How (not) to sell living annuities (aka income drawdown accounts)

A recent campaign by an investment house in South Africa has touted the benefits of their living annuities. The potential retiree can get up to “31 years” of additional retirement income with their “high equity living annuity” which has lower fees, as opposed to standard living annuities which charge higher fees and only have a “medium” equity content. This illustrates a very real problem in the sale of financial products: how do you tell consumers about risk?

Source: Microsoft Clipart

A living annuity or income-drawdown account is essentially a pot of money (invested in some portfolio of assets) from which the pensioner can draw a monthly income, either until they die or the money runs out. The latter is a very real possibility. I fear that financial marketing can encourage consumers to take on this risk without giving sufficient awareness of it.

The company’s radio campaign doesn’t mention risk at all. It merely contrasts these two options: high equity with low costs and medium equity with high costs. This creates the impression that high equity content is supposedly better than low equity content. (It also creates the false impression that cost is somehow linked to equity content, but I will not discuss this further). The company’s press release goes even further, saying (not quite directly) that a high equity portfolio is less risky than a medium equity portfolio.[1] Their retirement calculator reinforces this: by choosing a high equity annuity, I can increase how long my projected income will last.

I have not seen the internals of the calculations. However, it is well known that equities have a larger volatility than fixed interest and do not offer a steady income stream. The risk of a substantial decline in an equity portfolio along with a continued drawdown depleting a pensioner’s funds should be much larger with more equities. That being said, equities do grow more quickly over the long term and thus, for someone willing to take on the risk, it can allow either for a higher income or a longer-lasting income. This must, however, come at the cost of additional risk.

One possible justification for a move to higher equity living annuities is that pensioners are living longer and are thus able to absorb larger market fluctuations in the short term. This does, however, require a well-chosen (and well-managed) drawdown amount to avoid depleting capital in the short term.  Some equity content will also be needed to provide an income that increases with inflation (it is debatable just how much this should be – the inflation-beating returns of equities come with additional risks as already mentioned).

The above should highlight that risk is not a simple issue. Even well-meaning and established companies struggle with it and consumers are not always willing to listen. Benefits are easy to talk about and easy to put into ad campaigns. Risks, however, require you to face uncertainty. It is an unfortunate fact that benefits get slogans and diagrams and colour, whereas risks get plain text and footnotes, if they’re mentioned at all.

[1] Ostensibly because pensioners with a high equity portfolio before the crisis would still have done better today, even drawing an income. This may or may not be true, but stronger evidence than this is needed.


False advertising

I recently ventured into a bit of consumer activism and want to relate the experience – not to blow my own trumpet, but to encourage others to be more vigilant. I was looking at the webpage of a well-respected investment house in South Africa last year and read about their investment approach. Buried in this “investment philosophy” was the following sentence:

The word “pragmatic” refers to the Charles Peirce School of investment philosophy, which advocates taking a practical approach to matters with reference to historical events.

I found this rather interesting – I had never heard of the “Charles Peirce School” and I immediately googled it. Nothing. I could find not a single reference to any school, movement, or theory of investment associated with Charles Peirce. The man, Charles Peirce, however, did exist. He was a very influential philosopher, founder of a movement called “pragmatism”, which, as far as I could tell, had absolutely nothing to say about investments.
I sent the investment house an email querying their use of the term, asking they either provide a valid explanation of the term or remove it from their marketing material. Here is the full message I sent (square brackets included ex-post):

Dear Sir/Madam
I am hoping to get some clarification regarding ideas expressed on your website. 
On this [link removed] page describing your investment philosophy you use the following words "the Charles Peirce School of investment philosophy, which advocates taking a practical approach to matters with reference to historical events." 
Would you please clarify your justifications for using this particular terminology, providing supporting documentation. My own cursory search has revealed 
1. Charles Peirce was a philosopher, not an investment theorist
2. Though he found [sic] a school of philosophy called pragmatism this does not appear to have anything to do with investing and the use of the word pragmatism in this context is very specialised and not the same as the everyday use of the word
3. no evidence for a recognised school of thought termed the "Charles Peirce School of investment philosophy" and thus no evidence that they espouse "a practical approach to matters with reference to historical events" 
Please let me know if any of the above points are incorrect. For instance, if there are a group of investment practitioners or theorists that identify themselves as the Charles Peirce School, please point them out to me. 
If, however, my points are correct then you must either remove the reference to Charles Peirce or make it very clear that you are reinterpreting his ideas (this probably cannot be done without a publication with proper citations justifying this reinterpretation).

I received the following reply (which I have censored somewhat):
Thank you for your email. Our investment CIO and strategist have considered your questions and have the following response: 
You are correct in saying thaCharles Peirce was a philosopher and not investment theorist and that there is no official Charles Peirce school of investment philosophy. 
The explanation on the website was intended to refer back to his philosophy to convey the fact that we apply pragmatism to investments.  So instead of applying the theory of value investing dogmatically [explanation removed] 
We plan to adjust the wording of the website copy to address your concerns and thank you for kindly taking the time to give us your feedback.

To the company’s credit, they did remove the reference to the Charles Peirce School and the current version is better. I should also make it clear that this investment company is well-respected for a reason. I think this was a glitch in their marketing machine, no more.

As such my little quest to remove just a few sentences from the company’s electronic marketing material may seem a little quaint – was there really any harm? I believe there was. Essentially, the reference to the Charles Peirce School was a lie. It created the false impression that the company was associated with some well-researched and respected group of investment theorists, whereas in fact this group did not exist. It would have been far wiser simply to explain the idea behind their investment philosophy (which is what they now do), rather than inventing fancy references to make their approach seem sophisticated.

The financial markets are extremely hard for consumers to grapple with. The environment is complex enough without companies trying to baffle consumers with misleading marketing. There is a large degree of mysticism in active investment – the investment manager is a kind of black box: money goes in, magic happens, huge returns come out. This is a false impression, especially as the work done by many fundamental managers is really just a combination of a simple philosophy, common sense and experience. We cannot allow consumers to be taken advantage of in this manner. The only way this will stop is if educated consumers take the time to respond to misleading practices where they find them.

[To avoid potential problems, I have decided not to mention the name of the investment house in this post. I am not, however, under the illusion that I have guaranteed their anonymity.]


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.


  • 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