1. Introduction

This contribution is the second in a series of papers (of which [1] was the first) on modeling issues related to bank capital regulation and the subprime mortgage crisis (SMC) in a discrete-time framework. Some of the world's top banking experts spent nearly a decade designing regulation in the form of the Basel II Capital Accord in order to ensure the health of the global banking industry. However, the following question still remains: What if some of their suppositions were inaccurate?

The possibility that this may be answered in the affirmative was debated during the SMC. This crisis that started unraveling from 2007 onwards and was initiated by subprime mortgage loan losses in the U.S. brought into question the effectiveness of global macroeconomic policy, financial stability, and financial regulation such as the new Basel II capital adequacy framework for banks (see, e.g., [2]).

Basel II aims to address weaknesses in the Basel I capital adequacy framework for banks by incorporating more detailed calibration of credit risk and requiring the pricing of other forms of risk such as operational risk. However, the 2007–2008 implementation of Basel II corresponded to major losses suffered by some of the world's major banks due to financial crises. Furthermore, the risk models that underpin Basel II are similar to the ones many of those banks use. Under the Basel II framework, regulators allow large banks with sophisticated risk management systems to use risk assessment based on their own models in determining the minimum amount of capital they are required to hold by the regulators as a buffer against unexpected losses. Of concern is that by the end of 2008, in the U.S., non-risk-weighted capital adequacy ratios (CARs) were near historically low levels of about . Naturally, these facts challenge the usefulness of important elements in the Basel II accord.

The SMC fallout did not originate from lightly-regulated hedge funds, but from banks regulated by governments. For instance, in the fourth quarter of 2007, Citigroup Inc. had its worst-ever quarterly loss of  9.83 billion and had to raise more than  20 billion in capital from outside investors, including foreign-government investment funds. This was done in order to augment the depleted capital on its balance sheet after bad investments in mortgage-backed securities (MBSs) and collateralized debt obligations (CDOs). According to the Federal Deposit Insurance Corporation (FDIC), at the time, Citigroup held  80 billion in core capital on its balance sheet to protect against its  1.1 trillion in assets. In the second half of 2007, Citigroup wrote down about  20 billion. Interestingly, at the end of 2007, major U.S. banks like J.P. Morgan Chase and Co., Wachovia Corp., Washington Mutual Inc., and Citigroup lobbied for leaner, European-style capital cushions. These banks urged the U.S. government

“to help ensure U.S. banking institutions remain strong and competitive, the federal banking agencies should avoid imposing domestic capital regulation that provides an advantage to non-U.S. banks.”

They argued that tighter rules would make it tougher for them to compete globally, since more of their money would be tied up in the capital cushion. Eventually, in July 2008, the U.S. Federal Reserve and regulators acceded to the banks' requests by allowing them to follow rules similar to those in Europe. That ruling potentially could enable American banks to hold looser, European-style capital. Ironically, by then, cracks in the global financial system were already spreading rapidly.

