2005 27 50 290
2006 50 100 468
2007 30 70 330
2008 30 70 330

1.2.4. Preliminaries about Subprime Mortgage Models

We introduce a subprime mortgage model with default to encapsulate the key aspects of mortgage securitization.

Figure 1 presents a subprime mortgage model involving nine subprime agents, four subprime banks, and three types of markets. As far as subprime agents are concerned, we note that circles 1a, 1b, 1c, and 1d represent flawed independent assessments by house appraisers, mortgage brokers, CRAs rating SPVs, and monoline insurers being rated by CRAs, respectively. Regarding the former agent, the process of mortgage origination is flawed with house appraisers not performing their duties with integrity and independence. According to [17], this type of fraud is the “linchpin of the house buying transaction” and is an example of operational risk. Also, the symbol X indicates that the cash flow stops as a consequence of defaults. Before the SMC, appraisers estimated house values based on data that showed that the house market would continue to grow (compare with 1A and 1B). In steps 1C and 1D, independent mortgage brokers arrange mortgage deals and perform checks of their own while originators originate mortgages in 1E. Subprime mortgagors generally pay high mortgage interest rates to compensate for their increased risk from poor credit histories (compare with 1F). Next, the servicer collects monthly payments from mortgagors and remits payments to dealers and SPVs. In this regard, 1G is the mortgage interest rate paid by mortgagors to the servicer of the reference mortgage portfolios, while the interest rate 1H (mortgage interest rate minus the servicing fee) is passed by the servicer to the SPV for the payout to investors. Originator mortgage insurers compensate originators for losses due to mortgage defaults. Several subprime agents interact with the SPV. For instance, the trustee holds or manages and invests in mortgages and SMPs for the benefit of another. Also, the underwriter is a subprime agent who assists the SPV in underwriting new SMPs. Monoline insurers guarantee investors' timely repayment of bond principal and interest when an SPV defaults. In essence, such insurers provide guarantees to SPVs, often in the form of credit wraps, that enhance the credit rating of the SPV. They are so named because they provide services to only one industry. These insurance companies first began providing wraps for municipal bond issues, but now they provide credit enhancement for other types of SMP bonds, such as RMBSs and CDOs. In so doing, monoline insurers act as credit enhancement providers that reduce the risk of subprime mortgage securitization.

The originator has access to mortgage investments that may be financed by borrowing from the lender, represented by 1I. The lender, acting in the interest of risk-neutral shareholders, either invests its deposits in Treasuries or in the originator's mortgage projects. In return, the originator pays interest on these investments to the lender, represented by 1J. Next, the originator deals with the mortgage market represented by 1O and 1P, respectively. Also, the originator pools its mortgages and sells them to dealers and/or SPVs (see 1K). The dealer or SPV pays the originator an amount which is slightly greater than the value of the reference mortgage portfolios as in 1L. A SPV is an organization formed for a limited purpose that holds the legal rights over mortgages transferred by originators during securitization. In addition, the SPV divides this pool into sen, mezz, and jun tranches which are exposed to different levels of credit risk. Moreover, the SPV sells these tranches as securities backed by mortgages to investors (see 1N) that is paid out at an interest rate determined by the mortgage default rate, prepayment and foreclosure (see 1M). Also, SPVs deal with the SMP bond market for investment purposes (compare with 1Q and 1R). Furthermore, originators have SMPs on their balance sheets, that have connections with this bond market. Investors invest in this bond market, represented by 1S and receive returns on SMPs in 1T. The money market and hedge fund market are secondary markets where previously issued marketable securities such as SMPs are bought and sold (compare with 1W and 1X). Investors invest in these short-term securities (see, 1U) to receive profit, represented by 1V. During the SMC, the model represented in Figure 1 was placed under major duress as house prices began to plummet. As a consequence, there was a cessation in subprime agent activities and the cash flows to the markets began to dry up, thus, causing the whole subprime mortgage model to collapse.

We note that the traditional mortgage model is embedded in Figure 1 and consists of mortgagors, lenders and originators as well as the mortgage market. In this model, the lender lends funds to the originator to fund mortgage originations (see, 1I and 1J). Home valuation as well as income and credit checks were done by the originator before issuing the mortgage. The originator then extends mortgages and receives repayments that are represented by 1E and 1F, respectively. The originator also deals with the mortgage market in 1O and 1P. When a mortgagor defaults on repayments, the originator repossesses the house.

1.2.5. Preliminaries about Subprime Risks

The main risks that arise when dealing with SMPs are credit (including counterparty and default), market (including interest rate, price, and liquidity), operational (including house appraisal, valuation, and compensation), tranching (including maturity mismatch and synthetic), and systemic (including maturity transformation) risks. For sake of argument, risks falling in the categories described above are cumulatively known as subprime risks. In Figure 2 below, we provide a diagrammatic overview of the aforementioned subprime risks.

The most fundamental of the above risks is credit and market risk. Credit risk involves the originator's risk of mortgage losses and the possible inability of SPVs to make good on investor payments. This risk category generally includes counterparty risk that, in our case, is the risk that a banking agent does not pay out on a bond, credit derivative or credit insurance contract. It refers to the ability of banking agents—such as originators, mortgagors, servicers, investors, SPVs, trustees, underwriters, and depositors—to fulfill their obligations towards each other. During the SMC, even banking agents who thought they had hedged their bets by buying insurance—via credit default swap (CDS) or monoline insurance contracts—still faced the risk that the insurer will be unable to pay.

In our case, market risk is the risk that the value of the mortgage portfolio will decrease mainly due to changes in the value of securities prices and interest rates (see, e.g., Sections 2.1 and 4.2). Interest rate risk arises from the possibility that SMP interest rate returns will change. Subcategories of interest rate risk are basis and prepayment risk. The former is the risk associated with yields on SMPs and costs on deposits which are based on different bases with different rates and assumptions. Prepayment risk results from the ability of subprime mortgagors to voluntarily (refinancing) and involuntarily (default) prepay their mortgages under a given interest rate regime. Liquidity risk arises from situations in which a banking agent interested in selling (buying) SMPs cannot do it because nobody in the market wants to buy (sell) those SMPs. Such risk includes funding and credit crunch risk. Funding risk refers to the lack of funds or deposits to finance mortgages and credit crunch risk refers to the risk of tightened mortgage supply and increased credit standards. We consider price risk to be the risk that SMPs will depreciate in value, resulting in financial losses, markdowns and possibly margin calls. Subcategories of price risk are valuation risk (resulting from the valuation of long-term SMP investments) and reinvestment risk (resulting from the valuation of short-term SMP investments). Valuation issues are a key concern that must be dealt with if the capital markets are to be kept stable, and they involve a great deal of operational risk.

Operational risk is the risk of incurring losses resulting from insufficient or inadequate procedures, processes, systems or improper actions taken. As we have commented before, for mortgage origination, operational risk involves documentation, background checks and progress integrity. Also, subprime mortgage securitization embeds operational risk via misselling, valuation and IB issues. Operational risk related to mortgage origination and securitization results directly from the design and intricacy of mortgages and related structured products. Moreover, investors carry operational risk associated with mark-to-market issues, the worth of SMPs when sold in volatile markets and uncertainty involved in investment payoffs. Also, market reactions include increased volatility leading to behavior that can increase operational risk such as unauthorized trades, dodgy valuations and processing issues. Often additional operational risk issues such as model validation, data accuracy, and stress testing lie beneath large market risk events (see, e.g., [17]).

Tranching risk is the risk that arises from the intricacy associated with the slicing of SMPs into tranches in securitization deals. Prepayment, interest rate, price and tranching risk involves the intricacy of subprime SMPs. Another tranching risk that is of issue for SMPs is maturity mismatch risk that results from the discrepancy between the economic lifetimes of SMPs and the investment horizons of IBs. Synthetic risk can be traded via credit derivatives—like CDSs—referencing individual subprime RMBS bonds, synthetic CDOs or via an index linked to a basket of such bonds.

In banking, systemic risk is the risk that problems at one bank will endanger the rest of the banking system. In other words, it refers to the risk imposed by interlinkages and—dependencies in the system where the failure of a single entity or cluster of entities can cause a cascading effect which could potentially bankrupt the banking system or market.

1.3. Main Questions and Outline of the Paper

In this subsection, we identify the main questions solved in and give an outline of the paper.

1.3.1. Main Questions

The main questions that are solved in this paper may be formulated as follows.

Question 1 (modeling of profit under subprime mortgage securitization). Can we construct discrete-time subprime mortgage models that incorporate default, monoline insurance, costs of funds and profits under mortgage securitization? (see Sections 2.1 and 3.2).

Question 2 (modeling of risk under subprime mortgage securitization). Can we identify the risks associated with the different components of the subprime mortgage models mentioned in Question 1? (see Sections 2.1 and 3.2).

Question 3 (subprime mortgage securitization intricacy and design leading to information problems, valuation opaqueness and ineffective risk mitigation). Was the SMC partly caused by the intricacy and design of mortgage securitization that led to information (asymmetry, contagion, inefficiency and loss) problems, valuation opaqueness and ineffective risk mitigation? (see Sections 2, 3, 4, and 5).

Question 4 (optimal valuation problem under subprime mortgage securitization). In order to obtain an optimal valuation under subprime mortgage securitization, which decisions regarding mortgage rates, deposits and Treasuries must be made? (see Theorems 2.1 and 3.1 of Sections 2.3 and 3.4, resp.).

