Dynamic balance sheet model with liquidity risk. No 1896 / April 2016.
Date: 2016-04-29
Institutions: European Central Bank
Subjects: bank, banking, banking policy, financial institution, financial market

Working Paper Series 

Grzegorz Hałaj  

Dynamic balance sheet model with 
liquidity risk 

No 1896 / April 2016 

Note: This Working Paper should not be reported as representing the views of the European Central Bank (ECB). 
The views expressed are those of the author and do not necessarily reflect those of the ECB. 

 
 
 
Abstract

Theoretically optimal responses of banks to various liquidity and solvency shocks are mod-
elled. The proposed framework is based on a risk-adjusted return portfolio choice in multiple
periods subject to the default risk related either to liquidity or solvency problems. Performance
of the model and sensitivity of optimal balance sheet structures to some key parameters of the
model are illustrated in a speciﬁc calibrated setup. The results of the simulations shed light on
the eﬀectiveness of the liquidity and solvency regulation. The ﬂexible implementation of the
model and its semi-analytical solvability allows for various easy applications of the framework
for the macro-prudential policy analysis.

Keywords: asset structure, optimal portfolio, banking
JEL Classiﬁcation: G11, G21, C61

ECB Working Paper 1896, April 201611 Non-technical summary

Banks’ balance sheets change their composition due to market conditions implied by a generic growth
of credit and debt in the economy and valuation changes, but also due to bank’s strategic actions.
The strategic driver is particularly interesting and challenging from the modeling perspective since
it combines the changes in risk and trends in the economy, the impact of the regulatory regime and
the asset and liability management (ALM). These strategic responses to the changing market and
regulatory parameters have to be well understood in order to gauge the inﬂuence and eﬀectiveness
that macro-prudential policies can have on banking system.

Banks optimise their asset and liability structure and although the goals are multi-dimensional
the main theme of risk-adjusted return maximisation prevails in their objectives. Banks need to
stay proﬁtable given an acceptable level of risk. The proﬁtability and adequate risk-taking increase
the capital base of banks and their resilience to the shocks and have a positive signaling eﬀect for
investors buying shares or debt issued. Moreover, the regulatory regime imposes on banks the need
to retain suﬃcient amount of capital to withstand the shocks and enough liquid assets to meet
their obligations in the majority of plausible scenarios that can materialise in a given horizon. The
regulation deﬁnes rigid limits to the expansion of the balance sheet and concentration of exposures
relative to capital base and liquidity buﬀers. All these conditions make the managing of banks’
balance sheets a challenging process which in consequence is also challenging from the modeling
perspective. This complications notwithstanding, practitioners and the research community are
seeking for the right framework to correctly model the sensitivity of the balance sheet structure to
the changing market conditions.

We propose a multi-period model of a bank maximising its risk-adjusted return on capital given
liquidity and solvency constraints. Solvency and liquidity limits are modelled in a ’worst case’
manner. It means that only those strategies of banks are admissible that guarantee with a very high
probability that the bank remains solvent and liquid. This approach to the risk measurement and
limits is reﬂected in the existent regulatory regime by the Value-at-Risk (VaR) concept, both in the
solvency context (Basel II VaR capital constraint or measurement of banks’ economic capital) and in
the liquidity context (Liquidity-at-Risk internal models of banks’ cash-ﬂow distributions, see Matz
and Neu (2006)). The proposed model is an extension of the classical stochastic programming tools
`a la Kusy and Ziemba (1986) which were developed to support the asset and liability management
process and incorporates the tail risk measures of liquidity and solvency.

The framework that we propose captures the main features of the ALM decision making process
under regulatory and internal risk constraints and is computationally tractable and easy to solve and
simulate. The model is semi-analytical, i.e. some parts of it have closed form solutions that improve
the speed of the Monte-Carlo simulations which are necessary to solve the optimisation program of
a bank (using the standard dynamic programming techniques). Therefore, it is applicable in the
stress testing context to endogenize bank’s reactions to adverse economic and ﬁnancial conditions
to understand re-balancing of banks’ asset portfolios following in particular credit risk or funding
risk shocks. Moreover, its tractability can be helpful in improving the treatment of banks’ decision
problems in the DSGE framework.

The simulations performed in a stylised setup of the parameters, yet realistic and reﬂecting
the most important trade-oﬀs that bank is confronted with (e.g. return vs risk or proﬁtability vs
solvency and liquidity conditions), provide with the insight into the relationship between the optimal
investment choice and the changes in the credit, market and funding risk parameters and in the risk
constraints. The results suggest some very non-linear eﬀects that changes in return and volatility
parameters can produce for the bank lending behaviour. These eﬀects are related to the changes in
relative risk-adjusted return characteristics of the diﬀerent balance sheet categories, the correlation
structure and the constraints binding for some threshold values of the parameters.

The results also give some insight into the policy of the liquidity and solvency regulation. The
regulation of the banks’ risk taking at the same time in the liquidity and solvency dimension may
not be eﬀective. More speciﬁcally, an improvement of solvency conditions may induce banks to
expose themselves to more liquidity risk. This trade-oﬀ revealed in our model supports a general

ECB Working Paper 1896, April 20162statement about regulation that potential consequences of regulatory policies have to be assessed
and weighted against each other with caution.

ECB Working Paper 1896, April 201632 Introduction

The aim of the study is to propose an analytical framework to analyse banks’ theoretically optimal
responses to changing liquidity and solvency conditions. Liquidity is understood either as a poten-
tial to liquidate assets at their book value or to roll-over the maturing funding sources. Notably,
the recent ﬁnancial crises showed that the liquidity shocks can have signiﬁcant impact on banks’
behaviours. Funding risk is actively managed by banks which means that banks try to ensure that
they have enough liquid sources to meet the obligations, funding outﬂows or collateral downgrades.
The active management determines the funding mix and it determines the volume of the liquid
assets that are used as a counterbalancing capacity against funding outﬂow and that creates a
potential for credit supply. Therefore, it is one of the most important factors of the investment
strategy; banks have to remain both liquid and proﬁtable. The solvency conditions have equally
important role in constraining the investment strategies of banks. Banks need to keep suﬃcient
capital to withstand the potential losses that they may incur. However, we emphasise the liquidity
dimension of our optimisation model since the solvency risk is a more standard component of the
micro-founded models of the bank behaviours.1

The fact that banks’ balance sheets tend to change dynamically may be particularly problematic
for ﬁnancial analysts and banking regulators. A usual static balance sheet assumption (e.g. taken
for the stress testing exercises) may be valid only in some very special cases of shocks of the low
magnitude and in relatively short periods (in cases it is reasonable to assume that the adjustment
may need time for a preparation and coordination of actions by the management of a bank). The
dynamic balance sheet feature of stress testing exercises was explained by Henry et al. (2013) and
RTF (2015).

There are at least four policy relevant aspects of the relationship between investment strategies

and funding conditions.

1. Macro-prudential policy analysis. The macro-prudential policies try to create incentives for
ﬁnancial system to smooth the processes of the real sector, in particular to provide enough
bank credit for the corporate sector and households. Eﬃcient policy tools have to take into
account the complicated dynamics of banks’ balance sheets. The proposed framework allows
for analysing the possible reaction functions of banks to funding shocks, solvency constraints,
credit risk impacting risk-return characteristics of the loan portfolios and the regulatory pa-
rameters (e.g. additional capital and liquidity buﬀers). The macro-prudential policies that
take into account banks’ optimisation of their balance sheets can replace the usually applied
rules of thumb.

2. Dynamic solvency conditions. Although a bank may be solvent from an accounting perspec-
tive (possessing enough capital at a given point in time to cover expected, and even part of
unexpected losses) it may not run any sustainable business. Funding sources may be volatile
enough and diﬃcult to roll-over carrying a substantial risk of ﬁre-sales of assets covering an
outﬂow of funding. The proposed model can help to detect balance sheet structures that may
lead to a substantial increase of insolvency risk in the future irrespective of currently applied
strategies.

3. Credit supply conditions. A bank may be forced to increase the share of the liquid assets in its
balance sheet to protect against volatility of funding sources. A preventive tactic to increase
liquidity buﬀer may have a negative externality of reduced potential to lend to the economy.

4. Eﬀectiveness of liquidity management. Risk based capital regulation has a strong and well-
established position in the traditional banking regulation.
It is less so in case of liquidity
rules imposed on banks. In Basel II they had a form of a code of good practice rather than
liquidity indicators and benchmarks. Such an approach was justiﬁed by a belief that setting
liquidity limits based on very aggregate supervisory data cannot be an eﬀective tool to control

1See for instance Danielsson et al. (2002); Hilberink and Rogers (2002); Cuoco and Liu (2006).

ECB Working Paper 1896, April 20164banks’ liquidity conditions. Contrary to solvency measures, aggregation for liquidity pur-
poses gives a very imprecise picture. Therefore Basel II related liquidity standards focussed
on promoting investment in internal liquidity management systems satisfying some high level
principles. Basel III and CRDIV departed from that approach and imposed minimal liquidity
ratios (short-term Liquidity Coverage Ratio (LCR) and longer-term Net Stable Funding Ratio
(NSFR)). This regulation addresses the causes of the crisis that erupted in 2007 and tries to
capture these characteristics of balance sheet items that are important from a liquidity per-
spective (e.g. stability of funding sources, generally expected haircuts on certain asset classes
or operational relationships with customers). This notwithstanding, the LCR and NSFR liq-
uidity rules also simplify the liquidity risk measurement and cannot replace a fully-ﬂedged
statistical and behavioural cash management, collateral management and sustainability of
funding sources. We apply in our framework the Value-at-Risk based liquidity measures.

To model the rational choices of banks about the structuring of their balance sheets some port-
folio optimisation techniques can be used. The deliberate actions taken by banks that change the
composition of the balance sheet have their rational economic goals. The complexity of the balance
sheet management is related to the fact that it is a multi-criteria problem with goals changing in
time depending on the liquidity and solvency outlook. Banks, as all other ﬁrms try maximise their
proﬁts but also have to build adequate buﬀers against possible ﬂuctuations of their funding, espe-
cially given the high leverage of the banks’ business model. Nonetheless, the optimisation tools,
especially adapted to the portfolio choice problems in ﬁnancial mathematics, provide a rich, the-
oretically well-founded toolkit to describe the process of the risk-adjusted proﬁt maximisation in
which banks are involved on a regular basis in their risk management and ALM activities.

