What do job-application aptitude tests really measure?

It is common practice for companies (including financial companies) to use aptitude tests (mostly numerical and verbal reasoning) in order to assess candidates. I myself have done such tests and though I do not begrudge companies that use them, I have started to question their validity. Do they actually measure something useful? I explore this theme in a blog post on my life-related blog Meditations of Lambchop. Here is an extract from the post

This excellent essay, A Mathematician’s Lament, by Paul Lockhart has convinced me of something I suspected since learning actual mathematics at university (and not even, at first, within official lectures): our schooling ruins mathematics for children. The same kind of thinking that has created the education system appears to have infected human resource departments in most major companies. I refer, of course, to the ubiquitous use of aptitude tests (a subset of psychometric assessments).  It seems to me that our schools are satisfied with teaching arithmetic rather than mathematics and HR is satisfied with testing “skills” that no candidate will ever need. As with schooling it seems hardly anyone questions the current system. Read more here.

Don't spot the pattern

If I write 2 then 4 then 6, then we feel good because we know that next comes 8. We can foresee it. We are not in the hands of destiny. Unfortunately, however, this has nothing to do with truth. – Arthur Seldom, The Oxford Murders (movie)

The series 2, 4, 8, could obviously be followed by 16, but also by 10 or 7004.
It's always possible to find a rule, a justification which allows a series to be continued by any number. It all depends on how complicated the rule is. – Arthur Seldom, The Oxford Murders (movie)

I remember getting questions at school of the form “which number comes next?” At the time I thought these questions were perfectly normal. I now think they are nonsensical. As such it troubles me to see similar questions (with diagrams rather than numbers) are being used in psychometrics assessments. For instance here are ones called diagrammatic reasoning tests. Whether numbers or diagrams, the idea is the same and it should be put to an end.

These questions expect you to extrapolate from a finite set of data. The problem is, as with the above quotes, there are infinitely many ways to do this. The only difference between them is that some “feel” more right than others. They are intuitive, they are “simple”. But both of these things are in fact rather subjective. And so while these questions pretend to have only one right answer, they really do not.

Here is an example. The series 1 2 3 5 …

This could be “all integers with at most one factor”, i.e. all the primes and the number 1 – then the next number is 7. It could also be the Fibonacci sequence, but starting at 1 2 instead of 1 1 – then the next number is 8.  Of course one could think up infinitely many rules for completing this sequence. Another simple rule is to assume it is periodic 1 2 3 5 1 2 3 5…. - then the next number is 1. Of course if you looked only at the first three elements in the series you would probably guess the next number is 4.

The question, then is not really “what is the next number?” it is: Find a function from the natural numbers to the natural numbers which has the given sequence as its first mappings. The function should be “simple”, meaning it should be described (possibly as a recurrence relation) only with addition, subtract, exponentiation, etc. and should be the one function that whoever is marking the question would think is the simplest.

The problem is that the problem is never actually stated like this. The ways in which you are allowed to describe your function are not enumerated and there is no objective means of determining what is
“simple”. Thus for any above mediocre mind, the problem is not to find the next number, but to determine how far beyond the standard set of descriptions for functions they should allow their mind to search.

Thus the question really only does the following: it forces you to confine your search to what is expected already. It hinders the ability to think beyond this and it penalises anyone who happens to think differently from the standard. It creates a false impression of truth and limits human creativity. The only way to ask these questions (if you have to ask them at all) is to give a precise description of the form of function allowed and then make sure only one function in this set satisfies the requirements. The same is true for diagrammatic questions.

Stop confusing clever with lucky

I get very annoyed when I see idiotic journalism, such as this article (admittedly Business Insider does not exactly maintain high standards of journalism, but in this case they just copied CNBC). To recap: one 22-year old kid put all his savings into Tesla shares (and later options on Tesla shares) and today has made quite a hefty profit. But when his friends told him he was crazy, they were right.

It is stupid to present this kid’s outstanding luck as success. He was not successful – he was blindly, stupidly, and ignorantly lucky. There was absolutely no way of knowing that Tesla would outshine all the other technology companies. There must be thousands of investors who have similarly bet heavily in some particular stock and found themselves losing everything. I am, of course, not saying anything new here. Taleb argued just these lines in his book Fooled by Randomness, which everyone should read.
But we don’t hear about the investors who lose everything. We only hear about the ones who strike it lucky. Because we think they are special, that they had some special foresight. Mostly they were just lucky.

I admit that markets are not efficient. Some people may have the ability, through research, to increase their chances of doing well. Most active managers think they have this ability (and all of them claim to have it). But only the stupid ones would invest most of their capital in one stock. There is too much room for error.