1.3.2. Outline of the Paper

Section 2 contains a discussion of an optimal profit problem under RMBSs. To make this possible, capital, information, risk and valuation for a subprime mortgage model under RMBSs is analyzed. In this regard, a mechanism for mortgage securitization, RMBS bond structure, cost of funds for RMBSs, financing, adverse selection, monoline insurance contracts for subprime RMBSs as well as residuals underly our discussions. Section 3 is analogous to Section 2 by investigating an optimal profit problem under RMBS CDOs. Section 4 discusses aspects of the relationship between subprime mortgage securitization and Basel regulation. Also, Section 5 provides examples of aspects of the aforementioned issues, while Section 6 discusses important conclusions and topics for future research. Finally, an appendix containing additional information and the proofs of the main results is provided in the appendix.

2. Profit, Risk, and Valuation under RMBSs

In this section, we provide more details about RMBSs and related issues such as profit, risk, and valuation. In the sequel, we assume that the notation Π, 𝑟𝑀, 𝑀, 𝑐𝑀𝜔, 𝑝𝑖, 𝑐𝑝, 𝑟𝑓, 𝑟𝑅, 𝑟𝑆, 𝑆, 𝒞, 𝐶(𝐄[𝑆(𝒞)]), 𝑟𝐵, 𝑐𝐵, 𝐵, 𝑟𝚃, 𝚃, 𝑃𝚃(𝚃), 𝑟𝐷, 𝑐𝐷, 𝐷, 𝑟𝙱, 𝑐𝙱, 𝙱, Π𝑝, 𝐾, 𝑛, 𝐸, 𝑂, 𝜔(𝒞), 𝜔𝐵, 𝑓𝑀, 𝑚𝑉𝑎𝑅, and 𝙾 corresponds to that of Section 1.2. Furthermore, the notation 𝑟𝑆Σ represents the loss rate on RMBSs, 𝑓Σ is the fraction of 𝑀 that is securitized and 𝑓Σ denotes the fraction of 𝑀, realized as new RMBSs, where 𝑓Σ𝑓Σ.

The following assumption about the relationship between the investor's and originator's profit is important for subsequent analysis.

Assumption 1 (relationship between the originator and investor). We suppose that the originator and investor share the same balance sheet in terms of 𝐵, 𝚃, 𝐷, 𝙱 and 𝐾 (compare with (1.2)). Furthermore, we assume that the investor's mortgages can be decomposed as 𝑀=𝑓Σ𝑀+(1𝑓Σ)𝑀. Finally, we suppose that the investor's profit can be expressed as a function of the variables in the previous paragraph and the securitization components 𝙴, 𝙵, 𝑟𝑟, and 𝑉𝑎 (see Section 1.2 for more details).

This assumption enables us to subsequently derive an expression for the investor's profit under RMBSs as in (2.1) from the originator's profit formula given by (1.1). We note that important features of Section 2 are illustrated in Sections 5.1, 5.2, and 5.3.

The key design feature of subprime mortgages involves the ability of mortgagors to finance and refinance their houses based on capital gains due to house price appreciation over short horizons and then turning this into collateral for a new mortgage or extracting equity for consumption. As is alluded to in Section 2, the unique design of mortgages resulted in unique structures for their securitizations (response to Question 3). During the SMC, CRAs were reprimanded for giving investment-grade ratings to RMBSs backed by risky mortgages. Before the SMC, these high ratings enabled such RMBSs to be sold to investors, thereby financing and exacerbating the housing boom. The issuing of these ratings were believed justified because of risk-reducing practices, such as monoline insurance and equity investors willing to bear the first losses. However, during the SMC, it became clear that some role players in rating subprime-related securities knew at the time that the rating process was faulty. Uncertainty in financial markets spread to other subprime agents, increasing the counterparty risk which caused interest rates to increase. Refinancing became almost impossible and default rates exploded. All these operations embed systemic risk which finally caused the banking system to collapse (compare with Section 2.1).

Clearly, during the SMC, the securitization of credit risks was a source of moral hazard that compromised global banking sector stability. Before the SMC, the practice of splitting the claims to a reference mortgage portfolio into tranches was a response to this concern. In this case, sen and mezz tranches can be considered to be senior and junior debt, respectively. If originators held equity tranches and if, because of packaging and diversification, the probability of default, that is, the probability that reference portfolio returns do not attain the sum of sen and mezz claims, were (close to) zero, we would (almost) be neglecting moral hazard effects. How the banking system failed despite the preceding scenario is explained next (compare with Section 2.1). Unfortunately, in reality, both ifs in the statement above were not satisfied. Originators did not, in general, hold the equity tranches of the portfolios that they generated. In truth, as time went on, ever greater portions of equity tranches were sold to external investors. Moreover, default probabilities for sen and mezz tranches were significant because packaging did not provide for sufficient diversification of returns on the reference mortgage portfolios in RMBS portfolios (see, e.g., [16]).

2.1. Profit and Risk under RMBSs

In this subsection, we discuss a subprime mortgage model for capital, information, risk, and valuation and its relation to retained earnings.

2.1.1. A Subprime Mortgage Model for Profit and Risk under RMBSs

In this paper, a subprime mortgage model for capital, information, risk, and valuation under RMBSs can be constructed by considering the difference between cash inflow and outflow. In period 𝑡, cash inflow is constituted by returns on the residual from mortgage securitization, 𝑟𝑟𝑡𝑓Σ𝑡𝑓Σ𝑡𝑀𝑡, SMPs, 𝑟𝑀𝑡𝑓(1Σ𝑡)𝑓Σ𝑡𝑀𝑡, unsecuritized mortgages, 𝑟𝑀𝑡(1𝑓Σ𝑡)𝑀𝑡, unsecuritized mortgages that are prepaid, 𝑐𝑝𝑡𝑟𝑓𝑡(1𝑓Σ𝑡)𝑀𝑡, 𝑟𝚃𝑡𝚃𝑡, (𝑟𝐵𝑡𝑐𝐵𝑡)𝐵𝑡, as well as 𝐶(𝐄[𝑆(𝒞)]), and the present value of future gains from subsequent mortgage origination and securitizations, Π𝑡Σ𝑝. On the other hand, in period 𝑡, we consider the average weighted cost of funds to securitize mortgages, 𝑐𝑀Σ𝜔𝑓Σ𝑡𝑓Σ𝑡𝑀𝑡, losses from securitized mortgages, 𝑟𝑡𝑆Σ𝑓Σ𝑡𝑓Σ𝑡𝑀𝑡, forfeit costs related to monoline insurance wrapping RMBSs, 𝑐𝑡𝑖Σ𝑓Σ𝑡𝑓Σ𝑡𝑀𝑡, transaction cost to originate mortgages, 𝑐𝑡𝑡𝑓(1Σ𝑡)𝑓Σ𝑡𝑀𝑡, and transaction costs from securitized mortgages 𝑐𝑡𝑡Σ𝑓(1Σ𝑡)𝑓Σ𝑡𝑀𝑡 as part of cash outflow. Additional components of outflow are weighted average cost of funds for originating mortgages, 𝑐𝑡𝑀𝜔(1𝑓Σ𝑡)𝑀𝑡, mortgage insurance premium for unsecuritized mortgages, 𝑝𝑖(𝒞𝑡)(1𝑓Σ𝑡)𝑀𝑡, nett losses for unsecuritized mortgages, (1𝑟𝑅𝑡)𝑟𝑆𝑡(1𝑓Σ𝑡)𝑀𝑡, decreasing value of adverse selection, 𝑎𝑓Σ𝑡𝑀𝑡, losses from suboptimal SPVs, 𝙴𝑡 and cost of funding SPVs, 𝙵𝑡. From the above and (1.1), we have that a subprime mortgage model for profit under subprime RMBSs may have the formΠΣ𝑡=𝑟𝑟𝑡𝑐𝑡𝑀Σ𝜔𝑟𝑡𝑆Σ𝑐𝑡𝑖Σ𝑓Σ𝑡𝑓Σ𝑡𝑀𝑡+𝑟𝑀𝑡𝑐𝑡𝑡𝑐𝑡𝑡Σ𝑓1Σ𝑡𝑓Σ𝑡𝑀𝑡+𝑟𝑀𝑡𝑐𝑡𝑀𝜔𝑝𝑖𝑡𝒞𝑡+𝑐𝑝𝑡𝑟𝑓𝑡1𝑟𝑅𝑡𝑟𝑆𝑡1𝑓Σ𝑡𝑀𝑡𝑎𝑓Σ𝑡𝑀𝑡+𝑟𝚃𝑡𝚃𝑡𝑟𝙱𝑡+𝑐𝙱𝑡𝙱𝑡+𝑟𝐵𝑡𝑐𝐵𝑡𝐵𝑡𝑟𝐷𝑡+𝑐𝐷𝑡𝐷𝑡𝐄𝑆𝒞+𝐶𝑡𝑃𝚃𝚃𝑡+Π𝑡Σ𝑝𝙴𝑡𝙵𝑡,(2.1) where Π𝑡Σ𝑝=Π𝑝𝑡+ΠΣ𝑡. Furthermore, by considering 𝜕𝑆(𝒞𝑡)/𝜕𝒞𝐵𝑡<0 and (2.1), ΠΣ is an increasing function of RMBS credit rating 𝒞𝐵, so that 𝜕ΠΣ𝑡/𝜕𝒞𝐵𝑡>0. Furthermore, the monoline insurance forfeit cost term, 𝑐𝑖Σ, is a function of SPV's monoline insurance premium and payment terms.

From (2.1), it is clear that bank valuation under RMBSs involves the valuing of the RMBSs themselves. In general, valuing such a vanilla corporate bond is based on default, interest rate and prepayment risks. The number of mortgagors with mortgages underlying RMBSs who prepay, increases when interest rates decrease because they can refinance at a lower fixed interest rate. Since interest rate and prepayment risks are related, it is difficult to solve mathematical models of RMBS value. This level of difficulty increases with the intricacy of the interest rate model and the sophistication of the prepayment-interest rate dependence. As a consequence, to our knowledge, no viable closed-form solutions have been found. In models of this type numerical methods provide approximate theoretical prices. These are also required in most models which specify the credit risk as a stochastic function with an interest rate correlation. Practitioners typically use Monte Carlo method or Binomial Tree numerical solutions. Of course, in (2.1) and hereafter, we assume that the RMBSs can be valued in a reasonably accurate way.