The optimisation approach to the banks’ asset and liability management dates back to Kusy
and Ziemba (1986).2 The whole strand of research uses the stochastic programming techniques
from operations research (Consigli and Dempster, 1998; Klaassen, 1998; Robert and Weissensteiner,
2011), which in practice are well-suited for risk management problems operating with granular
portfolios and requiring many decision variables. Bank balance sheet problems can be tackled
with approaches of optimal portfolio choice with transaction costs (see Davis and Norman (1990);
Hilberink and Rogers (2002)). Transaction costs introduce stickiness in the rebalancing of the portfo-
lio structure by modeling of costly liquidation of a position. This approach is particularly appealing
for banks’balance sheets containing illiquid positions like loans or oﬀ-the-counter corporate bonds.
However, the theory of portfolio choice with transaction costs easily produces computationally un-
tractable problems. They work in a very simpliﬁed set-up not allowing for a usual heterogeneity
of balance sheet categories of banks’ books managed under various stringent capital and liquidity
rules. Another interesting, potentially applicable strand of research is related to the real option the-
ory of illiquid investment opportunities. In liability-driven asset management it has already been
employed by Ang et al. (2013). Some recent studies apply the robust optimisation tools (G¨ulpinar
and Pachamanova, 2013) that reduce variability of portfolio structures and are computationally
tractable.

Banks’ balance sheet optimal choice became part of many applications of the optimal portfolio
choice theory to the micro-foundation of banks behaviour and regulation. There are two broad
groups of approaches in this strand of literature. First, the DSGE models try to characterise the
equilibrium behaviours of diﬀerent stylised types of market players in the economy including banks
with the intermediation function (Darracq Paries et al., 2011; Aoki and Sudo, 2013; Clerc et al.,
2014). In this approach it is possible to endogenize the parameters of portfolio choice model taking
into account the aggregate eﬀects of the decisions of banks and other market players. In contrast,
we take the parameters as given but are able to operate in a tractable way with a richer structure
of the optimisation problem. Second, Zhu (2008); Acharya (2003) examine the impact of regulation
on banks’ investment and funding decisions to inform the optimal regulatory regime and assess the
credit supply conditions. The models in this group are similar in the setup to the one we propose

2In fact, the quantitative approaches to asset-liability management are more advanced in insurance context, see

Iyengar (2010). Some papers present very applicable solutions, e.g. Hilli et al. (2007).

ECB Working Paper 1896, April 20165but we focus on the shock transmission channels and nonlinearities related to regulatory as well as
to the tail risk constraints.

Our model is similar to Birge and J´udice (2013). The main diﬀerences of our approach stem from
inclusion of risky funding volumes, explicit treatment of ﬁre-sales and liquidity counterbalancing
buﬀers. We introduce the regulatory capital constraint to our model by means of the capital ratio
(i.e. the risk-weighted assets over Core Tier 1 capital base). In this respect, our modelling approach
is also closely related to the dynamic balance sheet model of Ha laj (2013). However, we introduce
also risk sensitive capital and liquidity limits deﬁned by the Value-at-Risk (VaR) of a bank’s capital
and Liquidity-at-Risk (LaR) of the funding sources. The concept of the VaR of the capital is
closely related to the VaR constraint embedded into the regulatory regime of Basel II (and Basel
III) inducing banks to structure their balance sheet in such a way that they should remain solvent
with a high (predetermined) probability. LaR is similar in philosophy to VaR used for the solvency
regulation (Matz and Neu, 2006). It limits bank’s exposures to the liquidity risk by allowing for
taking only these liquidity positions that with high probability ensure that the bank can meet its
obligations in the near future. Alternatively, the liquidity risk limits could be approached via the
regulatory LCR constraint (Balasubramanyan and VanHoose, 2013).
It is easy to conceptually
extend our model to cover the LCR limits but it would be diﬃcult to parameterise this type of a
constraint for the very aggregate balance sheet that we operate with.

In the traditional banking literature (Baltensperger, 1980; Boyd and Nicol´o, 2005; Pelizzon and
Schaefer, 2005) banks are assumed to take investment decisions under the risk neutrality assump-
tion.3 Risk impacts banks’ decisions only via regulatory constraints. We follow an approach from a
diﬀerent strand of literature (Howard and Matheson, 1972; Danielsson et al., 2002; Cuoco and Liu,
2006) where decisions are risk sensitive. Banks are assumed to be risk averse in our setup what
follows the capital management practice in banks where RAROC4 and RARORAC are common
indicators for managing accepted levels of the risk exposure and are the standard part of the ALM
indicators (Adam, 2008).5
In fact, Baltensperger (1980) admits that joint modelling of loan or
deposit volumes and diversiﬁcation within these two portfolios can be approached rather by utility
maximisation (implying risk sensitive decisions) than expected proﬁt maximisation.

Summing up, the main contribution of our paper to the banking literature is to provide a tool
to study endogenous balance sheet structures implied by both funding conditions and investment
opportunities. It has a reduced form that improves tractability of a constrained optimisation with
VaR-based risk measurement. Therefore, despite covering features of decision marking process under
tail risk constraints it is inexpensive to solve. On the potential application side, the framework
can be useful in the stress testing context when macro-ﬁnancial shocks have to be translated into
changes of the balance sheet structure. Moreover, it allows for the detection of tensions between the
demand for credit and the supply side conditions of banks’ balance sheets (insuﬃcient or excessively
volatile funding, a poor quality of the counterbalancing capacity or the credit risk accumulated in
the outstanding loan portfolios) that may hamper the credit expansion. Moreover, its insight into
banks’ theoretically optimal responses to some capital shocks and the liquidity requirements may
be helpful for the macro-prudential policy assessment, whereby changes in the capital requirements,
in the liquidity and credit risk parameters of banks’ balance sheets can be translated into the credit
supply drivers of some general tendencies in credit growth in the economy.6

A sensitivity analysis of the optimal strategies to the parameters of the model that we performed
shows some interesting patterns in the potential lending behaviours of banks and also some inter-
esting features of the model itself. First, the correlation between the risk drivers is a very important
factor that may change the lending behaviour, especially when the correlation in absolute terms

3See also Elyasiani et al. (1995); Balasubramanyan and VanHoose (2013).
4Risk-Adjusted Return on Capital and Risk-Adjusted Return on Risk-Adjusted Capital.
5The literature of the banking theory is not equivocal about the risk averseness of banks. The traditional strand
of literature builds on the assumption of risk neutrality (Baltensperger, 1980; Boyd and Nicol´o, 2005; Pelizzon and
Schaefer, 2005). In contrast, (Howard and Matheson, 1972; Danielsson et al., 2002; Cuoco and Liu, 2006) models
banks’ decisions as risk sensitive.

6E.g. it can help answering a question of the impact of more stringent capital requirements on banks’ propensity

to lend in given liquidity conditions and proﬁtability and riskiness of loans.

ECB Working Paper 1896, April 20166increases (particularly visible in case of the correlation between credit and market risk). It also
aﬀects the volatility of the capital level at the optimum. Second, changes in lending volumes are
in a non-linear fashion related to the volatility of securities and the correlation of the risks as an
outcome of the interplay of risk constraints (non-linear by their virtue), bank’s attempts to stay
proﬁtable and bank’s preferences of the risk-taking. For instance, the theoretically optimal lending
volumes rise as the volatility in the securities portfolio increases until the Value-at-Risk constraint
starts to be binding and the lending starts to decline following the maximum lending allowed by
the capitalisation of the bank. Third, lending is very sensitive to the relative proﬁtability of the
securities and loans – more sensitive than to the relative riskiness of the two asset classes. Fourth,
a change in some of the parameters of the balance sheet that improve the solvency conditions of a
bank may at the same time deteriorate its exposure to the liquidity risk. That trade-oﬀ renders an
eﬀective regulatory policy diﬃcult to calibrate.

Notably, the model in this reduced form cannot eﬀectively support the the risk management
process of a bank. It operates on a very aggregate balance sheet. The computational complexity
of the optimisation based approaches to the assets and liability management is still a problem
for current risk management support systems to allow for a comprehensive modelling of balance
sheet in its usually observed granularity. Although, there are prototypes of models relying on the
risk-return optimisation of banks’ exposures used for ALM7, the increasing power of IT systems is
needed to gain an eﬃciency acceptable in Management Information Systems. Moreover, the model
treats parameters as exogenous and several of them (e.g. these referring to the credit quality of
lending portfolios) may depend on the choice of banks. It may be particular simplifying if banks’
individually optimal portfolio choices are aggregated across the ﬁnancial system. To address the
shortcoming, a DSGE framework can be employed and our computationally tractable model oﬀers
an alternative to improve the usually very stylised optimisation of banks’ balance sheet structures.
In the following sections we present details (also quite technical) of the modelling approach
(section 3) and in section 4 we derive a dynamic programming equation for a 2-period model.
Section 5 provides some illustrative examples of parameterisations and sensitivity analysis of the
optimal lending strategies and corresponding solvency positions.

3 Model

3.1 Modelling approach

This section presents the details of a model of the asset structure choice of a bank.

In general, we work in a setup of the risk-adjusted return maximisation. The risk is related to
uncertain funding sources (the risk of an outﬂow of deposits), the credit risk in the loan portfolio
(outstanding and new volumes treated separately) and the volatile prices of the liquid securities.
All these types of risk are accumulated in the balance sheet of a bank that tries to maximise risk
adjusted proﬁt given the capital and liquidity constraints. The bank operates in a multi-period
(T -period in general, 2-period in the implementation) time frame facing the risk of:

ˆ falling into ill-liquidity when investing too much in loans and exposing itself to a risk of having

insuﬃcient liquid funds to meet the obligations at the end of each period;

ˆ falling into insolvency if losses (loan losses and devaluation of securities) erode capital base.

3.2 Risky funding sources

Assumption 1. There is only one type of funding sources available. It is described by an
autoregressive, one factor model. Funding requires to pay a non-random interest rate.
Funding risk is correlated, in particular with the value of securities portfolio.

7See Adam (2008).

ECB Working Paper 1896, April 20167Bank’s funding volumes are assumed to be homogenous (i.e. consisting of only one type of funding
sources) and follow a simple autoregressive risky process. We deﬁne risk in a probability space with
ﬁltration:
describing a shock to the
funding sources F , with the following dynamics:

t)t=1,2,....8 Let η be a process on

(R), F), F = (

= (R, P,

F

P

P

B

F (t + 1) = F (t) + γF (t) + η(t + 1)

(1)

The initial funding of a bank is denoted F (0).

