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The role of central bank capital – a simple model

The role of central bank capital – a simple model

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Central banks and public institutions as investors

main purpose of BMW had been to show how central bank capital may
matter for the achievement of the central bank’s policy tasks. The mechanisms by which central bank capital can impact on a central bank’s ability
to achieve price stability were illustrated in this paper by a simple model in
which there is a kind of dichotomy between the level of capital and inflation
performance. The model is an appropriate starting point to derive the actual
reasons for the relevance of central bank capital in the most transparent
way. The starting point of the model specification is the following central
bank balance sheet.
Stylised balance sheet of a central bank


Monetary policy operations (‘M’)
Other financial assets (‘F’)

Banknotes (‘B’)
Capital (‘C’)

Banknotes are assumed to always appear on the liability side, while the
three other items can be a priori on any side of the balance sheet. For the
purpose of the model, a positive sign is given to monetary policy and other
financial assets when they appear on the asset side and a positive sign to
capital when it appears on the liability side. The following assumptions are
taken on each of these items:
 Monetary policy operations can be interpreted as the residual of the balance
sheet. This position is remunerated at iM per cent, the operational target
interest rate of the central bank. Assume that the central bank, when
setting, follows a kind of simplified Taylor rule of the type iM,t ¼ 4 þ 1.5
(pt À 1À2). According to this rule, the real rate of interest is 2 per cent and
the inflation target is also 2 per cent.12 An additional condition has also
been introduced in the Taylor rule, namely that in case it would imply
pushing expected inflation in the following year into negative values, the
rule is modified so as to imply an expected inflation of 0 per cent. It will
later be modelled that for profitability/capital reasons, i.e., reasons not
relating directly to its core task, the central bank may also deviate from
this interest rate setting rule.
 Other financial assets contain foreign exchange reserves including gold
but possibly also domestic financial assets clearly not relating to monetary
policy. Assume it is remunerated at iF per cent. The rate iF per cent may

See e.g. Woodford 2003 for a discussion of the properties of such policy rules.


Bindseil, U.

be higher or lower than iM per cent, which depends inter alia on the yield
curve, international imbalances in economic conditions, the share (if any)
of gold in F, etc. Also, F can be assumed to produce revaluation gains/
losses each year. One may assume that iF,t ¼ iM,t þ q þ xt with normally,
but not necessarily, q>0, implying that the rate of return on F would tend
to be higher than the interest rate applied to the monetary policy
instruments, and xt is a random variable with zero mean reflecting the
associated risks. F can in principle be determined by the central bank,
but it may also be partially imposed on the central bank through its
secondary functions or ad hoc requests of the Government. Indeed, F
may include, especially in developing countries, claims resulting from
bank bailouts or from direct lending to the Government, etc. Typically,
such assets are remunerated at below market interest rates, such that one
would obtain q > 0. The model treats financial assets in the most
simplistic way, but this is obviously where the traditional central bank
risk management would be very differentiated about (while ignoring the
three other balance sheet items).
 Banknotes are assumed to depend on inflation and normally follow some
increasing trend over time, growing faster when inflation is high. Assume
that Bt ¼ BtÀ1 þ BtÀ1 ð2 þ pt Þ=100 þ BtÀ1 et , whereby pt is the inflation
rate, ‘2’ is the assumed real interest or growth rate and et is a noise term.
It is assumed that the real interest rate is exogenous. Despite the
development of new retail payment technologies over many years and
speculation that banknotes could vanish in the long run, banknotes have
continued to increase in most countries at approximately the rate of
growth of nominal GDP. Our stylized balance sheet does not contain
reserves (deposits) of banks with the central bank, but it can be assumed
alternatively that reserves are implicitly contained in banknotes (which
may thus be interpreted as the monetary base). The irrelevance of the
particular distribution of demand between banknotes in circulation and
reserves with the central bank would thus add robustness to this
assumption on the dynamics of the monetary base.13
 Capital depends on the previous year’s capital, the previous year’s profit
(or loss), and the profit sharing rule between the central bank and the
Government. In the basic model setting, it is assumed that the profit

A switch from banknotes holdings to reserve holdings would imply that seignorage revenues would in the first case
stem from a general tax to the holders of banknotes, while in the second case they would be comparable to a tax on
the banking sector.


