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As elements of the financial system, banks perform a variety of functions, including that of operators maintaining clients’ accounts, funds transfers, securities trading, currency exchange etc. However the main function banks perform as financial intermediaries is transforming the attracted (borrowed) funds in the loans required by real economy and people.

In the context of economic instability, the role of banking risk management becomes of paramount importance. According to GARP, the five core categories of banking risks include: *credit* risk, *market* risk, *liquidity *risk, *operational risk* and *event* risk.

The banking risks literature pays most of its attention to *credit* risks. A wide range of mathematical models of credit risks that became commercial products (*CreditMetrics ^{TM}, EDFCalc*

^{â}*, CreditRisk+, CreditPortfolioView*) are in place, but credit risk modeling, prediction, analysis and management still remain in the highlight [].

^{TM}*Market* risk modeling has quite a long history dating back to studies of Markowitz (*Markowitz,* 1952) and Roy (*Roy*, 1952), and is related with optimizing asset portfolios in risk/yield terms, when the most common risk management measures include either portfolio variance, or various indicators generated by *Lower Partial Moments*, *LPM _{k}*(t) of order k relative to t) in different combinations of k and t, such as default probability (DP), mean absolute semi deviation (MASD), standard semi deviation (SSD) etc.

*Operational* risks are mostly referred to bank management operations, HR management and information technology processes, while *event* risks are related with external shocks that can be treated as exogenous scenarios in stress testing practices.

Particular attention should be given to *liquidity *risks, whose fully-fledged analysis and management, especially stress testing, require to approach the bank as a dynamical system. At the same time, the existing approaches to analysis and adequacy of bank liquidity rely mostly on static single-period models [1-6]. Not so much experience has been gained by the banks in what concerns multi-period models [7], and the simulation models they underlie [].

It’s also worth noting that depending on the model focus and the scope of solved tasks, the employed mathematical tools vary considerably. Thus, credit risks are simulated using a variety of probability (stochastic) models, linear and non-linear programming models are used for risk-based asset optimization, multi-period models of assets / liabilities management are discrete-time recurrent (difference) equations. In this paper, we will use continuous-time models based on differential equations.

** Substantiation of the Model Aggregation Method**

In general terms, loan dynamics x(t,t) is described with the transfer equation[1]

¶x/¶t+¶x/¶t = -ex + u(t, t) (1)

where t is the current time, t is the time counted down from the date of loan disbursement, u(t,t) is the loan disbursement function, e(t) is the velocity (rate) of loan repayment

Since loans are usually granted on some standard term T_{k} (one day, one week, one month, three months, six months, one year etc.), the equation (1) may be presented as a set of same-type equations k=1,2,3,…

¶x_{k}/¶t+¶x_{k}/¶t = -e_{k}x_{k} (2)

with boundary conditions of x_{k}(t,0) = u_{k}(t), each of which has an analytical solution

x_{k}(t,t)= u_{k}(t-t)exp(-e_{k}t) (3)

Therefore, on the date t, total loans disbursed for the term T_{k} are equal to

x_{k}(t)=òx_{k}(t,t)dt= òu_{k}(t-t)exp(-e_{k}t)dt, (4)

and their dynamics is described as

dx_{k}/dt=u_{k}(t) — e_{k}x_{k} — u_{k}(t-T_{k})exp(-e_{k}T_{k}) (5)

Then, the loan portfolio dynamics х(t)

dx/dt=u(t) — e^{*}x — Su_{k}(t-T_{k})exp(-e_{k}T_{k}) (6)

where х(t) is the loan portfolio value, u(t)= Su_{k}(t) is the total flow of disbursed loans, e^{*} =(Se_{k}x_{k})/x is the weighted average rate (velocity) of loan repayment

х(t)=Sx_{k}(t)= Sòu_{k}(t-t)exp(-e_{k}t)dt (7)

Let’s present the output flow as

u_{k}(t-T_{k})exp(-e_{k}T_{k})= u^{*}_{k}(t) + Du_{k}(t) (8)

where u^{*}_{k}(t) is the current average amount of loans disbursed for the term T_{k} taking into account their repayment (amortization), Du_{k}(t) is the loan deviation from the mean value

u^{*}_{k}(t) =x_{k}(t)/T_{k} (9)

It follows from (7) and (9) that the loan portfolio may be presented as

x(t)=Su^{*}_{k}(t)T_{k} (10)

Let’s define the loan portfolio turnover time T_{x} as

T_{x} =x(t)/Su_{k}(t-T_{k})exp(-e_{k}T_{k})= Su^{*}_{k}(t)T_{k}/[Su^{*}_{k}(t)+ SDu^{*}_{k}(t)] (11)

While Du^{*}_{k}(t) values can vary within a wide range, the total deviation SDu^{*}_{k}(t) from the average flow is insignificant in the stable bank, i.e.

