#### 3.1. Background

According to

Stein (

2012b), the United States financial crisis in 2008 was caused by excessive financial obligations/mortgages of private households (i.e., bubbles in the mortgage market defined as unsustainable debt/income ratios), while in the 1980s, the financial crisis was related to the business sector. Essentially, Stein argued that, although debt problems may have originated in either the public or private sectors in different nations, the result was still declining asset values, and the mechanisms at work resulted in a contagion effect either from the United States to Europe and/or from one European nation to another depending on the debtor-to-debtee relationship under examination. Of course, in each scenario, Stein made it clear that the primary source of the problem was not the presence of debt but excess debt within the country/countries under analysis.

Stein derived an optimal-debt ratio and built on it to identify an early warning signal (EWS) of a debt crisis, which is defined as the excess debt of households (actual-debt ratio less than the optimal-debt ratio). As the excess-debt level rises, the probability of a debt crisis increases. It has been shown that rising house prices since the late 1990s led to above-average capital gains for households, thereby increasing owner equity. The supply of mortgages increased, and consequently, financial obligations as a percentage of disposable income increased for private households. At the same time, the quality of loans declined (subprime mortgages). Of course, this process was not sustainable. As capital gains

1 dropped below the interest rate, debtors could not service their debts any longer, and foreclosures led to a collapse in the value of financial derivatives.

Before delving into the theoretical model and empirical analysis presented in this paper, it is important to briefly provide background information regarding Islamic banks’ mortgage-loan operations. As mentioned, their operations do not involve interest rates; rather, they have a proprietary program called the LARIBA (interest-free) model, and they use equity-participation systems or profit loss sharing. They offer two types of mortgage loans. First, profit sharing (murabaha) is offered in which the bank does not loan money to the buyer to purchase the home or other property; rather, the bank buys the home itself and then resells it to the buyer at a profit. The buyer typically pays a fairly large down payment, and the price at which the bank buys the property is disclosed to the end buyer. The second method is decreasing rent (ijara). The bank purchases the home and resells it to the buyer; however, unlike the first method, in this case, the home remains in the bank’s name until the total price is paid. The buyer takes up residence immediately and makes payments to the bank on the purchase price, but in addition to the payments, the buyer also pays a fair market rent. This method is preferred in countries such as the United Kingdom to avoid double payments of taxes (

Khan 2010).

Dr. Abdul-Rahman, Chairman and CEO of the Bank of Whittier, said during an interview that they had few nonperforming mortgages in 2008, and this was due to their riba-free (RF) discipline

2 He explained that the bank tracks home prices in US dollars relative to more stable commodities, such as gold, silver, wheat, or rice, thereby allowing for the discovery of real-estate bubbles. He added that, on the basis of the used strategy, the bank had an early warning signal of a macro-bubble forming in the real-estate sector. In addition, they marked the property to the market, meaning that they researched the rental value of a similar property in the same neighborhood. He stated that the inputs of their model are the amount to be invested (financed), the number of years of financing, and the rent. The unknown is the rate of return on investment

3. Their internal business model allowed them to survive the recent financial crisis.

#### 3.2. Theoretical Model

I introduce a model of optimal leverage that helps us to define overleveraging. The model sketched here is a low-dimensional stochastic variant of a model of banking leveraging, and it follows

Stein (

2008,

2012a) and

Brunnermeier and Sannikov (

2014).

Overall, my model is very similar to those of

Brunnermeier and Sannikov (

2014) and

Stein (

2012a). Both models have leveraging and payouts as choice variables, and net worth as a state variable. Moreover, both models are stochastic. Similar to this study,

Brunnermeier and Sannikov (

2014) specifically focused on the banking sector; however, the setting is more general compared to the one used in this paper. There are households that save, and financial experts representing financial intermediaries that invest in capital assets owned by households and financial intermediaries. Both have different discount rates. I focus solely on the behavior of financial intermediaries.

In this model, I use preferences in the objective function and Brownian motions as state variables similar to both studies. The

Stein (

2012a) model, assuming certain restrictions, uses log utility and allows to exactly compute excess leveraging. Capital return is also stochastic due to capital gains, and the interest rate is stochastic as well, similar to my model and in contrast with that of

Brunnermeier and Sannikov (

2014); where only the capital return is stochastic, and the interest rate is taken as constant. Both

Brunnermeier and Sannikov (

2014) and

Stein (

2012a) employed a continuous time version, but the problem in this paper is formulated as a discrete time variant with a discounted instantaneous payout and an optimal leveraging function.