Below we roughly attempt to associate different risk types to different cash inflow and outflow terms in (2.1). We note that the cash inflow terms 𝑟𝑟𝑡𝑓Σ𝑡𝑓Σ𝑡𝑀𝑡 and 𝑟𝑀𝑡𝑓(1Σ𝑡)𝑓Σ𝑡𝑀𝑡 embed credit, market (in particular, interest rate), tranching and operational risks while 𝑟𝑀𝑡(1𝑓Σ𝑡)𝑀𝑡 carries market (specifically, interest rate) and credit risks. Also, 𝑐𝑝𝑡𝑟𝑓𝑡(1𝑓Σ𝑡)𝑀𝑡 can be associated with market (in particular, prepayment) risk while (𝑟𝐵𝑡𝑐𝐵𝑡)𝐵𝑡 mainly embeds market risk. 𝐶(𝐄[𝑆(𝒞)]) and Π𝑡Σ𝑝 involve at least credit (particularly, counterparty) and market (more specifically, interest rate, basis, prepayment, liquidity and price), respectively. In (2.1), the cash outflow terms 𝑐𝑀Σ𝜔𝑓Σ𝑡𝑓Σ𝑡𝑀𝑡, 𝑐𝑡𝑡𝑓(1Σ𝑡)𝑓Σ𝑡𝑀𝑡 and 𝑐𝑡𝑡Σ𝑓(1Σ𝑡)𝑓Σ𝑡𝑀𝑡 involve credit, tranching and operational risks while 𝑐𝑡𝑀𝜔(1𝑓Σ𝑡)𝑀𝑡 and 𝑝𝑖(𝒞𝑡)(1𝑓Σ𝑡)𝑀𝑡 carry credit and operational risks. Also, 𝑟𝑡𝑆Σ𝑓Σ𝑡𝑓Σ𝑡𝑀𝑡 embeds credit, market (including valuation), tranching and operational risks and 𝑐𝑡𝑖Σ𝑓Σ𝑡𝑓Σ𝑡𝑀𝑡 involves credit (in particular, counterparty), tranching and operational risks. Furthermore, (1𝑟𝑅𝑡)𝑟𝑆𝑡(1𝑓Σ𝑡)𝑀𝑡 and 𝑎𝑓Σ𝑡𝑀𝑡 both carry credit and market (including valuation) risks. Finally, 𝙴𝑡 and 𝙵𝑡 embed credit (in particular, counterparty and valuation) and market and operational risks, respectively. In reality, the risks that we associate with each of the cash inflow and outflow terms in (2.1) are more complicated than presented above. For instance, these risks are interrelated and may be strongly correlated with each other. All of the above risk-carrying terms contribute to systemic risk that affects the entire banking system.

In the early 80s, house financing in the US and many European countries changed from fixed-rate (FRMs) to adjustable-rate mortgages (ARMs) resulting in an interest rate risk shift to mortgagors. However, when market interest rates rose again in the late 80s, originators found that many mortgagors were unable or unwilling to fulfil their obligations at the newly adjusted rates. Essentially, this meant that the interest rate (market) risk that originators thought they had eradicated had merely been transformed into counterparty credit risk. Presently, it seems that the lesson of the 80s that ARMs cause credit risk to be higher, seems to have been forgotten or neglected since the credit risk would affect the RMBS bondholders rather than originators (see, e.g., [16]). Section 2.1 implies that the system of house financing based on RMBSs has some eminently reasonable features. Firstly, this system permits originators to divest themselves from the interest rate risk that is associated with such financing. The experience of the US Savings & Loans debacle has shown that banks cannot cope with this risk. The experience with ARMs has also shown that debtors are not able to bear this risk and that the attempt to burden them with it may merely transform the interest rate risk into counterparty credit risk. Securitization shifts this risk to a third party.

A subprime mortgage model for profit under RMBSs from (2.1) and (2.4) reflects the fact that originators sell mortgages and distribute risk to investors through mortgage securitization. This way of mitigating risks involves at least operational (including valuation and compensation), liquidity (market) and tranching (including maturity mismatch) risk that returned to originators when the SMC unfolded. Originators are more likely to securitize more mortgages if they hold less capital, are less profitable and/or liquid and have mortgages of low quality. This situation was prevalent before the SMC when originators' pursuit of yield did not take decreased capital, liquidity and mortgage quality into account. The investors in RMBSs also embed credit risk which involves bankruptcy if the aforementioned agents cannot raise funds.

2.1.2. Profit under RMBSs and Retained Earnings

As for originator's profit under mortgages, Π, we conclude that the investor's profit under RMBSs, ΠΣ, are used to meet its obligations, that include dividend payments on equity, 𝑛𝑡𝑑𝑡. The retained earnings, 𝐸𝑟𝑡, subsequent to these payments may be computed by usingΠ𝑡=𝐸𝑟𝑡+𝑛𝑡𝑑𝑡+1+𝑟𝑂𝑡𝑂𝑡.(2.2) After adding and subtracting (𝑟𝑀𝑡𝑐𝑡𝑀𝜔𝑝𝑖𝑡+𝑐𝑝𝑡𝑟𝑓𝑡(1𝑟𝑅𝑡)𝑟𝑆𝑡)𝑀𝑡 from (2.1), we obtainΠΣ𝑡=Π𝑡+𝑟𝑟𝑡𝑐𝑡𝑀Σ𝜔𝑟𝑡𝑆Σ𝑐𝑡𝑖Σ𝑟𝑀𝑡+𝑐𝑡𝑡+𝑐𝑡𝑡Σ𝑓Σ𝑡𝑓Σ𝑡𝑀𝑡+𝑐𝑡𝑀𝜔+𝑝𝑖𝑡+1𝑟𝑅𝑡𝑟𝑆𝑡𝑐𝑡𝑡𝑐𝑡𝑡Σ𝑐𝑝𝑡𝑟𝑓𝑡𝑓𝑎Σ𝑡𝑀𝑡𝙴𝑡𝙵𝑡+ΠΣ𝑡.(2.3) If we replace Π𝑡 by using (2.2), ΠΣ𝑡 is given byΠΣ𝑡=𝐸𝑟𝑡+𝑛𝑡𝑑𝑡+1+𝑟𝑂𝑡𝑂𝑡+𝑟𝑟𝑡𝑐𝑡𝑀Σ𝜔𝑟𝑡𝑆Σ𝑐𝑡𝑖Σ𝑟𝑀𝑡+𝑐𝑡𝑡+𝑐𝑡𝑡Σ𝑓Σ𝑡𝑓Σ𝑡𝑀𝑡+𝑐𝑡𝑀𝜔+𝑝𝑖𝑡+1𝑟𝑅𝑡𝑟𝑆𝑡𝑐𝑡𝑡𝑐𝑡𝑡Σ𝑐𝑝𝑡𝑟𝑓𝑡𝑓𝑎Σ𝑡𝑀𝑡𝙴𝑡𝙵𝑡+ΠΣ𝑡.(2.4) From (1.7) and (2.4), we may derive an expression for the investor's capital of the form𝐾Σ𝑡+1=𝑛𝑡𝑑𝑡+𝐸𝑡ΠΣ𝑡+Δ𝐹𝑡+1+𝑟𝑂𝑡𝑂𝑡+𝑟𝑟𝑡𝑐𝑡𝑀Σ𝜔𝑟𝑡𝑆Σ𝑐𝑡𝑖Σ𝑟𝑀𝑡+𝑐𝑡𝑡+𝑐𝑡𝑡Σ𝑓Σ𝑡𝑓Σ𝑡𝑀𝑡+𝑐𝑡𝑀𝜔+𝑝𝑖𝑡+1𝑟𝑅𝑡𝑟𝑆𝑡𝑐𝑡𝑡𝑐𝑡𝑡Σ𝑐𝑝𝑡𝑟𝑓𝑡𝑓𝑎Σ𝑡𝑀𝑡𝙴𝑡𝙵𝑡+ΠΣ𝑡,(2.5) where 𝐾𝑡 is defined by (1.2).

In Section 2.1.2, ΠΣ is given by (2.4), while 𝐾Σ has the form (2.5). It is interesting to note that the formulas for ΠΣ and 𝐾Σ depend on Π and 𝐾, respectively, and are far more intricate than the latter. Defaults on RMBSs increased significantly as the crisis expanded from the housing market to other parts of the economy, causing ΠΣ (as well as retained earnings in (2.2)) to decrease. During the SMC, capital adequacy ratios declined as 𝐾Σ levels became depleted while banks were highly leveraged. As a consequence, methods and processes which embed operational risk failed. In this period, such risk rose as banks succeeded in decreasing their capital requirements. Operational risk was not fully understood and acknowledged which resulted in loss of liquidity and failed risk mitigation management (compare with Question 3).