SDu^{*}_{k}(t)/Su^{*}_{k}(t)<<1 (12)

Then, to a first approximation, the turnover time T_{x} is a weighted average of loan term T_{k}

T_{x} = Su^{*}_{k}(t)T_{k}/Su^{*}_{k}(t) (13)

The turnover time T_{k} in its meaning is equivalent to duration D_{k}(t), which is the weighted average time before asset or liability is discharged, but it is calculated much easier.

It follows from the model (2)-(3) for the loans disbursed for the term T_{k}:

D_{k}(t) = [1/x_{k}(t)]ò(T_{k}-t)x_{k}(t,t)dt = T_{k} — [1/x_{k}(t)]òtu_{k}(t-t)exp(-e_{k}t)dt (14)

In case of constant flow of payments u^{*}_{k}(t) and e_{k}=0, the duration is obvious to be equal to one half of the turnover time

D_{k}(t)= T_{k}/2 (15)

In case of payment flow, which is decreasing as it nears the time of repayment, duration is growing. Thus, in case of e_{k}T_{k}=1, i.e., when the debt is reducing by the end of the loan term by e=2.72 times, D_{k} = 0,582T_{k}.

Expressions similar to (1)-(13) can be found when describing dynamics of the deposits provided for the term y(t,t) with the only difference being that the negative member e_{k}x_{k} meaning loan repayment is replaced with the positive member meaning accrual of deposit interest r_{k}y_{k}.** **

**The Aggregated Model of the Bank**

To present the logics of operations of the banking institution, let us consider the simplest high-level model of dynamics of the core financial flows, which, nevertheless, describes key aspects of its operations. In the most compact form, which is convenient for a mathematical study, this model is further stated as a system of ordinary differential equations.

When choosing the state vector let’s limit ourselves with five aggregated balance sheet items, only four of which are independent in accordance with the principle of equality of assets and liabilities (Table 1). Shareholder’s (equity) capital usually acts as the balancing variable.

The exogenous variable – borrowed and attracted funds (term deposits and demand deposits of individuals and legal entities, clients’ account balances, loans from other banks) serves as the principal source of bank’s funds and the starting point of the model.

Dynamics of term deposits у_{1} and demand deposits y_{2} within the aggregated model is described with same-type equations (1), so for the sake of simplicity, these components of liabilities are combined y = у_{1}+y_{2}, while parameters T_{y}(t) and r_{y} are weighted average

dy/dt = v(t)-y/T_{y}(t)+r_{y}y (1)

where v(t) is a net inflow of funds to the deposit accounts, T_{y}(t) is the time of liabilities turnover (average time the funds are on the deposit accounts), r_{y} is the interest accrued on deposits. • Here it is suggested that the interest should be withdrawn together with the principal amount of the deposit at the end of the term, though the model may also use another approach, when the interest is withdrawn as far as it is accrued.

The main issue of liabilities simulation is that v(t) is a random process. In case of crisis developments, the inflow v(t) is decreased, and D_{y}(t) is reduced as a result of outflow of funds from client accounts and withdrawal of term deposits (if the latter is provided for under the agreement conditions).

One can state three approaches to prediction and simulation v(t).

*Scenario approach*. A set of possible (suggested) exogenous time-varying functions v(t) etc. (scenarios) are specified.*Statistic approach*. To build v(t), one of the methods of prediction (simulation) of time series is used.*Bayesian approach*. It is based on combining the scenario approach with one or multiple random variables. Depending on the value taken by this random variable, one or another scenario of inflow and outflow of funds takes place in the certain time interval.