The funding deposits pay an interest rate rF , which implies the interest expenses of C(t) =

rF F (t) due at the end of the period [t

1, t].

−

The change in funding may imply the need to ﬁre sale liquidation of part of the securities
portfolio. Fire-sales are triggered by the drop in the stock of funding. The inﬂow of funding
1))− implies the need to
(F (t)
“ﬁre-sale” part of the liquid assets.9 The loss due to the ﬁre sales is proportional to the liquidated
1))−, a bank needs to
volume which involves a haircut h – to cover an outﬂow of (F (t)
liquidate (F (t)

1))+ is favourable for banks. The outﬂow (F (t)

1))−/(1

F (t

F (t

F (t

F (t

h).

−

−

−

−

−

−

Parameters γ, rF and h are deterministic.

−

−

−

3.3 Loans

Assumption 2. The loan portfolio is homogenous and subject to default risk. Loans pay
deterministic interest rate. Loan portfolio is perfectly illiquid, i.e. only the maturing part
can be reinvested. The new business has its own default risk characteristics, correlated
with the default risk of the outstanding business (as well as with risk factors of securities
portfolio and funding).

Let ρ and ρN be some processes on
the outstanding loan portfolio L and the new origination.

, taking values from (

P

−∞

, 1], describing a credit quality of

L(t + 1) = (1

−

m)L(t)ρ(t + 1) + πL(t)ρN (t + 1)

(2)

−

−

m)L(t)(1

ρ(t + 1)) units of the outstanding volume of loans defaults between t
It means that (1
and t + 1. The new business volumes are also subject to a default risk: πL(t)(1
ρN (t + 1)) defaults
between t and t + 1. The parameters ρ and ρN can be interpreted as loss rates. In practice, they
are functions of default distribution (probability of default distribution) and a loss given default
(LGD). This observation is important for the application of the model – loss rates are estimated by
multiplying random default probability with an average LGD.

−

Notably, in the current setup of the model, the loan portfolio is homogenous implying the 1-
dimensional decision variable πL.
It can be relaxed to a multi-product loan portfolio but one
decision variable is enough to capture trade-oﬀs between the return potential, the liquidity risk and
the inﬂuences of the funding risk on the asset structures. Moreover, the current setup helps to keep
the computational cost of the optimisation programm low (deﬁned in section 3.6).

The interest income from loans is measured by the rate payment r multiplied by the end-of-period

volume of the loans, i.e.:

I(t + 1) = rL(t + 1)

Notably, the interest income of loans is aﬀected by the defaulted volume of loans which is reﬂected
by taking the volume of loans from the end of period [t, t + 1] to compute interests earned in that
period. An average volume L(t+1)+L(t)
could be an alternative option, also easy to implement, but
leading to a bit more complex formulas. The maturity proﬁle m is constant (and deterministic) and
the loan interest rate r is deterministic as well.

2

8B is a family of the Borel sets.
9We label this event “ﬁre-sales” for convenience but it reﬂects a likely price/volume impact, even though minor,
of selling of any given volume of securities for the liquidity purposes. This minor impact can simply be modelled by
relatively small h, e.g h = 0.1%.

ECB Working Paper 1896, April 20168Credit losses in period [t, t + 1], for simplicity directly and fully deducted from capital, are

denoted

3.4 Securities

∆L(t + 1) : = L(t + 1)

(1

−

−

m)L(t)

−

πL(t)

Assumption 3. A homogenous portfolio of securities, the value of which is governed
by one risk factor, matures at the end of every period, i.e. it is purchased (completely
renewed) at t and is maturing at t + 1. No short-selling is allowed.

The part of the value of the balance sheet that is not invested in the loan portfolio is allocated into
the securities portfolio. The total reinvestment potential is equivalent to the sum of the maturing
loans, the value of securities, the change in funding and the P&L impact of the ﬁre-sales. Notably,
the total reinvestment portfolio is impacted by the change in funding asymmetrically depending on
the sign of the change: the change of the reinvestment portfolio related to funding is equal to

∆FK (t) = F (t)

F (t

1)

−

−

−

1

(F (t)

F (t

−

−

h

1))−

h

−

In case of the funding outﬂow, the bank “ﬁre-sales” its securities to meet the obligations. The price
of securities is risky and driven by a stochastic process ǫ, adapted in

.

Therefore, the law of motion for the value of securities is described by

P

S(t + 1) = (mL(t) + S(t) + I(t)

C(t) + ∆FK (t)

πL(t))ǫ(t + 1)

−
It is assumed that the valuation process accounts for the interest payments. Therefore the P&L
impact of securities portfolio is the valuation change

−

(3)

∆S(t + 1) = (mL(t) + S(t) + I(t)

C(t) + ∆FK (t)

−

−

πL(t))(ǫ(t + 1)

1)

−

3.5 Capital

Assumption 4. Capital is a residual part of the balance sheet. At the end of each period
t > 0 its level changes according to the accrued net interest income generated in period
[t
1, t], to valuation changes of securities portfolio and to ﬁre-sales of securities in case
of the outﬂow of funding.

−

The dynamics of capital K is implied by the realised P&L with an assumption of no raising of
some new capital. The proﬁts of banks are assumed to consist of the interest income and expenses
(related to the loans and deposits), the credit losses, the revaluation of the securities portfolio
and the valuation changes related to the ﬁre-sales of the securities implied by a decline of funding
deposits. For t = 0

and for t > 0

K(0) = L(0) + S(0)

F (0)

−

K(t) = K(t

1) + rL(t)

−

−

rF F (t) + ∆L(t) + ∆S(t)

h

−

h

−

1

(F (t)

F (t

−

−

1))−

3.6 Goal

Assumption 5. Bank maximises the sum of discounted risk-adjusted returns from capital
subject to liquidity and capital constraints. At each time t, the liquidity constraint assures
αF , for small αF , bank is able to meet its obligations at t+1.
that with high probability 1
Capital constraint is based on a regulatory concept requiring a bank to hold enough capital
for the risk weighted assets (minimum 8% capital ratio is a baseline case) combined with
the economic capital concept based on VaR calculations. An illiquid or insolvent bank
incurs a penalty cost proportional to its volume of the assets.

−

ECB Working Paper 1896, April 20169A bank is supposed to maximise the expected return on equity, i.e. the expected value of:

R(t + 1) =

K(t + 1)

−
K(t)

K(t)

(4)

adjusted by the risk of that return and aggregated across periods.

There are two types of constraints imposed on the investment strategy: related to liquidity (LaR)
and solvency position (VaR). Liquidity is understood as the balance sheet composition that allows
for paying back due liabilities. We omit the cash ﬂow balance of interest paid by loans and funding
since we focus on liquidity shocks related to the ﬂuctuations of deposits. For the liquidity purposes
a short period ∆l is assumed – a holding period – in which the liquidity position cannot be adjusted.
The investment strategy should then keep enough liquid resources to cover an outﬂow of deposits in
αF fraction of scenarios, at a given conﬁdence level αF . The securities in the counterbalancing
1
capacity can be liquidated with a haircut h reﬂecting a discount that can be expected in case of the
liquidation (potentially quite high in a ‘ﬁre-sales’ mode). Putting formally, the LaR constraint has
the form:

−

VaRα

E

(1

−

h)S(t + ∆l) +

F (t + ∆l)

F (t)

−

t

|F

0

≥

(5)

(cid:0)
It can be interpreted as bank’s internal requirement to hold enough liquid securities to cover even
the αF worst outﬂow of the funding sources. Notably, the stock of liquid assets has to increase if
their volatility and correlation with the funding sources increases.

(cid:3)(cid:1)

(cid:0)

(cid:1)

(cid:2)

Solvency constraints have two forms. One has a very regulatory nature. For the risk weights ωL

and ωS, and minimum capital ratio κ (e.g. equal to 8%):

κ(ωL((1

−

m)L(t) + ¯π) + ωS(((m + r)L(t) + S(t) + ∆FK (t)

¯π)))

−

≤

K(t)

(6)

which translates into (assuming ωL > ωS):

K(t)/κ

ωL(1

−

−

m)L(t)

ωS((m + r)L(t) + S(t) + ∆FK (t))
−
ωL

ωS

¯π

≤

−
However, banks manage their investment portfolios taking worst case scenarios of capital position
in a given ∆K period into account.10 We consider ∆K period forward distribution of income and
require that the capital of a bank covers the losses in (1

100% of cases.

αK )

K(t) + VaRαK (E[K(t + ∆K)

−

−
K(t)

∗
t]) > 0

|F

(7)

Constraints on the strategies at t do not fully prevent the bank from default deﬁned as failing to
meet liquidity and solvency requirements at the end of [t, t + 1] period. We construct the liquidity
process:

and the solvency process

Liq(t + 1) = (1

−

h)S(t + 1) + (F (t + 1)

F (t)),

−

Solv(t + 1) = K(t + 1),

which is simply the process of capital evolution. We deﬁne a stopping time τ on (P, F) as

τ = inf

t

|

{

Solv(t) < 0

∨

Liq(t) < 0

}

(8)

portraying an event of a default. Illiquidity or insolvency implies some additional costs for the bank.
The costs are assumed to be proportional to the asset volume L(τ ) + S(τ ), with proportionality
coeﬃcient φ. They are paid once, at the time of default but only if default occurs before the end of
the investment horizon T .11

10E.g. under the ICAAP process.
11A precise estimate of the coeﬃcient φ may in general be diﬃcult and depending on the jurisdiction, resolution
schemes, investors behaviours and complexity of the ﬁnancial system (Hardy, 2013). For instance, the Lehman case
show the complications of the litigation costs due to uncertainty of who owned what to whom.

