The

**aggregate liquidity management model**gave us the following equation for what determines the interbank rate:

**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.

I used Equation A to create the following table,
which describes the central bank’s policy options for changing interest rates.
This is not in Bindseil’s book, but it directly follows from Equation A, and I
think it is a nice way to condense the equation’s implications. Each cell is
the result of keeping constant, changing, or not controlling some combination
of the variables in Equation A.

The columns represent the central bank’s options for controlling the standing facility rates

*i*and

_{B}*i*The CB can either actively change them over some period of time, leave them constant, or not even provide them. The rows represent how the CB can impact the probability of recourse to the standing facilities, which is done through OMOs. Similarly, the CB can either actively change them over some period of time, keep them constant, or not intervene at all. Note that keeping

_{D}*P*constant isn’t exactly analogous to keeping the standing facility rates constant. To keep the facility rates constant, the CB does not have to do anything more than leave them at some stated rate. In contrast, to keep

*P*constant, the CB will have to be constantly responding to liquidity shocks and changes in the market’s perceived probability density function of recourse (see

**Appendix**) via OMOs.

Below, I explain each policy option one-by-one.
Before doing so, I’d like to note the middle cell, which reads “static rate.”
I’ve greyed it out since this policy option actually isn’t a viable way for the
central bank to change rates. If the CB keeps

*i*, and_{B}, i_{D}*P*constant, then the interbank rate is therefore constant. As such, you can think of this option as what the CB does when it wants to stabilize rates. All the other options are ways for the central bank to change rates.*Symmetric Corridor*

I’ll
start by explaining the ‘symmetric corridor’ approach to steering rates, which
is employed by the ECB (Bindseil’s central bank) among many other central banks, and is
arguably the easiest to understand. The idea behind this approach is that the
interbank rate will be the average of the standing facility rates, and if the
central bank wants to alter the interbank rate, all it does is change the
standing facility rates in parallel. Examining Equation A, we can see there are
two key ingredients to making this happen:

- Maintaining
*P = 50%*at all times - Set
*i*and_{B}*i*such that the average of the two rates,_{D}*i = (i*, is equal to the target rate._{b}+i_{d})/2

*B – D = M – A*, then to achieve (1), the central bank just needs to set

*M*equal to expectations of

*A*, assuming the probability density function of

*A*is symmetric (see

**Appendix**). Since we assume that in the short-run,

*A*is exogenous and thus independent of the target rate (empirically a reasonable assumption), then all a rate change entails is changing rates, not quantities! Bindseil on the rates versus quantity debate:

Adding some balance, Bindseil also critiques people more likely to be on the rates side. He writes that even though “OMOs are not relevant to implementing a change of market rates under the symmetric corridor approach… [this] has nothing to do with the idea of ‘open mouth operations’ made popular by Guthrie and Wright (2000). To change market rates, it is indeed not enough for a central banker to open is or her mouth, but the central bank needs to effectively adjust rates of standing facility in relevant reserve maintenance period.” And I’d also add the central bank will need to continue to monitor“Thus the volume of outstanding open market operations is not affected by a change in the stance of monetary policy, which is of course the opposite of what proponents of reserve position doctrine had in mind.”

*A*: if the density function of

*A*changes away from being symmetric, this will require a change in the quantity

*M*. As noted above, however, we wouldn't expect this to occur from the rate change itself.

Here’s a
graphical representation of the symmetric corridor approach, which I take from

**this Fed paper.**I’d recommend reading that paper, with the model presented here in mind. They use many of the same qualitative explanations as Bindseil, but with their own spin, and indeed state that they are also coming from the Poole (1968) lineage. I had read this paper before but didn’t completely understand it. Now with Bindseil’s models in mind (particularly the**liquidity shock model**), it’s quite clear .*Floor/Ceiling System*

The floor/ceiling
system is similar to the symmetric corridor system in the sense that

*P*is kept constant and the interbank rate depends only on*i*and_{B}*i*. The main difference is that P is now pegged to either 0% or 100%, such that the interbank rate will be either_{D}*i*or_{B}*i*. Monetary policy in the U.S. currently is a floor system. Banks are flush with so many excess reserves that the probability of recourse to the deposit facility is virtually 100%. As such, the interbank rate is near the interest rate on deposits at the central bank’s deposit facility (it differs a bit due to some market and institutional irregularities that aren’t important here). Here is an illustration from the aforementioned Fed paper:_{D}
Note that there is now
a range of quantity M that can drive the interbank rate to floor, in a sense
“divorcing money from monetary policy,” as that paper states. This is different
from the symmetric corridor, where there is a particular value