Warren Buffett is often cited as an example of an investor that managed to beat the market with his foresight, and this may be true (considering his long track record). But he was also very lucky. When he was just starting he invested 75% of his worth in one stock, GEICO, which paid off handsomely. This was after just one conversation with an executive there (one who became CEO shortly afterward). Buffett was extremely lucky. If that single investment had not paid off (and how do you really know it will pay off after speaking to one person, who can’t control the direction of the entire company?) we would never have heard the name Buffett.

Taking stupid risks is fine if you have a safety net (for instance rich parents), because then you’re not really putting all your eggs in one basket. For the rest of us: ignore the media when it tells you people who risked everything were smart. They were lucky.

References

• Buffett, W. (2010). Letter to the shareholders of Berkshire Hathaway Inc. October. Retrieved from http://www.berkshirehathaway.com/letters/2010ltr.pdf
• Lebeau, P. (2013). College Student Put His Life Savings Into Tesla, Made A Killing. Business Insider. Retrieved August 28, 2013, from http://www.businessinsider.com/college-student-put-his-life-savings-into-tesla-made-a-killing-2013-8
• Taleb, N. N. (2007). Fooled by Randomness (2nd ed.). Penguin.

Volatility weighting primer

Volatility weighting is one common means of attempting to improve the risk profiles of strategies, that is to give them smoother returns. It consists of taking some asset (or the returns from a strategy) and dividing the investment you make by the (estimated) future volatility of the strategy. The goal of this is in fact not reduce the volatility of returns, per sé, but rather the volatility of volatility. In practice it seems that volatility weighting does seem to work and gives higher Sharpe ratios (a measure of the amount of return for every unit of risk taken).

Some simple maths (skip this bit if you fear maths)

Consider for instance an asset with returns

with (for simplicity) sigma σ and epsilon ε independent and epsilon ε is mean zero with variance 1. Here σ represents the volatility of the process. It is predictable (the value for the next period is known today) but it is random – it changes from period to period. The Sharpe ratio of this strategy is

Notice that this ratio is maximised if volatility is deterministic, that is if the coefficient of variation of volatility

is zero. Making volatility deterministic is, in this case, exactly what volatility weighting does.

What we need for volatility weighting to work

1.  To be able to forecast volatility. It is typical to model volatility as a predictable process, but in practice we cannot even observe volatility directly. There is, however, some evidence that volatility is sticky, so that one can predict it to some degree.
2. We need the portion of returns not depending on volatility not be too large. If this is not the case, then we do not have the multiplicative nature of the return series. It does, however, seem that the effect of this portion of returns is not so large as to completely invalidate the use of volatility weighting.

Why volatility weighting works

1.  I have already mentioned that there is an effect of stabilising the volatility and that this creates a more stable returns series. In particular we saw earlier that volatility weighting seems to work by reducing the coefficient of variation of volatility.
2. There is, however, another possible effect, volatility timing. If returns are negatively related to volatility then volatility weighting will mean investing less when volatility is high and more when it is low, which intuitively seems to be a good strategy. This won’t work if the conditions in the previous section are not met, though.

It is not clear which of the above two effects is more important in practice, or if they can even be separated in any meaningful manner. It is clear, though, that volatility weighting can work even where the relationship with volatility is positive (so the second effect is not then the most important). It is also true that for some strategies, for instance momentum, the relationship with volatility is often negative and so we can expect the timing effect to play a role as well.

Own volatility or underlying volatility

There are many ways to do volatility weighting. Here are two that I have looked at:

1. Weighting a strategy by its own volatility: you look at some investment strategy and estimate its volatility in some form and then scale how much you invest in the entire strategy.
2. Weighting the underlying assets: Here you consider some investment strategy based on a set of assets. Now replace the assets with a set of assets that have been volatility weighted. I call this using normalised returns. 

Both of the above forms of volatility weighting appear to be effective, the latter probably more so.

Momentum strategies

Momentum is an age-old feature of financial markets. It is perhaps the simplest and also the most puzzling of the “anomalies” discovered.  It is simply the tendency for assets (for example shares of some company) that did well (or poorly) in the past to continue to do so for a time in the future. It has been extensively examined in academia and has been found to be present in virtually all markets and going as far back as we have data. It has even persisted some decades after being extensively investigated for the first time. And still, it seems, we do not understand it very well. In today’s post I just want to highlight some different momentum strategies and their uses.