ECB Working Paper 1896, April 201610We assume that banks optimise the risk-adjusted return on capital, aggregated within the horizon

of the optimisation. The goal functional is the following:

J(l0, s0, f0, ρ0, π) = E

δt

R(t)

T

t=1
(cid:16)
X
ER(t))2

−

βV ar(R(t))

(cid:17)

−

δτ φ
(cid:16)

S(τ ) + L(τ )

I
{τ <T }

(9)

(cid:17)

where V ar(R(t)) = E[(R(t)
t−1] is a conditional variance process. The goal functional
incorporates a penalty cost of either illiquidity or insolvency event. In other words, the bank pays
some litigation costs and has to rebuild its reputation when breaching the solvency thresholds or is
incapable of covering the outﬂow of funds with the liquid assets. Consequently, it may loose some
potential revenues in the future.

|F

−

3.7 Towards tractability

The model does not have a closed form solution and can be cumbersome to solve numerically in its
general setup. The risk factors in the model are the main source of computational complexity. To
simplify the computation we assume that

– risk factors of securities and funding are normally distributed, i.e. factor ǫ has a normal
distribution
– risk factors of loans are log-normally distributed, i.e. for normally distributed Z and
Z N , ρ = 1

(µǫ, σǫ), η is normally distributed with mean µη and variance σ2
η;

N
exp(Z) and ρN = 1

exp(Z N )

−

−

The tractability is obtained with some trade-oﬀs, the most important being the negativity of the
processes ρ, ρN , S and F , although a proper and reasonable calibration may decrease the probability
of such an event to close to zero. Moreover, since the returns on ﬁnancial assets frequently exhibit
fat-tails they should be described for instance by means of the so-called stable distributions (Focardi
et al., 2013). Consequently, the normality assumption would have implications for underestimation
of the VaR of the portfolios. To address this problem, a smaller probability corresponding to the
worst outﬂow or the worst capital ratio than the one implied by the normal distribution with the
given estimated variance can be applied. For instance, if the worst case capital ratio is deﬁned
by a probability αK equal to 1% then the VaRαK can be approximated by a more stringent 5-σ
or 7-σ level instead of 2.33-σ. Under the normality assumption, the constraint 5 is signiﬁcantly
simpliﬁed. The conditional expectation yields a normally distributed random variable. Given that
the VaR periods of liquidity and solvency constraints are diﬀerent that the proﬁt cycle, the mean
and standard deviation parameters have to be scaled, i.e.

µX,l = ∆lµX

σX,l =

∆lσX

µX,K = ∆KµX

σX,K =

p

∆KσX

Let us denote

aǫ(t

−

1, t) = (m + r)L(t) + S(t)
bρ(t) = (1

m)L(t)

−

p

rF F (t) + ∆FK (t)

−

The mean and variance of the conditional expectation are:

M (t) =

Σ2(t) =

(1

h)¯πµǫ(t+1),l + (1

−

−
h)(aǫ(t
(1
+2cǫ(t+1),η(t+1)(1
(cid:0)

−

−

1, t)

−

−
h)(aǫ(t

h)aǫ(t

−
¯π)σǫ(t+1),l

−
2

1, t)µǫ(t+1),l + γF (t) + µη(t+1),l

+

ση(t+1),l

2

1, t)
(cid:1)

−

−

¯π)σǫ(t+1),lση(t+1),l
(cid:0)

(cid:1)

and the liquidity condition requires to solve

M (t)
Σ(t) ≤

−

Φ(αF )

ECB Working Paper 1896, April 201611where Φ is an inverse function of the standard normal distribution. For usually small αF , Φ(αF ) is
negative. It means that for ¯π such that M (t)

0, there is no solution. It implies that

≤

(1

−

¯π

≤

h)aǫµǫ,l + γF (t) + µη,l
h)µǫ,l

(1

−

= aǫ +

γF (t) + µη,l
h)µǫ,l
(1

−

(10)

if only (1

−

h)µǫ,l > 0 which is a reasonable assumption. Therefore, equivalently

Φ(αF )2Σ2(t) + M (t)2

−

0

≥

From the two VaR-based constraints, the one related to liquidity conditions is easier to tackle.
The liquidity constraint yields a quadratic inequality. A coeﬃcient at the ¯π2 term is positive for a
’reasonable’ parametrisation of the model. The inequality is solved by ¯π satisfying (for brevity, we
drop time indices)

¯π

∈ "−∞

B

, −

−

√B2
2A

−

4AC

−

B + √B2
2A

−

4AC

, +

∞#

# ∪ "

(11)

where

A =

B =

h)µǫ,l)2

ǫ,l + ((1
h)aǫµǫ,l + γF (t) + µη,l)

−

h)2σ2

−

Φ(αF )2(1
2(1

−
−
+2Φ(αF )2

−
h)µǫ,l ((1
(1

ǫ,laǫ + cǫ,η(1

−
h)2σ2
h)aǫµǫ,l + γF (t) + µη,l)2
ǫ + σ2
h)2σ2

ǫ,la2

(cid:2)
(1

−

η,l + 2(1

−
Φ(αF )2

−

−

h)σǫ,lση,l

−

C = ((1

Clearly, the coeﬃcient A is positive if

(cid:2)

(cid:3)
h)cǫ,ηaǫσǫ,lση,l

−

(cid:3)

(12)

For instance, if αF = 1% then Φ−2
the volatility parameter σ2

≃

ǫ,l has to be smaller than 40%.

Φ(αF )−2 >

σ2
ǫ,l
µ2
ǫ,l

.

0.18 and for return of securities close to 1 (e.g. µǫ,l

[0.9, 1.1])

∈

At t = 1, the bank needs to solve a quadratic maximisation problem, calculation of which also
applies to asset portfolio choice at t = 0. Let us denote by µρ and µρN the means of log-normal
distributions of default risk which are equal to:

µρ = 1

−

exp

µZ +

(cid:18)

σ2
Z
2

(cid:19)

µρN = 1

−

exp

µZN +

(cid:18)

σ2
ZN
2

(cid:19)

(13)

and standard deviations

σρ = exp

µZ +

(cid:18)

σ2
Z
2

exp(σ2

Z )

(cid:19) q

−

1 σρN = exp

µZN +

(cid:18)

σ2
ZN
2

exp(σ2

ZN )

(cid:19) q

1

−

(14)

To characterise the goal function, we need to calculate also correlation of ρ and ρN , and correlation
of log-normal ρ and ρN with normal variables ǫ and η. The former can be directly calculated and
equals

exp

µZ + µZN +

(cid:18)

(cid:18)

Z + σ2
σ2

ZN + 2cZ,ZN σZ σZN

2

µρµρN

/(σρσρZ )

−

(cid:19)

(cid:19)

The later can be obtained via Stein’s lemma yielding eg. for ρ and ǫ

cρ,ǫ =

−

exp

µZ +

(cid:18)

σ2
Z
2

(cid:19)

(cZ,ǫσǫσZ ) /σρσǫ

ECB Working Paper 1896, April 201612The impact of ﬁre-sales on capital creates another complication in the calculations. To represent
the mean and variance of the return process we need three types of integrals calculated in the
appendix A, i.e. for n

0, 1, 2

R:

, a

∈ {

}

∈

I (n)(a, B0, B1, B2) =

R, B2 < 0, B1 ∈
xn exp

a

R and B0 ∈
B2x2 + B1x + B0

dx

(15)

The conditional return Rc(t) ∆= E[R(t + 1)

(cid:0)

(cid:1)
t] has a distribution which is a mixture of normal,

truncated normal and log-normal random variables with mean and variance given by

−∞

Z

|F

µRc(t)

where

1), S(t), F (t

1), F (t), ¯π)

∆= µ(L(t), S(t
=

−

−
(1 + r)bρ(t)µρ(t+1) −
(cid:0)
rF (F (t) + γF (t) + µη(t+1)) +
−

bρ(t) + (1 + r)¯πµρN

h

h)√2Πση

(1

−

−

¯π + (ae(t

1, t)

−
−
η + I (1)
(γF (t)I (0)
η )

¯π)(µǫ(t+1) −
/K(t)

1)

(16)

η = I (n)(
I (n)

−

γF (t),

µ2
η
2σ2
η

,

µη
σ2
η

,

−

−

(2σ2

η)−1)

Σ2

Rc(t)

∆= Σ2(L(t), S(t), F (t

−

1), F (t), ¯π)

= K(t)−2

(1 + r)bρ(t)σρ(t+1)

2

+

(1 + r)¯πσρN (t+1)

!

2

(cid:17)

(cid:18)(cid:16)

(aǫ(t

1, t)

−

−
2h
h)√2Πση

+

+

(cid:16)

(1

−

¯π)σǫ(t+1)

2

(cid:17)
+

(cid:16)
rF ση(t+1)

2

(cid:17)

(cid:16)

(cid:17)
η + I (2)
η + 2γF (t)I (1)

η

(γF (t))2I (0)
(cid:16)

2

(cid:17)

2h
h)√2Πση

(γF (t)I (0)

η + I (1)
η )

−

(1

−  
+2cρ(t+1),ρN (t+1)(1 + r)2bρ(t)¯πσρ(t+1)σρN (t+1)
+2cǫ(t+1),ρN (t+1)(1 + r)(aǫ(t

1, t)

!

2cη(t+1),ρN (t+1)(1 + r)rF ¯πση(t+1)σρN (t+1)

−
+2cρ(t+1),ǫ(t+1)(1 + r)bρ(t)σρ(t+1)(aǫ(t

1, t)

−

−

−

2cρ(t+1),η(t+1)rF (1 + r)bρ(t)σρ(t+1)ση(t+1)
2cǫ(t+1),η(t+1)rF (aǫ(t

¯π)σǫ(t+1)ση(t+1)

1, t)
, (γF (t) + η(t + 1))−)

−

−

−
+2cov(
·

¯π)¯πσǫ(t+1)σρN (t+1)

−

¯π)σǫ(t+1)

−

(cid:1)

(17)

where the last component refers synthetically to the covariance of (γF (t) + η)− with all other risk
factors in the model (see Appendix A).

The unconstrained maximiser (dropping the time index for brevity) is given by (easy to solve

because of a quadratic form):

¯π∗ = arg max¯π∈R

µRc(t) −
The capital constraint based on the VaR formula remain complicated enough even in the “Gaus-
sian world”.
It does not have a closed form solution as in the case of the liquidity constraint.
However, we use the usual moment matching technique to yield a tractable formula for the con-
straint imposed on the lending choice. Notably, we match the moments of the conditional net
income

Rc(t)}

βΣ2

(18)

{

N It : = E[K(t + 1)

t] = E[R(t + 1)K(t)

K(t)

|F

−

t]

|F

ECB Working Paper 1896, April 201613with a normal distribution. The advantages of this particular choice of the distribution are twofold.
First, since N It is a mixture of normal and log-normal distributions both tails stretch to inﬁnity
which of course is true also for the normally distributed random variable. Second, percentiles of
the normal distribution are straightforward to compute. On the downside, the normal distribution
does not allow controlling for the skewness of N I which may have important consequences for
underestimation of the VaR numbers.

