Optimal Securitization in Banks : Agent Based Modelling Notes
Dr. Sheri Markose*# and Yan Dong (*#CCFEA and Economics Dept., Economics Dept. University of Essex)
Assume there are N banks, i= 1, …………N
Bank Balance Sheet:
Simple case
ASSETS(L)  LIABILITIES (L)      A_{t}  A_{0}: Initial Assets  D_{t}  D_{0}: Initial Deposit  Loans     Liquid Assets
        Equity Capital : E_{t} = A_{t}  L_{t}
Objective of the bank is to Maximize Capital net of Capital Requirement
Hence the liability at t+1 is given by adding on the interest rate payment (1) with is the initial value of deposit of the bank.
As for assets, traditionally, the growth in the value of assets is influenced by the following two factors: (i) Return on the asset: . (ii) Default on assets denoted by () which is the proportion of defaulted loans and hence is the survival value of assets from t.
Hence, we have (2) Moreover, to consider the impact of capital adequacy requirement on the bank’s balance sheet, let ε denotes the capital adequacy requirement ratio, and the amount of capital is the minimum capital required to be held on the balance sheet in the capital account.
2.2.2. Insolvency Analysis We analyze the condition of capital injection, insolvency and bankruptcy by using the following formulation.
Defining the bank’s equity as: If 3a Bank is solvent 3b => Bank is still solvent, but capital injection is required 3c Bank is bankrupt.
Hence, when the value of total assets in the bank is greater than total liabilities, and the equity capital of the bank is greater than the capital requirement, the bank is solvent. It is the state described by equation 3a. However, in equation 3b, if the value of the total assets is greater than the value of total liabilities, but the capital requirement is not fulfilled, then the bank needs to sell some assets to increase equity capital. The bankruptcy scenario is represented by equation 3c, where capital adequacy requirement is also violated. Since the bank does not have extra assets to sell, it will be bankrupt.
Role of securitization and its benefit on assetliability management: α : the proportion of securitized assets, we assume the securitized assets are held as liquid assets or cash. Consequently, this part of total assets will not be constrained in relation to the capital adequacy requirement^{1}. Hence the capital required can be reduced to . If the capital adequacy requirement is greater than the equity capital in the bank, the capital injection is needed. Otherwise, the capital accumulation can be realized. M in equation (4) defines the condition for capital injection/accumulation.
(4) Hence if M > 0 Capital injection is needed M < 0 Capital accumulation
The asset accumulation process with securitization can be expressed as: (5) At time t, the nonsecuritized assets are , and these assets are subject to a survival rate, which is . As a result is the amount of assets that will survive from period t to period t+1. Then at the same time, the bank keeps as liquid assets that have been raised from securitization. Moreover, bank will get the investment return from the surviving asset, which is ^{2}in equation (5). In addition, if the equity capital falls below the capital requirement, the bank needs to sell M amount of assets to replenish the capital account, hence “M” has to be subtracted from equation (5) is the cost of securitization, which is needed to be subtracted from the total benefit.
Substituting equation (4) in equation (5), we have: (6) where
(7) and . ^{3}
From equation (6), q can be treated as the asset accumulator, which cause asset to grow at the rate of q. Since is usually greater than 0; q is a positive function of α. If α increases, q will increase, and assets will start accumulating. Clearly, if ε is too high, it can put a brake on the asset accumulation . Solving equation (6) recursively, we have (8) hence, the equation can be rewritten as: (9) It is clear that from equation (6), q is a positive function of α. When α increases, the asset value grows.
2.2.4. Optimal securitization ratio
In the previous analysis, we find that securitization can help the bank to increase its profit. However, there is no consideration about the cost of securitization.
Case 1 Linear cost In previous research on securitization, usually the cost of securitization is set as a constant ratio; hence the cost is always a fixed proportion of the value of securitized assets. The cost function is usually written as , which is we assumed in the previous analysis. Since, The maximum capital accumulation for T=1 can be realized when capital injections are minimized or by maximizing the –M w.r.t α. Then we can have the optimal α as
Figure I.5 Capital accumulation and α analysis I (T=5) (linear Cost)
It’s clear that if the cost of securitization is lower than the profit of securitization, the asset accumulation will increase. However, if the cost is higher, the bank will stop securitizing its assets.
However, it is not quite the truth in the financial market. Since the quality of the bank’s assets is varied, the cost of securitization would be varied as well. ^{4} Therefore, if the cost of securitization is not linear, the bank can not securitize their assets without any constraints.