2.2. Valuation under RMBSs

If the expression for retained earnings given by (2.4) is substituted into (1.8), the nett cash flow under RMBSs generated by the investor is given by𝑁Σ𝑡=ΠΣ𝑡Δ𝐹𝑡=𝑛𝑡𝑑𝑡+𝐸𝑡𝐾Σ𝑡+1+1+𝑟𝑂𝑡𝑂𝑡+𝑟𝑟𝑡𝑐𝑡𝑀Σ𝜔𝑟𝑡𝑆Σ𝑐𝑡𝑖Σ𝑟𝑀𝑡+𝑐𝑡𝑡+𝑐𝑡𝑡Σ𝑓Σ𝑡𝑓Σ𝑡𝑀𝑡+𝑐𝑡𝑀𝜔+𝑝𝑖𝑡+1𝑟𝑅𝑡𝑟𝑆𝑡𝑐𝑡𝑡𝑐𝑡𝑡Σ𝑐𝑝𝑡𝑟𝑓𝑡𝑓𝑎Σ𝑡𝑀𝑡𝙴𝑡𝙵𝑡+ΠΣ𝑡.(2.6) We know that valuation is equal to the investor's nett cash flow plus exdividend value. This translates to the expression𝑉Σ𝑡=𝑁Σ𝑡+𝐾Σ𝑡+1,(2.7) where 𝐾𝑡 is defined by (1.2). Furthermore, the stock analyst evaluates the expected future cash flows in 𝑗 periods based on a stochastic discount factor, 𝛿𝑡,𝑗 such that the investor's value is𝑉Σ𝑡=𝑁Σ𝑡+𝐄𝑗=1𝛿𝑡,𝑗𝑁Σ𝑡+𝑗.(2.8) When US house prices declined in 2006 and 2007, refinancing became more difficult and ARMs began to reset at higher rates. This resulted in a dramatic increase in mortgage delinquencies, so that RMBSs began to lose value. Since these mortgage products are on the balance sheet of most banks, their valuation given by (2.8) in Section 2.2 began to decline (see, also, formulas (2.6) and (2.7)). Before the SMC, moderate reference mortgage portfolio delinquency did not affect valuation in a significant way. However, the value of mortgages and related structured products such as RMBSs decreased significantly due incidences of operational, tranching, and liquidity risks during the SMC. The yield from these structured mortgage products decreased as a consequence of high default rates (credit risk) which caused liquidity problems with a commensurate rise in the instances of credit crunch and funding risk (see Section 1.2.5 for more details about these risks).

The imposition of fair value accounting for mortgages and related SMPs such as RMBSs enhances the scope for systemic risk that involves the malfunctioning of the entire banking system. Under this type of accounting, the values at which securities are held in banks' books depend on the prices that prevail in the market (see formulas (2.6), (2.7), and (2.8) for valuations of banks holding such securities). In the event of a change in securities prices, the bank must adjust its books even if the price change is due to market malfunctioning and it has no intention of selling the security, but intends to hold it to maturity. Under currently prevailing capital adequacy requirements, this adjustment has immediate implications for the bank's financial activities. In particular, if market prices of securities held by the bank have decreased, the bank must either recapitalize by issuing new equity or retrench its overall operations. The functioning of the banking system thus depends on how well credit markets are functioning. In short, impairments of the ability of markets to value mortgages and related structured products such as RMBSs can have a large impact on bank valuation (compare with Section 2.2).

2.3. Optimal Valuation under RMBSs

In this subsection, we make use of the modeling of assets, liabilities and capital of the preceding section to solve an optimal valuation problem. The investor's total capital constraint for subprime RMBSs at face value is given by𝐾Σ𝑡=𝑛𝑡𝐸𝑡1+𝑂𝑡𝜔𝜌𝑀𝑀𝑡𝒞+𝜔𝐵𝑡𝐵𝑡+12.5𝑓𝑀,(𝑚𝑉𝑎𝑅+𝙾)(2.9) where 𝜔(𝒞𝐵𝑡) and 𝜔𝑀 are the risk weights related to subprime RMBSs and mortgages, respectively, while 𝜌—Basel II pegs 𝜌 at approximately 0.08—is the Basel capital regulation ratio of regulatory capital to risk weighted assets. In order to state the investor's optimal valuation problem, it is necessary to assume the following.

Assumption 2 (subprime investing bank's performance criterion). Suppose that the investor's valuation performance criterion, 𝐽Σ, at 𝑡 is given by 𝐽Σ𝑡=ΠΣ𝑡+𝑙𝑏𝑡𝐾Σ𝑡𝜔𝜌𝑀𝑀𝑡𝒞+𝜔𝐵𝑡𝐵𝑡+12.5𝑓𝑀(𝑚𝑉𝑎𝑅+𝙾)𝑐𝑡𝑑𝑤𝐾Σ𝑡+1𝛿+𝐄𝑡,1𝑉𝐾Σ𝑡+1,𝑥𝑡+1,(2.10) where 𝑙𝑏𝑡 is the Lagrangian multiplier for the total capital constraint, 𝐾Σ𝑡 is defined by (2.9), 𝐄[] is the expectation conditional on the investor's information in period 𝑡 and 𝑥𝑡 is the deposit withdrawals in period 𝑡 with probability distribution 𝑓(𝑥𝑡). Also, 𝑐𝑡𝑑𝑤 is the deadweight cost of total capital that consists of common and preferred equity as well as subordinate debt, 𝑉 is the value function with a discount factor denoted by 𝛿𝑡,1.

2.3.1. Statement of the Optimal Valuation Problem under RMBSs

The optimal valuation problem is to maximize investor value given by (2.8). We can now state the optimal valuation problem as follows.

Question 5 (statement of the optimal valuation problem under RMBSs). Suppose that the total capital constraint and the performance criterion, 𝐽Σ, are given by (2.9) and (2.10), respectively. The optimal valuation problem under RMBSs is to maximize the investor's value given by (2.8) by choosing the RMBS rate, deposits, and regulatory capital for 𝑉Σ𝐾Σ𝑡,𝑥𝑡=max𝑟𝐵𝑡,𝐷𝑡,ΠΣ𝑡𝐽Σ𝑡,(2.11) subject to RMBS, balance sheet, cash flow, and financing constraints given by 𝐵𝑡=𝑏0+𝑏1𝑟𝐵𝑡+𝑏2𝒞𝐵𝑡+𝜎𝐵𝑡,𝐷(2.12)𝑡=𝐵𝑡+𝑀𝑡+𝚃𝑡𝙱𝑡𝐾𝑡,1𝛾(2.13) equations (2.1) and (2.5), respectively.

2.3.2. Solution to an Optimal Valuation Problem under RMBSs

In this subsection, we find a solution to Question 5 when the capital constraint (2.9) holds as well as when it does not. In this regard, the main result can be stated and proved as follows.

Theorem 2.1 (solution to the optimal valuation problem under RMBSs). Suppose that 𝐽Σ and 𝑉Σ are given by (2.10) and (2.11), respectively. When the capital constraint given by (2.9) holds (i.e., 𝑙𝑏𝑡>0), a solution to the optimal valuation problem under RMBSs yields an optimal 𝐵 and 𝑟𝐵 of the form 𝐵𝑡=𝐾Σ𝑡𝒞𝜌𝜔𝐵𝑡𝜔𝑀𝑀𝑡+12.5𝑓𝑀(𝑚𝑉𝑎𝑅+𝙾)𝜔𝒞𝐵𝑡,𝑟(2.14)𝐵𝑡1=𝑏1𝑏0+𝑏2𝒞𝐵𝑡+𝜎𝐵𝑡𝐵𝑡,(2.15) respectively. In this case, the investor's optimal deposits and provisions for deposit withdrawals via Treasuries and optimal profits under RMBSs are given by 𝐷Σ𝑡=11𝛾𝐷+𝐷𝑟𝑝𝑡𝑟𝚃𝑡+𝑟𝙱𝑡+𝑐𝙱𝑡+𝑟𝐵𝑡𝑐𝐵𝑡1𝑟1𝛾𝐷𝑡+𝑐𝐷𝑡+𝐾Σ𝑡𝒞𝜌𝜔𝐵𝑡𝜔𝑀𝑀𝑡+12.5𝑓𝑀(𝑚𝑉𝑎𝑅+𝙾)𝜔𝒞𝐵𝑡+𝑀𝑡𝐾𝑡𝙱𝑡,𝚃(2.16)Σ𝑡=𝐷+𝐷𝑟𝑝𝑡𝑟𝚃𝑡+𝑟𝙱𝑡+𝑐𝙱𝑡+𝑟𝐵𝑡𝑐𝐵𝑡1𝑟1𝛾𝐷𝑡+𝑐𝐷𝑡Π,(2.17)Σ𝑡=𝐾Σ𝑡𝜌𝜔𝑀𝜔𝒞𝐵𝑡𝐵𝑡+12.5𝑓𝑀(𝑚𝑉𝑎𝑅+𝙾)𝜔𝑀×𝑓Σ𝑡𝑓Σ𝑡𝑟𝑟𝑡𝑐𝑡𝑀Σ𝜔𝑟𝑡𝑆Σ𝑐𝑡𝑖Σ𝑟𝑀𝑡+𝑐𝑡𝑡+𝑐𝑡𝑡Σ+𝑓Σ𝑡𝑐𝑡𝑀𝜔+𝑝𝑖𝑡+1𝑟𝑅𝑡𝑟𝑆𝑡𝑐𝑡𝑡𝑐𝑡𝑡Σ𝑐𝑝𝑡𝑟𝑓𝑡+𝑟𝑎𝑀𝑡𝑐𝑡𝑀𝜔𝑝𝑖𝑡+𝑐𝑝𝑡𝑟𝑓𝑡1𝑟𝑅𝑡𝑟𝑆𝑡+𝐾Σ𝑡𝒞𝜌𝜔𝐵𝑡𝜔𝑀𝑀𝑡+12.5𝑓𝑀(𝑚𝑉𝑎𝑅+𝙾)𝜔𝒞𝐵𝑡×1𝑏1𝐾Σ𝑡𝒞𝜌𝜔𝐵𝑡𝜔𝑀𝑀𝑡+12.5𝑓𝑀(𝑚𝑉𝑎𝑅+𝙾)𝜔𝒞𝐵𝑡𝑏0𝑏2𝒞𝐵𝑡𝜎𝐵𝑡𝑐𝐵𝑡𝑟𝐷𝑡+𝑐𝐷𝑡1+1𝛾𝐷+𝐷𝑟𝑝𝑡𝑟𝚃𝑡+1𝑏1𝐾Σ𝑡𝒞𝜌𝜔𝐵𝑡𝜔𝑀𝑀𝑡+12.5𝑓𝑀(𝑚𝑉𝑎𝑅+𝙾)𝜔𝒞𝐵𝑡𝑏0𝑏2𝒞𝐵𝑡𝜎𝐵𝑡𝑐𝐵𝑡+𝑟𝙱𝑡+𝑐𝙱𝑡1𝑟1𝛾𝐷𝑡+𝑐𝐷𝑡𝑟𝚃𝑡𝑟𝐷𝑡+𝑐𝐷𝑡1𝑟1𝛾𝐷𝑡+𝑐𝐷𝑡1𝑀1𝛾𝑡𝐾𝑡𝙱𝑡𝑟𝙱𝑡+𝑐𝙱𝑡𝙱𝑡𝐄𝑆𝒞+𝐶𝑡𝑃𝚃𝚃𝑡+Π𝑡Σ𝑝𝙴𝑡𝙵𝑡,(2.18) respectively.