To approximate the VaR capital constraint we use the mean µRc(t) and standard deviation of
return σRc(t) deﬁned in formulas 16 and 17 but with mean µX and risk σX parameters replaced by
µX,K and σX,K respectively. The approximation of the VaR constraint yields

VaRαK

NIt

K(t)µRc(t)

−
K(t)σRc(t)

> −

(cid:19)

K(t)

K(t)µRc(t)

−

K(t)σRc(t)

(cid:18)

Φ2(αK ) <

(K(t) + K(t)µK(t))2

K 2(t)σ2

Rc (t+1)

∧

K(t)(1 + µRc(t+1)) > 0

Φ2(αK )K 2(t)σ2

R(t) + (K(t) + K(t)µK(t))2 > 0

−

For a notational convenience let us decompose the mean and variance of R into terms at the optimal
strategy and the residual terms.

Aµ
0 = (1 + r)bρ(t)µρ(t+1),K −

bρ(t) + aǫ(t

rF (F (t) + γF (t) + µη(t+1),K) +

−

Aµ
π = (1 + r)µρN ,K −

1

(µǫ(t+1),K −

−

(1

1)

1, t)(µǫ(t+1),K −

h

h)√2Πση(t+1),K

−

−

1)

(γF (t)I (0)

η + I (1)
η )

(19)

Aσ

0 =

(1 + r)bρ(t)σρ(t+1),K

1, t)σ2

ǫ(t+1),K +

rF ση(t+1),K

2

+ a2

ǫ (t

−

+

(cid:16)

(1

−

h2

h)2√2Πση,K

(cid:16)

(cid:17)

(γF (t))2I (0)

η + 2γF (t)I (1)

(cid:16)
η + I (2)
η

2

(cid:17)

2

(cid:17)

h

(1

h)√2Πση,K

−  
!
+ 2cρ(t+1),ǫ(t+1)(1 + r)bρ(t)σρ(t+1),K aǫ(t

−

(γF (t)I (0)

η + I (1)
η )

2cρ(t+1),η(t+1)rF (1 + r)bρ(t)σρ(t+1),K ση(t+1),K
1, t)σǫ(t+1),Kση(t+1),K

2cǫ(t+1),η(t+1)rF aǫ(t

1, t)σǫ(t+1),K

−

−

−

−

+

+

−
(1 + r)bρ(t)
√2Πση,K

(1 + r)bρ(t)
√2Πση,K
(1 + r)bρ(t)
√2Πση,K
haǫ(t
√2Πση,K (1

−

h

−
h

−
h

−

1

1

1

h

h

h

−
γF exp(c2

2/2)I (0)(

exp(c2

2/2)I (1)(

−

−

γF, c0 −
µ2
2σ2

γF, c0 −

η,K

µ2
(cid:1)
2σ2

η,K

, c1 +

µη,K
σ2

η,K

,

−

1
2σ2

η,K

)

η,K

η,K

, c1 +

µη,K
σ2

η,K

,

−

1
2σ2

η,K

)

µρ,K (γF (t)I (0)

η + I (1)
η )

1, t)

¯c0γF I (0)
h

h)

−

η + ¯c0I (1)

η + ¯c1γF I (1)

η + ¯c1I (2)

η −

µǫ,K

γF (t)I (0)

η + I (1)

η

(cid:16)

(cid:17)i

(20)

ECB Working Paper 1896, April 201614Aσ

2aǫ(t

π =
+ 2cǫ(t+1),ρN (t+1)(1 + r)aǫ(t

−

−

1, t)σ2

ǫ(t+1),K + 2cρ(t+1),ρN (t+1)(1 + r)2bρ(t)σρ(t+1),K σρN (t+1),K

1, t)σǫ(t+1),KσρN (t+1),K

−

2cη(t+1),ρN (t+1)(1 + r)rF ση(t+1),K σρN (t+1),K
2cρ(t+1),ǫ(t+1)(1 + r)bρ(t)σρ(t+1),K σǫ(t+1),K

−
−
+ 2cǫ(t+1),η(t+1)rF σǫ(t+1),Kση(t+1),K

γF exp(c2

2/2)I (0)(

exp(c2

2/2)I (1)(

−

(cid:1)
γF, c0 −
−
µ2
2σ2

γF, c0 −

η,K

µ2
2σ2

η,K

, c1 +

µη,K
σ2

η,K

,

−

1
2σ2

η,K

)

η,K

η,K

, c1 +

µη,K
σ2

η,K

,

−

1
2σ2

η,K

)

µρN ,K(γF (t)I (0)

η + I (1)
η )

1 + r
√2Πση,K
1 + r
√2Πση,K
1 + r
√2Πση,K
h

h

−
h

−
h

−

1

1

1

h

h

h

−

−

+

−

Aσ

ππ =

√2Πση,K (1

¯c0γF I (0)
h
(1 + r)σρN (t+1),K

+ σ2

h)

−

2

(cid:16)

(cid:17)

η + ¯c0I (1)

η + ¯c1γF I (1)

η + ¯c1I (2)

η −

µǫ,K

γF (t)I (0)

η + I (1)

η

(cid:16)

(cid:17)i

ǫ(t+1),K −

2cǫ(t+1),ρN (t+1)(1 + r)σǫ(t+1),K σρN (t+1),K

Consequently, the VaR constraint can be reformulated to

Φ2(αK)(Aσ

0 + Aσ

π ¯π + Aσ

ππ ¯π2) + (K(t) + Aσ

0 + Aµ

π ¯π)2 > 0

−

and by grouping of terms to

Φ2(αK )Aσ

ππ + (Aµ

π)2

¯π2 +

Φ2(αK)Aσ

π + 2Aµ

π(K(t) + Aµ
0 )

¯π +

Φ2(αK )Aσ

0 + (K(t) + Aµ

0 )2

> 0

(cid:2)

(cid:3)

−

−
(cid:2)
(cid:3)
For many reasonable parameterisations of the model, the coeﬃcient at ¯π2 is negative. It means
that the admissible investment would be bounded from above. Notably, it is theoretically possible
that the strategy is bounded from below by a positive number meaning that the set of admissible
strategies can be disjoint; the bank is not allowed to invest too much in the risky loans but may be
forced to allocate a minimum part of the reinvestment portfolio for proﬁtability reasons (i.e. bank
could be solvent only if earning a suﬃcient income from proﬁtable loans).

−

(cid:3)

(cid:2)

4 2-period setup

The general multi-period model can be costly in terms of computational time. We focus on the
applications of a 2-period model. It only requires approximate (Monte-Carlo) methods to be imple-
mented in the ﬁrst period whereas at the second (and ﬁnal) period the solution is explicit. Notably,
it preserves all the important features of T -period model; the inter-temporal eﬀects resulting in a
trade-oﬀ between investing more in proﬁtable loans now and facing risk of illiquidity or generating
less income but increasing survival probability.

The optimisation problem is solved numerically by means of the dynamic programming. It is a
convenient way to derive the optimal portfolios in a backward manner. The value function of the
2-period model at time t = 2 is equal to 0, i.e.

V2 ≡

0

(21)

since this is the end of investment horizon and, by deﬁnition, no income is generated afterwards.
At t = 1 the problem is solved analytically and the optimal portfolio choice at t = 0 is calculated
as value π∗
0 that maximises the sum of average of the value function over the sampled portfolios at
t = 1 and the utility of loan investment π∗
0.

ECB Working Paper 1896, April 201615At t = 0 we deﬁne random functions (randomness implied by ǫ(1), ρ(1), ρN (1) and η(1)):

l(l0, ¯π) = (1

m)l0ρ(1) + ¯πρN (1)

−

s(s0, l0, f0, f0, ¯π) = ml0 + s0 + r((1
f (f0) = f0 + γf0 + η(1)

m)l0ρ(1) + ¯π) + f0 −

f0 −

−

¯π)ǫ(1)

k(s0, l0, f0, f0, ¯π) = (l0 + s0 −
The value function at t = 0 satisﬁes the following dynamic programming formula

rF f (f0) + ∆L(t) + ∆S(t)

f0) + rl(l0, ¯π)

−

−

1

h

−

(f (f0)

h

−

f0)−

V0(l0, s0, f0) = max
¯π∈A(0)

E[R(1)

βV ar(R(1))]+

−

n
δφ(l(l0, ¯π) + s(s0, l0, f0, f0, ¯π))I

−

and applying 16 and 17 it can be transformed to

{τ =1} + δV1(l(l0, ¯π), s(s0, l0, f0, f0, ¯π), f0, f (f0))

o

µ(l0, s0, f0, f0, ¯π)

V0(l0, s0, f0) = max
¯π∈A(0)
(l(l0, ¯π) + s(s0, l0, f0, f0, ¯π))I
h

−
+δEV1(l(l0, ¯π), s(s0, l0, f0, f0, ¯π), f0, f (f0))

δφE

−

n

βΣs(l0, s0, f0, f0, ¯π)

{k(s0,l0,f0,f0,¯π)<0 ∨ (1−h)s(s0,l0,f0,f0,¯π)<f (f0)−f0}

i
(22)

The function V0 is well-deﬁned and well-behaved since V1 is continuous and it can be shown that
{τ =1} is also continuous.12 Hence, max is attained
g : (0, +
∞
on a compact set

R+ such that g(l0, s0, f0, ¯π) : = EI

(0).

→

]4

The form of equation 22 is not very speciﬁc to the 2-period setup of the model.

It can be
straightforwardly generalised to a multi-period case, whereby the value function at any t < T is
identical to 22 after replacing 0 and 1 time indices with t and t + 1 respectively.

A

o

5 Simulations

We choose a speciﬁc parametrisation of the model summarised in the table 1. The two important
comments to the table are related to the selection of the LaR and VaR horizon (∆l and ∆K
respectively) and to the credit risk parameters. First, we assume that ∆l = 0.08(3), corresponding
to 30 days in a 360-day year investment horizon, and ∆K = 0.25 which corresponds to 90 days for
the solvency constraint. Second, the parameters for the average and standard deviation of defaults
on the new loans (ρN ) having by assumption the log-normal distribution are presented in terms of
the means and standard deviation of normal distribution generating that log-normal distribution.
After transformation (given by formulas 13 and 14), the mean and standard deviation of ρN are 5.3%
and 8.0% respectively. Taking into account the interest rate on loans, the Sharpe ratios of loans
and securities can be calculated to compare the relative risk-adjusted return of both investment
opportunities available to the bank (the ration for loans amounts to 0.59 against 0.50 for securities
so the risk-return proﬁle of the two categories is relatively balanced).
Case 1: Optimal asset structure for diﬀerent levels of funding. The optimal lending volume is
calculated for funding volumes ranging between 93.0 and 93.8, all other parameters unchanged
except implicitly changing capital level from 7.0 to 6.8. This can be interpreted as an increase in
leverage ratio. Figure 1 presents a distribution of CAR projected one period ahead using the optimal
lending strategy for diﬀerent levels of funding. The upper-left pane of the graph plot the admissible
region of investment in new loans against diﬀerent levels of funding. The region is delimited by the
intersection of CAR-based, VaR-based and liquidity constraints. A straightforward observation is

12In fact, a suﬃcient condition for g to be continuous is that ǫ, η, ρ and ρN have continuous cumulative distribution

functions.