Case 2 Nonlinear Cost The nonlinear cost function can be written as: (11)^{5} In this formula, if α increases, the total cost will increase at nonlinear rate. Hence equation (7) can be rewritten as: (12) and . ^{6} nd substitute equation (12) to (4), we can get: (13)
The maximum capital accumulation for T=1 can be realized when capital injections are minimized or by maximizing the –M w.r.t α. Then we can have the optimal α as (15)^{7}
In equation (15), α is expressed as an equation of β, γ, ε, θ and . Here γ is negatively related to α, while ε is positively related to α^{8}. It means if the bank is subject to higher credit rate risk (lower survival rate) and higher capital adequacy requirement, it tends to securitize more assets to obtain the higher profit.
For the multiperiod optimization, we can set T > 1, and numerically solve the optimal value for α ^{9}. Figure 6 plots the relationship between the capital accumulation and α when θ=0.05 when T=5
Figure I.6 Capital accumulation and α analysis I (T=5) (Nonlinear Cost)
The optimal values for from T=1 to T=5 is plotted in the Figure 7. It is shown that when the decision horizon T is greater, the optimal value for is smaller. The reason behinds this phenomenon is that for a longer decision horizon, the bank has more time to achieve the maximum profit. Figure I.7
Note that normally we assume the optimal value of securitization 0<<1, but in certain circumstance^{10}, it can be greater than 1. Figure I.8 and α analysis I (T=5) (Nonlinear Cost)
In Figure 6 and 8, it is clearly shown that securitization can help the bank to increase the capital accumulation, as well as the ROE; even the bank is facing the worst default rate risk (0.07). However, the benefit from securitization will be partially offset by the increasing nonlinear cost. It will not be beneficial to the bank to securitize any more assets than the optimal level.
Moreover, Figure 9 shows the results of the sensitivity analysis on the capital accumulation for different parameters in the cost functions (θ=0.05, 0.075 and 0.15 separately) when T=5. Figure I.9: Capital accumulation and α analysis II (T=5) (Nonlinear Cost)
Figure 9 implies that securitization can help the bank to increase the capital accumulation. . Moreover, when the parameter for the cost function θ is increasing, the margin for the bank to reach higher profit is decreasing. As it is shown in the Figure 8, when θ is increased from 0.05 to 0.1, the capital accumulation is continuously declining and same to optimal α. Therefore, if the cost for securitization increases, bank should securitizes less and the capital accumulation will be reduced.
2. The Model:
2.1. Objective of the model
The bank’s activity on maximizing profit is constrained by insolvency risks in the course of operations. Bank insolvency risks arise from a number of sources, which are mainly credit or bad debt risk, and the impact of interest rate risk^{11}.
Credit risk constitutes the most important risk among all the financial risks in terms of its potential losses. It is defined by the losses in the event of default of the borrower, or in the event of a deterioration of the borrower’s credit quality. It is captured by the default probability with which some assets are destroyed from one period to the next. Since credit risk is the most common cause of bank failures, supervisory community set strict regulations to control the risk^{12}.
Interest rate risk is a consequence of the sensitivity of capital and income to changes in interest rates, which originates in the changes in the slope and shape of the yield curve. When interest rates fluctuate, a bank’s earnings and expenses exhibit changes, as do the economic value of its assets, liabilities, and offbalance sheet positions. Interest rate risk is modeled by the wellknown asymmetry in the duration between assets and liabilities. The duration gap and hence interest rate risk is measured (approximately) by the difference in the respective derivatives of the present values of assets and liability visàvis the interest rates^{13}.
Further the bank is operating under the assumption that there is capital adequacy requirement and capital has to be replenished by the bank from its operating profits if at each period its assets fall below its liabilities plus the capital requirement.
In order to protect banks from these former risks, a certain amount of capital is required as a buffer against unexpected losses. Moreover, the amount of capital held by a bank must be commensurate with its level of risk^{14}. Therefore capital adequacy requirement is a constraint imposed to the bank as a requirement regarding the building of a safety net. The current credit risk capital regime for banking institutions was established by the 1988 Basle Capital Accord, which was an agreement by international banking regulators to impose a common system of capital requirements on commercial banks^{15} The Accord was also subsequently adopted more widely and has become the global benchmark for regulatory credit risk capital standards. Within the European Union, the Capital Accord was first adopted for banks through two directives: the Solvency ratio and Own Funds Directives, and was later also applied to investment firms^{16}. With the adopting of The Accord, the risk management in the bank is largely improved.
However, the cost and amount of this required capital will largely affect the profit of the bank. An important benefit of securitization is that it can permit the bank to reduce the capital adequacy requirement imposed by the regulatory provisions. After securitization, the securitized assets will be transformed into liquid assets, which are subject to a much lower or even zero capital requirements. By doing so, the bank can reinvest its capital in other projects and increase its profit.
The features of the model are: It uses a dynamical way to assess the effectiveness of securitization on profit maximization and interest rate risk immunization. The credit risk is captured by the default probability with which some assets are destroyed from one period to the next. It is based on the stylized facts of the FDIC insured banks^{17}; and the interest rate risk is modeled by the wellknown asymmetry in the duration between assets and liabilities; the bank is risk neutral.