Proof. A full proof of Theorem 2.1 is given in Appendix A.

The next corollary follows immediately from Theorem 2.1.

Corollary 2.2 (solution to the optimal valuation problem under RMBSs (slack)). Suppose that 𝐽Σ and 𝑉Σ are given by (2.10) and (2.11), respectively and 𝑃(𝒞𝑡)>0. When the capital constraint (2.9) does not hold (i.e., 𝑙𝑏𝑡=0), a solution to the optimal valuation problem under RMBSs posed in Question 5 yields optimal RMBS supply and its rate 𝐵Σ𝑛𝑡=23𝑏0+𝑏2𝒞𝐵𝑡+𝜎𝐵𝑡+𝑏13×𝑟𝑀𝑡𝑐𝑡𝑀𝜔𝑝𝑖𝑡𝒞𝑡+𝑐𝑝𝑡𝑟𝑓𝑡+2𝑐𝐵𝑡1𝑟𝑅𝑡𝑟𝑆𝑡+𝑟𝐷𝑡+𝑐𝐷𝑡+𝑟(1𝛾)𝑟𝑡𝑐𝑡𝑀Σ𝜔𝑟𝑡𝑆Σ𝑐𝑡𝑖Σ𝑟𝑀𝑡+𝑐𝑡𝑡+𝑐𝑡𝑡Σ𝑓Σ𝑡𝑓Σ𝑡+𝑐𝑡𝑀𝜔+𝑝𝑖𝑡𝒞𝑡𝑐𝑝𝑡𝑟𝑓t+1𝑟𝑅𝑡𝑟𝑆𝑡𝑐𝑡𝑡𝑐𝑡𝑡Σ𝑓𝑎Σ𝑡,𝑟(2.19)𝐵Σ𝑛𝑡1=3𝑏1𝑏0+𝑏2𝒞𝐵𝑡+𝜎𝐵𝑡+13𝑟𝑀𝑡𝑐𝑡𝑀𝜔𝑝𝑖𝑡𝒞𝑡+𝑐𝑝𝑡𝑟𝑓𝑡+2𝑐𝐵𝑡1𝑟𝑅𝑡𝑟𝑆𝑡+𝑟𝐷𝑡+𝑐𝐷𝑡+𝑟(1𝛾)𝑟𝑡𝑐𝑡𝑀Σ𝜔𝑟𝑡𝑆Σ𝑐𝑡𝑖Σ𝑟𝑀𝑡+𝑐𝑡𝑡+𝑐𝑡𝑡Σ𝑓Σ𝑡𝑓Σ𝑡+𝑐𝑡𝑀𝜔+𝑝𝑖𝑡𝒞𝑡𝑐𝑝𝑡𝑟𝑓𝑡+1𝑟𝑅𝑡𝑟𝑆𝑡𝑐𝑡𝑡𝑐𝑡𝑡Σ𝑓𝑎Σ𝑡,(2.20) respectively. In this case, the corresponding 𝑇𝑡, deposits and profits under RMBSs are given by 𝚃Σ𝑛𝑡=𝐷+𝐷𝑟𝑝𝑡𝑟𝚃𝑡+𝑟𝙱𝑡+𝑐𝙱𝑡+𝑟𝐵𝑡𝑐𝐵𝑡1𝑟1𝛾𝐷𝑡+𝑐𝐷𝑡,(2.21)𝐷Σ𝑛𝑡=11𝛾𝐷+𝐷𝑟𝑝𝑡𝑟𝚃𝑡+𝑟𝙱𝑡+𝑐𝙱𝑡+𝑟𝐵𝑡𝑐𝐵𝑡1𝑟1𝛾𝐷𝑡+𝑐𝐷𝑡+𝐵Σ𝑛𝑡+𝑀𝑡𝐾𝑡𝙱𝑡,(2.22)ΠΣ𝑛𝑡=𝑀𝑡𝑟𝑀𝑡𝑐𝑡𝑀𝜔𝑝𝑖𝑡𝒞𝑡+𝑐𝑝𝑡𝑟𝑓𝑡1𝑟𝑅𝑡𝑟𝑆𝑡+𝑟𝑟𝑡𝑐𝑡𝑀Σ𝜔𝑟𝑡𝑆Σ𝑐𝑡𝑖Σ𝑟𝑀𝑡+𝑐𝑡+𝑐𝑡𝑡Σ𝑓Σ𝑡𝑓Σ𝑡+𝑐𝑡𝑀𝜔+𝑝𝑖𝑡𝒞𝑡+1𝑟𝑅𝑡𝑟𝑆𝑡𝑐𝑡𝑐𝑡𝑡Σ𝑐𝑝𝑡𝑟𝑓𝑡𝑓𝑎Σ𝑡+23𝑏0+𝑏2𝒞𝐵𝑡+𝜎𝐵𝑡+𝑏13𝑟𝑀𝑡𝑐𝑡𝑀𝜔𝑝𝑖𝑡𝒞𝑡+𝑐𝑝𝑡𝑟𝑓𝑡+2𝑐𝐵𝑡1𝑟𝑅𝑡𝑟𝑆𝑡+𝑟𝐷𝑡+𝑐𝐷𝑡(+𝑟1𝛾)𝑟𝑡𝑐𝑡𝑀Σ𝜔𝑟𝑡𝑆Σ𝑐𝑡𝑖Σ𝑟𝑀𝑡+𝑐𝑡𝑡+𝑐𝑡𝑡Σ𝑓Σ𝑡𝑓Σ𝑡+𝑐𝑡𝑀𝜔+𝑝𝑖𝑡𝒞𝑡𝑐𝑝𝑡𝑟𝑓𝑡+1𝑟𝑅𝑡𝑟𝑆𝑡𝑐𝑡𝑡𝑐𝑡𝑡Σ𝑓𝑎Σ𝑡×13𝑟𝑀𝑡𝑐𝑡𝑀𝜔𝑝𝑖𝑡𝒞𝑡+𝑐𝑝𝑡𝑟𝑓𝑡+2𝑐𝐵𝑡1𝑟𝑅𝑡𝑟𝑆𝑡+𝑟𝐷𝑡+𝑐𝐷𝑡+𝑟(1𝛾)𝑟𝑡𝑐𝑡𝑀Σ𝜔𝑟𝑡𝑆Σ𝑐𝑡𝑖Σ𝑟𝑀𝑡+𝑐𝑡𝑡+𝑐𝑡𝑡Σ𝑓Σ𝑡𝑓Σ𝑡+𝑐𝑡𝑀𝜔+𝑝𝑖𝑡𝒞𝑡𝑐𝑝𝑡𝑟𝑓𝑡+1𝑟𝑅𝑡𝑟𝑆𝑡𝑐𝑡𝑡𝑐𝑡𝑡Σ𝑓𝑎Σ𝑡13𝑏1𝑏0+𝑏2𝒞𝐵𝑡+𝜎𝐵𝑡𝑐𝐵𝑡𝑟𝐷𝑡+𝑐𝐷𝑡1+1𝛾𝐷+𝐷𝑟𝑝𝑡𝑟𝚃𝑡13𝑏1𝑏0+𝑏2𝒞𝐵𝑡+𝜎𝐵𝑡13𝑟𝑀𝑡𝑐𝑡𝑀𝜔𝑝𝑖𝑡𝒞𝑡+𝑐𝑝𝑡𝑟𝑓𝑡+2𝑐𝐵𝑡1𝑟𝑅𝑡𝑟𝑆𝑡+𝑟𝐷𝑡+𝑐𝐷𝑡+𝑟(1𝛾)𝑟𝑡𝑐𝑡𝑀Σ𝜔𝑟𝑡𝑆Σ𝑐𝑡𝑖Σ𝑟𝑀𝑡+𝑐𝑡𝑡+𝑐𝑡𝑡Σ𝑓Σ𝑡𝑓Σ𝑡+𝑐𝑡𝑀𝜔+𝑝𝑖𝑡𝒞𝑡𝑐𝑝𝑡𝑟𝑓𝑡+1𝑟𝑅𝑡𝑟𝑆𝑡𝑐𝑡𝑡𝑐𝑡𝑡Σ𝑓𝑎Σ𝑡+𝑐𝐵𝑡+𝑟𝙱𝑡+𝑐𝙱𝑡1𝑟1𝛾𝐷𝑡+𝑐𝐷𝑡𝑟𝚃𝑡𝑟𝐷𝑡+𝑐𝐷𝑡1𝑟1𝛾𝐷𝑡+𝑐𝐷𝑡1𝑀1𝛾𝑡𝐾𝑡𝙱𝑡𝑟𝙱𝑡+𝑐𝙱𝑡𝙱𝑡𝐄𝑆𝒞+𝐶𝑡𝑃𝚃𝚃𝑡+Π𝑡Σ𝑝𝙴𝑡𝙵𝑡,(2.23) respectively.