ECB Working Paper 1896, April 201616Table 1: Example of parametrisation of the model

Parameter
m
r
rF
h
γ
κ
αF , αK
β
wL
wS
δ
L
S
F
φ

Value Description

0.10
0.07
0.03
0.05
0.02
0.08
0.01
5.00
0.85
0.20
0.70
70.00
30.00
93.00
0.00

maturity
interest rate of L
cost of funding
haircut
trend of funding
capital constraint
liq., solv. constraints
risk aversion
risk weight (L)
risk weight (S)
discount factor
loans
securities
funding
penalty rate

Parameter
µǫ
σǫ
µZ
σZ
µZN
σZN
µη
ση
cǫ,Z
cǫ,ZN
cǫ,η
cZ,ZN
cZ,η
cZN ,η

Value Description

1.03
0.06
-3.55
1.10
-3.20
1.25
0.00
10.00
0.00
0.00
0.00
0.00
0.00
0.00
.

mean return on S
std of return on S
mean default factor (outstanding L)
std default factor (outstanding L)
mean default factor (new L)
std default factor (new L)
mean of funding risk
std of funding risk
corr(securities,default of outstanding L)
corr(securities,default of new L)
corr(securities,funding)
corr(outstanding L, new L
corr(default of outstanding L, funding)
corr(default of new L, funding)

Source: own calculations

that the lending remains stable until the VaR-based capital constraint becomes a binding constraint.
Clearly, the increasing leverage implies that a bank can invest less in VaR consuming loans. The
whole distribution of the capital ratios shifts to the left (upper-right pane). Notably, due to the
ﬁre-sales the distribution is also skewed to the left.
Case 2: Optimal asset structure for diﬀerent levels of funding risk. An analogous simulation to
the Case 1 was conducted assuming that the volatility parameter of the funding (σF ) was varied
in the range of 10 to 20.
It impacts the maximum allowed reinvestment into loan portfolio by
the liquidity constrain, whereas the CAR capital constraint remains ﬂat (insensitive to the risk
parameters on the funding side). For the low volatility of the funding, as the volatility starts to
increase the optimal lending portfolio slides down slowly and stays below the potential level (implied
by liquidity and CAR constraints). At a certain level of volatility (13 units in the example), liquidity
constraint becomes binding and afterwards the optimal lending follows the path implied by the
liquidity condition. Interpreting the results form the angle of the credit provision to the economy,
if the volatility of funding is relatively low then the optimal lending is immune to the changes of
the volatility of the funding sources until the funding risk becomes a binding constraint from the
solvency perspective.
Case 3: Riskiness of securities portfolio. Portfolio allocation may vary substantially if the relative
riskiness of loan and securities (liquidity) portfolio changes (between 6% and 11.5%). We veriﬁed
the theoretically optimal structure of banks assets under diﬀerent volatility of the securities (see 3).
Apparently, rising volatility leads to a shift of allocation towards the loan portfolio. The classical
Sharp ratio matters for the choice and the liquidity constraint seems not to be binding in this case.
However, the VaR-based capital constraint prevails severely reducing the lending potential of the
bank as the volatility grows. For σS below 10.0% the constraints are not binding and allocation to
the loan portfolio increases but above the 10.0% threshold the VaR constraint becomes binding and
the optimal investment in the new loans slides down slightly along the VaR solvency constraint.
Notably, the decrease in lending is more sensitive to changes in the volatility of securities if the
correlation between returns from securities and loan losses is positive. The graph 4 shows that
the correlation ampliﬁes the combined risk in the banks’s assets and high volatility of S translates
into lower potential for lending. Moreover, the higher market and credit risk ﬁnd a compensation
in the rising lower bound for investment in the new loans which oﬀer higher expected interest
payments. Consequently, the CAR oscillates more as well (the density of the distribution ﬂattens
on the upper-right pane).

It is interesting to see the impact of the correlation between securities and funding. A highly
positive correlation between returns from securities and the funding volumes – a more unfavorable
conﬁguration from the liquidity perspective – does not change much the picture qualitatively (5).
Though, the capital constraint is less restrictive and allows loan portfolios to grow more in high

ECB Working Paper 1896, April 201617Figure 1: Sensitivity of the optimal balance sheet structure and its capitalisation to leverage ratio

op t L
ca r
Va R
liq
lowe r b

4 5

4 0

3 5

3 0

2 5

2 0

1 5

1 0

5

0
9 3 .0 9 3 .1 9 3 .2 9 3 .3 9 3 .4 9 3 .5 9 3 .6 9 3 .7 9 3 .8
(a ) Fu n d in g  volu m e

0 .1 8

0 .1 6

0 .1 4

0 .1 2

0 .1 0

0 .0 8

0 .0 6

0 .0 4

0 .0 2

0 .0 0

2 0

3 0

1 8

1 6

1 4

1 2

1 0

8

6

4

2

0
0 .0 0

0 .2 5

0 .2 0

0 .1 5

0 .1 0

0 .0 5

0 .0 5

0 .1 0

0 .1 5

0 .2 0

0 .2 5

(b ) Proje ction : CAR

9 3 .8
9 3 .7
9 3 .6
9 3 .5
9 3 .4
9 3 .3
9 3 .2
9 3 .1
9 3 .0

1 5

6 0

5 0

4 0

7 0
(c) Proje ction : loa n s

8 0

9 0

0 .0 0

− 1 0

0

− 5
(d ) Proje ction : ca p ita l

1 0

5

Note: opt L: optimal lending volume; car: maximum lending volume allowed by the CAR ratio with a given
level of capital; liq: maximum lending allowed by the liquidity constraint; reinv: reinvested part of the
balance sheet, i.e. (1 − m)L + S. The shaded area shows the region of admissible investment in the new
loan portfolio volumes. The density plots present kernel density estimates of projected at the end of the
ﬁrst period, optimal levels of (b) CAR, (c) loan volumes and (d) capital
Source: own calculations, graphs generated using Matplotlib in Python

ECB Working Paper 1896, April 201618Figure 2: Capital Adequacy Ratio distribution for various levels of funding risk

4 5

4 0

3 5

3 0

2 5

2 0

1 5

1 0

5

0
1 0

1 2

0 .1 6

0 .1 4

0 .1 2

0 .1 0

0 .0 8

0 .0 6

0 .0 4

0 .0 2

0 .0 0

2 0

3 0

1 4

1 6
(a ) Fu n d in g  vola tility

1 8

6 0

5 0

4 0

7 0
(c) Proje ction : loa n s

op t L
ca r
Va R
liq
lowe r b

1 6

1 4

1 2

1 0

8

6

4

2

2 0

2 2

0
0 .0 0

0 .2 0

0 .1 5

0 .1 0

0 .0 5

8 0

9 0

0 .0 0

− 1 0

0 .0 5

0 .1 0

0 .1 5

0 .2 0

0 .2 5

(b ) Proje ction : CAR

2 0

1 8

1 6

1 4

1 2

1 0

1 5

0

− 5
(d ) Proje ction : ca p ita l

1 0

5

Source: own calculations, graphs generated using Matplotlib in Python

ECB Working Paper 1896, April 201619Figure 3: Capital Adequacy Ratio distribution for various levels of volatility of securities

4 5

4 0

3 5

3 0

2 5

2 0

1 5

1 0

5

0
0 .0 6

0 .1 6

0 .1 4

0 .1 2

0 .1 0

0 .0 8

0 .0 6

0 .0 4

0 .0 2

op t L
ca r
Va R
liq
lowe r b

1 8

1 6

1 4

1 2

1 0

8

6

4

2

0 .0 7

0 .0 8

0 .0 9
(a ) Se cu ritie s vola tility

0 .1 0

0 .1 1

0 .1 2

0
0 .0 0

0 .2 5

0 .2 0

0 .1 5

0 .1 0

0 .0 5

0 .0 5

0 .1 0

0 .1 5

0 .2 0

0 .2 5

(b ) Proje ction : CAR

0 .1 1

0 .1 0

0 .0 9

0 .0 8

0 .0 7

0 .0 6

1 5

0 .0 0

2 0

3 0

6 0

5 0

4 0

7 0
(c) Proje ction : loa n s

8 0

9 0

0 .0 0

− 1 0

0

− 5
(d ) Proje ction : ca p ita l

1 0

5

Source: own calculations, graphs generated using Matplotlib in Python

ECB Working Paper 1896, April 201620Figure 4: Capital Adequacy Ratio distribution for various levels of volatility of securities and positive
correlation between loan losses and returns from securities

op t L
ca r
Va R
liq
lowe r b

4 5

4 0

3 5

3 0

2 5

2 0

1 5

1 0

5

0
0 .0 6 0 0 .0 6 5 0 .0 7 0 0 .0 7 5 0 .0 8 0 0 .0 8 5 0 .0 9 0
(a ) Se cu ritie s vola tility