Table 1. Standard Structure of the Aggregated Balance Sheet of a Commercial Bank, %

Assets | Liabilities | ||

Loans disbursed to legal entities and individuals x(0) | 60 | Equity c(0) | 10 |

Debt securities b(0) | 15 | Deposits (term and demand deposits, client accounts) and borrowed funds y(0) | 90 |

Correspondent accounts, reserves, cash s(0) | 25 | ||

Total assets | 100 | Total liabilities | 100 |

The bank’s loan portfolio is generated with the attracted (borrowed) funds

dx/dt = u(t) — x/T_{x}(t) — ex (2)

where u(t) is the net outflow of funds in the form of disbursed loans, T_{x}(t) is the assets turnover time, e is the loan amortization factor (loan repayment speed).

Usually, when a loan is disbursed, a deposit account (loan facility) is opened at the same time on the liabilities side, with the borrower withdrawing funds in installments as required from this account, but for the sake of simplicity only the resultant flows are taken into account in the model.

Borrowing demand g(t) can either exceed the funds at the bank’s disposal h(t), or be insufficient. That is why

u(t) = min{g(t), h(t)} (3)

where g(t) is the borrowing demand, h(t) is the bank’s funds planned to be disbursed as loans.

Approaches to simulation of the borrowing demand g(t) are similar to that described above for deposit inflow simulation v(t).

Pursuant to the banking risk management policy, only part g_{x}<1 of the available funds is allocated for lending

h(t) = g_{x}(t)q(t), (4)

where q(t) is estimated available funds of the bank (inflow less outflow of funds).

Other bank funds are spent to purchase other earning assets, or can be allocated to increase funds in the correspondent accounts and as cash s(t) thus used as the reserves aimed to mitigate liquidity risks.

Most part of available funds of the bank, including non-demanded funds intended for lending max{0, h(t)-g(t)} is placed by the bank in the portfolio assets – investment securities, mostly in the dated bonds, and traded risk assets (shares). At the same time, available securities are paid off or sold. This mechanism can be described as follows

db/dt = w(t) + max{0, h(t)-g(t)} — b/T_{b}(t), (5)

where b(t) is investment in securities, w(t) is bank’s funds planned to be used for purchasing portfolio assets

w(t) = g_{b}(t)q(t) (6)

where g_{b} is a part of the funds spent on purchasing the securities, T_{b}(t) is the turnover time of the securities portfolio.

The key issue of asset management is the algorithm of channeling the bank’s funds that, in case of reasonable management, is supposed to depend on the estimated net inflow q(t).

This algorithm may be presented as follows. The available investment resources of the bank q(t) are calculated as the resultant between the inflow (released funds, interest income, deposit growth, redemption of securities) and output flow (growth of reserves, interest expenses, bad debts, operating and other expenses)

q(t) = dy/dt – dr/dt + (1- x)x/T_{x}(t) + b/T_{b}(t) + ex + r_{x}x + r_{b}b — r_{y}y – z(t) (7)

where r_{x}, r_{y}, r_{b} are yield rates, of loans, deposits and securities, respectively (coupon yield), r(t) is the reserve fund, z(t) is the planned operating expenses and other bank payments, 0<x(t)£1 is a random process that characterizes bank loss from the bad debts.

Let’s provide explanations on certain equation elements (7).

The principal part of the funds attracted by the bank must be secured with legal reserves. In Russia, the legal reserves are withdrawn from the banks, placed in non-interest bearing account in the Bank of Russia and can be used to cover the liquidity shortage, only if the set of conditions is met (averaging mechanism). Besides, the bank must establish additional reserves for possible bad debts and as security of current payments. Primary reserves, as combined with the government stock (secondary reserves), create the required liquidity cushion that ensures bank stability against adverse changes of the external conditions.

Further, as a separate component, we’ll single out the reserves available to support liquidity as percent of the attracted funds, with this percent (above the legal requirement) may be regulated by the bank itself

r=ay (8)

where a is the cash reserve ratio.

