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For example, in this post, it is asked where one can practice for trader's mental math tests.

I can understand why as a trader you'd practice mental health test, but, in this day and age, can someone please explain to me what the use is of being able to calculate, e.g., -$\frac{8}{2.3}$ without a calculator? Is it to train you to calculate your year end bonus quickly, or something else?

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    $\begingroup$ I used to trade the EUR/CZK cross currency basis, this is quoted as a spread on the EUR leg, and used to be heavily negative: say -55bps. When asked for a quote, I quickly had to come up with my bid-offer: here the bid would be to "pay" the basis (so pay EUR, receive CZK) and offer vice-versa. So a quote could be -60 / -50 (which is "counterintuitive, cause you wanna bid "less" than your offer, but here "less" means more negative). Say someone hit my bid, and then I go to broker to hedge, he tells me market moved up by 2.5 bps, and now it's -57.5 / - 52.5. $\endgroup$ Jun 9, 2023 at 16:57
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    $\begingroup$ Continued: say when I got hit I traded 12,500 DV01: if I now hit the brokers bid at -57.5, what's my PnL? I am paying -60 bps and will be receiving -57.5 bps => so I can make some money here. How much? 2.5 * 12,500. Sure i can grab a calculator, but if the market is active I wanna hit that quote fast, so at least should be able to process these operations in my head quickly. Btw: I wasn't particularly good at this aspect of the job and if I am honest, especially at the beginning struggled to quickly quote bid/offer in the negative numbers. Just sharing as an example... :) $\endgroup$ Jun 9, 2023 at 17:01
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    $\begingroup$ @JanStuller Yeah okay, I guess there are some benefits to it, although I still think making it into a full blown test a la Optiver or Flow traders is a bit too much. As buy side index eqd PM I never really needed it, but that's probably because var/volswaps are 'slow' trades compared to IR swaps or FX. $\endgroup$
    – Frido
    Jun 9, 2023 at 17:25
  • $\begingroup$ yep, spot FX and especially IR Swaps move fast. EM IR swaps are the worst, because these can gap on a bad day by up to 10pbs. Hungary is prob the worst in the CEE, nightmare to quote. $\endgroup$ Jun 12, 2023 at 8:59

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I'm not in the financial industry but, as a college math instructor, feel like I could share some insight. This is a type a question that we frequently need to answer regularly, so we wind up with some practice at it. E.g., SE Math Educators has fielded multiple questions on the (vaguely similar) issue of what's the value of learning times tables?

My top (but not sole) response to the OP's thought experiment is that one needs to be able to double-check the results of their technology output for reasonability. Specific example: Let's say you're in a fast-paced, high-pressure situation, look down to a calculator to compute -8/2.3, and get the result of -0.3478 (and change). Do you execute on that? Or what if you got +3.478 (and change)?

Both of these are of course in error, resulting from a single keypress mistake on the calculator. These physical errors on a simple calculator are relatively common, and I'd expect that you need the human in the loop to be able to immediately reject them on sight them as obviously insane results.

Now, here's where my couple decades of experience as a community college remedial math instructor comes in: Most of the population cannot do this. Obviously, I'd expect that everyone reading this page can do so. And my observations are frequently rejected as completely unbelievable by people who've gone to high-quality schools, surrounded by students and colleagues of above-average intelligence. But roughly one-third of our community college students can take a remedial arithmetic class, and be asked to practice estimations like this every day for a year, and at the end still be entirely unable to grok what we're talking about here, or what's wrong with that simple example.

It's also pretty common to hear about clerks mis-keying things in a cash register and being unpersuadable that the final bill is clearly incorrect. Or embarrassing public cases like the repeated phone company dollars-vs-cents decimal math error.

While there are many skills one could assume could be up-skilled on the job, basic math fluency is arguably not one of them. The real-life experiences and research seems to be that if someone can't become numerically fluent by middle school, they likely never get there. So to me it does seem reasonable that a company would want to use such a test as one of their simple filters.