Moreover, both

Brunnermeier and Sannikov (

2014) and

Stein (

2012a) stated that a shock to asset prices creates a vicious cycle through the balance sheets of banks. In other words, risk taking and excessive borrowing occur when asset prices are volatile. They defined what they referred to as the volatility paradox as the shock to asset prices that negatively affects the banks’ balance sheets and subsequently disrupts the real sector. Thus, when the prices of banks’ assets decrease, and thus their equity value and net worth decrease, margin-loan requirements increase. For financial intermediaries to remain liquid, they take haircuts and deleverage. Consequently, a fire sale of assets begins, further decreasing the asset price, and net worth declines, thus triggering an endogenous jump in volatility and a risk for all, which generates a downward spiral. This is in accordance with the findings of this paper.

The asset-price channel through which the banking system’s instability is triggered was also studied by

Mittnik and Semmler (

2012,

2013). In this model, the unconstrained growth of capital assets through excessive borrowing, facilitated by the lack of regulations imposed on financial intermediaries, is considered the main cause for banking-sector instability

4. On the other hand, large payouts with no “skin in the game” affect banks’ risk-taking behaviors, equity development, and leveraging. The higher the payout is, the more leveraged the bank becomes, which increases the aggregate risk and risk premia for all. In summary, the increased risk spreads, and risk premia, especially at a time when defaults begin, expose banks to vulnerabilities and financial stress triggered by security-price movements.

To derive an optimal-debt ratio, Stein used stochastic optimal control (SOC). A hypothetical investor selects an optimal-debt ratio, $f\left(t\right)$, to maximize the expectation of a concave function of net worth, $X\left(t\right)$, where T is the terminal date. The model assumes that the optimal-debt/net-worth ratio significantly depends on the stochastic process concerning the capital-gain variable. The expected growth of net worth is also maximal when the debt ratio is at the optimal level.

Optimal leverage is given by:

such that

where

$r\text{}$ is the bank’s capital gain/loss;

$\text{}i\text{}$ is the credit cost of banks;

$\beta $ is the productivity of capital;

$y\left(t\right)$ is the deviation of capital gain from its trend; σ

^{2} is the variance; and

$\rho $ represents the negative-correlation coefficient between interest rate and capital gain

5. Through the presented model, Stein could determine excess debt and an early warning signal of a potential crisis. As mentioned, it is this mechanism that played a role in the decreasing net worth of individuals, households, and institutions in the United States, and that was amplified by the increased leverage and pricing volatility of complex securities.

To measure the excess leveraging of banks, the introduced and defined Stein model was followed with a focus on the solution of the dynamic version of the model, which allowed for using time-series data on banks. One difference from Stein is that, in this case, each bank’s productivity of capital was not assumed to be deterministic or constant as in the Stein model; rather, it was calculated for the years of 2000–2016.

The optimal-debt level was calculated for the years 2000 until 2016; thus, excess debt, which is the measure of overleveraging in this paper, was estimated. To calculate the banks’ optimal-debt ratios, data on the banks’ capital gain/loss, market interest rates, and the productivity of capital were collected. Using these variables, the risk and return components of the model were then calculated. Using the abovementioned variables, the optimal-and actual-debt ratios were calculated for a sample of twenty banks, 10 Islamic and 10 conventional banks, in five different countries

6. The full calculations are presented in 20 tables

7 with 18 columns each. Data were derived from the banks’ annual reports, balance sheets, and financial statements, in addition to some data received directly from bank managers. The calculations are summarized below.

Column 1 consists of capital gain/loss that represents the return in percentage to the investors of the bank from capital appreciation or loss in a particular year. This capital gain/loss is calculated by dividing the change in each bank’s stock-market cap by the beginning market cap at each period. The market caps were Hodrick–Prescott (HP)-filtered to eliminate the effects of daily stock-market swings. HP Filter is a data-smoothing technique frequently applied on time series data to remove short-term fluctuations associated with the business cycle. For example, to calculate the National Bank of Bahrain capital gain of 0.05% during 2006 shown in

Table A3, I divided the change in the bank’s market cap from 31 December 2005 to 31 December 2006 by the market cap on 31 December 2005.

Column 2 represents the market interest rate. The 10-year treasury yield was used to represent the market interest rate, and is therefore presented in percentage

8.