At the beginning of 2008, it appeared that the SMC had caused a higher degree of problems for non-U.S. financial institutions. The write-downs that British, European, and Asian institutions had to make on U.S. subprime mortgage debt were something that some analysts attributed to Basel II that gave institutions some carte blanche when it came to raising capital for securities with top credit ratings. For instance, on Friday, 14 September 2007, Northern Rock, the U.K.'s fifth-biggest mortgage lender, started experiencing a bank run after it was revealed that the bank was having trouble raising liquidity. Within one day, customers had withdrawn an estimated  1 billion resulting in the first bank run in the U.K. since 1866. Earlier, applying Basel II principles, Northern Rock announced it would boost its shareholder dividend by —a step that depleted its capital even as regulators warned about the lender's condition. Adam Applegarth, Northern Rocks CEO at the time of the crisis, defended the lender's dividend boosting before a parliamentary inquiry on the bank run. He argued that because of the high credit quality of its mortgage loans in Basel II terms, the lender could opt to hold less capital to cover potential losses. To finance its expansion, Northern Rock began to borrow heavily in global financial markets, rather than relying as much on traditional customer bank deposits. In fact, its deposits-to-total liabilities and equity ratio had decreased from at the end of 1997 to at the end of 2006, less than half the level of most fellow mortgage lenders. This meant that Northern Rock had access to insufficient cash when its own liquidity dried up after investors decided to stop financing its growth. Eventually, Northern Rock was nationalized by the British government. Even in Switzerland, where Basel II was devised, regulators have questioned capital regulation. For instance, UBS AG wrote down  18 billion in losses due to risk mismanagement and exposure to subprime mortgages and other risky assets. In December 2007, UBS disclosed plans to boost its capital with a  12.1 billion injection from the Government of Singapore Investment Corp. and an unidentified Middle Eastern investor. By Friday, 10 October 2008 (Black Friday), the Japanese company, Yamato Life, filed for bankruptcy becoming what is viewed as the first direct casualty in Japan from the fallout of the SMC. Yamato Life had  2.7 billion in liabilities at the time of the filing. Some analysts believed it to be a foreshadowing of things to come in Japan as the credit crisis began to affect the country. In short, the Basel rules are being questioned for containing inadequate prescriptions for monitoring a bank's liquidity; in other words, its ability to readily sell assets, or borrow affordably, to cover obligations. In principle, Basel II regulators worldwide are required to track a bank's risk-taking and to check how the bank monitors itself. Traditionally, it is not their task to prescribe to banks about the size and type of risks they can take. In the U.S., some regulators have recently shown a willingness to tighten Basel capital regulation. They are motivated by the failure of more than 1000 banks amid unforeseen risks related to interest rates and real estate during the 1980's S and L crisis. In addition, regulators both in the U.S. and abroad are gearing themselves to amend Basel II. In all likelihood, this will involve enforcing a higher level of capital against assets that may be construed to be risky in the wake of the SMC.

1.1. Relation to Previous Literature

In this subsection, we discuss terse literature reviews of subprime mortgage loans, Basel capital regulation as well as the SMC. More comprehensive reviews of some of these topics have already been done in [1] (see, also, [3]).

1.1.1. Brief Literature Review of Subprime Mortgage Loans

In [4], light is shed on the subprime mortgagor, the workings of a typical subprime mortgage loan, and the historical performance of subprime mortgage credit. In order to keep the discussion from becoming too abstract, they find it useful to frame many of these issues in the context of a real-life example. Using loan-level data, [5] analyzes the quality of subprime mortgage loans by adjusting their performance for differences in borrower characteristics, loan characteristics, and macroeconomic conditions. They find that the quality of loans deteriorated for six consecutive years before the SMC and that securitizers were, to some extent, aware of it.

The mortgage lenders that retained credit risk from subprime mortgages were the first to be affected by the SMC, as borrowers became unable or unwilling to make payments. Major banks and other financial institutions around the world have reported losses of approximately US  435 billion as of 17 July 2008 (see [6, 7]). Owing to a form of financial engineering called securitization, many mortgage lenders had passed the rights to the mortgage payments and related credit/default risk to third-party investors via mortgage-backed securities (MBSs) and collateralized debt obligations (CDOs). Corporate, individual and institutional investors holding MBSs or CDOs faced significant losses, as the value of the underlying mortgage assets declined. Stock markets in many countries declined significantly.

It is a widely accepted fact that subprime mortgages exhibit cyclical tendencies. The fact that credit ratings (profitability) behave procyclically by rising (falling) during economic booms and falling (rising) during recessions (see, e.g., [1, 8, 9]) is incorporated in our models. (Procyclicality is a normal feature of financial systems in which asset price increases and credit expansions support the business cycle and contribute to economic growth.) In particular, credit rating agencies (CRAs) are now under scrutiny for having given investment-grade ratings to CDOs and MBSs based on subprime mortgage loans during the boom. These high ratings were believed justified because of risk reducing practices, including overcollateralization (pledging collateral in excess of debt issued), credit default insurance, via, for instance, credit default swaps as well as the intervention of equity investors willing to bear the first losses such as for CDOs.