*M*that needs to be set for*P*to equal 50%. I would argue, however, that the symmetric corridor approach also divorces money from monetary policy, in the sense that a range of rates can be achieved with the same quantity*M*. This was explained above, with Bindseil using it as argument against monetarist-style thinking.
The ceiling system ~~is
something I came up with - or at least I haven't seen it explicitly discussed elsewhere ~~(see comment section below. Also, Bindseil notes that this approach was practiced by the German Reichsbank from 1876 until at least 1914). ~~Theoretically, ~~you could make it certain that banks
would end up short of reserves at the end of the maintenance period, such that
the interbank rate equals the borrowing rate. ~~I am not sure if this has ever
been implemented, but it seems theoretically possible.~~

*Asymmetric Corridor*

I am actually not sure
if “asymmetric corridor” is a widely accepted term, but I’ll proceed with the following
definition: there is a fixed spread between the interbank rate and borrowing
facility rate, but the deposit facility rate stays constant. An example of this
was the U.S. prior to the 2008, where discount window loans were some fixed
spread above the target interbank rate, but there were no interest on reserves
(

*i*was fixed at 0%). As such, the distance between the floor and target rate would change if the central bank changed its target rate. Here’s a graphical illustration of this, although the linkage between the target rate and the ceiling is not made explicit:_{D}
In order to change the
interest rate under this system (again assuming the ceiling is tied to the
target rate), all variables in Equation A would have to change. For instance,
suppose the target interbank rate was 5% and the spread to

*i*was 1%. In this case,_{B}*i*_{t}_{ }= 5%,*i*=0%,_{D}*i*=6%. Examining Equation A, then then_{B}*P*would have to equal 5/6 to achieve the target rate of 5% (6%*5/6 + 0%*1/6 = 5%). Suppose the CB then changed the target rate to 4%, such that*i*would now equal 5%. In this case, to achieve that target,_{B}*P*would have to be equal to 4/5 (5%*4/5 + 0*1/5 = 4%). To change*P*from 5/6 to 4/5 requires the Fed to alter liquidity conditions in the market by changing the quantity*M*via OMOs (presumably by soaking up a bit of reserves to make them a bit more scarce).*Other Theoretical Systems*

One could imagine a
variety of other theoretical systems by mixing and matching the various
combinations in the table.

Suppose the standing
facility rates were held constant but

Another theoretical approach would be to set

*P*was allowed to be manipulated through OMOs. If the borrowing rate was detached from the interbank rate, in contrast to the example above, then rate changes could be achieved by just varying*P*through OMOs and leaving the standing facility rates constant. This would be a pure quantity-oriented approach to rate changes,*assuming that the corridor would be wide enough for the central bank to enact a range of rate changes*. You can refer to any of the above graphs to imagine this, whereby the bank would be changing quantity to move rates along the demand curve.Another theoretical approach would be to set

*P*at some value other than 50% (which pegs rates at the corresponding position within a given corridor), and rate changes could be implemented by changing the floor and ceiling. This would be similar to the symmetric corridor approach, except that*P*would not equal 50%. The added challenge here is that the central bank has to be more in tune with the actual shape of the probability density function to get*M*such that*P*will equal 30%. With the symmetric approach, the central bank just cares if the distribution is roughly symmetric through time, allowing it to set*M*equal to expected*A*. Here, it’s more complicated since the higher order moments of the density function constantly matter for the central bank to attain the proper value of*M*.
Here’s a pure rates
oriented approach. Suppose the central bank doesn’t conduct OMOs at all, and
thus doesn’t exert influence on

*P.*The CB could still control*i*by setting*i*_{B}_{ }and*i*_{D}_{ }to the proper levels given the market’s perceived value of*P*without OMOs.*Bounded Volatility*

If the central bank
left the standing rates constant but didn’t intervene in markets with OMOs,
then

*P*would fluctuate based solely on market conditions. As a result,*i*would fluctuate within the corridor set by the standing facilities.*?*

I’ve labeled instances where the central bank does
not provide a standing facility with a question mark. What I had in mind
initially was that the CB does not offer a borrowing facility, thus abandoning
their ‘lender of last resort role.’ As such, the central bank is not placing a
ceiling on rates, which greatly increases the potential for high volatility and
bank failures. A passage from Bindseil’s book applies here:

“One big question mark exists, however, which is that we need to define what happens at the end of the maintenance period if a bank does not have sufficient reserves. If the rule is that the bank gets shut down, then rates indeed would likely be highly volatile. But this seems like a ridiculous hypothetical to me. More plausible is that there’d be some sort of costly penalty instead. But in that case, we’re essentially back to a borrowing facility, which really is just a penalty.This is in line with Bagehot’s (1873: 58) early insight regarding the inherent instability of money markets to the effect that only the central bank could limit the implied volatility of rates, for instance by offering standing facilities...”