Central banks and public institutions as investors

sharing rule is as follows: if profit is positive, i.e. Pt À 1>0 then
Ct ¼ Ct À 1 þ aPt À1 (with 0 < a < 1) else Ct ¼ Ct À 1 þ Pt À 1, and a is set
to 0.5. Profits depend on the returns on the different balance sheet
positions and on operating costs. With regard to operating costs, q, it
may be assumed that they grow over time at the inflation rate. Profit and
thus Capital is likely to contain a further random element, which reflects
that extraordinary costs may arise to the central bank when the
Government manages to assign additional duties to the bank. In the
less industrialized countries, these costs could typically be the support of
insolvent banks, or the forced granting of credit to the Government. As
mentioned above, such factors can also be modelled as affecting the
remuneration rate of financial assets.
An equation that explains the evolution across time of the inflation rate
completes the model. A Wicksellian relationship between inflation tomorrow and inflation today is assumed, i.e. ptþ1 ¼ pt þ bð2 þ pt À iM;t Þ þ lt ,
i.e. inflation normally accelerates if interest rates on monetary policy
operations are below the sum of the real rate on capital (2 per cent) and the
current inflation rate. The noise term lt means that inflation is never fully
controlled. The equation also implies that there is a risk of ending in a
deflationary trap: when pt < À 2, then, due to the zero constraint to interest
rates, prices should start falling further and further, even if interest rates are
zero. If lt $ N ð0; r2l Þ, this can always happen theoretically, but of course
the likelihood decreases rapidly when the sum of the present inflation and
of the real rate is high. Adding a time index t for the year, the time series are
thus determined as follows over time:14
pt ¼ ptÀ1 þ bð2 þ ptÀ1 À iM ;tÀ1 Þ þ lt


qt ¼ ð1 þ pt =100ÞqtÀ1


Ft ¼ F


if PtÀ1 ! 0 then Ct ¼ CtÀ1 þ aPtÀ1 ðwith 0 < a < 1Þ;
else Ct ¼ CtÀ1 þ PtÀ1
Bt ¼ BtÀ1 þ BtÀ1 ð2 þ pt Þ=100 þ et


The order of the equations, although irrelevant from a conceptual point of view, reflects how the eight variables can
be updated sequentially and thus how simulations can be obtained.


Bindseil, U.

þ 2 þ ptÀ1 then
þ 2 þ ptÀ1
¼ maxð4 þ 1:5ðptÀ1 À 2Þ; 0Þ; else iM ¼

if maxð4 þ 1:5ðptÀ1 À 2Þ; 0Þ <


iF;t ¼ iM;t þ q þ xt


Mt ¼ Bt þ Ct À Ft


Pt ¼ iM ;t Mt þ iF;t Ft À qt


This simple modelling framework captures all basic factors relevant for the
profit situation of a central bank and the related need for central bank capital.
It can also be used to analyse the interaction between the central bank balance
sheet, interest rates and inflation. It should be noted that, from equation (1.1)
and iM;t ¼ 4 þ 1:5ðptÀ1 À 2Þ, a second-order differences equation can be
derived of the form ptþ1 À ð1 þ bÞptÀ1 þ 1:5bptÀ2 ¼ b þ lt . Disregarding
the stochastic component, l, this equation has a non-divergent solution
whenever À2=3 < b < 2=3. The constant solution pt ¼ 2; 8t, is a priori a
solution in the deterministic setting. However, it has probability 0 when
considering again the shocks lt.
Simulations can be performed to calculate the likelihood of profitability
problems arising under various circumstances. The model can be calibrated
for any central bank and for any macroeconomic environment. The impact
of capital on the central bank’s profitability and hence financial independence is now briefly discussed. First, as long as bankruptcy of the central
bank is excluded, by definition, negative capital is not a problem per se.
Indeed, as long as the central bank can issue the legal tender, it is not clear
what could cause bankruptcy. By substitution, using the balance sheet
identity, one obtains the profit function:
Pt ¼ iM ;t ðBt þ Ct Þ þ ðiF À iM Þ:Ft À qt


Therefore, a higher capital means higher profits since it increases the size of
the (cost-free) liability side. For given values of the other parameters, one
may therefore calculate a critical value of central bank capital, which is


Central banks and public institutions as investors

needed to make the central bank profitable at a specific moment in time:
Pt > 0 ) Ct > À

