Applied Macro

First Order Approximation

I used this phrase in a post two weeks ago (Models – How do computers play chess?), and in causing some debate, has made me realise how it has both subtle and important differences in meanings. This has implications for how we approach problems and links to some of the root causes of mutual incomprehension that I often come across.

Some of the different meanings:

1). General usage
It is the same as “first approximation” or a “ball park estimate” and used interchangeably. It is an educated guess with a few assumptions and likely a simple model. I think this is the most common usage. It tells you nothing about the nature of the model as this will vary by context.

2). Engineering/physics
Often used as an indication of level of accuracy, like significant figures, and this accuracy improves with higher orders of approximation. For example, if estimating the number of residents of a town the answers could be:

1st order approximation / 1 sf 40,000
2nd order approximation / 2s.f. 37,000
etc.

3). Mathematics
1st order often refers to linearity. For example, a Taylor series of a function of the form:

1st order approximation a + bx (also called a linear approximation)

2nd order approximation a + bx + cx^2 (also called quadratic)

Order also exists in statistics, with arithmetic mean and variance known as the first and second order statistics of a sample. Skew and kurtosis are third and fourth order and can be thought of as shape parameters telling how far from the normal distribution you are.

4). Financial derivatives pricing
For option pricing, it refers to the order of differentiation, so is helpful when thinking about sensitivities in the change in price:

1st order approximation Delta (1st derivative of price with respect to underling price) 2nd order approximation Gamma (2nd derivative of price)

For bond pricing, second order approximation is also called convexity adjustment which again is used to help understand the non-linearity of bond prices.

5). It can refer to the number and importance of the variables in a model.

A first order approximation may only deal with primary drivers. A second order model would include secondary drivers used to refine the estimate.

For example, a first order approximation of the time taken for a ball to drop would be to use Newton’s second law, F= ma. A second order approximation might include some appreciation of wind resistance.

The meanings may not align

These definitions might appear to be much the same thing. You can easily argue that a simple linear model using only the most important drivers will produce a decent ball-park estimate to one significant figure.

But this apparent similarity of definition means that a common trap is to not notice when they are different.

A first order approximation can be quadratic

To estimate the height of a cannon ball after firing we need to draw a parabola not a straight line. Often the non-linearity is so important that any linear model is awful.

(see How Not to be Wrong: The hidden Maths of Everyday Life by Jordan Ellenberg).

Making estimates more “accurate”, using higher power terms may make the model worse
In maths or physics textbooks, this approach always works out given that you already know the mathematical function or have an underlying relative which is stable. But in the world of economics and finance it can lead to a huge methodological error, thinking that the better our model fits the data the better the model. I worked with many analysts who have struggled with this and kept producing models with wonderful correlations and R^2. This leads them to think they have a model which “explains” the price action as well as possible. But these models are invariably useless, have no predictive value and need to be “recalibrated” to make them refit new data as it comes in.

It takes judgement to know which variables to use.
For instance, in the example with dropping an object, Newton’s second law will do an excellent job on a ball bearing from 10m but a pretty poor job on a parachutist. Which drivers will be important in financial markets varies over time and it takes a lot of flexibility to stay open-minded as to potential outcomes.

Money 1 – A Creation Myth

In this piece, I take “Macroeconomics” by Greg Mankiw, the bestselling undergraduate textbook, as the source for this story.

A history of money

In the beginning, there was barter…
and a rudimentary economy was based on it. This is extremely inefficient as you have to walk around all day carrying lots of goods, hoping you bump into someone who has something you need who at the same time wants something of yours and will trade you for it.

Efficiency demanded the use of commodities….
Given how poorly organised this world would have been, “it is not surprising that in any society, no matter how primitive, some form of commodity money arises to facilitate exchange”. “Most societies in the past have used a commodity with some intrinsic value for money”. We can see this because archaeologists have found lots of gold, silver and copper coins from previous civilisations.

A really nice example of recent commodity money is the use of cigarettes for currency in a POW camp in WW2. This is an excellent example of why commodity currencies existed and how they operate.

As society evolved thus did “fiat money”….
A modern development in the history of money is the development of “fiat money” which is “money that has no intrinsic value”. This occurs via a process of “evolution from commodity to fiat money”. The process by which this happens is rather mysterious but “in the end the use of money in exchange is a social convention: everyone values fiat money because they expect everyone else to value it.”

Modern money

“Money is the stock of assets that can be readily used to make transactions” and it can be defined by its uses which are:

Store of value
Unit of account
Medium of exchange

Between history and mythology

Unfortunately, as so often is the case with creation myths, none of this is actually true. Understanding what money is and why economists are taught its history in such a strange way is important. In fact, I would say it is central to understanding current economic policy and also how best to invest.

Myth #1 In the beginning there was barter

There is no evidence of any society has ever used barter as their primary means of exchange. This should be unsurprising as it would be horrifically inefficient.