All this is to say that the right column may not be
a very useful construct if we’re thinking about a central bank in the first
place. But it’s interesting in the sense that, at least for me, it drives home
the message that the ‘lender of last resort’ function is a fundamentally core
feature of central banking – practically inseparable. I can almost imagine a
central bank that doesn’t conduct OMOs more easily than one that doesn’t
provide standing facilities. Perhaps more on this in a future post on rates
versus quantities.

Great summary. We're definitely thinking on the same lines.

ReplyDelete"The ceiling system is something I came up with - or at least I haven't seen it explicitly discussed elsewhere. Theoretically, you could make it certain that banks would end up short of reserves at the end of the maintenance period, such that the interbank rate equals the borrowing rate. I am not sure if this has ever been implemented, but it seems theoretically possible."

I think that the Federal Reserve implemented something like this in the 1920s. (See Reifler-Burgess doctrine). The Fed did not have a deposit facility, but it did have lending facility, or discount window. The Fed would conduct open market sales in order to drive banks to borrow at the discount window, or open market purchases in order to reduce use of the discount window. Rates weren't always at the ceiling, but when Fed sales had reduced reserves to a sufficiently low level then the ceiling probably served as the upper bound.

I like the point on open mouth operations.

The phrase “divorcing money from monetary policy" is a touchy one to me. Variations in the interest rate portion of money's return and/or its convenience yield (controlled by its quantity) give monetary policy a bite. While I realize that your phrase is meant to indicate that variations in its convenience yield can become unimportant, it remains the fact that there is something that money must "do" or "provide" if monetary policy is to work, even if the return it provides is only the pecuniary interest rate. So strictly speaking, money can't be divorced from monetary policy.

"I can almost imagine a central bank that doesn’t conduct OMOs more easily than one that doesn’t provide standing facilities." ...and... "This would be a pure quantity-oriented approach to rate changes, assuming that the corridor would be wide enough for the central bank to enact a range of rate changes."

Wasn't the pre-2004 Fed a pure quantity approach? It had no deposit facility, and going to the discount window was really not an option since the stigma created by Fed borrowing meant that the true borrowing rate was far higher than the official discount rate.

Thanks for the thoughtful and insightful comments!

Delete“Wasn't the pre-2004 Fed a pure quantity approach? It had no deposit facility, and going to the discount window was really not an option since the stigma created by Fed borrowing meant that the true borrowing rate was far higher than the official discount rate.”

So let me think through this. The ingredients for the ‘pure qty approach’ as I defined it were a large enough but fixed spread between i_b and i_d (here i_d = 0) such that only OMOs are used to enact rate changes. If during this period of time, the Fed altered its target i without altering i_b, then it would seem that this period of time would fit my definition of the ‘pure qty approach.’ I am inferring that you made your point about stigma because stigma makes the spread effectively quite large. In other words, let’s say the advertised i_b = x, but stigma costs = s. The effective i_b* is then (x+s), and so the spread is (x+s) as opposed to just x. “S” thus gives the CB that much more wiggle room to target rates via just OMOs before having to also increase the stated i_b rate.

However, one potential wrinkle is that the Fed may have been charging a separate penalty for reserve deficiencies that was less costly than going to the discount window (x+s). If that was the case, then we should consider that reserve deficiency penalty i_b* instead, and redo the analysis from there.

Honestly, I’m not too familiar with the pre-2004 Fed approach. Do you know a good primer? If, like today, the Fed back then also moved the stated i_b (or, if applicable and effective, the reserve deficiency penalty) in parallel with the target rate, then it doesn’t quite fit the ‘pure qty approach,’ and we’re back to something more like the asymmetric corridor. However, if it was the case that s >>> x, such that any realistic delta x didn’t cause a meaningful change in i_b*, then we are mimicking the ‘pure qty approach’ pretty well. If the reserve deficiency penalty was effective instead, then it all depends on if and how much that penalty was altered with target rate changes.