A property and a strategy

Momentum is a property of asset prices in markets and momentum strategies try to benefit from this property. One way of understanding momentum is to consider different momentum strategies and the profits they make, which gives an indirect means of understanding how asset prices work. For investors, of course, this is perhaps the most convenient way to study momentum as they are inherently interested in the strategies. They only care about momentum as a property if they can exploit it. The distinction between momentum as a property and as a strategy is not always clear because academics have not yet, I think, deemed it important to make the distinction explicit and thus both are simply called momentum. 

How to construct a momentum strategy

Momentum strategies come in all shapes and forms. Basically all of technical analysis is some kind of momentum strategy. A very general way of thinking about constructing a momentum strategy is depicted in the picture below. One starts by identifying some kind of trend (or signal) for each of the assets you are considering. This gives the direction of the momentum for the asset (for instance up or down). One can then assign a strength (or score) to this signal, which can be related to the magnitude of the momentum or the confidence you place in it. Then based on the signal and strength one makes an allocation decision – you decide how to bet in order (hopefully) to profit.

Time-series and cross-sectional momentum

Momentum strategies come in two main forms (though they are related). The first is to consider momentum for individual assets – the tendency for an asset’s price to go up if it went up in the past.  Here the signal and strength are evaluated for assets in isolation. This is time-series momentum. This form of momentum can be contrasted with cross-sectional momentum, which considers the momentum of assets relative to each other, e.g. the tendency of one asset to perform better than other assets if it also did so in the past, for instance. Here the signal and strength depends on how assets compare to each other.

Time-series momentum (strategy) tends to do well if an asset’s return is related positively related to its own past (property), for instance in what is called an AR(1) process:

Thus a higher return in the past predicts a higher return in the future.

Cross-sectional momentum (strategy) tends to do well if one asset’s return is negatively related to the past return of another asset (property), for instance if (numbering the assets 1 and 2)

This means that a high return on the one asset predicts a lower return for the other asset in the future.

Some simple strategies

Here are some simple strategies, based on a simple taxonomy:

Signed time-series momentum: buy any asset that went up in the past; sell any asset that went down.

Signed cross-sectional momentum: this is analogous to the above, but now invest in deviations from the average return or the market return. For instance the deviation of asset i’s return from the average is

If the asset did better than the average, buy the asset and sell the market and do the opposite if it did worse. This is a bet that assets that had above average performance in the future will continue to do so in the future.

Linear time-series strategy: again buy any asset that went up and sell any asset that went down, but invest more in assets with larger returns (invest proportionally to the asset’s past return)

Linear cross-sectional strategy: the same as above, but for deviations from the average (or market) return.

Quantile cross-sectional strategy: buy, for instance, the top third of assets and sell the bottom third.

In practice only the signed time-series and quantile cross-sectional strategies are used. The other strategies are, however, useful in formulating theory. For instance the linear strategies are easier to cope with mathematically, but amplify volatility too much to make them useful in practice.

Some references

My thesis:

• Du Plessis, J. (2013). Demystifying momentum: time-series and cross-sectional momentum, volatility and dispersion. University of Amsterdam. Retrieved from http://www.johandp.com/downloads/johandp_momentum_final.pdf?attredirects=0

Some empirical articles looking at momentum strategies:

• Jegadeesh, N., & Titman, S. (1993). Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency. The Journal of Finance, 48(1), 65–91. doi:10.1111/j.1540-6261.1993.tb04702.x
• Lewellen, J. (2002). Momentum and Autocorrelation in Stock Returns. (C. H. Schiller, Ed.)Review of Financial Studies, 15(2), 533–564. doi:10.1093/rfs/15.2.533
• Moskowitz, T. J., Ooi, Y. H., & Pedersen, L. H. (2012). Time series momentum. Journal of Financial Economics, 104(2), 228–250. doi:10.1016/j.jfineco.2011.11.003

Statistical intuition fails

Warning: this post contains some mathematics. However, non-technical readers may ignore the equations and focus on the concepts, which are far more important in any case.

I am ashamed to say that my statistical intuition has failed me. I was doing some brainteasers and came across this one here:

In a country in which people only want boys every family continues to have children until they have a boy. If they have a girl, they have another child. If they have a boy, they stop. What is the proportion of boys to girls in the country?

My initial thought was there should be more girls than boys. But this is completely false. As long as all births are independent and a boy or a girl remains equally likely, the ratio of boys to girls in the whole population will be 1 (assuming the population is large). The choice the parents make of when to stop having babies has no influence on the overall distribution of boys or girls. This is actually obvious if you think about it, but you can also prove it mathematically as follows:

Let X₁… X_n be the number of girls in each household in the country. These have a iid geometric distributions and the ratio of boys to girls is

By the law of large numbers this tends to
,
which is 1.

