In investing, one must find some security to hold for a relatively long duration. Trading, however, is a bit different. In trading, one must find something that is under- or over- priced for quick (often relative value) market corrections. In particular, traders try to find opportunities which are (1) hard for others to spot, (2) hard for others to exploit, or hard for others to exploit (3) as quickly as the traders can exploit the opportunities.
Speaking generally, in both investing and trading one uses tools to reason which investments / trades will act accordingly. In both cases, one also filters choices based on standardized descriptions. But in trading choices happen fast and relative value needs to be found more and more deeply every year. As implied above, mathematics is important because it:
- is a language for common understanding
- gives rise to standardized filtering which allows for very quick decision making
- allows for standard methods of information discovery (so you can use the same tools for different products)
- is a set of tools for reasoning (to see deeply)
- is expertise to avoid misunderstanding and to avoid false conclusions (to see clearly)
- is much more, but not relevant to the post
Expertise to avoid misunderstanding
- Some math is considered so basic that we take it for granted that we can apply strict methods to it. 1+1 = 2 right? If that is true, then how many chewing gums do you get by adding one to another? How many empty sets do you have if you add 5 empty sets together? Can you show me half a piece of chalk? This math is basic enough that you would never make this mistake with these examples. The point is that there is an assumption made even with something as basic as counting.
- Some math is used so commonly that we forget the changes implicitly made when moving from fractions to decimals and back. Take for example averaging.
- Alice ate 1/2 an apple. Varun ate 1/4 of an apple. What is the average number of apples eaten?
- (.5 + .25) / 2 = 3/8 Simple.
- A baseball player had a hitting average of .5 in year one and .95 in year two.
- Simple method: (.5 + .95) / 2 = .725
- Actually, it turns out he hit 15 balls in year one and 95 balls in year two. So 15/30 + 95/100= 110/130, well over 80%. We forgot that the decimals no longer had the weighting information.
-----Examples for the general populace------------------
Risk free investments are risk free- You are promised a risk-free, 10% return after a year (there is no chance of default). You reason that you should be about 5% richer after half a year (or a little less if there is a high frequency of compounding). Unexpectedly you need the money so you decide that you want to take your profits in the middle of the year by selling your risk-free investment to someone else. The only problem is that the market is now offering 20% "risk-free" and you are only able to sell at a major loss. The assumption that you wouldn't have to pay for opportunity cost is unreasonable (thanks to Stephanie for pointing out the correct way to say this). Your investment is not as risk-free as you thought.
---------Examples for Finance majors (chosen because these are so accepted)-------
Diversification leads to higher returns - Why should diversification lead to higher returns? Why doesn't diversification just water down the skill you put into choosing your investments, as Buffet proposes? The answer is to most is simple: either you have no skill or your skill isn't better than the value of diversification. Who knew that the nice guys telling you to diversify were actually implying that you suck. In mathematics, one says geometric returns lag arithmetic returns. The more the lag (which happens with high volatility), the more to be had from diversifying.
Least Squares Regression for Hedge Ratio- Many people, sadly including myself, use "math" blindly. Think of the last time you ran a regression to determine a hedge ratio. What happened? Which security did you choose to be the independent variable? Based on what algorithm did the computer fit the best line?
What would happen if you chose the other security to be the independent variable? Ans: Using financial data, you would get a different hedge ratio. That should raise a major red flag. This anomaly happens because your standard linear regression assumes that you have no error in determining the value of the independent variable. You should be using an "orthogonal least squares." Check the assumptions here. [More in a later post]
What's the point?!
The point is that not only do we use math blindly, but also we use blind math often. Somewhat rationally, you might argue to yourself that you should specialize in finance. That means, of course, you cannot waste time learning theoretical math because you can just use tools developed by others.
Due to our preference for "specializing," you, me, and the next finance major are the people using math incredibly blindly and not even knowing how blind we are. It is a mistake to feel that the immense precision of math means that one is accurate to some nice decimal places (accuracy vs precision). Not only is there a difference between accuracy and precision (ok, I tricked you into that last one), but often the precision is way overstated as well.
You won't find the term 'mathematical blindness' in Wikipedia and you may be asking, what exactly is mathematical blindness? Blind math is the usage of math without the understanding of the assumptions undertaken to make this math work. To some degree, this blind math is the applied math you've always done! It's not your fault! You followed examples to do what your teachers taught you to do.
Does it then follow that the person who knows the most math is the least blind?
Perhaps not. Quant funds, run by people with incredible mathematical understanding, have a reputation for getting killed in the marketplace. However, many quant funds also have a reputation for making a killing in the marketplace. Math makes them confident (and tells them to lever up). What is more dangerous than a person willing to stake his life on something? But then again, who is more bound for success than the confident?
Sometimes blindness will be mathematical and other times will be caused by assumptions in another field (eg. financial assumptions). The blindness is in misunderstanding anything! It is the magnitude of assumptions made from all sources.
Blind financial math (sometimes called model risk) comes about from (but not limited to):
- applying mathematical tools to the wrong types of problems
- Running the wrong regression
- Mistaking covariance for correlation or vice-versa
- applying unrealistic (but sometimes necessary) financial assumptions to the data
- The future will look like the past
- There is a distribution to this random element
- using the wrong data
- Should you keep the outliers?
- Should you take out certain days?
- Are the more recent days more important?
- Does the time of the year matter?
- not keeping track of assumptions / not challenging assumptions often enough
- You diversify like you are told [The assumption is that your skill in selection isn't better than the value of further diversification]
- You take over a trading spot and inherit plenty of assumptions you didn't even know were made