I now kind of regret writing ‘pure qty approach.’ Those words have to be taken in context of the fixed standing facility corridor. Which brings me to your quotation of my “imagine…” sentence. The words ‘pure qty approach’ would arguably be more fitting in a world without standing facilities. But I’m not sure we can say Fed pre-2004 was a world without standing facilities. It was just that i_b* might have been larger than the stated i_b – either due to stigma costs or if the reserve deficiency penalty was effective instead. More to the point, if we buy the model that spits out Equation A, then you have to have *something* for i_b – whether it’s a standing facility rate or a reserve deficiency penalty etc. Even if we go with the individual shock model instead, you need something for i_b. So that’s maybe a better way to demonstrate why it’s easier to imagine a central bank without OMOs than one without standing facilities (of some sort). But I now realize that sentence would be stronger if I substituted something else for ‘without standing facilities’, since you might not consider reserve deficiency penalties etc. as being sourced from a ‘facility.’ So I should sharpen up that idea…

I should probably edit that section on the ceiling system. In addition to your erudite point regarding Reifler-Burgess (which I wasn’t familiar with), the ceiling system was the (an?) answer to the Bank of England’s question of “how to make Bank rate effective” in the 1800s era of discounting. Presumably they used it from time to time?

Delete“The phrase “divorcing money from monetary policy" is a touchy one to me...”

Can you please direct me to your posts where you define these concepts (convenience yield, distinguishing from the pecuniary interest rate, etc.)? Like JKH, it’s hard for me to follow you here, in large part due to definitions. I know I’ve run across some of the posts, but want to make sure I’m looking at everything I need. I wanted to get through a good portion of Bindseil before I thought hard about your conceptualization, but now I think I’m ready.

"I am inferring that you made your point about stigma because stigma makes the spread effectively quite large."

DeleteYep, that was my point.

"“S” thus gives the CB that much more wiggle room to target rates via just OMOs before having to also increase the stated i_b rate."

Exactly.

Take a look at this graph.

http://research.stlouisfed.org/fred2/graph/?g=oAy

It shows movements in the fed funds target, the fed funds rate, and the discount rate during the 80s and 90s. First of all, the target and the ff rate are usually above the discount rate, which is odd. It seems to indicate that the discount rate isn't holding the overnight rate down. Is some sort of stigma effect widening the spread (your S term)? Maybe, but the premium of the funds rate over the discount rate isn't constant, it fluctuates quite widely. Would stigma really account for such wide fluctuations? Lastly, the fed funds target, and the fed funds rate, often move without a corresponding increase/decrease in the discount rate. Which indicates a pure quantity effect, at least some of the time. Anyways, just food for thought.

All my stuff on the convenience yield is here. Leave a comment if you have questions.

Great graph and great questions. I'd just add to the research agenda that we'd have to look into the reserve deficiency penalty. Also (and this is something I'll need to post about and look more into), as insightful/useful as Equation A is (in my opinion), it might not be exactly what we need for the U.S. Equation A technically only applies if banks can average reserves around 0 through the maintenance period. But this isn't true in the U.S., since the maintenance period is many days long, but there are costly overnight overdrafts. So we need to incorporate that, and I'm not exactly sure how that changes things. Maybe it means we need to blend the aggregate and individual shock models, etc. Bindseil actually alludes to this, and cites a paper or two, but says that modeling this becomes exceedingly complex...

DeleteThanks for the link. Will start to dive in.

Also, I remember reading one of Mehrling's papers that critiques the accuracy of Equation A. I'll need to revisit his point as well.

DeleteAll this to say that while I think this model is really insightful, it may not be the end of the line.

BTW, curious to get your thoughts on the following. If the blogosphere roughly agreed with models/Equations A and B, then it seems like that’d go some way to clarifying massive debates such as these: http://www.interfluidity.com/v2/3763.html. You simply look at the variables in the equation and decide which you want to change, which then tells you what the central bank will have to do.

ReplyDeleteInstead, we all tend to get lost in these lengthy qualitative descriptions, all which have differing assumptions. Of course, the qual is necessary to develop the model, but once you have a (pared-down) model, high-level discussion can be greatly simplified.

What I’m driving to is I imagine reality is that many (most) of those debate participants aren't very familiar with or even aware of anything like Equation A – including some of the professors – and thus the resulting chaos. That’s not their fault; this area of research seems relatively unpopular. This was my motivation for learning this stuff and then putting it on the internet for all to see and discuss. So I'm curious about the extent to which you agree, and if you do somewhat, it’d be cool to draw more attention to this somehow.

I think it would be useful. The blogosphere doesn't seem to be a a big fan of equations, though. It's always easier to get an idea to spread on blogs if you can explain things in a few concise words without losing too much meaning. That's one reason why Sumner's NGDP targeting is so popular.

Delete