1.1.2. Brief Literature Review of Basel Capital Regulation

The most significant innovation of Basel II is the departure from a sole reliance on capital adequacy ratios. Basel II consists of three mutually reinforcing pillars, which together should contribute to safety and soundness in the financial system (see, e.g., [2]). To ensure that risks within an entire banking group are considered, Basel II is extended on a consolidated basis to holding companies of banking groups. The main objective of the Basel II Capital Accord is to promote standards for measurement and management of financial and operational risk in banking. Its approach to such risk issues has been severely criticized in the literature, inevitably leading to doubts about its practical implementation. In particular, many investigations have warned against the procyclicality induced by the IRB capital formula (see, e.g., [2]). Since the release of the Second Consultative Paper [10], many studies have assessed empirically the magnitude of procyclicality in the IRB capital formula (see, e.g., [11]). Also, there is overwhelming evidence to suggest that the movements of subprime mortgage loan extension, loan loss provisioning, capital and profitability are strongly correlated with the business cycle. While not providing an in-depth discussion of the first of the aforementioned problems, our contribution focusses strongly on issues related to subprime mortgage loan pricing and regulatory capital in both the Basel I and II paradigms. Also, we suggest that bank capital is less variable under Basel II than Basel I (see, e.g., [2, 10]).

1.1.3. Brief Literature Review of the Subprime Mortgage Crisis

It is generally accepted that the SMC resulted from the bursting of the U.S. housing bubble (see, e.g., [12]). Coupled to this was high default rates on “subprime” and adjustable rate mortgages (ARMs). Loan incentives, such as easy initial terms, in conjunction with an acceleration in rising housing prices encouraged borrowers to assume difficult mortgages in the belief they would be able to quickly refinance at more favorable terms. However, once housing prices started to drop moderately in 2006-2007 in many parts of the U.S., refinancing became more difficult. Defaults and foreclosure activity increased dramatically, as easy initial terms expired, home prices failed to go up as anticipated, and ARM interest rates reset higher. Foreclosures accelerated in the U.S. in late 2006 and triggered a global financial crisis through 2007 and 2008. During 2007, nearly  1.3 million U.S. housing properties were subject to foreclosure activity, up from 2006 (see [13] for more details).

The working paper [5] makes several novel contributions to the literature on the SMC. Firstly, the authors, Demyanyk and Van Hemert, quantify how much different determinants have contributed to the observed high delinquency and foreclosure rates for vintage 2006 loans, which led up to the 2007 SMC. Their data analysis suggests that different loan-level characteristics as well as low house price appreciation were quantitatively too small to explain the bad performance of 2006 loans. Secondly, the authors uncover a downward trend in loan quality, determined as loan performance adjusted for differences in loan and borrower characteristics as well as subsequent house price appreciation. They further show that there was a deterioration of lending standards and a decrease in the subprime-prime mortgage rate spread during the 2001–2006 period. Together these results provide evidence that the rise and fall of the subprime mortgage market follows a classic lending boom-bust scenario, in which unsustainable growth leads to the collapse of the market. Thirdly, Demyanyk and Van Hemert show that continual deterioration of loan quality could have been detected long before the crisis by means of a simple statistical exercise. Fourth, securitizers were, to some extent, aware of this deterioration over time, as evidenced by changing determinants of mortgage rates. Furthermore, paper [5] documents that the poor performance of the vintage 2006 loans was not confined to a particular segment of the subprime mortgage market. In this regard, fixed-rate, adjustable-rate, purchase-money, cash-out refinancing, low-documentation, and full-documentation loans originated in 2006 all showed substantially higher delinquency and foreclosure rates than loans made in the preceding five years. As a consequence, [5] contradicts a widely held belief that the SMC was mostly confined to adjustable-rate or low-documentation mortgages.