In Column 3, beta (β) represents the productivity of capital. The beta is calculated as the bank’s annual gross revenue divided by total capital. The total capital here is calculated as the shareholder equity plus half of both short-term and total long-term debt

9. To determine shareholder equity, I obtained the annual value of each bank’s shareholder equity from the balance sheet. Short-term debt comprises all the banks’ current liabilities that are usually due within 12 months. Long-term debts, on the other hand, are calculated as the combinations of long-term liabilities and other liabilities in banks’ balance sheets. These are basically all bank liabilities due in more than a year’s time. Therefore, each bank’s productivity of capital is calculated for the years of 2000–2016, and not constant as in

Stein (

2012b).

Columns 4 through 9 are the risk elements in the model (

Stein 2012a). Column 4 represents beta variance that calculated as the difference between each year’s beta from the mean beta for the years 2000 and 2016, representing the deviation of each period’s beta from the mean. Column 5 is also a component of the risk element. This is calculated as one-half of the square of the capital-gain variable. Column 6 is the statistical correlation between interest-rate and capital-gain variables over the period from 2000 to 2016

10. Columns 7 and 8 are the variance for the interest-rate and capital-gain variables, respectively. Each period’s variance is calculated as the deviation of that period’s value from the mean. Therefore, interest-rate variance is the difference between each year’s 10-year treasury yield and the mean interest rate from 2000 to 2016. Similarly, capital-gain variance is the difference between each year’s capital gain/loss and the mean capital gain from 2000 to 2016.

Column 9, which is the product of the correlation between the stochastic variables (interest rate and capital gain), and interest-rate and capital-gain variance, represents an additional component of the risk element. It is calculated as the product of the correlation factor of the stochastic variables (Column 6), interest-rate variance (Column 7), and capital-gain variance (Column 8)

11. Columns 10–12 are used to determine the risk-investors bear when they decide to hold equity in the bank, and this is a key issue for the investors’ decision making.

Columns 10 and 11 represent the standard deviations of the interest rate and capital gains, respectively. Therefore, Column 10 is the standard deviation of values in Column 2, while Column 11 is the standard deviation of values in Column 1. Here, standard deviations are constant over the periods, as in the Stein model.

Column 12, on the other hand, is calculated as twice the value of variances of the two stochastic variables and their correlation. This is, therefore, calculated as 2 multiplied by Column 9.

Column 13 is, hence, the risk of an investor holding the equity of the bank at each time period as in Equation (A2). The risk is calculated using Columns 10–12. The risk is calculated by adding the standard deviations of the interest rate (Column 10) plus the standard deviation of capital gain (Column 11) minus the risk component in Column 12.

In the model, the optimal-debt ratio maximizes the difference between net return and risk term. Therefore, only if the net return exceeds the risk premium does the optimal-debt ratio become positive. The optimal-debt ratio, therefore, is not a constant, as Stein also noted

Stein (

2012a), but rather varies directly with net return and risk.

In Column 14, I then calculated, using all the above-mentioned variables, Stein optimal-debt ratio, f*(t). Debt ratios were normalized to remove the effects of seasonality. Therefore, normalized f*(t) measures the deviation of the optimal-debt ratio away from the mean. Negative values in Column 14 represent lower optimal-debt ratios away from the mean ratio during the applicable periods. The components of the optimal-debt ratio are, therefore, primarily the capital gains for equity holders of the bank’s stock, the market interest rate, and the risk term. The optimal-debt ratio maximizes the difference between mean return and risk term. The formula for calculating optimal-debt ratio using the above column numbers is: ((Column 1 − Column 2) + Column 3 − Column 4 − Column 5 + Column 9)/Column 13 (

Ebisike 2014). This reiterates what was mentioned above that optimal-debt ratio is positive only if the net return is greater than the risk premium, and this can intuitively be seen.

In Column 15, I calculated normalized optimal-debt ratios using Column 14, the mean and standard deviation of the optimal-debt values

12. In addition to calculating the optimal-debt ratio, I calculated the banks’ actual-debt ratio in order to calculate the excess-debt ratio. The actual-debt ratio of the banks was equal to long-term debt divided by total assets, which are given in the banks’ annual reports as well. Actual-debt ratios are also normalized in the same way as optimal-debt ratios are, and are presented in Column 16. After optimal- and actual-debt ratios are calculated as discussed above, excess debt is calculated in the last columns as normalized actual minus optimal debt. The graphs of the two ratios, namely, actual- and optimal-debt ratios, are presented in

Figure 1,

Figure 2,

Figure 3,

Figure 4 and

Figure 5 below, followed by empirical analysis.