Proof. A full proof of Corollary 2.2 is given in Appendix B.

In Section 2.3, the investor's valuation performance criterion, 𝐽Σ, at 𝑡 is given by (2.10). During the SMC, when valuation was a major issue, 𝐽Σ was difficult to compute since the valuation of components such as 𝐵 was not easy to determine. In addition, before the SMC, CRAs used idiosyncratic valuation techniques to give investment-grade ratings to RMBSs despite the fact that the mortgage face value, mortgage rate, the investor's optimal deposits and provisions for deposit withdrawals via Treasuries of the forms (2.14), (2.15), (2.16), and (2.17), respectively, computed in Theorem 2.1—subject to RMBS, balance sheet, cash flow and financing constraints given by (2.12), (2.13), (2.1), and (2.5), respectively—were clearly suboptimal.

When the capital constraint given by (2.9) holds (i.e., 𝑙𝑏𝑡>0), a solution to the optimal valuation problem under RMBSs yields optimal profit under RMBSs of the form (2.18). With hindsight, it is clear that the aforementioned subprime parameters did not compare favorably with their optimal counterparts. Also, during the SMC, the financing constraint was violated with not enough capital being held. When the capital constraint (2.9) does not hold (i.e., 𝑙𝑏𝑡=0) and 𝑃(𝒞𝑡)>0, then optimal RMBS supply and its rate, (2.19) and (2.20), respectively, are solutions to the optimal valuation problem stated in Corollary 2.2.

3. Profit, Risk, and Valuation under RMBS CDOs

In this section, we discuss the relationships between the SMC and profit, risk as well as valuation under RMBS CDOs. In the sequel, we assume that the notation Π, 𝑟𝑀, 𝑀, 𝑐𝑀𝜔, 𝑝𝑖, 𝑐𝑝, 𝑟𝑓, 𝑟𝑅, 𝑟𝑆, 𝑆, 𝒞, 𝐶(𝐄[𝑆(𝒞)]), 𝑟𝐵, 𝑐𝐵, 𝐵, 𝑟𝚃, 𝚃, 𝑃𝚃(𝚃), 𝑟𝐷, 𝑐𝐷, 𝐷, 𝑟𝙱, 𝑐𝙱, 𝙱, Π𝑝, 𝐾, 𝑛, 𝐸, 𝑂, 𝜔(𝒞), 𝜔𝐵, 𝑓𝑀, 𝑚𝑉𝑎𝑅, 𝙾, 𝑟𝑆Σ, 𝑓Σ, and 𝑓Σ, corresponds to that of Sections 1 and 2. Further suppositions about notation are that 𝑟𝑟, 𝑐𝑀Σ𝜔, 𝑐𝑖Σ, 𝑐𝑡, 𝑐𝑡Σ, 𝑎, ΠΣ𝑝, 𝙴, and 𝙵 denote the same parameters as in Section 2.

3.1. More Background to RMBS CDOs

In the sequel, we concentrate on RMBS CDOs where the reference assets of the CDO portfolios are mainly RMBSs. The chain formed by subprime mortgages, RMBSs, and RMBS CDOs is given in Figure 3 below.

Note that as we proceed from left to right in Figure 3, subprime mortgages are securitized into RMBSs that, in turn, get securitized into RMBS CDOs. As far as the latter is concerned, it is clearly shown that RMBS bonds rated AAA, AA, and A constitute high grade CDO portfolios. On the other hand, the BBB-rated RMBS bonds are securitized into a mezz CDO, since its portfolio mainly consists of BBB-rated RMBSs and their tranches. At the end, if bonds issued by mezz CDOs are put into CDO portfolios, then a type of CDO is known as CDO squared or CDO2.

Assumption 3 (senior tranches of RMBSs). We assume that risky marketable securities, 𝐵, appearing in the balance sheet (1.2), consist entirely of the senior tranches of RMBSs that are wrapped by a monoline insurer. Also, the investor has an incentive to retain an interest in these tranches.

This assumption implies that CDO structure depends on the securitization of senior tranches of RMBSs in particular. We note that important features of Section 3 are illustrated in Sections 5.1, 5.2, and 5.3.

3.2. Profit and Risk under RMBS CDOs

In this subsection, we investigate a subprime mortgage model for profit under RMBS CDOs and its relationship with retained earnings.

3.2.1. A Subprime Mortgage Model for Profit and Risk under RMBS CDOs

In this paper, a subprime mortgage model for profit under RMBS CDOs can be constructed by considering the difference between cash inflow and outflow. For this profit, in period 𝑡, cash inflow is constituted by returns on the residual from RMBSs securitization, 𝑟𝑡𝑟𝑏𝑓𝑡Σ𝑏𝑓𝑡Σ𝑏𝐵𝑡, securitized subprime RMBSs, 𝑟𝐵𝑡𝑓(1𝑡Σ𝑏)𝑓𝑡Σ𝑏𝐵𝑡, unsecuritized securities, 𝑟𝐵𝑡(1𝑓𝑡Σ𝑏)𝐵𝑡, Treasuries, 𝑟𝚃𝑡𝚃𝑡, and mortgages, 𝑟𝑀𝑡𝑀𝑡, as well as the recovery amount, 𝑅𝑡, monoline insurance protection leg payments, 𝐶(𝑆(𝒞𝑡)), and the present value of future gains from subsequent RMBS purchases and their securitizations, Π𝑡Σ𝑝. On the other hand, we consider the average weighted cost of funds to securitize RMBSs, 𝑐𝑀Σ𝜔𝑏𝑓𝑡Σ𝑏𝑓𝑡Σ𝑏𝐵𝑡, losses from securitized RMBSs, 𝑟𝑡𝑆Σ𝑏𝑓𝑡Σ𝑏𝑓𝑡Σ𝑏𝐵𝑡, forfeit costs related to monoline insurance wrapping RMBS CDOs, 𝑐𝑡𝑖Σ𝑏𝑓𝑡Σ𝑏𝑓𝑡Σ𝑏𝐵𝑡, transaction cost to sell RMBSs, 𝑐𝑡𝑡𝑏𝑓(1𝑡Σ𝑏)𝑓𝑡Σ𝑏𝐵𝑡, and transaction costs from securitized RMBSs 𝑐𝑡𝑡Σ𝑏𝑓(1𝑡Σ𝑏)𝑓𝑡Σ𝑏𝐵𝑡 as part of cash outflow. Additional components of outflow are weighted average cost of funds for selling RMBSs, 𝑐𝑡𝑀𝜔𝑏(1𝑓𝑡Σ𝑏)𝐵𝑡, fraction of the face value of unsecuritized RMBSs corresponding to 𝑓𝑡Σ𝑏(1𝑓𝑡Σ𝑏)𝐵𝑡, monoline insurance premium for unsecuritized RMBSs losses, 𝑝𝑡𝑖𝑏(1𝑓𝑡Σ𝑏)𝐵𝑡, decreasing value of adverse selection, 𝑎𝑏𝑓𝑡Σ𝑏𝐵𝑡, the all-in cost of holding RMBSs, 𝑐𝑀𝑡𝑀𝑡, interest paid to depositors, 𝑟𝐷𝑡𝐷𝑡, the cost of taking deposits, 𝑐𝐷𝐷𝑡, provisions against deposit withdrawals, 𝑃𝚃(𝚃𝑡), while 𝑟𝙱 and 𝑐𝙱 are the borrower rate and marginal cost of borrowing, respectively, losses from suboptimal SPVs, 𝙴𝑡 and costs for funding RMBS securitization, 𝙵𝑡. From the above, we have that model for profit under subprime RMBS CDOs may have the formΠ𝑡Σ𝑏=𝑟𝑟𝑡𝑐𝑡𝑀Σ𝜔𝑟𝑡𝑆Σ𝑐𝑡𝑖Σ𝑓Σ𝑡𝑓Σ𝑡𝑀𝑡+𝑟𝑀𝑡𝑐𝑡𝑡𝑐𝑡𝑡Σ𝑓1Σ𝑡𝑓Σ𝑡𝑀𝑡+𝑟𝑀𝑡𝑐𝑡𝑀𝜔𝑝𝑖𝑡+𝑐𝑝𝑡𝑟𝑓𝑡1𝑟𝑅𝑡𝑟𝑆𝑡1𝑓Σ𝑡𝑀𝑡𝑎𝑓Σ𝑡𝑀𝑡+𝑟𝑡𝑟𝑏𝑐𝑡𝑀Σ𝜔𝑏𝑟𝑡𝑆Σ𝑏𝑐𝑡𝑖Σ𝑏𝑓𝑡Σ𝑏𝑓𝑡Σ𝑏𝐵𝑡+𝑟𝐵𝑡𝑐𝑡𝑡𝑏𝑐𝑡𝑡Σ𝑏𝑓1𝑡Σ𝑏𝑓𝑡Σ𝑏𝐵𝑡+𝑟𝐵𝑡𝑐𝑡𝑀𝜔𝑏𝑝𝑡𝑖𝑏+𝑐𝑡𝑏𝑝𝑟𝑡𝑓𝑏1𝑟𝑡𝑅𝑏𝑟𝑡𝑆𝑏1𝑓𝑡Σ𝑏𝐵𝑡𝑎𝑏𝑓𝑡Σ𝑏𝐵𝑡+𝑟𝚃𝑡𝚃𝑡𝑟𝙱𝑡+𝑐𝙱𝑡𝙱𝑡𝑟𝐷𝑡+𝑐𝐷𝑡𝐷𝑡𝐄𝑆𝒞+𝐶𝑡𝑃𝚃𝚃𝑡+Π𝑡Σ𝑝𝙴𝑡𝙵𝑡,(3.1) where Π𝑡Σ𝑝=Π𝑝𝑡+ΠΣ𝑡. Furthermore, by considering 𝜕𝑆(𝒞𝑡)/𝜕𝒞𝐵𝑡<0 and (3.1), we know that ΠΣ𝑏 is an increasing function of RMBS credit rating, 𝒞𝐵.