Since mid 2007, role players in the financial industry have blamed the Basel II Capital Accord for certain aspects of the SMC. In this regard, the adequacy of capital levels in the banking industry, the role of credit rating agencies in financial regulation, the procyclicality of minimum capital requirements, and the fair-value assessment of banking assets have become the most studied topics. Paper [14] poses the following related questions. Is Basel II guilty of causing the subprime mortgage crisis? Is it appropriate to judge Basel II on the basis of features that are unlikely to have caused the subprime mortgage crisis? Should Basel II be completely abandoned or should an attempt rather be made to overcome its shortcomings? Paper [14] attempts to provide some answers to the questions raised above. After a short review of the main features of the financial crisis as well as of the rationale behind the Basel II rules, the authors try to describe the actual role played by the new prudential regulation in the crisis and discuss the main argument raised in the current debate. They conclude that, while aspects of Basel II need strengthening, there are not good enough reasons for abandoning the accord in its entirety.

1.2. Preliminaries

In this subsection, we provide preliminaries about balance sheets, subprime mortgage loans, bank profit, the Basel Capital Accords, and the SMC.

1.2.1. Preliminaries about Bank Balance Sheets

As is well known, the bank balance sheet consists of assets (uses of funds) and liabilities (sources of funds) that are balanced by bank capital (see, e.g., [15]) according to the well-known relation In period , the main on-balance sheet items in (1.1) can specifically be identified as where and are the market value of short- and long-term loans, cash, short- and long-term securities, bonds, Treasuries, reserves, outstanding debt, number of shares, market price of the bank's common equity, subordinate debt, and loan loss reserves, respectively.

1.2.2. Preliminaries about Subprime Mortgage Loans

Subprime mortgage loans are loans whose interest rate repayment is below the prime rate. A study by the U.S. Federal Reserve found that the average difference between subprime and prime mortgage interest rates (the “subprime markup”) declined from 280 basis points in 2001, to 130 basis points in 2007. In other words, the risk premium required by lenders to offer a subprime loan declined. This occurred even though the credit ratings of subprime borrowers, and the characteristics of subprime loans, both declined during the 2001–2006 period, which should have had the opposite effect. The combination is common to classic boom and recession credit cycles (see [5]).

The 2001 Interagency Expanded Guidance for Subprime Lending Programs defines the subprime borrower as one who generally displays a range of credit risk characteristics, including one or more of the following:

A diagrammatic overview of cash flows for subprime mortgage loans may be represented as shown in Figure 1.

Figure 1: Diagrammatic overview of cash flows for subprime mortgage loans.

From Figure 1, we note that 1A represents the cash flow from the mortgage lender for financing the portfolio of subprime mortgage loans. Also, 1b denotes the interest rate, received from this subprime portfolio. The arrow numbered 1C is the mortgage cash payments by subprime mortgagors.

1.2.3. Preliminaries about Bank Profits

As far as profit, is concerned, we closely follow paper [1] by using the basic fact that profits can be characterized as the difference between income and expenses that are reported in the bank's income statement. In our case, income is solely constituted by the return on intangible assets, the return on subprime mortgage loans, and the return on Treasuries, In this regard, and denote the rates of return on intangible assets, subprime mortgage loans (that may include a component for provisions for expected loan losses), and Treasuries, respectively. Furthermore, we assume that the level of macro economic activity is denoted by As expenses, in period we consider the cost of monitoring and screening of subprime mortgage loans and capital, interest paid to depositors, the cost of taking deposits, the cost of deposit withdrawals, the value of subprime mortgage loan losses, and total loan loss provisions, Here and are the deposit rate and cost of deposits, respectively. We assume all the aforementioned costs would sum to operating costs so that profit, can be expressed as

1.2.4. Preliminaries about the Basel Capital Accords and the Subprime Mortgage Crisis

Many of the preliminaries presented in this subsection are directly related to the contents of [1, Sections   to ]. Basel capital regulation has its roots in the 1980s, when bank regulations varied dramatically from country to country, making it tough for banks to compete across borders. Central bankers from around the globe congregated in Switzerland to agree upon basic standards, which were unveiled in 1988. Subsequently, Basel II focused on expanding those regulations in order to protect financial systems against complex new investment products being introduced by banks. Basel II went into effect in European countries in 2007 and 2008.