ðiF À iM Þ
Ft þ qt À Bt


Unsurprisingly, the higher the monetary policy interest rates, the lower the
critical level of capital required to avoid losses, since the central bank does
not pay interest on banknotes (or excess reserves, i.e. reserve holdings in
excess of the required reserves). A priori this level of capital can be positive
or negative, i.e. positive capital is neither sufficient nor necessary for a
central bank to be profitable. It would also be possible for a central bank
with positive capital to suffer losses over a long period, which could
eventually result in negative capital. Likewise, a central bank with negative
capital could have permanent profits, which would eventually lead to
positive capital. Moreover, when considering the longer-term profitability
outlook of a central bank in this deterministic set-up, it will turn out that
initial conditions for capital and other balance sheet factors are irrelevant
and the only crucial aspect is given by the growth rate of banknotes as
compared with the growth rate of operating costs. The intuition for this
result (stated in proposition 1 below) is that, when considering only the
long term, in the end the growth rate of banknotes needs to dominate the
growth rate of costs, independently of other initial conditions.
When running Monte Carlo simulations of the model (see Bindseil et al.
2004a section 4), the starting value of the array (M0, F0, B0, C0, p0, i0) as well
as the level of the parameters (a, b, q, a2e, r2x, r2l) will be crucial for
determining the likelihood that a central bank will be at a certain moment
in time in the domain of positive capital and profitability.
Having shown that in the model above, a perfect dichotomy exists
between the central bank’s balance sheet and its monetary performance,
BMW continue by asking how one explains the observation, made for
instance by Stella (2003), that many financially weak central banks are
associated with high inflation rates. It is likely that there is another set of
factors, related to the institutional environment in which the central bank
exists, that is causing a relationship between the weakness in the central
bank’s financial position and its inability to control inflation. BMW argue
that the relevance of capital for the achievement of price stability can be
explained by considering what exactly happens in case the privilege to issue
legal tender is withdrawn from the central bank. If the central bank lost the
right to issue currency, it would still need to pay its expenses (salaries, etc.)


Bindseil, U.

in a new legal tender that it does not issue. Also, banknotes and outstanding
credits would need to be redeemed in the new currency at a certain fixed
exchange rate. Consider the two cases of central banks with positive and
with negative capital with a very simple balance sheet consisting only in
Capital, Banknotes, and monetary policy operations.
Two central banks, before their right to issue legal tender is withdrawn

Positive Capital Central Bank
Monetary policy


Negative Capital Central Bank


Monetary policy

After the withdrawal of the right to issue legal tender, both central banks
become normal financial institutions. After liquidating their banknotes and
monetary policy operations, their balance sheets take the following shape:
Two central banks, after their right to issue legal tender is withdrawn

Positive Capital (former) Central Bank


Negative Capital (former) Central Bank

Financial debt

Obviously, the second institution is bankrupt, and the holders of its
banknotes and of liquidity absorbing monetary policy operations are not
likely to recover their claims. Also, the institution will immediately have
to stop paying salaries and pensions, etc. In case of a positive probability
of withdrawal of the right to issue legal tender, central bank capital and
profitability will thus matter. In the case of negative capital, the staff and
decision-making bodies of the central bank thus have incentives to get out
of the negative capital situation by lowering interest rates below the neutral
level, which in turn triggers inflation, and eventually an increase of the
monetary base up till positive capital is restored.
One may thus conclude that the higher the likelihood of a central bank to
lose its right to issue legal tender, the more important central bank capital
becomes. As the likelihood of such an event will however never be zero,
central bank capital will always matter. Once this conclusion is drawn, one
can start deriving, through simulations, which level of central bank capital
is adequate to ensure a monetary policy aiming exclusively at maintaining


Central banks and public institutions as investors

price stability. Assuming that the central bank will thus normally care about
profitability and positive capital, one may, in the case of negative capital,
substitute the interest rate generated by the Taylor rule iM,t by an interest
rate ~iM;t determined as follows (with h<0 a constant):
~iM;t ¼ minð4 þ h; iM;t Þ
The functional form given to the capital term in this equation is, of course,
ad hoc. It implies that if capital is negative, the central bank no longer reacts
to an increase of inflation (reflected in the suppression of the inflation term)
and even reduces rates further, by an amount corresponding to h. Assuming
that central banks will thus follow inflationary policies when having negative capital, and introducing the possibility of large negative shock to profit
(due e.g. to a foreign exchange revaluation or ‘contingent liabilities’ as
formulated by Blejer and Schumacher (2000)) in the simple model above,
allows deriving a positive relationship between capital and inflation performance. One may then calculate the ‘value at risk’ of the central bank and
determine a capital that with, say, a 95 per cent probability ensures that
within one year capital will not be exhausted. This is the approach basically taken by Ernhagen et al. (2002) without however the comprehensive
modelling framework proposed by BMW.