Myth #2 Commodity money was the primary form of money for most of history

This myth is more serious and way more pervasive.
However the evidence from the existence of coins far from backs it up, I think it is good evidence of the opposite.

Imagine we are in ancient Rome and we have a Denarius coin in front of us
(Deni from Latin “containing ten” originally was the value of 10 asses)

It has a nice picture of Hadrian on it, “he” of the wall.
It was made of silver and so has an intrinsic value from its weight in silver
(there are examples of use of gold in coins too – the history is interchangeable)

Consider this, let:

A= intrinsic melt-down value of the coin

B= face value on the coin

Then scenarios are:

A > B the coin would not exist, it would be melted down.

A = B why mint it in first place? Why bother calling it a Denarius at all and put the Emperor’s face on it? It would be simpler just to weigh it. There is no benefit for the government to go to the trouble and expense of minting these things.

B > A Now there is a reason to mint it – a profit! The difference (B–A) is known as “seignorage”. We know this was a main source of income for monarchs for centuries from records. But if B>A then there is no strong link between the intrinsic value of the metal and the value of the money. It sets a lower limit but nothing more.
So what is the difference to fiat money? Not much. History indeed has little evidence for a prevalence of commodity money.

But what about the example of cigarettes in the POW camp?
I love this example because it is correct, and utterly misleading. There is an important reason why a commodity currency was used. It is because there was no way to enforce an obligation as the members of the economy were not in control of their society (see below for why this matters).

Myth #3 Money is an asset

Money is not a thing, or an asset like any other asset in the economy. It is much more special than that. It is a ledger item which always consists of an asset and a liability which come into existence at the same time. There is nothing else like it and it is central to the functioning of the economy. I will delve into this in the following posts.

A common error when struggling with such an abstract concept, it is often much easier and more natural to think in tangible terms. An analogy for this is units of measurement. It is now “obvious” that the concept of measurement is conceptually separate from any physical object. I can separate the concept of “1 metre” from the physical reality of a “piece of metal 1 metre long”. Although it is hard to imagine than this was not always obvious for humans, it was certainly not the case in ancient societies. In fact, it is striking how well these societies were able to operate, before the concept of number being separable from their physical objects, allowed formal arithmetic.

Myth #4 Money can be defined by its uses

This myth is again common but is a non-unique definition for money. There are many, many assets which could be used for the functions:

Store of value
Unit of account
Medium of exchange

For example: dollars, gold, bitcoin, cigarettes, diamonds, canned food, oil etc.
In fact, anything non-perishable as bananas would not store well. The concept that in a mainstream economics they assume that anything can be used for money is important. Economic theories have developed from it, often containing the hidden assumption that money is not special and can largely be ignored. It is an asset like any other asset, is priced in the same way as any other asset. Therefore we should not be surprised that all the output from these models show no important role for money in the economy. I would have hoped the financial crisis would have exposed this as a myth.

Myth 5 Fiat money is a modern development

In fact, it is the oldest form of money.
I would prefer to say that “fiat money” means “money” and that “commodity money” is best defined simply as a “commodity”.

Conclusion

The story of money taught to economics students contains many a myth.
Next, I will tell an alternative story of money. The ideas I will present are not difficult.

But as Keynes said, “The difficulty lies not so much in developing new ideas as in escaping from old ones.”

The misuse of Significance

Definition

What does the word “significant” mean?

Dictionaries most often suggest a range of closely related definitions.
In a more everyday sense:

Importance e.g. this new discovery is a significant development
Meaningful e.g. the significance of the message was not lost on John

In mathematics, you get the example of:

Significant figures – e.g. 1.524658 is 1.5 to 2 sig fig

This use of the word is mathematical jargon with a precise meaning, but it also tallies with our general use of the word. We only want to look at the digits which are important and mean something.

In statistics:

“significant” means probably true (not due to chance)

Some issues arise from this

Something statistically significant may not be important

A result may be true and therefore significant when backed up by statistics, it doesn’t however mean it is important in the more standard English usage sense. I think this statistical interpretation can easily come into conflict with the everyday meaning and is fraught with danger.

When you jump out of a plane without a parachute it is likely that holding up an umbrella has a “significant” effect on your speed. I doubt you would think that this effect was important when you hit the ground.

I’m sure you can think of many things that are probably true but not important!

Statistical relationships are not transitive

An example from medicine, drugs for the most part are tested against a placebo rather than against each other. Drug A may perform better in tests against a placebo than Drug B. (ie has more significant results) However, that does not mean you know that Drug A will perform better in tests against Drug B. Unfortunately, current medical practice makes this implicit assumption when approving drugs.

This is a common misconception that you can use simple logic to infer other relationships. Unfortunately, this is not true. There is a similarly confused relationship with correlation. Statistical relationships like this are not transitive. https://iase-web.org/documents/papers/isi56/CPM80_CastroSotos.pdf

The 5% threshold for statistical significance is arbitrary

When you say that one result is significant and another is not because one has a 4.9% chance of being random and the other has 5.1%. This is the correct usage of the technical term but people ascribe more meaning to the word than that. One of the ideas is held to be “true” and the other is discarded.

