# Some thoughts on (negative) interest rates and inflation from a micro perspective, Part I

the next part is here

Being a theoretical physicist by training and a derivatives quant later on I tend to see things through a particular lens, and over the last few years I have had the impression that some of the apparatus built up there might be helpful to understand inflation and interest rates, in particular negative ones. This is still work in progress, so I’ll see where this is going. In this medium post (or posts, as the case may be) I’ll try to avoid mathematics where ever I can, the main reason being that medium does not do formulas. I am in parallel working on some more technical notes, and once I figure out how to publish those I’ll do so.

#### Numeraires and symmetries

So let me start with the fundamental idea that I want to develop: a key concept when pricing derivatives is that of a numeraire, or unit of account, and a lot of the analytical pricing solutions rely on a smart choice of that numeraire to make calculation tractable. For example, when pricing European options on any underlying the natural numeraire is the forward contract with the maturity that coincides with the option expiry, and as if by magic all the nasty stuff like dividends or interest rates disappears (well, it does not disappear, but it is all absorbed into the forward, which is why it is the natural numeraire to look at options). However — just because the forward numeraire is the one that is the most easy to use in this case does not mean it is the right one. In fact, there is no right numeraire: all numeraires are equally valid, and all numeraires are bound to give the same results on the observable values. In other words, we have something that a physicist would call a symmetry: there is a group of transformations that we can apply to our system, and that changes its behaviour in a predictable manner.

#### Symmetries in physics

Probably the simplest symmetry in physics is that of rotation: assume we are in an infinite Euclidean plane (…flat Earth…) and if we rotate our apparatus by a certain angle the physics does not change. Note that this does not mean that the measurements don’t change. Some don’t indeed, and those are called invariants. Speed and distance are such invariants: if I shoot a cannon to the north or to the east, the cannon ball will fly with the same speed, and — assuming a homogenous environment — will land the same distance from the cannon. However, my measurements will usually not be invariant, but rather covariant (meaning I know how to transform them). For example, assume a cartesian coordinate system with the cannon at (0,0), and if I shoot to the north the cannon ball will end up at the point (100,0). If I turn the cannon by 90 degrees the ball will end up at (0,100), and if I only turn it by 45 degrees the ball will end up at (71,71). Those coordinates are not invariant, the are covariant. What is invariant however is the distance from the origin which is 100=71*sqrt(2).

Note that in this case we assumed that were actually turning the cannon in the real world so things did change. There is a dual view to this one which is that we keep the apparatus the same, but we change our coordinate system. So instead of saying “we turned the cannon from point north to pointing north east” we say “we first call the direction into which the cannon is shooting ‘north’, ie (1,0) and then we call it ‘east’, ie (0,1)”. The mathematics in those two cases is the same, but the interesting point is that as far as an external observer is concerned, nothing changed. Anyone without access to my notes will not be able to assert whether I think the cannon shot to the north or to the east, but everyone will agree that the cannon ball ended up 100m away from the cannon.

#### The efficient market for exchangeables

Back to finance an numeraires. For the time being let’s consider a snapshot, taken at one particular point in time. In finance people are often talking about assets, but I want to use the term exchangeables instead to denote loosely anything that I can trade for something else. Note that this includes assets in the usual sense (eg financial one like stocks, and real one like factory buildings) but also consumables (eg bread, flowers, chicken and eggs) and even services (eg cleaning, air transport etc). In my idealised environment the following conditions hold

• the exchangeables can be grouped into a (finite, or countably or uncountably infinite) number of fungibility classes; everyone agrees on what those classes, and everyone considers all items within one class perfect substitutes for one another
• the market for exchangeables is frictionless and arbitrage free, meaning that any closed chain of exchanges ends up with exactly the original amount

Fungibility classes. I don’t want to discuss fungibility classes in too much detail here, as I think they are intuitively clear, but rather complex in practice: an egg…class A…produced by company X…bought in London…near Bank station…on January 5th…at five to midnight…. You get the picture, but let’s not go down that rabbit hole, just trust me that we can deal with this issue if we have to, and that it is not central for what we are trying to do.

Frictionless and arbitrage free. The more interesting point is that of being friction and arbitrage free. Let’s look at a few closed chains. Say I have one litre of wine, and I sell it for coal, and I sell the coal for wine. That is a closed chain, and as per the requirement above I must end up with exactly one litre of wine. This in particular implies that there is no transaction fee, hence the term frictionless.

Let’s look at a longer chain: I have wine, sell it for coal, sell the coal for cheese, and the cheese for wine, and I have to end up with the same amount. Alternatively and equivalently I could say that if I exchange my wine to cheese, or if I exchange my wine to coal and the coal to cheese I end up with the same amount of cheese at the end, therefore I can not make money trading, and therefore the whole thing can be called arbitrage free.

#### Market prices

It is pretty easy to prove that with the above conditions for every two fungibility classes i and j there is a unique exchange ratio Rij that determines how many units of type j I can get for one unit of type i. The closed chain condition implies that whenever I have a product of exchange ratios that ends up where it started then the whole thing is one. For example

Rij *Rji = 1

or

Rij * Rjk * Rki =1

It is easy to prove that if those conditions hold for all i,j,k then there exists a vector of market prices (Pi) such that

Rij = Pi / Pj

whatever i and j. It is easy to see that the prices imply that closed chains collapse to unity: whenever a chain is closed all prices appear in the numerator and denominator and therefore cancel out. To go the other way we can simply define the price of item i as its exchange ratio with the item 1, ie we define

Pi := R1i

and it is easy to see that those satisfy the condition.