2.2. The First Model – Securitization and Profit Maximization in the Bank
2.2.1 Bank Assets and Liability Management: In order to analyze the effect of securitization on the asset value, in term of the balance sheet variables, we assume that the bank’s liabilities are composed of a fixed value of deposits at time t. Hence the liability at t+1 is given by adding on the interest rate payment (1) with is the initial value of deposit of the bank.
As for assets, traditionally, the growth in the value of assets is influenced by the following two factors: (i) Return on the asset: . (ii) Default on assets denoted by () which is the proportion of defaulted loans and hence is the survival value of assets from t. Hence, we have (2) Moreover, to consider the impact of capital adequacy requirement on the bank’s balance sheet, let ε denotes the capital adequacy requirement ratio, and the amount of capital is the minimum capital required to be held on the balance sheet in the capital account.
2.2.2. Insolvency Analysis We analyze the condition of capital injection, insolvency and bankruptcy by using the following formulation.
Defining the bank’s equity as: If 3a Bank is solvent 3b => Bank is still solvent, but capital injection is required 3c Bank is bankrupt.
Hence, when the value of total assets in the bank is greater than total liabilities, and the equity capital of the bank is greater than the capital requirement, the bank is solvent. It is the state described by equation 3a. However, in equation 3b, if the value of the total assets is greater than the value of total liabilities, but the capital requirement is not fulfilled, then the bank needs to sell some assets to increase equity capital. The bankruptcy scenario is represented by equation 3c, where capital adequacy requirement is also violated. Since the bank does not have extra assets to sell, it will be bankrupt.
Now we will discuss the role of securitization and its benefit on assetliability management. Denoting α as the proportion of securitized assets, we assume the securitized assets are held as liquid assets or cash. Consequently, this part of total assets will not be constrained in relation to the capital adequacy requirement^{18}. Hence the capital required can be reduced to . If the capital adequacy requirement is greater than the equity capital in the bank, the capital injection is needed. Otherwise, the capital accumulation can be realized. M in equation (4) defines the condition for capital injection/accumulation.
(4) Hence if M > 0 Capital injection is needed M < 0 Capital accumulation
The asset accumulation process with securitization can be expressed as: (5) At time t, the nonsecuritized assets are , and these assets are subject to a survival rate, which is . As a result is the amount of assets that will survive from period t to period t+1. Then at the same time, the bank keeps as liquid assets that have been raised from securitization. Moreover, bank will get the investment return from the surviving asset, which is ^{19}in equation (5). In addition, if the equity capital falls below the capital requirement, the bank needs to sell M amount of assets to replenish the capital account, hence “M” has to be subtracted from equation (5) is the cost of securitization, which is needed to be subtracted from the total benefit.
Substituting equation (4) in equation (5), we have: (6) where (7) and . ^{20}
From equation (6), q can be treated as the asset accumulator, which cause asset to grow at the rate of q. Since is usually greater than 0; q is a positive function of α. If α increases, q will increase, and assets will start accumulating. Clearly, if ε is too high, it can put a brake on the asset accumulation . Solving equation (6) recursively, we have (8) hence, the equation can be rewritten as: (9) It is clear that from equation (6), q is a positive function of α. When α increases, the asset value grows. Now we present a preliminary graphical analysis of the impact of securitization on the growth of asset value. Firstly, we assume no securitization through the condition α=0. Figure2 shows the results of the sensitivity analysis on the value of total assets for different default rates when there are not securitization activities. The chosen default alternative rates are ^{21}. (assuming, , , , , , )
In the Figure 2, Line 4 represents the total value of liabilities from period 1 to period 8. If the value of asset is below Line 4, the bank is bankrupt. It can be observed from Line 1(default rate is 0.07), where the bank is bankrupt in period 4. Only in Line 3 and Line 2 scenarios, which have lower default rates (0.05, 0.006 respectively), there is no bankruptcy.. Therefore when the default rate is higher than 5%, the bank shuts down.
Figure I.2 Asset value growth for different survival rates (without securitization) Source: Calculated by the author,
Then we precede this exercise by considering the worst scenario where the default rate is 7%, but assuming securitization is taken place. This scenario is analyzed in Figure 3. Setting α equal to 0.1, 0.2, and 0.3 separately, which means the bank chooses to securitize 10%, 20% or 30% of its assets. By holding all other variables unchanged, we have Line 1, 2 & 3 respectively to represents the different assets values. Line 5 shows the value of liability at different periods. It can be found that securitization can relieve losses caused by the default risk in the bank. Line 3 shows that when the bank securitizes 30% of its assets, not only it will survive to the next period, remaining solvent, but also to boost the growth of asset value^{22}. Then thinner lines show the capital injection needed for the bank. Line 8 shows that when α =30%, M becomes negative, which means the capital held by bank is greater than the requirement, hence no injection is needed.