Taking into account that the bank may modify the reserve percentage in a flexible manner

dr/dt=ady/dt+yda/dt (9)

Formula (7) allows to design the criterion of and assess solvency of the bank. Reduced resources q(t) alert solvency reduction. This can occur, when the deposit outflow starts exceeding their inflow, i.e. dy/dt becomes negative, the amount of bad debts x grows and operating expenses z(t) increase. If q(t) becomes negative, it means that the bank starts shifting to reduced liquidity, which can finally lead to insolvency and bankruptcy, when the equity becomes negative. (7), taken together with (8), returns the necessary condition (lower limit) of the financial stability

(1- x)x/T_{x}(t) + b/T_{b}(t) + ex + r_{x}x + r_{b}b > z(t) + r_{y}y — (1-a)dy/dt (10)

We’ll define the financial stability headroom of the bank by the ratio c that may be used as the stability criterion and scale

c(t) = [(1- x)x/T_{x}(t) + b/T_{b}(t) + ex + r_{x}x + r_{b}b]/[z(t)+ r_{y}y -(1-a)dy/dt] (11)

As an expert evaluation, we can offer the following scale: 1<c<1,5 – low stability, 1,5<c<3 – medium stability, c>3 – high stability. Apparently, this parameter is fluctuating throughout the bank operations going down during economic recession featuring reduced lending demand and outflow of funds from depositors’ account.

To make current payments, the bank needs available cash in the correspondent account of the correspondent banks and in its cash office. These most liquid components of the assets (primary reserves), including bad debt reserves, are united in variable s(t).

As shown above, under pressure, in case of economic shocks or the bank’s high-risk lending policy, the value q(t) may turn out to be negative thus resulting in termination of loan business, suspension of acquiring other assets and reduction of s(t). When the bank’s financial situation improves, including as a result of state support measures, resolution and capitalization increase, the flow of resources reverses sign and the liquidity adequacy s(t) must be restored.

With this taken into account, both previously introduced variables – lending cash flows h(t) and portfolio investment cash flows w(t) should be adjusted as follows

h(t) = g_{x}(t)max{0; sgn(0, s-r)}max{0, q(t)}, (12)

w(t) = g_{b}(t) max{0; sgn(0, s-r)}max{0, q(t)}, (13)

and in the cash dynamics equation s(t) it is necessary to provide possible switching between the modes of expenditure and replenishment

ds/dt = sgn(0, r-s)max[0, q(t)] + min[0, q(t)] + dr/dt (14)

as mentioned above, the bank’s equity is a balancing variable, i.e.

c=x+s+b-y (15)

dc/dt = ds/dt + dx/dt + db/dt — dy/dt = r_{x}x — r_{y}y + r_{b}b – z(t) — xx/T_{x}(t)

+ min[0, q(t)] (16)

Bank’s equity grows due to profit (less the income tax and dividends paid to the shareholders). For the sake of simplicity, taxes are not accounted in this model. The dividends are also considered not to be distributed, and all profit is allocated to increase the equity value.

The capital adequacy ratio is used as the main structural constraint. In this model, the constraint takes on the form as follows

c(t)/[(1-f)A] = c(t)/{(1-f)[c(t)+y(t)]} ³ q (17)

where f is the share of the risk-free assets, q is the capital adequacy ratio (q=0.08 according to the Basel Committee guidelines, q=0.1 for Russian banks).

Therefore

c(t) ³ {(1-f)q/[1-(1-f)q]}y(t) (18)

Further, k ratio is more convenient to use as the adequacy ratio

k = (1-f)q/[1-(1-f)q] (19)

For example, f=0.3, q=0.1, then k= 0.075.

The built model describes dynamics of the main variables of the bank’s condition, allows to simulate mechanisms of management and transformation of cash flows and study sensitivity of the balance sheet items and bank profit to the management efforts and external factors, including stress. Thus, it can be considered as a backbone for the theoretical and analytical research. At the same time, aggregating balance sheet items, use of the integral parameters of turnover of assets and liabilities, and the assumption of the continuous smooth character of the used functions prevent from showing some important aspects of the bank’s operations. The next step in enhancing adequate description of the bank’s operations is using the distributed parameter models [2].

**Structural Constraint Impact on the Bank Performance**

Just like any other financial organization attracting funds of people and companies, every bank acts in the context of tight restrictions imposed by the external regulator (in Russia it is the Bank of Russia, in the USA it is the Federal Reserve System), and internal rules. These restrictions are aimed to maximize mitigation of various banking risks, but at the same time they considerably affect the structure and performance of assets.

Suppose that the bank is stable for some period of time, i.e., its amount and structure of assets and liabilities remain unchanged, while profit is fully distributed and its equity does not grow. In this case, one can analytically study impact of different parameters on bank’s financial performance, provided the restrictions imposed on the balance sheet structure by the supervisory body are met.