Banks rely on capital to cushion against loan losses and, ultimately, failure. Under pre-Basel II rules, setting the level of this capital was a simple process. Banks held a specific amount of capital, which was calculated based on the types of assets they hold. For example, MBSs did not require much capital because they were considered to be relatively safe. Basel II changes that by allowing banks to calculate their level of capital reserves based in part on their own assessments of risk and the opinion of CRAs. In support of Basel II, the U.S. Federal Reserve argued that its standards give banks incentives to bolster their risk management. In addition, Basel II requires institutions to maintain a safety net of capital to protect against trouble in “off balance sheet” assets they may have an issue that had largely escaped prior regulatory oversight. Some analysts claim that the SMC will strengthen Basel II by giving banks valuable new data to factor into their models.

1.3. Main Problems and Outline of the Paper

Undergirded by the analysis in [1], we extend aspects of the literature mentioned in Section 1.1 in several important directions. Firstly, in a Basel I framework, we investigate the dynamics of bank capital and credit and their sensitivity to changes in the level of macroeconomic activity. Here asset risk-weights are kept constant and we only consider credit and market risk. Next, we repeat the above in a Basel II paradigm where subprime mortgage loan risk-weights vary while risk-weights for intangible assets and shares remain constant. Also, á la Basel II, we consider credit and market risk as well as operational risk. Thirdly, we include a discussion of subprime mortgage loans and their reduced risk premiums. Finally, we consider the effect that Basel capital regulation has had on the SMC.

1.3.1. Main Problems

The main questions that we pose in this paper may be stated as follows.

Problem 1 (Bank Credit and Capital Under Basel I). What is the effect of changes in the level of macroeconomic activity on bank credit and capital when risk-weights are constant under Basel I? (see Theorem 3.1 as well as Propositions 3.2 and 3.3 of Section 3).

Problem 2 (Bank Credit and Capital Under Basel II). What is the effect of changes in the level of macroeconomic activity on bank credit and capital under Basel II? (see Theorem 4.1 as well as Propositions 4.2 and 4.3 of Section 4).

Problem 3 (The Basel Accords and the SMC). Did the Basel accords exacerbate the subprime mortgage crisis? (see Section 5).

1.3.2. Outline of the Paper

In the current subsection, an outline of our contribution is given. In Section 2, we present discrete-time stochastic models for subprime banking activities. Banks respond differently to shocks that affect subprime mortgage loan demand, when the minimum capital equirements are calculated by using risk-weighted assets. In the Hicksian case, these responses are usually sensitive to macroeconomic conditions that are related to the term in (2.3). Subprime mortgage loan defaults are independent of the capital adequacy paradigm that is chosen. In this regard, empirical evidence supports the opinion that enhanced levels of macroeconomic activity reduce the subprime mortgage loan default rate and thus the loan marginal cost.

Section 3 contains a discussion of the cyclicality of bank credit and capital where, as in Basel I, we assume that risk-weights are constant. In addition, borrowers are allowed to default on their subprime mortgage loan repayments. Furthermore, we present a result about the effects of changes in the level of macroeconomic activity on the quantity and price of subprime mortgage loans (see Theorem 3.1 of Section 3.1). Moreover, we establish the impact of changes in the business cycle on subprime mortgage loans and their loan rates when the capital constraint holds and when it does not (see, e.g., Proposition 3.2 of Section 3.2 and Proposition 3.3 of Section 3.3; also compare with (3.2).