In real life, and in financial markets, our statistical intuition also fails us. One of the reasons I think I will never be able to be a trader (or at least not a good one) is that I will never trust any decision I make. It’s only through meticulous analysis, by slowing down, that you can overcome your biases and that is exactly the opposite of trading.

Self-critique

I recently finished my masters thesis and I would now like to do something that most people never do. I am going to critique my work. I believe that being self-critical is essential not only in research, but in life in general. In research it is necessary to further the truth. It is not enough that others are critical of you (though it is necessary). You must be critical of yourself – only then will you be willing to remedy your flaws, change your convictions and pursue truth and goodness rather than your own prejudiced agenda.

I write this post as much to myself as for any reader who may come across it. It is another public reminder of a lesson I fear I may forget in the future, and of which I may need to be reminded. Even if you are not interested in my thesis, some of the points of criticism here may be useful for your own writing. I found in writing this post that self-criticism is hard. It’s really hard to come up with anything but weak flaws in your own work. I reckon it will take time to cultivate a truly self-critical nature.

My thesis was about momentum. I looked at time-series and cross-sectional momentum and their relationship with volatility and cross-sectional dispersion, specifically considering volatility weighting as a means of improving momentum strategies. If this sounds like Greek, do not fear, for I will be explaining all of this in later posts. Today I just want to list my critiques, starting with the more serious ones.

Subjective conclusions drawn from mixed results

I often found when I ran some numbers that I got results that were not clear-cut for any any one conclusion. Then I had to be satisfied with making a qualified conclusion if I thought there was enough support for it in the data. “Enough support” is subjective and others may feel differently. It may be a wiser course of action not to draw any conclusion at all, but it may also be far too conservative.

A lack of focus

Good academic research tends to focus on one thing and examine it thoroughly. My thesis, I think, tried to look at somewhat too much, and as a result ended up being huge and with not one aspect being treated quite as it deserved.

Strict assumptions (that I violate)

In order to prove things easily you need to make assumptions. Often you end up assuming independence or normality where it is clearly not the case in the data. In my case I needed to make such assumptions to prove things about volatility weighting and in at least one case I am not even certain there is a non-trivial (that is to say, interesting) process that satisfies my assumptions. I have little choice but to violate my assumptions (there are for instance no volatility estimators that satisfy the assumptions I had to make).

Overly detailed results that are hard to interpret

I have lots and lots and lots of tables in my thesis.  They are big and make your eyes sore. Ideally one should find a way of presenting just the right numbers, without hiding ambiguity and evidence that doesn’t support your conclusions. It is hard to read text referring to specific numbers in very large tables and keep track of what is happening. Making a graph is even better as it gives an immediate impression (but you may still want to report the numbers so that people can check them). I had relatively few graphs as I could think of no good way to convert my tables into something visual. This is a weakness. You may think that academics should be able to read such dense material, that they should take the time and effort. It is, however, a simple fact that academics are human and that they do not. Even if they do,  they are less likely to get the right picture if the information is not presented in accessible manner.

Use of advanced techniques without necessarily having the appropriate understanding

I used what are called “Robust regressions” in my thesis in order to cope with the fact that financial data contains so many extreme values. I had never used robust regressions before and only briefly looked up what the robust regressions did, then used them. I did not take the time to get well acquainted with their theory (as this would have been quite a task, I think) and I simply used a standard weighting function with a standard parameter.  Most likely this is still better than simply using OLS regressions (which I think are absolutely a no-go in financial research, except as a baseline comparison), but it is still possible that the version of robust regressions I used were not the most appropriate (deciding what is appropriate is of course more an art than a science) and it would have been preferable to have had more training in using them.

Linearity

I used linear models for the theoretical and empirical investigations. One thing that is clear from finance, though, is that nothing is linear and so results from linear models can be misleading. We have, however, I think, only poor substitutes and thus linearity is still common in academia. This is thus only a weak criticism on my part, but I would like to see a move away from linear models, if only we could find an accessible and preferably tractable alternative.

Little thought for practicalities

I did not consider that I was basing my results on markets that closed at different times (essentially I assumed they closed at the same time); I did not include transaction costs, commissions, taxes, etc. This is not unreasonable. To include all these things meticulously would detract from the main purpose of the study. But they are important and their inclusion could potentially change the nature of the relationships found (though this is unlikely).

If you read my thesis and you think there are other criticisms, then please let me know. Perhaps I'll include them in a further post.