8. Integrated risk management for public investors
8.1 Integrated financial risk management in general
Integrated risk management is the holy grail of risk management in any
bank. It means essentially being comprehensive and consistent in terms of
the risk–return analysis and management of all of the institution’s activities.
Often, the term is also used for corporates, and it is stressed that it includes
not only financial risks, but all other sorts of risks, like business or operational risks. Sometimes, the term integrated risk management is also associated with ‘best practice’ concepts and ‘firm-wide risk management’ such
as done by Jorion (2003, chapter 27). Accordingly, integrated / firm-wide
risk management would rest on three pillars, namely best practice policies
(clear mission statement, well-defined risk tolerance and philosophy,
responsible policies endorsed and understood by the board of directors);
best practice methodologies (analytical methods to measure, control, and
manage financial risks), and best practice infrastructures (IT systems,


Bindseil, U.

organizational design). Focusing here on the issue of integrating financial
risk management in the narrow sense, one may structure the key inputs to
this approach in the following somewhat theoretical three categories:
(1) The starting point of an integrated risk management of a bank is a
business model of the bank and of the relevant business lines. For
each business line, a sort of ‘production function’ has to be assumed,
which maps input factors into outputs, and which allows, knowing
input and output prices, to calculate a profit contribution function and
eventually an optimal size of the different activities.
(2) Risk factors have to be introduced into this, whereby these may
concern both market prices and the production processes themselves.
A description of the stochastics includes ideally the joint probability
distributions of all risk factors, or, more pragmatically, some descriptive
parameters like variances and covariances.
(3) The relationship between overall risk taking, leveraging, capital and
refinancing costs has to be established. Choosing certain values of these
variables, taking into account the relationship between them, means at
the same time accepting a certain probability of default, which is also
relevant for the relationship with other, non-financial stakeholders.
On the basis of these inputs, one can then in theory derive simultaneously
the following elements of an optimum: First, one may establish the efficient
frontier of the company in the expected profit–risk plane. Second, by
matching the efficient frontier with the risk–return preferences of the
company’s owners (or other stakeholders), one may find the point on the
efficient set which maximizes the utility function of the owner (and/or other
stakeholders). In this, taxation considerations should be taken into account
as well, as taxation is normally not linear (but convex), and therefore makes
expected profits shrink when volatility of gross profits increases. Third, in
line with the chosen point on the efficient frontier, one obtains optimal
amounts of business activities (or asset sizes etc.) in the different business
lines, and, accordingly allocates a risk budget to those business lines and
activities. Moreover, one may implement specific tools of integrated risk
management, such as RAROC (risk-adjusted return on capital), which
allows to check ex post whether capital (or the risk budget) allocation is
optimal or not, and which can be used for evaluation and compensation of
business units and staff. Finally, another element of the optimum is a degree
of local costly risk-mitigating measures in each of the business activities.
Integrated risk management is certainly a tough challenge for any company.
In view of its complexities and methodological problems (e.g. sub-additivity


Central banks and public institutions as investors

of risks), it will in practice always be based on a number of ‘heroic’ and
questionable assumptions in particular regarding risk factors. Moreover, it
will be opposed by business lines that would be negatively affected from
applying its conclusions.
8.2 Integrated risk management issues for public investors
For public investors, additional difficulties arise from the predominance
of policy goals against pure return (risk) considerations, the perceived
importance of reputation risks, the difficulties to derive from basic business
economics an overall risk tolerance, etc. Consider first the three main inputs
to integrated risk management, such as applying specifically to public
(1) Business model and ‘production function’ of the public institution
and of the relevant business lines. Public institutions have often only
a very limited number of activities which may be deemed to be of
a ‘business’ nature. One of them is the investment of financial assets as
far as unconstrained by policy requirements.16 The overall extent of
this activity is given by the amount of funds available, i.e. there is little
perceived freedom of deciding on the overall scope of investment
activities. However, there is of course room for decisions at the subbusiness lines level, like how much to invest into which currency, what
asset types to invest in, etc. On a first look, all this may appear to be a
relatively simple portfolio optimization problem. However if one takes
into account set-up costs per asset type and risk management activities
per asset type, the analysis has to be enriched with elements from a
more standard business decision of a corporate (or a bank), which has
to decide on what businesses to go into or to specialize into. When
making the list of a public institution’s activities to be considered in an
integrated risk management, one should also not forget the policy tasks
discussed in Section 6.
(2) Risk factors. The risk factors relevant for the public investor are mainly
the classical financial risk factors like domestic and foreign interest rate,
spread, credit, foreign exchange rate and commodity risk. In addition,

See Sangmanee and Raenkhum 2000 for a first paper on integrated central bank risk management.
Others might be (1) provision of payments or securities settlement systems; (2) provision or reserve management
services for other central banks or public institutions; (3) cash handling services. In any case, the correlation
structure of these other business lines with investment management by the central bank are of limited relevance such
that it is fair to analyse the investment issues separately.