In Section 4, results analogous to those obtained in Section 3 are given for situations where subprime mortgage loan losses and loan risk-weights are a function of the level of macroeconomic activity (i.e., risk-weights vary with changes in business cycle phases). These situations are cast within the framework offered by the Basel II Capital Accord. Analogously, we study situations where the capital constraint holds and where it does not (see Theorem 4.1, Proposition 4.2, and Proposition 4.3 in Sections 4.1, 4.2, and 4.3, resp.). In Section 4.4, we discuss the cyclicality of subprime mortgage loans and their rates under Basel II in subsequent time periods.

In Section 5, we consider the relationship between the Basel Capital Accords and the SMC. In particular, we try to answer the main question posed by the paper, namely, whether the capital accords exacerbated the mortgage crisis.

Section 6 provides concluding remarks and a discussion about possible future research. Subsequent to the reference list we insert an appendix that contains a table summarizing the main results as well as two tables outlining the main differences between the Basel I and II Capital Accords.

2. Subprime Banking Models

This section is an adaptation of the models in [1] to the subprime case. Also, we present an optimization result achieved in preparation for Section 3.

2.1. Subprime Mortgage Loans

We suppose that, after providing liquidity, the bank lends in the form of th period subprime mortgage loans, at the bank's subprime loan rate, This loan rate, for profit maximizing banks, is determined as follows: where is the base rate and is the risk premium. As was mentioned before, the “subprime markup” declined considerably for the period 2001 to 2007. In other words, the risk premium required by lenders to offer a subprime loan declined. This occurred even though the credit ratings of subprime borrowers, and the characteristics of subprime loans, both declined during this period. Next, we introduce the generic variable, that represents the level of macroeconomic activity in the bank's loan market. We suppose that follows the first-order autoregressive stochastic process where denotes zero-mean stochastic shocks to macroeconomic activity.

In our case, the bank faces a Hicksian demand for subprime mortgage loans given by We note that the subprime mortgage loan demand in (2.3) is an increasing function of and a decreasing function of Further, we suppose that is the random shock to the subprime mortgage loan demand with support that is independent of an exogenous stochastic variable, to be characterized below. Also, we assume that the subprime mortgage loan supply process, follows the first-order autoregressive stochastic process where and denotes zero-mean stochastic shocks to subprime mortgage loan supply.

An initial observation is that subprime mortgage loan losses are also dependent on macroeconomic activity. As a consequence, for the value of subprime mortgage loan losses, and the default rate, we set where increases when macroeconomic conditions deteriorate according to We note that the above description of the subprime mortgage loan loss rate is consistent with empirical evidence that suggests that bank losses on subprime mortgage loan portfolios are correlated with the business cycle under any capital adequacy regime (see, e.g., [1, 8]).

2.2. Reserves

Bank reserves are the deposits held in accounts with the central bank of a country plus money that is physically held by banks (vault cash). Such reserves constitute money that is not lent out but is earmarked to cater for withdrawals by depositors. Since it is uncommon for depositors to withdraw all of their funds simultaneously, only a portion of total deposits may be needed as reserves. As a result of this description, we may introduce a reserve-deposit ratio, for which The bank uses the remaining deposits to earn profit, either by issuing subprime mortgage loans or by investing in assets such as Treasuries and stocks.

2.3. Risk-Weighted Assets

We consider risk-weighted assets (RWAs) that are defined by placing each on- and off-balance sheet item into a risk category. The more risky assets are assigned a larger weight in this study. As a result, RWAs are a weighted average of the various assets of the banks. In the sequel, we denote the risk-weights on intangible assets, subprime mortgage loans, Treasuries and reserves by and respectively. With regard to the latter, we can identify a special risk-weight on subprime mortgage loans, that is a decreasing function of current macroeconomic conditions so that This is in line with the procyclical notion that during booms, when macroeconomic activity increases, the risk-weights will decrease. On the other hand, during recessions, risk-weights may increase because of an elevated probability of default and/or loss given default on subprime mortgage loans.

