**Introduction**

In a

**previous post**, we explored how the central bank’s balance sheet evolves over the reserve maintenance period. In this post, we’ll review one of two models**Bindseil**presents for how the central bank controls the interbank rate. Here, we focus on the “aggregate liquidity management model.”
Bindseil notes that there is a rich and
heterogeneous literature related to these ideas, but that most of it places an
emphasis on examining the empirical properties of the interbank rate, rather
than modeling how the central bank controls it. So this is a bit "exclusive," if you will. The modeling approach that Bindseil focuses on has

**Poole (1968)**at its origins.

The Aggregate Liquidity Management Model with Certainty

The Aggregate Liquidity Management Model with Certainty

The “aggregate liquidity management model" focuses on the banking system as a whole, as opposed to taking an individual bank's perspective. The other model I'll review in a future post adopts t

**he latter****approach**.
To begin, assume perfect interbank markets. Also assume the following
type of reserve maintenance period, which is one-day long and permits intraday overdrafts (which
can be thought of as reserve averaging around 0):

The idea is that the central bank conducts the open market operation (OMO) at the beginning of the day, banks make trades in the interbank market for overnight funds based on all available information at that time, the interbank market closes, a stochastic (random) liquidity shock occurs at the end of the day, and banks seek recourse to the standing facilities as appropriate. This sequence of events should be familiar to you from

**this previous post on the central bank's balance sheet**. A multi-day reserve period would work as well, assuming multi-day reserve averaging around 0 and daily OMOs:

With that set-up, let’s consider the following statement
from Bindseil:

“Like any financial market, the market for reserves in the aggregate liquidity model is interesting owing to its uncertainty.”

To understand this, first suppose that there is

*no*uncertainty regarding any of the values on the right side of Equation 6 from the**balance sheet post**.**Equation 6:**

*B*

_{a}*– D*

_{a}= RR + XSR - M - B_{i}+ D_{i}+ A (all variables as averages over the maintenance period)
That
means that banks will know the direction of the aggregate recourse to the
standing facilities at the end of the maintenance period. This knowledge should
necessarily determine the price of reserves in the market, since it tells the
market what the marginal value of reserves will be at the end of the
maintenance period.

To represent this idea more formalistically, assume for simplicity that B

_{i}, D

_{i}, XSR, and RR all equal 0. Equation 6 then reduces to

*B – D = M – A*: aggregate recourse is determined by the extent to which OMOs offset autonomous liquidity shocks. Let’s also introduce the variables

*i*and

_{t=1}…i_{n}, i_{D},*i*, which stand for the overnight market interest rate on reserves at time t of the reserve maintenance period, the deposit standing facility rate, and the borrowing standing facility rate, respectively. Therefore:

_{B}

If M > A, B = 0 and D = M – A; i

If M < A, D = 0 and B = A – M; i

If M > A, B = 0 and D = M – A; i

_{1}= i_{2 }= … i_{n}= i_{D}If M < A, D = 0 and B = A – M; i

_{1}= i_{2 }= … i_{n}= i_{B}If it weren’t, then arbitrage opportunities would emerge. To understand this, suppose on average through the maintenance period, it is known that“Because holding reserves at any point during the maintenance period contributes equally to fulfilling reserve requirements, the overnight interest rate should be constant throughout the reserve maintenance period.”

*A*and

*M*will be equal to 10 and 5, respectively. This means that the banking system will be short of reserves (

*M – A = -5 = B*) at the end of the period, implying that recourse to standing facility B will be 5 and that the marginal interbank rate at that time will be

*i*= 5% (all of these are of course arbitrary numbers I have picked). If at some time before this, the interbank rate is less than 5%, then banks will borrow at the cheaper rate until the rate is bid up to 5%, either looking to lend them later or acquire them at a cheaper price than will be available later. Conversely, there should be no reason that the interbank rate goes above 5%, since the borrowing facility is always there (assuming minimal “stigma” costs associated with the borrowing facility). In competitive market equilibrium, these arbitrage opportunities should not emerge, thus ensuring that the interest rate remains at

_{B}*i*throughout the maintenance period.*

_{B}I think there may be something profound about this, regarding the rates versus quantities debate.