In programming, an analog might be the fizz buzz test; again, this challenge is so shockingly simple that proficient coders can't imagine anyone could possibly fail at it. But hiring managers say a relatively large proportion of coders with higher CS degrees fail to do so in a hiring test.

Related: My brother-in-law is a smart guy and works as a manager at a financial services company. But fairly late into his career (certainly not applying for a starting job), he felt that he was falling behind in certain meetings and conversations where numbers were being thrown around rapidly (i.e., people were not stopping a meeting so people could double-check claims with a calculator). So on his own accord he got a math-skills app on his phone and was drilling himself on a daily basis to get better at these same kinds of mental math, as he felt it was important to continue excelling at his job in practice.

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    $\begingroup$ This is a very good answer. Knowing timetables indeed makes a big difference in everyday life as well (cash register example). I cannot comment on the traders tests but frequently the coding is a lot more complex than writing fizzbuzz. I agree that a simply test like fizzbuzz doesn't require preparation, (if you are used to coding) and still helps to differentiate. $\endgroup$
    – AKdemy
    Jun 10, 2023 at 6:34
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    $\begingroup$ Agree this is a good answer. But note that the mental arithmetic tests for some of the firms Frido is referring to have very high cutoffs. They seem to want people in some very top percentile in this ability. $\endgroup$
    – fes
    Jun 11, 2023 at 9:52
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Years ago when I interacted by phone with floor traders at a Chicago options exchange I was very impressed by their mental skills (which I felt I didn't match). Not so much math operations (that too) but memory for figures and speed in retrieval and comprehension of complex trading instructions. Whether this skill can be tested with the "mental math" test that you talk about I don't know. But clearly these traders have a special mental ability with numbers. I would write the orders on a piece of paper first and read from the paper during the phone call because I was afraid of making a mistake. The traders, with one phone in each ear and talking to two different people would somehow always immediately understand, repeat the order correctly and carry it out. Hours later they would remember my previous order and be able to tell me if the quote had changed, by how much, etc. Very impressive, I could never have done that job.

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  • $\begingroup$ Hi nbb02, see my comment also regarding floor traders which I placed under Hans' answer. $\endgroup$
    – Frido
    Jun 9, 2023 at 15:19
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    $\begingroup$ I knew a guy on the floor back in the heyday who didn't loose his first 10k; they eventually bought him a corvette, because he was an idiot savant that pushed on pull doors, but he had the ability of retrieval and comprehension of complex trading instructions, +1. $\endgroup$
    – Mazura
    Jun 10, 2023 at 1:16
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I can offer a personal anecdote for why these skills are sometimes needed.

Stemming from my career as a professional trader making trades by hand--though this is far in the past and before I ran my own funds--I still retain most of the mental math skills required back then. Nowadays, if a partner or I have to step in and trade by hand due to the size of the trade being too large or the market we are executing in too illiquid for our execution algorithms, these skills come back into play.

Particularly when we have large hedged positions that offset each other at different prime brokers and are unwinding them, this is, generally, a delicate process that we prefer to accomplish towards the end of the day. We can't afford to make a mistake, leave an account exposed due to a decimal in the wrong place, and have markets close on us because we were using a calculator--things move fast!

Yes, there are several ways of mitigating this risk, but they involve allowing third parties to see more information about our aggregate positions than we care to divulge.

To your original point, this is not common but occasionally comes into play.

I hope this helps a little bit.

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    $\begingroup$ Your and others' inputs are definitely helpful and most of all interesting. As mentioned in another comment, I see how in particular situations it can come in handy but I find it overkill to make mental math a kind of entrance exam. It's always something that you can train on the job if/when necessary. $\endgroup$
    – Frido
    Jun 9, 2023 at 17:29
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    $\begingroup$ @Frido, I agree 100%. It shouldn't be part of an entrance exam in this day and age. More of a skill that a trader should demonstrate to be added to a special situations trading team or something like that. $\endgroup$
    – amdopt
    Jun 9, 2023 at 17:56
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There are two reasons that I can offer:

  • A mundane reason is that the companies are testing for it in their interviews and to get the job you have to pass it. In that sense it is similar to Leetcode for software engineering interviews. It offers a (possibly weak) signal on aptitude which allows companies to filter the candidate pool down quickly to handle the scale of incoming applications.
  • Some people when coming from a university where they studied mathematics (including me) are not that numerically inclined in that they barely did any coursework where it is necessary to have that skill (as you say, calculators exist) and they even might have been actively discouraged to train that skill during their studies. (Physicists are on average probably more numerically inclined.) However, a certain basic feeling of numbers might still be necessary on the job when talking to traders or when it is necessary to follow certain conversations quickly and certain hedge funds (Cantab Capital for example) specifically noted that they had a problem when hiring from prestigious academic programs, when they did not filter for this skill.
  • As to your specific example, you are right that one can always reach to a calculator for this on the job. However, for the reasons outlined above I believe that it is much more about testing for numerical literacy rather than any specific example which might be close to what you need to calculate quickly on the job.
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  • $\begingroup$ I applied for Optiver actually shortly after graduation (in the 1990s) and they had two tests: an analytical test (calculate integrals, differentiation, if a circle has radius 1 and you walk around it 0 times how far have you traveled, etc) which I passed, and a mental math test which I flunked miserably. $\endgroup$
    – Frido
    Jun 9, 2023 at 15:18
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    $\begingroup$ continued.. At that time, having studied something useless like physics, I was completely surprised there was such a thing as a test that required you to calculate things which a simple calculator can do for you, and unless you're a floor trader I still think it's at best a weak filter and at worst a complete waste of everybody's time. But then again, I never did become a trader :) $\endgroup$
    – Frido
    Jun 9, 2023 at 15:19
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I never worked as a trader, which means I cannot answer the specific question. However, I have seen (from both sides) plenty of similar applications like HackerRank, Leetcode and the like. Personally, my experience makes me question the usefulness of such tests in finding valuable candidates.

That said, I think these tests are quite useful in getting rid of false positives (candidates one thinks are good but turn out unreliable in the end). Although I think it is at the expense of the candidates who have to study something that is often rather useless for their job. These test may also create plenty of false negatives (eliminating very good candidates) but if there are enough applicants, this is not a big issue (for the hiring firm). The good thing is that the exceptionally gifted candidates (which I am certainly not) will almost surely pass such tests as well.

If I can quickly find all strings formed from characters mapped to digits of a number, or correctly return an array of all the palindrome pairs of words or write all sorts of linked lists is more often than not just hard practice. Prior to COVID (currently one needs to be happy to find job applicants across the spectrum in my experience), there were a lot of applicants for few available jobs. Using screening methods will leave only a few potential candidates. In reality, plenty of companies use computerized HR tools to pre-screen already before that (with thresholds for degrees, languages, experience etc). If your application doesn't tick enough boxes, you never actually get in touch with a person. Insofar, these tests are in designed to get rid of false positives.

The downside is that these tests tell you very little about the actual performance of a candidate. The scenario that @amdopt points out will involve real stress. If it were a simple multiplication within some range, there would be no need for an experienced partner to step in.

A pass mark could simply mean a candidate spent a lot of time studying these types of questions. While one might say that being dedicated is an important trait, I think these tests still tell you little about the actual performance in a work environment. If you have done the CFA or something similar, you will probably have realized that lots of people with little actual, applicable knowledge can pass these exams, provided they studied a lot (of practive questions). If one robotically memorizes these questions (or practices multiplying random numbers in her head for weeks), it is indeed a waste of time and resources, although it may actually end up helping you find a job. However, that time could have been spent more wisely (in my humble opinion).

The CRO at the bank I currently work at once told me after the first training sessions I did with a new employee, that he never really cares much about the immediate performance. In his opinion, one can only tell after a year or two if someone has potential or not, which by his definition means that one can independently carry out a reasonable amount of work, with the right amount of detail and without too many errors. In the words of Nick Patterson, “Do you notice when your results are obviously rubbish?”

It is one thing to be able to explain how a digital is replicated with a tight spread, or to code a quick app doing this (the interview type question), and another to realize if the value you get makes sense (if your inputs are sound...).

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