2.4. Capital

For the purposes of our study, regulatory capital, is the book value of bank capital defined as the difference between the accounting value of the assets and liabilities. More specifically, Tier 1 capital is represented by period 's market value of the bank equity, where is the number of shares and is the period market price of the bank's common equity. Tier 2 capital mainly consists of subordinate debt, that is subordinate to deposits and hence faces greater default risk and loan loss reserves, Subordinate debt issued in period is represented by a one-period bond that pays an interest rate, Also, we assume that loan loss reserves held in period changes at the rate, Tier 3 capital is not considered at all. In the sequel, we take the bank's total regulatory capital, in period to be For given by (2.9), we obtain the balance sheet constraint We define the regulatory capital constraint by the inequality where and . If we assume that the risk-weights associated with intangible assets, shares, cash, bonds, Treasuries, reserves, and subprime mortgage loans may be taken to be , and respectively, then (2.11) becomes the capital constraint

2.5. Profit

We assume that (2.5) holds. If we now add and subtract from (1.3) and use the fact that we obtain This is the cash flow constraint for a bank and will be used later. Furthermore, by considering and (2.14), we suspect that profit, is an increasing function of current macroeconomic conditions, so that This is connected with procyclicality where we expect profitability to increase during booms, when macroeconomic activity increases. By contrast, profitability may decrease during recessions because of, among many other factors, an increase in provisioning (see (2.14).

To establish the relationship between bank profitability and retained earnings, a model of bank financing is introduced that is based on [16]. We know that bank profits, are used to meet the bank's commitments that include dividend payments on equity, and interest and principal payments on subordinate debt, The retained earnings, subsequent to these payments may be computed by using In standard usage, retained earnings refer to earnings that are not paid out in dividends, interest, or taxes. They represent wealth accumulating in the bank and should be capitalized in the value of the bank's equity. Retained earnings are also defined to include bank charter value income. Normally, charter value refers to the present value of anticipated profits from future lending.

2.6. Valuation

In each period, banks invest in fixed assets (including buildings and equipment) which we denote by The bank is assumed to maintain these assets throughout its existence so that the bank must only cover the costs related to the depreciation of fixed assets, These activities are financed through retaining earnings and the eliciting of additional debt and equity, so that We can use (2.16) and (2.17) to obtain an expression for bank capital of the form where is defined by (2.9).

If the expression for retained earnings given by (2.16) is substituted into (2.17), the nett cash flow generated by the bank for a shareholder is given by In addition, we have the relationship

This translates to the expression where is defined by (2.9). Furthermore, the stock analyst (acting in the interest of a shareholder) evaluates the expected future cash flows in periods based on a stochastic discount factor such that the value of the bank is

2.7. An Optimal Subprime Mortgage Loan Pricing Problem

In this subsection, we present the main features of the optimal subprime mortgage loan pricing problem solved in [1].

2.7.1. Statement of the Optimal Loan Pricing Problem

In the sequel, suppose that the bank's performance criterion, at is given by where is the Lagrangian multiplier for the total capital constraint (2.13), is defined by (2.9), is the expectation conditional on the bank's information at time and is the deposit withdrawals in period with probability distribution Also, is the deadweight cost of total capital consisting of debt and equity. We are now in a position to formally state the optimal valuation problem for banks that we solve in the sequel.

Problem 4 (Statement of the Optimal Loan Pricing Problem). Suppose that the total capital constraint and the performance criterion, are given by (2.13) and (2.23), respectively. The optimal subprime mortgage loan pricing problem is to maximize the value of the bank given by (2.22) from the point of view of a stock analyst, by choosing the subprime mortgage loan rate, deposits and regulatory capital for subject to the subprime mortgage loan demand, balance sheet, cash flow, and financing constraints given by (2.3), (2.10), (2.14) and (2.18), respectively.