From the above, we note that in (3.1) the cash inflow terms 𝑟𝑡𝑟𝑏𝑓𝑡Σ𝑏𝑓𝑡Σ𝑏𝐵𝑡 and 𝑟𝐵𝑡𝑓(1𝑡Σ𝑏)𝑓𝑡Σ𝑏𝐵𝑡 carry credit, market (in particular, interest rate), tranching and operational risks, while 𝑟𝐵𝑡(1𝑓𝑡Σ𝑏)𝐵𝑡 embed credit (in particular, counterparty) and market (in particular, interest rate) risks. In (3.1), the cash outflow terms 𝑐𝑀Σ𝜔𝑏𝑓𝑡Σ𝑏𝑓𝑡Σ𝑏𝐵𝑡,𝑟𝑡𝑆Σ𝑏𝑓𝑡Σ𝑏𝑓𝑡Σ𝑏𝐵𝑡,𝑐𝑡𝑖Σ𝑏𝑓𝑡Σ𝑏𝑓𝑡Σ𝑏𝐵𝑡, 𝑐𝑡𝑡𝑏𝑓(1𝑡Σ𝑏)𝑓𝑡Σ𝑏𝐵𝑡, and 𝑐𝑡𝑡Σ𝑏𝑓(1𝑡Σ𝑏)𝑓𝑡Σ𝑏𝐵𝑡 involve credit (for instance, counterparty), market (specifically, liquidity and valuation), tranching and operational risks. Also, 𝑐𝑡𝑀𝜔𝑏(1𝑓𝑡Σ𝑏)𝐵𝑡 and 𝑝𝑡𝑖𝑏(1𝑓𝑡Σ𝑏)𝐵𝑡 carry credit, market (particularly, liquidity) and operational risks while 𝑎𝑏𝑓𝑡Σ𝑏𝐵𝑡 embeds credit and market (in the form of liquidity and valuation) risks. As before, the risks that we associate with each of the cash inflow and outflow terms in (3.1) are less straightforward. For instance, strong correlations may exist between each of the aforementioned risks. Also, the risk-carrying terms found in (3.1) affect the entire banking system via systemic risk.

3.2.2. Profit under RMBS CDOs and Retained Earnings

We know that profits, Π𝑡, are used to meet its obligations, that include dividend payments on equity, 𝑛𝑡𝑑𝑡. The retained earnings, 𝐸𝑟𝑡, subsequent to these payments may be computed by using (2.2). After adding and subtracting (𝑟𝐵𝑡𝑐𝑡𝑀𝜔𝑏𝑝𝑡𝑖𝑏+𝑐𝑡𝑝𝑏𝑟𝑡𝑓𝑏(1𝑟𝑡𝑅𝑏)𝑟𝑡𝑆𝑏)𝐵𝑡 from (3.1), we getΠ𝑡Σ𝑏=Π𝑡+𝑟𝑟𝑡𝑐𝑡𝑀Σ𝜔𝑟𝑡𝑆Σ𝑐𝑡𝑖Σ𝑟𝑀𝑡+𝑐𝑡𝑡+𝑐𝑡𝑡Σ𝑓Σ𝑡𝑓Σ𝑡𝑀𝑡+𝑐𝑡𝑀𝜔+𝑝𝑖𝑡+1𝑟𝑅𝑡𝑟𝑆𝑡𝑐𝑡𝑡𝑐𝑡𝑡Σ𝑐𝑝𝑡𝑟𝑓𝑡𝑓𝑎Σ𝑡𝑀𝑡+𝑟𝑡𝑟𝑏𝑐𝑡𝑀Σ𝜔𝑏𝑟𝑡𝑆Σ𝑏𝑐𝑡𝑖Σ𝑏𝑟𝐵𝑡+𝑐𝑡𝑡𝑏+𝑐𝑡𝑡Σ𝑏𝑓𝑡Σ𝑏𝑓𝑡Σ𝑏𝐵𝑡+𝑐𝑡𝑀𝜔𝑏+𝑝𝑡𝑖𝑏+1𝑟𝑡𝑅𝑏𝑟𝑡𝑆𝑏𝑐𝑡𝑡𝑏𝑐𝑡𝑡Σ𝑏𝑐𝑡𝑝𝑏𝑟𝑡𝑓𝑏𝑎𝑏𝑓𝑡Σ𝑏𝐵𝑡+𝑟𝐵𝑡𝑐𝑡𝑀𝜔𝑏𝑝𝑡𝑖𝑏+𝑐𝑡𝑝𝑏𝑟𝑡𝑓𝑏1𝑟𝑡𝑅𝑏𝑟𝑡𝑆𝑏𝐵𝑡𝙴𝑡𝙵𝑡+ΠΣ𝑡.(3.2) Replace Π𝑡, by using (2.2). In this case, Π𝑡Σ𝑏 is given byΠ𝑡Σ𝑏=𝐸𝑟𝑡+𝑛𝑡𝑑𝑡+1+𝑟𝑂𝑡𝑂𝑡+𝑟𝑟𝑡𝑐𝑡𝑀Σ𝜔𝑟𝑡𝑆Σ𝑐𝑡𝑖Σ𝑟𝑀𝑡+𝑐𝑡𝑡+𝑐𝑡𝑡Σ𝑓Σ𝑡𝑓Σ𝑡𝑀𝑡+𝑐𝑡𝑀𝜔+𝑝𝑖𝑡+1𝑟𝑅𝑡𝑟𝑆𝑡𝑐𝑡𝑡𝑐𝑡𝑡Σ𝑐𝑝𝑡𝑟𝑓𝑡𝑓𝑎Σ𝑡𝑀𝑡+𝑟𝑡𝑟𝑏𝑐𝑡𝑀Σ𝜔𝑏𝑟𝑡𝑆Σ𝑏𝑐𝑡𝑖Σ𝑏𝑟𝐵𝑡+𝑐𝑡𝑡𝑏+𝑐𝑡𝑡Σ𝑏𝑓𝑡Σ𝑏𝑓𝑡Σ𝑏𝐵𝑡+𝑐𝑡𝑀𝜔𝑏+𝑝𝑡𝑖𝑏+1𝑟𝑡𝑅𝑏𝑟𝑡𝑆𝑏𝑐𝑡𝑡𝑏𝑐𝑡𝑡Σ𝑏𝑐𝑡𝑝𝑏𝑟𝑡𝑓𝑏𝑎𝑏𝑓𝑡Σ𝑏𝐵𝑡+𝑟𝐵𝑡𝑐𝑡𝑀𝜔𝑏𝑝𝑡𝑖𝑏+𝑐𝑡𝑝𝑏𝑟𝑡𝑓𝑏1𝑟𝑡𝑅𝑏𝑟𝑡𝑆𝑏𝐵𝑡𝙴𝑡𝙵𝑡+ΠΣ𝑡.(3.3) For (3.3) and (1.7), we obtain an expression for capital of the form𝐾Σ𝑏𝑡+1=𝑛𝑡𝑑𝑡+𝐸𝑡Π𝑡Σ𝑏+Δ𝐹𝑡+1+𝑟𝑂𝑡𝑂𝑡+𝑟𝑟𝑡𝑐𝑡𝑀Σ𝜔𝑟𝑡𝑆Σ𝑐𝑡𝑖Σ𝑟𝑀𝑡+𝑐𝑡𝑡+𝑐𝑡𝑡Σ𝑓Σ𝑡𝑓Σ𝑡𝑀𝑡+𝑐𝑡𝑀𝜔+𝑝𝑖𝑡+1𝑟𝑅𝑡𝑟𝑆𝑡𝑐𝑡𝑡𝑐𝑡𝑡Σ𝑐𝑝𝑡𝑟𝑓𝑡𝑓𝑎Σ𝑡𝑀𝑡+𝑟𝑡𝑟𝑏𝑐𝑡𝑀Σ𝜔𝑏𝑟𝑡𝑆Σ𝑏𝑐𝑡𝑖Σ𝑏𝑟𝐵𝑡+𝑐𝑡𝑡𝑏+𝑐𝑡𝑡Σ𝑏𝑓𝑡Σ𝑏𝑓𝑡Σ𝑏𝐵𝑡+𝑐𝑡𝑀𝜔𝑏+𝑝𝑡𝑖𝑏+1𝑟𝑡𝑅𝑏𝑟𝑡𝑆𝑏𝑐𝑡𝑡𝑏𝑐𝑡𝑡Σ𝑏𝑐𝑡𝑝𝑏𝑟𝑡𝑓𝑏𝑎𝑏𝑓𝑡Σ𝑏𝐵𝑡+𝑟𝐵𝑡𝑐𝑡𝑀𝜔𝑏𝑝𝑡𝑖𝑏+𝑐𝑡𝑝𝑏𝑟𝑡𝑓𝑏1𝑟𝑡𝑅𝑏𝑟𝑡𝑆𝑏𝐵𝑡𝙴𝑡𝙵𝑡+ΠΣ𝑡,(3.4) where 𝐾𝑡 is defined by (1.2).

A subprime mortgage model for profit under subprime RMBS CDOs has the form (3.1) given in Section 3.2. Under RMBS CDOs, Π𝑡Σ𝑏 is given by (3.3), while capital is of the form (3.4). In this regard, before the SMC, investors sought higher profits than those offered by US Treasury bonds. Continued strong demand for RMBSs and RMBS CDOs began to drive down lending standards related to originating mortgages destined for reference portfolios in securitization. RMBS CDOs lost most of their value which resulted in a large decline in the capital of many banks and government-sponsored enterprises (GSEs), with a resultant tightening of credit globally.