#### Market price symmetry

We have seen above that, given the set of exchange ratios, we can define a set of market prices that allow is to generate those exchange ratios simply with reference to the exchange ratio with our item 1. Of course there is nothing magic about item 1, we could have chosen 2, or 3, or any other one, provided we choose the same item everywhere. What we are doing is that we choose this item as our numeraire exchangeable (which by definition has unity price) and we define all other prices with reference to it.

But the group of symmetries is actually significantly larger. It is easy to see that in fact the complete symmetry group of market prices is that multiplicative group of the positive real numbers: whenever we have a vector of prices, then we can multiply all of the prices with the same number, and we have an equivalent vector of prices: when calculating the exchange ratios (which are the invariants in our bartering world) the multiplicative factor appears in the numerator and the denominator and cancels out.

This is important, so let’s do an example. Say we have wine, coal, and cheese, and our initial price vector is (1, 0.5, 3), implying that 1 unit of wine buys 0.5 units of coal or 3 units of cheese. We could for example use coal as numeraire, which would yield the price vector (2, 1, 6). We can also multiply our price vector with an arbitrary number, say 55, yielding a price vector of (55, 27.5, 165). We can easily check that in all three cases for example the exchange ratio of wine to cheese is

R31 = 3/1 = 6/2 = 165/55 = 3

meaning that our observables, the exchange ratios, are invariant.

#### Money, money, money

So far we were bartering and whilst we actually defined prices we still do not know what money is. In order to get closer to that one let’s consider a particular trading set-up:

• Trading happens only during a well defined period
• Exchange ratios are exogenous (and constant throughout the period), and satisfy the frictionless / no-arbitrage conditions define above, which implies that they can be calculated from a vector of prices
• The price vector is not unique because of the aforementioned scaling symmetry; however, there is a central entity — that we call a bank — that fixes the symmetry at a point of their choosing and publishes the price vector for this particular period
• The bank issues tokens — called money — and it undertakes to lend those against exchangeables; what this means is that any market participant can deposit exchangeables with the bank and gets a tokens in return, the number being determined by the price of the exchangeables deposited; those tokens must be returned to the bank at the end of the trading period, which then returns the exchangeables; traders who dont return all their tokens at the end of a period will be terminated
• During the trading period the market participants can trade tokens against exchangeables and vice versa at the published prices (they are not allowed to deviate from those prices).

The idea here is that at the beginning of the trading period the bank creates a certain supply of tokens — the money supply — that is being provided to the traders. Those tokens don’t have any intrinsic value; however, because the bank undertakes to exchange them against the originally provided exchangeables at the end of the period, those tokens become an exchangeable of their own right, albeit a special one that is disappearing at the end of the period (and that has by construction unity value).

Those tokens allow to simplify the trading process greatly: an agent who has wine and would like to exchange it against cheese and coal does not need to find someone who wants to do the equal and opposite trade. Instead, the agent sells (some of) his wine against tokens to someone who wants his wine, and then uses the tokens so obtained to purchase cheese and coal from the respective sellers.

Note that not everyone actually has to borrow tokens from the bank. For example, someone following the above process (ie first selling then buying exchangeables) does not need to borrow. However, this does not work on a whole-system basis because not everyone in the system can sell exchangeables against tokens before doing the reverse.

Note however that in principle it is enough to inject a single token into the system that can be exchanged back and forth until all exchangeable trades have been executed; because prices are exogenous and fixed the end result will not depend on the number of tokens injected, aka the money supply. It is clear however that operationally a large money supply is better, because it allows for the operations to happen in parallel rather than sequentially, following the token around.

#### Inflation and deflation

In the previous section we discussed a one-period barter system that was operationally optimised by introducing a central entity (a bank) that issued tokens (money). The important characteristics of those tokens are that

• the tokens are only valid during the particular trading period to which they belong: all are issued at the beginning and returned at the end
• the value of the tokens is arbitrary fixed with respect to the market prices of that period (note that market prices are exogenous and are held constant during that period)

We now extend this model to multiple trading periods, and we assume that bona-fide exchangeables (ie all but money) can be carried from one period to the next. We further assume that different trading periods can have different exchange ratios (still exogenously provided) but that there is a certain continuity of those ratios meaning they can not change too much from one period to the next (of course, over time changes can accumulate and therefore exchange ratios can change substantially).

However, at the beginning of each period, the bank is completely and utterly free to choose the overall price level in its publication. What I mean with this is whilst the bank must respect the exchange ratios (those are exogenous) it can choose which point to use for unity numeraire. In one period it might be coal, in another one wine, and in yet another one 55 times wine. Whilst the bank’s choice of numeraire does not impact any of the exchange ratios amongst exchangeables it does impact the value of the tokens it issue, because by definition they are of unity value.

For simplicity let’s assume that none of the exchange ratios have changed, and that the previous numeraire was ‘2 units of wine’ and the new numeraire is ‘1.8 units of wine’. To rephrase this, in the previous period, one token would buy 2 units of wine, and in this period it would by 1.8 units of wine. Or, to put in price terms: one unit of wine was previously 0.5 tokens, and it is now 0.556 tokens. This is what we might call a nominal inflation: the price of wine (expressed in tokens) went up by about 11%, from 0.5 to 0.556.

It is also clear that the nominal inflation here is completely and utterly meaningless, because it is based on a completely arbitrary choice by the bank. The bank could have as well opted for say 10% deflation (numeraire is 2.2 units of wine) or price stability (numeraire remains at 2 units) and in the real world absolutely nothing would have changed.

I’ll leave it at that for today. To be continued here

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