2.7.2. Solution of the Optimal Loan Pricing Problem

In this section, we find a solution to Problem 4 when the capital constraint is binding (see, e.g., [17]). In this regard, the main result can be stated and proved as follows.Theorem 2.1 (Solution to the Optimal Loan Pricing Problem). Suppose that and are given by (2.23) and (2.24), respectively, and When the capital constraint given by (2.13) is binding (i.e., ), a solution to the optimal subprime mortgage loan pricing problem stated in Problem 4 yields an optimal bank subprime mortgage loan supply and loan rate of the form respectively. In this case, the corresponding optimal deposits, provisions for deposit withdrawals, and profits are given by respectively.In the proof of Theorem 2.1, we note that the first-order conditions are given by Here is the cumulative distribution of the shock to the subprime mortgage loans.

In the case where the constraint (2.13) does not hold, the following corollary follows directly.

Corollary 2.2 (solution to the optimal loan pricing problem (slack)). Suppose that and are given by (2.23) and (2.24), respectively, and When the capital constraint (2.13) does not hold (i.e., ), a solution to the optimal loan pricing problem stated in Problem 2.1 yields the optimal bank subprime mortgage loan supply and its rate respectively. In this case, the corresponding , deposits and profits are given by respectively.

3. Bank Credit and Capital under Basel I (Constant Risk-Weights)

In this section, we use the models developed in Section 1 to discuss the cyclicality of bank credit and capital in a Basel I paradigm. Two standing assumptions are that borrowers may default on subprime mortgage loans and that their behavior may depend on the phase of the business cycle. We note, for instance, that in line with the prescripts of Basel I, risk-weights are kept constant and operational risk is not considered as in Basel II. In this situation, the capital constraint (2.13) may be adapted to become Furthermore, since risk-weights on subprime mortgage loans, intangible assets, and shares are kept constant, that is, (3.1) has the simpler form Where applicable, in the remainder of this section, we take the binding capital constraint to be (3.2).

3.1. Quantity and Price of Loans and Bank Capital under Basel I

In this subsection, under Basel I, we examine how subprime mortgage loan quantity and pricing as well as bank capital are affected by changes in the level of macroeconomic activity,

Theorem 3.1 (cyclicality of bank capital under Basel I). Suppose that and (i.e, risk-weights are constant). It follows that (1)if    then (2)if    then

Proof. In order to prove Theorem 3.1, we have to appeal to the implicit function theorem. Before using this theorem, we explicitly determine the critical shock to the demand for subprime mortgage loans, such that the total capital constraint (3.2) will just hold, that is, . By equating the optimal subprime mortgage loans from the two problems (with and ), we obtain Solving for , we get Using the Euler condition for the bank's subprime loan rate, we have that Furthermore, substituting provisions for deposit withdrawals we obtain By substituting and into the expression above, it follows that Using (2.24) to find the partial derivative of the value function with respect to bank capital yields By substituting the above expression into the optimal condition for total capital given by we obtain If we denote the left-hand side of the above expression by then it follows that By the implicit function theorem we have In order to calculate we utilize (3.11) to obtain where Therefore From the above, we may conclude that in this case , where This concludes the proof of Theorem 3.1.

3.2. Subprime Mortgage Loans and Loan Rates under Basel I (Constraint Slack)

In this subsection, we establish the impact of changes in the level of macroeconomic activity, on subprime mortgage loans, and the bank's subprime loan rate, We first present a result in the case where the capital constraint (3.2) is slack.

Proposition 3.2 (subprime mortgage loans and loan rates under Basel I (constraint slack)). If then

Proof. In order to prove Proposition 3.2 under a slack constraint, we have to determine the partial derivatives of the optimal subprime mortgage loan supply, and loan rate, with respect to of (2.32), respectively. Of course, we do not have to include operational risk as in Basel II. We also consider the condition Next, we can calculate