A significant result may have happened by random chance

Saying that a certain outcome would only occur 1 time in 20 if it were random sounds good. But what if you ran 20 sets of analysis? By random chance you should expect one of them to pass the “significance” test.

Was the test constructed properly?

This relates to a supremely important point that often statistics are quoted in situations they are not supposed to be used or have been not properly applied

How many relationships did you test?
In finance, all analysts look at lots of different data sets, over different time periods in search of something “significant”.

Did you look at any of the data before choosing what test to run?
I cannot imagine how someone could not fall into this trap. We only run tests on things we think might work. But the reason we think they might work is that we have done some rough statistical work already e.g. looked at a picture or perhaps just subconsciously noted some signs of a relationship. This means that the data has been mined and your choice of test is not independent.
How many people are trying to find these relationships?
Let’s say that you are extremely careful in how you do your statistics. Let’s imagine that everyone else in the firm you work at is similarly careful. Then when you produce a “significant” result you may reasonably think it is meaningful. After all you only ran one test and it worked! You then show your boss. Should she be impressed? Maybe not.

How many failed tests are not shown?
In my experience, analysts do not show me large quantities of research they have done which they think is completely meaningless. Highly trained with great degrees, they want to show me “good” work with “good” results. This means that the 19 analysts that did not find anything today do not show me anything. From the perspective of the individual the result appears to be strongly non-random. From my perspective, it looks entirely consistent with being random.

Is it meaningless?

No. it just means exactly what the equation says it means. You should remain aware of the context if you want to use it. My interaction with professionals of all types is that they are enormously well trained in the complexity of statistical methods and woefully under trained in the limitations of them. In fact, their high proficiency with manipulating the data and the methods makes them even more prone to methodological error of this type as they have essentially been trained in the art of data-mining.

Conclusion

I am yet to read a research piece from a bank which presents data demonstrating that their hypothesis is has no statistical significance. We should remember that this is significant.

Framework for valuing equities Part V – Relationships between Components

In Framework for Equity Valuation Part II I laid out this approach.

Breakdown of the Equity Price

Using some very simple algebra, I split the equity price into components:

Price = (Price/ Earnings) * Earnings

Price = (Price/ Earnings) * ( Earnings / Nominal GDP) * Nominal GDP

In this work, there was an assumption that the components are independent. I will now examine if this is sensible.

Are P/E and E/GDP independent?

I can find no consistent relationship between the two.
There appears to be a mild negative correlation overall, but at times there can be extended periods of both components falling, such as 1967-74 and 2000-2003, or both rising such as 1994-2000.

An intuitive relationship occurs when there is an expectation of a large rise or fall in earnings and the equity price rises or falls in anticipation. This means the PE ratio would rise in anticipation of earnings rising, and then fall back down as earnings expectations are realised. In this situation, I see earnings as the driver and the PE ratio as a passive variable.

A situation where PE was the independent driver was in the 1980s, when a broad fall in yields meant the PE ratio rose without any need for an expectation of a change in earnings. This supports the approach that we can look at the two factors as independent drivers.

Are Growth and PE ratio related?

This is a relationship that is often assumed to exist as we think periods of low growth or recession are associated with low confidence and high awareness of risk. This high “risk premium” means low PE ratio.

But the evidence to support this idea is not so clear. Of the past 9 recessions, the PE ratio only fell twice. There is some evidence to support the idea that the PE ratio falls in the year before the recession in anticipation of an earnings drop, then recovers quickly as those expectations are realised. This happened in 5 of the last 9 recessions so it is still a fairly mild effect.

Are Growth and earnings related?

I find the chart below intuitive and compelling. The reason that recessions drive equity markets down is because recessions drive corporate earnings down. If we look at earnings as a share of GDP from 1 year before the recession to the low during the recession they fell each time. The average fall was 21% with the smallest still a 9% fall and the largest (2008) down a massive 42%.

The rationale for this comes from thinking about the breakdown of national income in the NIPA data (Framework for Equity Valuation Part III Earnings Outlook). If there is downward pressure on nominal GDP whilst wages remain sticky, then the impact is felt in a magnified way in corporate earnings.

The magnitude of changes in earnings are very large during recessions and early recovery, so it is during these periods we should be especially alert when forming an equity outlook. The impact of whether growth is 2.5% or 2.8% is imperceptible by comparison. Lots of work by economists, strategists and asset managers is done to fine tune these types of economic forecast but a) it is not possible for them to be that accurate b) even if you could, the relationship to market prices is so loose as to make it useless information.

Conclusion

There is one important interrelationship we need to be very aware of. In previous recessions, earnings as a share of GDP have fallen rapidly and normally bottomed at around 7%. If that were repeated in the next recession, earnings would need to fall by 40% from current levels.