Bindseil, U.

there are the idiosyncratic policy-related risk factors, such as those
described in Section 6 for central banks.
(3) Relationship between total risk budget, leveraging, capital and
refinancing costs. In the case of commercial banks, a common way to
think is that for a certain business model, there is an optimal rating, e.g.
AA, which is associated with a certain probability of default. The bank
has thus to look at capital costs, and the risk–return profile of business
opportunities, to come to an optimal amount of capital and activities
leading to an overall risk–return profile being compatible with the
envisaged probability of default. For public investors, the first step of
such an approach, namely to set total risks as a function of the desired
rating and the capital cost function, seems to be the most difficult one.
For instance central banks have, virtually regardless of the risks in their
balance sheet, the rating of the relevant central government. Indeed,
they can normally cope with substantial negative capital for a long time,
as they should in the long run normally return to positive capital (see
Section 7). Before simply accepting an ad-hoc risk constraint, one could
aim at the following indirect approach, which relies on two assumptions, namely that (1) an implicit franchise capital of central banks can
be calculated and (2) that this franchise capital should correspond to
economic capital needs for ensuring the relevant sovereign issuer rating.
Franchise capital can be calculated from the discounted expected
income of a central bank due to its franchise to issue banknotes over
a certain period, say ten years. Of course, the choice of the parameters
and horizons underlying such a calculus will appear ad hoc, and it
should thus more be considered as an illustration of the issue than as an
approach to be followed.
Eventually, the company’s efficient frontier needs to be matched with the
risk–return preferences of the company’s owners. For individuals, risk–
return preferences are normally derived from the concavity of their utility
function, risk aversion following from Jensen’s law. For commercial banks,
as for institutions in general, assuming a utility function would be too ad
hoc. Instead, risk–return preferences should be derived from the business
model of the company, the environment in which it operates, and preferences of stakeholders. Typical sources of risk aversion with regard to public
institution’s profits may be: (i) Taking the specific perspective of the residual
claimant, the Government: The Government has an interest in the stability of
the transfer payments from the central bank for budget purposes. In particular, it will dislike the case that by surprise, the public institution’s


Central banks and public institutions as investors

payments will be zero in some year, or even more that it would have to
recapitalize the public institution. (ii) Taking the specific perspectives of the
Board of the public institution. Typically, risk aversion of companies also
stems from the profit–loss asymmetries implied by a progressive taxation
schedule (as average taxes paid increase with the volatilities of profits). In
the case of public institutions, a progressive profit transfer function has
similar implications: losses are typically kept by the public institution, and a
small fixed amount of profits can be kept for some provisions or reserves,
but profits in excess of some threshold are distributed fully to the Government. A public institution wishing to increase its capital in the wide
sense (including reserves and provisions) over time will try to ensure that it
always earns enough to accumulate capital as much as it can, but would not
care about how high its profits are beyond that threshold (although it has in
practice an interest to keep the Government happy for the sake of its
independence). (iii) For companies in general, risk aversion may be implied
by Financial distress in case of large losses: liquidity costs of fire asset sales,
financing premia for replacing capital, general demotivation of stakeholders
when the probability of default increases beyond the optimum for the
business model. For public institutions, this is probably a less relevant
source of risk aversion, since financial distress tends to remain remote.
(iv) Reputation costs being associated with large losses. This holds for any
company, but maybe even more for a public institution, for which the
public or the Government may assume that any large losses are due to
irresponsible behaviour.
As mentioned before, the relevance of reputation risks for public institutions will drive apart the apparent risk preferences of public institutions
for tasks assigned to them directly through their statutes, and indirectly
derived tasks reflecting a largely unconstrained choice. For the former, only
large losses affecting the Government’s finances in a substantive way should
matter and drive risk aversion, while for the latter, even very small losses are
painful. The general aversion of central banks against credit exposures
illustrates the issue: a default event affecting a corporate exposure, even
if underweighted, is perceived by central bankers to be associated with
headline risk, which is often quoted as reason to avoid such exposures. It is
not clear how to handle reputation risks in an integrated central bank
risk management framework. One could try to quantify the reputation risk
associated to the different financial risks, and to formulate one overall risk
budget and allocate it in an optimal way. Alternatively, one could argue that
reputation risks after all cannot be quantified well and that therefore, for