As we have seen before, subprime risks and profit play a key role in Section 3.2. In this regard, before the SMC, CDOs purchased subprime RMBS bonds because it was profitable. At first, lower-rated BBB tranches of subprime RMBS were difficult to sell since they were thin and, hence, unattractive. Later the thickness of these tranches increased and investment in them became more alluring for investors (compare with the examples contained in Sections 5.2 and 5.3). Despite this, a purchasing CDO may not be aware of the subprime risks inherent in the RMBS deal, including credit and synthetic risk. In this way, risks were underestimated and mortgages and structured mortgage products were overrated. In particular, tranching added intricacy to securitization. This assertion has resonance with the main hypothesis of this contribution involving the intricacy and design of subprime structured mortgage products and their role in information and opaqueness problems as well as risk mismanagement (compare with Question 3).

3.3. Valuation under RMBS CDOs

If the expression for retained earnings given by (3.3) is substituted into (1.8), the nett cash flow generated for a shareholder is given by𝑁𝑡Σ𝑏=Π𝑡Σ𝑏Δ𝐹𝑡=𝑛𝑡𝑑𝑡+𝐸𝑡𝐾Σ𝑏𝑡+1+1+𝑟𝑂𝑡𝑂𝑡+𝑟𝑟𝑡𝑐𝑡𝑀Σ𝜔𝑟𝑡𝑆Σ𝑐𝑡𝑖Σ𝑟𝑀𝑡+𝑐𝑡𝑡+𝑐𝑡𝑡Σ𝑓Σ𝑡𝑓Σ𝑡𝑀𝑡+𝑐𝑡𝑀𝜔+𝑝𝑖𝑡+1𝑟𝑅𝑡𝑟𝑆𝑡𝑐𝑡𝑡𝑐𝑡𝑡Σ𝑐𝑝𝑡𝑟𝑓𝑡𝑓𝑎Σ𝑡𝑀𝑡+𝑟𝑡𝑟𝑏𝑐𝑡𝑀Σ𝜔𝑏𝑟𝑡𝑆Σ𝑏𝑐𝑡𝑖Σ𝑏𝑟𝐵𝑡+𝑐𝑡𝑡𝑏+𝑐𝑡𝑡Σ𝑏𝑓𝑡Σ𝑏𝑓𝑡Σ𝑏𝐵𝑡+𝑐𝑡𝑀𝜔𝑏+𝑝𝑡𝑖𝑏+1𝑟𝑡𝑅𝑏𝑟𝑡𝑆𝑏𝑐𝑡𝑡𝑏𝑐𝑡𝑡Σ𝑏𝑐𝑡𝑝𝑏𝑟𝑡𝑓𝑏𝑎𝑏𝑓𝑡Σ𝑏𝐵𝑡+𝑟𝐵𝑡𝑐𝑡𝑀𝜔𝑏𝑝𝑡𝑖𝑏+𝑐𝑡𝑝𝑏𝑟𝑡𝑓𝑏1𝑟𝑡𝑅𝑏𝑟𝑡𝑆𝑏𝐵𝑡𝙴𝑡𝙵𝑡+ΠΣ𝑡.(3.5) We know that valuation is equal to the nett cash flow plus exdividend value. This translates to the expression𝑉𝑡Σ𝑏=𝑁𝑡Σ𝑏+𝐾Σ𝑏𝑡+1,(3.6) where 𝐾𝑡 is defined by (1.2). Furthermore, under RMBS CDOs, the analyst evaluates the expected future cash flows in 𝑗 periods based on a stochastic discount factor, 𝛿𝑡,𝑗, such that the investor's value is𝑉𝑡Σ𝑏=𝑁𝑡Σ𝑏+𝐄𝑗=1𝛿𝑡,𝑗𝑁Σ𝑏𝑡+𝑗.(3.7) In the above, we note that investor value under RMBS CDOs is given by (3.7). In this regard, to our knowledge, there is no standardization of triggers across CDOs with some having sequential cash flow triggers while others have OC trigger calculations based on ratings changes. As far as performing valuations is concerned, in reality, each RMBS CDO must be separately valued which may not be possible (compare with formula (3.5) for nett cash flow under RMBS CDOs). During the SMC, this played a role in the problems investors faced when they attempted a valuation of CDO tranches. Furthermore, RMBS CDOs, widely held by dealer banks and investors, lost most of their value during this period. Naturally, this led to a dramatic decrease in the investor's valuation from holding such structured mortgage products which, in turn, increased the subprime risks in mortgage markets (refer to formulas (3.6) and (3.7) for the investor's valuation under RMBS CDOs).

3.4. Optimal Valuation under RMBS CDOs

In this subsection, we make use of the modeling of assets, liabilities, and capital of the preceding section to solve an optimal valuation problem.

3.4.1. Statement of Optimal Valuation Problem under RMBS CDOs

Suppose that the investor's valuation performance criterion, 𝐽Σ𝑏, at 𝑡 is given by𝐽𝑡Σ𝑏=Π𝑡Σ𝑏+𝑙𝑏𝑡𝐾𝑡Σ𝑏𝜔𝜌𝑀𝑀𝑡𝒞+𝜔𝐵𝑡𝐵𝑡+12.5𝑓𝑀(𝑚𝑉𝑎𝑅+𝙾)𝑐𝑡𝑑𝑤𝐾Σ𝑏𝑡+1𝛿+𝐄𝑡,1𝑉𝐾Σ𝑏𝑡+1,𝑥𝑡+1,(3.8) where 𝑙𝑏𝑡 is the Lagrangian multiplier for the total capital constraint, 𝐾Σ𝑡 is defined by (2.9), 𝐄[] is the expectation conditional on the investor's information in period 𝑡 and 𝑥𝑡 is the deposit withdrawals in period 𝑡 with probability distribution 𝑓(𝑥𝑡). Also, 𝑐𝑡𝑑𝑤 is the deadweight cost of total capital that consists of equity.

The optimal valuation problem is to maximize the value given by (3.7). We can now state the optimal valuation problem as follows.

Question 6 (statement of optimal valuation problem under RMBS CDOs). Suppose that the total capital constraint, 𝐾Σ𝑏, and the performance criterion, 𝐽Σ𝑏, are given by (2.9) and (3.8), respectively. Investor's optimal valuation problem is to maximize its value given by (3.7) by choosing the RMBS rate, deposits, and regulatory capital for 𝑉Σ𝑏𝐾𝑡Σ𝑏,𝑥𝑡=max𝑟𝐵𝑡,𝐷𝑡,Π𝑡Σ𝑏𝐽𝑡Σ𝑏,(3.9) subject to RMBS, balance sheet, cash flow, and financing constraints given by (2.12), (2.13), (3.1), and (3.4), respectively.

3.4.2. Solution to an Optimal Valuation Problem under RMBS CDOs

In this subsection, we find a solution to Question 6 when the capital constraint (2.9) holds as well as when it does not. In this regard, the main result can be stated and proved as follows.

Theorem 3.1 (solution to an optimal valuation problem under RMBS CDOs). Suppose that 𝐽Σ𝑏 and 𝑉Σ𝑏 are given by (3.8) and (3.9), respectively. When the capital constraint given by (2.9) holds (i.e., 𝑙𝑏𝑡>0), a solution to the optimal valuation problem yields an optimal RMBS supply and rate given by (2.14) and (2.15), respectively. In this case, optimal deposits, provisions for deposit withdrawals via Treasuries, and profits under RMBS CDO securitization are given by 𝐷Σ𝑏𝑡=11𝛾𝐷+𝐷𝑟𝑝𝑡𝑟𝚃𝑡+𝑟𝙱𝑡+𝑐𝙱𝑡1𝑟1𝛾𝐷𝑡+𝑐𝐷𝑡+𝐾𝑡Σ𝑏𝒞𝜌𝜔𝐵𝑡𝜔𝑀𝑀𝑡+12.5𝑓𝐵(𝑚𝑉𝑎𝑅+𝙾)𝜔𝒞𝐵𝑡+𝑀𝑡𝐾𝑡𝙱𝑡,(3.10)𝚃Σ𝑏𝑡=𝐷+𝐷𝑟𝑝𝑡𝑟𝚃𝑡+𝑟𝙱𝑡+𝑐𝙱𝑡1𝑟1𝛾𝐷𝑡+𝑐𝐷𝑡,(3.11)ΠΣ𝑏𝑡=𝐾𝑡Σ𝑏𝜌𝜔𝑀𝜔𝒞𝐵𝑡𝐵𝑡+12.5𝑓𝑀(𝑚𝑉𝑎𝑅+𝙾)𝜔𝑀×𝑓Σ𝑡𝑓Σ𝑡𝑟𝑟𝑡𝑐𝑡𝑀Σ𝜔𝑟𝑡𝑆Σ𝑐𝑡𝑖Σ𝑟𝑀𝑡+𝑐𝑡𝑡+𝑐𝑡𝑡Σ+𝑓Σ𝑡𝑐𝑡𝑀𝜔+𝑝𝑖𝑡+1𝑟𝑅𝑡𝑟𝑆𝑡𝑐𝑡𝑡𝑐𝑡𝑡Σ𝑐𝑝𝑡𝑟𝑓𝑡+𝑟𝑎𝑀𝑡𝑐𝑡𝑀𝜔𝑝𝑖𝑡+𝑐𝑝𝑡𝑟𝑓𝑡1𝑟𝑅𝑡𝑟𝑆𝑡+𝐾𝑡Σ𝑏𝒞𝜌𝜔𝐵𝑡𝜔𝑀𝑀𝑡+12.5𝑓𝑀(𝑚𝑉𝑎𝑅+𝙾)𝜔𝒞𝐵𝑡×𝑓𝑡Σ𝑏𝑓𝑡Σ