*In a world without uncertainty, all OMOs do is swing the market between the two rates on the standing facilities, and these two rates only.*The direction, as opposed to the magnitude, of OMOs is what determines the rate. I’ll come back to this in a future post on rates versus quantities, about which I think we can hold a more intelligible conversation using the models in this post.

**The Aggregate Liquidity Management Model with Uncertainty**

If that is all understood, let’s now add

*uncertainty*with respect to future liquidity conditions in the reserve market, which makes our analysis much more realistic. To introduce what happens in this case, Bindseil actually quotes the

**Radcliffe Report from 1959**which “was the first to explicitly mention a lucid

*probabilistic*concept of the reserve balance as determining the short-term level of interest rates,” even before Poole (1968):

“The level of rates of interest in the money market therefore depends on the current level of the rate being charged by the Bank for loans from the Discount office, and on the market’s expectations as to the trend of the (discount) rate and as to the extent to which they are likely to be obliged to borrow from the Bank at this rate.”

Here is the
mathematical representation of this idea.

If you haven’t had an intro probability course with calculus, some of this
notation may look intimidating, but it’s really not difficult at all. See the

**Appendix**for an explanation. For now, here’s an interpretation of the equation in English:*“the overnight rate on any day will correspond to the weighted expected rate of the two standing facilities, the weights being the respective probabilities that the market will be ‘short’ or ‘long’ of reserves at the end of the maintenance period for having recourse to the standing facilities.*

**Equation 3.2 may be considered as the fundamental equation of monetary policy implementation**.”

If this
notation is still a bit too mystifying for you, we can get away with
simplifying it to this:

*Equation A**: i*

_{t}= i_{B}*(P) + i_{D}*(1-P)*where*

*i*is the interbank rate,

_{t}*i*

_{B}_{ }and

*i*are the expected standing facility rates at the end of the maintenance period,

_{D}*P*is the probability of recourse to the borrowing facility, and

*(1-P)*is the probability of recourse to the deposit facility. In other words,

*i*is a

_{t}*weighted average*of the standing facility rates.** The weights are determined by the probabilities of recourse to the standing facilities, which central banks can control by altering liquidity conditions via the variable

*M*(open market operations). By subtracting or adding reserves to the system, the central bank makes it more or less likely that the banking system will end the reserve maintenance period short or long, respectively.

If you’re not excited yet, you should be! Bindseil
calls this the "

*fundamental equation of monetary policy implementation."*Assuming you more or less buy the model, you finally have a coherent framework for thinking about how the central bank can set interest rates under the aforementioned assumptions.**Let’s next explore the options it offers the central bank with respect to steering interest rates**.
----

Footnotes:

*For
those interested in sprucing up their understanding of this with fancy-sounding
math terms, this is an example of the

**martingale property**. The martingale property states the conditional expected value of the next observation of a random variable*x*, given all the past observations_{t+1}*x*, is equal to the last observation_{1}…x_{t}*. In equation form, this is represented as:*__x___{t}*E(x*

_{t+1 }| x_{1}, x_{2}, … x_{t}) = x_{t}

_{}
In the
above example, the interbank rate should follow a martingale process, since the
current rate (

*i*) should be equal the expected value of the following time period rate (_{t}*i*) (all within the maintenance period)._{t+1}
**The
martingale property should still hold if there is uncertainty. That is, the
overnight right on any day should be the same as the expected overnight rate on
the following days. The difference from the case with uncertainty is that now
unanticipated news can emerge throughout the maintenance period, causing the
overnight rate to change. Nonetheless, these changes will not be systematically
predictable, so while rates will not be constant from an

*ex post*perspective, the property will still hold from an*ex ante*perspective.
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