0 .1 6

1 6

1 4

1 2

1 0

8

6

4

2

0
0 .0 0

0 .2 0

0 .1 5

0 .1 0

0 .0 5

0 .1 4

0 .1 2

0 .1 0

0 .0 8

0 .0 6

0 .0 4

0 .0 2

0 .0 0

2 0

3 0

0 .0 5

0 .1 0

0 .1 5

0 .2 0

0 .2 5

(b ) Proje ction : CAR

0 .0 8 5

0 .0 8 0

0 .0 7 5

0 .0 7 0

0 .0 6 5

0 .0 6 0

2 0

6 0

5 0

4 0

7 0
(c) Proje ction : loa n s

8 0

9 0

0 .0 0

− 1 0

0

− 5
5
1 5
(d ) Proje ction : ca p ita l

1 0

Note: Positive correlation was set by cǫ,ρN = −0.5
Source: own calculations, graphs generated using Matplotlib in Python

correlation regime.
Case 4: Correlation of the credit and market risk. The correlation between market risk (valuation
of securities) and credit risk have a substantial impact on the optimal allocation of banks’ assets.
Negative correlation narrows the investment opportunities for the banks in the model. Negative
correlation means that the devaluation of securities’ portfolios is likely to coincide with the increases
in loan losses. That translates to a binding VaR solvency constraint for very negative correlation.
Nevertheless, the rising negative correlation implies more investment in the loan portfolios. As the
dependence between market and credit losses becomes weaker the optimum indicate less lending
and more investment in the securities. Therefore, the relationship between propensity to lending
and correlation between asset classes is very nonlinear. Correlation is an extremely important factor
of the decision process.
Case 5: Return on securities. The return on securities, and in fact the relationship between returns
on securities and loans, is in our model a signiﬁcant driver of bank’s optimal choice of the asset
structure. As the returns on securities rise, the optimal allocation in the loan portfolio starts to
decrease when the return on securities reaches a certain level and steadily diminishes until the full
allocation in the securities portfolio is achieved. At the same time, capital level decline gradually
and become more oscillatory (bottom-right pane). However, capital adequacy increases since risk-
weighted assets drop materially (upper-right pane).
Case 6: Fire-sales. The outﬂow of deposits triggers a liquidation of the securities in the securities
portfolio S. The model allows for studying of the impact of the devaluation parameter (the haircut
h) on the optimal lending, on the bank’s proﬁtability and on the ﬁnancial soundness of a bank

ECB Working Paper 1896, April 201621Figure 5: Capital Adequacy Ratio distribution for various levels of volatility of securities, with
correlation of ǫ and η equal to 0.8

op t L
ca r
Va R
liq
lowe r b

4 5

4 0

3 5

3 0

2 5

2 0

1 5

1 0

5

0
0 .0 6 00 .0 6 50 .0 7 00 .0 7 50 .0 8 00 .0 8 50 .0 9 00 .0 9 50 .1 0 0
(a ) Se cu ritie s vola tility

0 .1 6

1 6

1 4

1 2

1 0

8

6

4

2

0
0 .0 0

0 .2 0

0 .1 5

0 .1 0

0 .0 5

0 .1 4

0 .1 2

0 .1 0

0 .0 8

0 .0 6

0 .0 4

0 .0 2

0 .0 0

2 0

3 0

0 .0 5

0 .1 0

0 .1 5

0 .2 0

0 .2 5

(b ) Proje ction : CAR

0 .0 9 5

0 .0 9 0

0 .0 8 5

0 .0 8 0

0 .0 7 5

0 .0 7 0

0 .0 6 5

0 .0 6 0

1 5

5 0

4 0
6 0
8 0
(c) Proje ction : loa n s

7 0

9 0

1 0 0

0 .0 0

− 1 0

0

− 5
(d ) Proje ction : ca p ita l

1 0

5

Source: own calculations, graphs generated using Matplotlib in Python

ECB Working Paper 1896, April 201622Figure 6: Capital Adequacy Ratio distribution for various levels of correlation between credit risk
and market risk

4 5

4 0

3 5

3 0

2 5

2 0

1 5

1 0

5

op t L
ca r
Va R
liq
lowe r b

2 5

2 0

1 5

1 0

5

0
− 0 .8− 0 .6− 0 .4 − 0 .2 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0
(a ) Corre la tion  of S a n d  L

0 .1 8

0 .1 6

0 .1 4

0 .1 2

0 .1 0

0 .0 8

0 .0 6

0 .0 4

0 .0 2

0 .0 0

2 0

3 0

0
0 .0 0

0 .0 5

0 .1 0

0 .1 5

0 .2 0

0 .2 5

(b ) Proje ction : CAR

0 .4 0

0 .3 5

0 .3 0

0 .2 5

0 .2 0

0 .1 5

0 .1 0

0 .0 5

0 .9
0 .7
0 .5
0 .3
0 .1
− 0 .1
− 0 .3
− 0 .5
− 0 .7

2 0

6 0

5 0

4 0

7 0
(c) Proje ction : loa n s

8 0

9 0

0 .0 0

− 1 0

0

− 5
5
1 5
(d ) Proje ction : ca p ita l

1 0

Source: own calculations, graphs generated using Matplotlib in Python

ECB Working Paper 1896, April 201623Figure 7: Capital Adequacy Ratio distribution for various levels of return on securities

4 5

4 0

3 5

3 0

2 5

2 0

1 5

1 0

5

op t L
ca r
Va R
liq
lowe r b

2 5

2 0

1 5

1 0

5

0
1 .0 1 01 .0 1 51 .0 2 01 .0 2 51 .0 3 01 .0 3 51 .0 4 01 .0 4 51 .0 5 01 .0 5 5
(a ) Se cu ritie s re tu rn

0 .2 5

0
0 .0 0

0 .0 5

0 .1 0

0 .1 5

0 .2 0

0 .2 5

0 .2 0

0 .1 5

0 .1 0

0 .0 5

0 .0 0

2 0

3 0

(b ) Proje ction : CAR

0 .2 5

0 .2 0

0 .1 5

0 .1 0

0 .0 5

1 .0 5 5
1 .0 5 0
1 .0 4 5
1 .0 4 0
1 .0 3 5
1 .0 3 0
1 .0 2 5
1 .0 2 0
1 .0 1 5
1 .0 1 0

1 5

6 0

5 0

4 0

7 0
(c) Proje ction : loa n s

8 0

9 0

0 .0 0

− 1 0

0

− 5
(d ) Proje ction : ca p ita l

1 0

5

Source: own calculations, graphs generated using Matplotlib in Python

ECB Working Paper 1896, April 201624(liquidity and solvency). As an illustration, the optimal program of the bank was solved for diﬀerent
parameters of the haircut – from the baseline 1% to 10%. The haircut of 10% reﬂects a sizeable
impact of the sell-oﬀ of the securities that would be required to meet the obligations related to an
outﬂow of deposits. The lending proﬁle is non-linear with respect to the haircut which is primarily
related to the solvency constraint which becomes binding for the suﬃciently large values of the
haircut, i.e. above 8%. Below that threshold the theoretically optimal volume of lending slightly
increases with the haircut.
It can be interpreted as seeking for relatively more proﬁtable loans.
Notably, the liquidity constraint is only to a very little extent aﬀected by the increasing haircut (and
is never binding in the speciﬁc example). The haircut does not signiﬁcantly aﬀect the capitalisation
of the banks (a stable distribution of CAR on Figure 8, bottom-right pane) even though the impact
on the valuation of the securities is visible (top-right pane of Figure 9). Conversely, the level of
haircut (i.e. liquidity of the counterbalancing securities portfolio) has a signiﬁcant inﬂuence on the
risk of bank’s liquidity and solvency default, illustrated on Figure 10. Intuitively, the solvency risk
rises as the cost of liquidation increases (implying higher erosion of the capital base). Analogously,
the bank is becoming more likely to default on the obligations towards the depositors until the
haircut renders the solvency a binding constraint. From that point the optimal lending volume
declines which is equivalent to a higher build-up of the liquid securities playing a role of the liquidity
buﬀer.

The liquidity brings a non-trivial dynamics to the model and provides an insight into the com-
plicated behaviors of banks. The interplay of the solvency and liquidity has some interesting policy
implications. Let us suppose that a regulator wants to improve market liquidity and reduces the
general liquidation haircut of securities from 10% to 8%. By assumption, these securities comprise
the liquidity portfolio of the bank characterised by the parameters in Table 1. Of course, it is a very
stylised and speciﬁc conﬁguration but the set of parameters is consistent and realistic. Consequently,
the bank is induced to increase lending at the optimum and its solvency improves in terms of the
probability that its capital ratio stays above the regulatory minimum. However, its liquidity buﬀer
declines and the bank become more prone to the liquidity risk (measured as the likelihood that the
outﬂow of the funding sources exceeds the liquidity buﬀer). The example shows that any policy
targeting either solvency or liquidity of the bank has to be calibrated and adopted with caution to
minimise the adverse externality it can cause – a regulatory action to contain one type of risk can
be harmful for a diﬀerent type.
Case 7: Bankruptcy costs. We have set the bankruptcy costs φ to 0 to distil the risk-adjusted return
eﬀects on the optimal portfolio reallocation from a more arbitrary estimate of the aversion against
insolvency and illiquidity risk. Intuitively, the Figure 11 shows that the optimal lending declines as
the fraction φ, representing the bankruptcy cost as a fraction of the total assets, rises. Though, the
sensitivity is rather moderate. Notably, the average capital ratio increases with the the penalty rate
but the dispersion of CAR as well meaning that inadequate values are more likely. From the policy
perspective this result indicates that the provision of credit is, to some extent sensitive to the bank
resolution scheme and its eﬃciency. It may inﬂuence the behaviour of banks and their shareholders
depending on the expected cost of bankruptcy.

6 Conclusions

The paper presents a stylised model of the optimal structure of a simpliﬁed balance sheet of a bank.
It models a balance sheet composed of two asset classes ﬁnance by homogenous risky funding. The
decision of a bank is to ﬁnd the level of lending to strike a balance between return form investment
into illiquid loans and liquid securities oﬀering counterbalancing capacity against funding risk (i.e.
ﬂuctuation of available funding volumes). The implementation of the model is comprehensive in
how it treats the return-vs-risk trade-oﬀ and regulatory aspects of the banking system. Moreover,
it is easy to solve by the Monte Carlo simulations and the complex, VaR-based risk constraints are
reduced to the closed-form, analytical formulas.