Then the derivates and a series of members in equations (1)-(2), (5), (14), (16) are set to zero, and one can completely define the balance sheet components via the model ratios.

The borrowed capital is determined by the product of the cash inflow rate by the modified turnover time T^{*}_{y}

y* = v/(1/T_{y}-r_{y}) = vT^{*}_{y} (20)

and the equity, pursuant to the constraint (18), must be at least

c* = ky*, (21)

where k=(1-f)q/[1-(1-f)q].

In the steady mode, pursuant to (14), non-performing assets are minimum and equal to liabilities

s* = r*= ay* (22)

Investment in low-income but reliable (low-risk) assets such as government stock are aimed to ensure financial stability of the bank and mitigate risks. The amount of these investments must correlate with the bank’s equity.

Then this component of the assets can be determined as

b* = nc* (23)

Further, we find the value of disbursed loans from the conditions of the balance sheet account,

x* = c* +y* — s* — b*, (24)

As a result, the bank’s balance-sheet may be presented analytically:

Table 2.

Bank’s Balance Sheet Based the Model Parameters

Assets A=(1+k)vT^{*}_{y} |
Liabilities L=(1+k)vT^{*}_{y} |

Loans disbursed to legal entities and individuals x* = [1+k(1-n)-a]vT^{*}_{y} |
Equity c* = kvT^{*}_{y} |

Funds (reserves, correspond account, cash) s* = avT^{*}_{y} |
Deposits (term and demand deposits), client accounts and borrowed funds y* = vT^{*}_{y} |

Securities b* = nkvT^{*}_{y} |

Interest income (margin) of the bank m taking into account the estimated loan loss ratio Ex is calculated as

m = r`_{x}x* + r_{b}b* — r_{y}y* = (r`_{x} + r_{b}nk — r_{y})y* (25)

where r`_{x} = r_{x} — (Ex)/T_{x}

Operation expenses z may be interpreted as some imputed rate r_{a} of the bank asset servicing

z=r_{a}A, (26)

then the pre-tax profit p amounts to

p = m – z = [r`_{x} + r_{b}nk — r_{y} — r_{a}(1+k)]y* (27)

Return on assets

ROA = p/A = [r`_{x} + r_{b}nk — r_{y} — r_{a}(1+k)]/(1+k) (28)

Return on equity

ROE = p/c* = ROA(1+k)/k (29)

**Conclusion **

The aggregated model of the bank as a dynamic system with lumped parameters allows to clearly show the transformation mechanism of core cash flows and formalize various rules of assets and liabilities management. Computer-aided implementation of this model may be used for computational studies of efficiency of different asset management algorithms.

Main management parameters of the bank’s balance-sheet that support choosing adequate combination of yield and liquidity risk include: T_{x} – the loan portfolio turnover time, T_{b} – the securities portfolio turnover time, r_{x} – the loan rate, r_{y} – the deposit rate, a – the cash reserve ratio.

In the near-stable situation it’s not difficult to derive simple analytical expressions allowing to research the impact of these parameters on the bank’s yield and liquidity risks. Thus, one can use formulae (28)-(29) to study impact of the control parameters, including a variety of ratios, on the yield, and correlate it with the loan portfolio risks denoted by Ex value in this model. The liquidity risk depends on the assets / liabilities turnover time ratio (T_{x} and T_{y}). Since T_{y} value is not used in expressions (28)-(29), then the ratio T_{x}/T_{y} is an independent parameter that can be used when analyzing the bank’s standing in risk/yield terms. Let us note that the loan and deposit yield rates affect the respective cash flows v(t) and g(t) and must be taken into account when simulating these random processes.

It is seen from (28)-(29) that k equity ratio to the amount of attracted and borrowed funds significantly impacts the return on equity, but barely affects the return on assets that depends mostly on their structure and interest margin.

The suggested model can be easily extended through drilling-down to the financing sources (demand / term / savings deposit etc.) and asset placement methods. To dramatically enhance the model adequacy, it is necessary to take into account the time structure of loans and term deposits [2].

*The research has been completed with financial support of the Ministry of Science and Higher Education of the Russian Federation in the discipline: “Fundamental and Applied Tasks of the Mathematical Simulation” No. 1.5169.2017/8.9* * *