The setup of the model straightforwardly suggests some interesting applications in the macro-

ECB Working Paper 1896, April 201625Figure 8: Capital Adequacy Ratio distribution for various levels of the ﬁre-sales haircut on securities

op t L
ca r
Va R
liq
lowe r b

3 5

3 0

2 5

2 0

1 5

1 0

5

0
0 .0 1 0 .0 2 0 .0 3 0 .0 4 0 .0 5 0 .0 6 0 .0 7 0 .0 8 0 .0 9 0 .1 0
(a ) h a ircu t (h ) on  se cu ritie s

0 .1 6

0 .1 4

0 .1 2

0 .1 0

0 .0 8

0 .0 6

0 .0 4

0 .0 2

0 .0 0

2 0

3 0

1 4

1 2

1 0

8

6

4

2

0
0 .0 0

0 .2 0

0 .1 5

0 .1 0

0 .0 5

0 .0 5

0 .1 0

0 .1 5

0 .2 0

0 .2 5

(b ) Proje ction : CAR

0 .1 0
0 .0 9
0 .0 8
0 .0 7
0 .0 6
0 .0 5
0 .0 4
0 .0 3
0 .0 2
0 .0 1

1 5

6 0

5 0

4 0

7 0
(c) Proje ction : loa n s

8 0

9 0

0 .0 0

− 1 0

0

− 5
(d ) Proje ction : ca p ita l

1 0

5

Source: own calculations, graphs generated using Matplotlib in Python

ECB Working Paper 1896, April 201626Figure 9: Decomposition of capital and P&L drivers for various levels of the ﬁre-sales haircut on
securities

1 2 0

1 0 0

8 0

6 0

4 0

2 0

0
0 .0 0

4 0

3 5

3 0

2 5

2 0

1 5

1 0

5

0
− 0 .2 0

0 .0 2

0 .0 4

0 .0 6

0 .0 8

0 .1 0

(a ) Proje ction : NII

− 0 .1 0

− 0 .1 5
0 .0 0
(c) Proje ction : loa n  los s e s

− 0 .0 5

0 .0 5

3 0

2 5

2 0

1 5

1 0

5

0 .0 4 0

0 .0 3 5

0 .0 3 0

0 .0 2 5

0 .0 2 0

0 .0 1 5

0 .0 1 0

0 .0 0 5

0 .0 0 0

0
− 0 .0 6 − 0 .0 4 − 0 .0 2

0 .0 0
(b ) Proje ction : va lu a tion

0 .0 2

0 .0 4

0 .0 6

0 .1 0
0 .0 9
0 .0 8
0 .0 7
0 .0 6
0 .0 5
0 .0 4
0 .0 3
0 .0 2
0 .0 1

Source: own calculations, graphs generated using Matplotlib in Python

4 0

8 0

6 0

1 0 0
(d ) Proje ction : tota l a s s e ts

1 2 0

1 4 0

1 6 0

ECB Working Paper 1896, April 201627Figure 10: Realised default frequency for various levels of the ﬁre-sales haircut on securities

0 .0 1 2
0 .0 1 1
0 .0 1 0
0 .0 0 9
0 .0 0 8
0 .0 0 7
0 .0 0 6
0 .0 0 5
0 .0 0 4

0 .0 1

0 .1 4 5

0 .1 4 0

0 .1 3 5

0 .1 3 0

0 .1 2 5

0 .1 2 0

0 .0 1

0 .0 2

0 .0 3

0 .0 4

0 .0 5

0 .0 6

0 .0 7

0 .0 8

0 .0 9

0 .1 0

liq. de fa ult proba bilitie s

0 .0 2

0 .0 3

0 .0 4

0 .0 5

0 .0 6

0 .0 7

0 .0 8

0 .0 9

0 .1 0

solv. de fa ult proba bilitie s

Source: own calculations, graphs generated using Matplotlib in Python

ECB Working Paper 1896, April 201628Figure 11: Capital Adequacy Ratio distribution for various levels of bankruptcy costs

opt L
car
VaR
liq
lower b

12

10

8

6

4

2

0.005

0.010

0.015

0.020

0.025

0
0.00

(a) penalty rate

0.20

0.15

0.10

0.05

45

40

35

30

25

20

15

10

5

0
0.000

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.00

20

0.05

0.10

0.15

0.20

0.25

(b) Projection: CAR

0.0225
0.0200
0.0175
0.0150
0.0125
0.0100
0.0075
0.0050
0.0025
0.0000

30

40

50

60

70

80

90

0.00

−10 −5

0

5

10

15

(c) Projection: loans

(d) Projection: capital

Note: A rather step-wise monotonicity of the optimal lending as a function of φ is a result of the numerical
approximation of the optimal solution and not a feature of the optimal solution per se.
Source: own calculations, graphs generated using Matplotlib in Python

ECB Working Paper 1896, April 201629prudential policy assessment; it can be used to analyse potential strategic responses of banks to
macro-prudential capital buﬀers, portfolio speciﬁc risk weights or liquidity requirements.

The current 2-asset model in 2 periods can be extended easily to n-asset model, however with
a loss of closed form portfolio strategy in the second period. This would massively complicate the
calculations. Consequently, some pure Monte-Carlo techniques would need to be applied and a
substantial increase in the computation time (although not unmanageable) should be expected.

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ECB Working Paper 1896, April 201631A Integrals

Let us solve the expression I (0)(a, B0, B1, B2). Applying the fact that for f
a2
a1 f ′(x)dx leads to
R

1, f (a1)

f (a2) =

−

∈ C

√2Π
2B2

√

−

B2
1
4B2

exp

(cid:18)

−

+ B0

Φ

a

(cid:19)

(cid:18)

2B2 −

−

p

B1
2B2 (cid:19)

−

√

(23)

I (0)(a, B0, B1, B2) =

Consequently,

I (1)(a, B0, B1, B2) =

[exp(B2x2 + B1x + B0)]

(cid:18)

=

exp(B2a2 + B1a + B0)

B1

−

√2Π

2B2

√

−

and again integrating by parts

a

−∞ −

a

B1

exp(B2x2 + B1x + B0)

/(2B2)

(cid:12)
(cid:12)
(cid:12)
exp

B2
1
4B2

−

(cid:18)

−∞

Z

+ B0

Φ

a

(cid:19)

(cid:18)

2B2 −

−

p

(cid:19)
B1
2B2 (cid:19)!

−

√

/(2B2)

I (2)(a, B0, B1, B2) =

xx exp(B2x2 + B1x + B0)dx

a

=

=

xeB2x2

a

1
2B2 (cid:18)
B1
2B2 Z
−
a
eB2a2
2B2

−∞

−∞

Z

+B1x+B0

a

a

−∞ −

−∞

Z

exp(B2x2 + B1x + B0)dx

(cid:19)

x exp(B2x2 + B1x + B0)dx

(cid:12)
(cid:12)
(cid:12)

+B1a+B0

−

(2B2)−1I (0)(a, B0, B1, B2)

B1
2B2

−

I (1)(a, B0, B1, B2)

Let us consider the covariance of terms involving

(γF (t) + η(t))− and ZN (or analogously Z).
Since ZN and η(t) are jointly Gaussian, ZN can be represented as an aﬃne combination of η(t) and
some independent, normally distributed variable u, Eu = 0 and σu = 1, as

−

Consequently,

Solving:

ZN = c0 + c1η(t) + c2u

Z = c2
η + c2
σ2
2
µZN = c0 + c1µη

1σ2

cZN ,η(t) =

cov(ZN , η(t))
σZN ση

=

c1ση
σZN

c0 = µZN −

cZN ,η

σZN
ση

µη,

c1 =

σZN
ση

cZN ,η,

c2 = σZN

1

q

c2
ZN ,η

−

ECB Working Paper 1896, April 201632 
Then:

(1 + r)eZN ,

cov(

−

h

−

−

1

h
−γF

= (1 + r)

1

=

−

−

−

+

(1 + r)

1 + r
√2Πση
1 + r
√2Πση
1 + r
√2Πση

h

−

1

E

h

h

h
−
h

h

h

h

1

1

1

−
h

−
h

−

(γF + η)−) = (1 + r)

cov(eZN , (γF + η)−)

h

−

1

h

(γF + y) exp(c0 + c1y + c2u) exp

(y

−

(cid:18)

µη)2

−
2σ2

η (cid:19)

dy

1
√2Πση

−∞ −
Z
EeZN E(γF + η)−

γF exp(c2

2/2)I (0)(

−

exp(c2

2/2)I (1)(

−

γF, c0 −
µ2
η
2σ2
η

γF, c0 −

µ2
η
2σ2
η

, c1 +

µη
σ2
η

,

1
2σ2
η

)

−

, c1 +

µη
σ2
η

,

1
2σ2
η

)

−

exp(µZN +

σ2
ZN
2

)(γF (t)I (0)

η + I (1)
η )

where I (n)
almost exactly the same representation.

η

is deﬁned in the context of equation 16. The covariance of

(γF (t) + η(t))− and Z has

−

The ﬁnal missing component of the covariance matrix is

we deﬁne

cov((aǫ(t

1, t)

−

−

¯π)ǫ(t + 1),

h

−

h

−

1

(γF (t) + η(t))−)

¯c0 = µǫ

cǫ,η

−

σǫ
ση

µη,

¯c1 =

σǫ
ση

cǫ,η,

¯c2 = σǫ

1

q

c2
ǫ,η

−

cov((aǫ(t

1, t)

¯π)ǫ(t + 1),

(γF (t) + η(t))−)

−
h(aǫ(t

−
1, t)
h

−
1

−

¯π)

E

−

−

1
√2Πση Z

−∞

h(aǫ(t

+

¯π)

−

Eǫ(t + 1)E(γF (t) + η(t))−)

1, t)
−
h
1
−
1, t)

=

=

−γF

(¯c0 + ¯c1z + ¯c2u)(γF + z) exp

(z

−

(cid:18)

µη)2

−
2σ2

η (cid:19)

dz

!

h(aǫ(t

−
√2Πση(1

¯π)

−
h)

−

¯c0γF I (0)
h

η + ¯c0I (1)

η + ¯c1γF I (1)

η + ¯c1I (2)

η −

γF (t)I (0)

η + I (1)

η

µǫ

(cid:16)

(cid:17)i

ECB Working Paper 1896, April 201633 
Acknowledgements 
The author is indebted to S. Dubecq, T. Hurd, D. Krusec, M. Leber, M. Rocco, D. Żochowski and the participants of the BIS RTF 
meeting (Vienna) for helpful discussions on the setup, calculations and results.  

Grzegorz Hałaj 
European Central Bank; email: grzegorz.halaj@ecb.int 

© European Central Bank, 2016 

Postal address 
Telephone 
Website 

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+49 69 1344 0 
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All rights reserved. Any reproduction, publication and reprint in the form of a different publication, whether printed or produced 
electronically, in whole or in part, is permitted only with the explicit written authorisation of the ECB or the authors.  

This paper can be downloaded without charge from www.ecb.europa.eu, from the Social Science Research Network electronic library at 
or from RePEc: Research Papers in Economics.  

Information on all of the papers published in the ECB Working Paper Series can be found on the ECB’s website. 

ISSN 
ISBN 
DOI 
EU catalogue No 

1725-2806 (online) 
978-92-899-2032-2 
10.2866/722676 
QB-AR-16-013-EN-N