Algebra

Oh great. It gets worse. "physics envy" is a postmodern insinuation deployed to disparage rigour and thoroughness of analysis even as postmodernists abuse physicsy-sounding language to do precisely what they imagine the competent academics are doing.

It's a capital error postmodernists make to assume that simply because they don't understand science and mathematics that no one does.

What's broken about the way finance is handled today is that a lot of arcane mathematics gets invoked which improves nothing but raises the barrier of entry. The innoculation to that ploy isn't less mathematics but better mathematics so that a graduate is able to recognise when maths is simply being used as a gimmick rather than as a tool.

What's broken about economics is... well, I'll let one of my FB status updates speak for itself:

"Economics is based on the idea that we're all rational consumers: that we weigh up a product and it's cost and we come to the right decision based on what will maximise our pleasure... and yet... I just bought this Justin Bieber singing toothbrush. O_o"

I've read the Munshi article.  Perhaps the argument here is distorted by a bit of hyperbole.  There are certainly situations where a simpler spread sheet worked out by brute force on a computer yields a more easily understandable presentation.  But, not always.  Sometimes it is helpful to understanding to use both methods and show that they yield the same result.  I love giving my younger kids a problem, letting them work on it and then showing them (or helping them derive) different ways of solving it. It teaches them that math need not be so mysterious.  Or as my dad used to say, "there's more than one way to skin a cat".

I agree with this sentence in the paper: "In this example the analytical solution is easier and more accurate but there are many situations where exactly the reverse is true. " So, sometimes the algebraic solution is better, sometimes the numerical solution is better. So we have to learn both methods!

It's a bit weird to tie "easier" with "more accurate". Unless I'm missing something, I'm going to need a specific example of where a numerical method can be more accurate than an analytical approach.

Numerical methods are important techniques where analytical solutions are either difficult or even impossible or computationally intractable. Like parachutes, you hope you wont need them but they're essential when you do.

They take on a special relevances if you're in the business of writing software tools for science, mathematics, and engineering. Or even if you're in the business of making critical decisions based on the results from such programs. You need to know under what kind of conditions a numerical method performs poorly. The importance of understanding that can't be stressed enough!

Basically, the algebraic solution is always better wherever it's possible but it's often not possible/feasible in a broad range of practical problems.

An algebriac solution to a problem is often not perfectly accurate. Sometimes we simplify a problem in order to make a problem algebriacally tractable. (Eg ignoring air resistance in the laws of motion.)  The algebra is perfect in "maths world" but the real world is often more complicated! A numerical approach might more easily address these complications.

That's not quite the same thing is it? Analytically solving a simpler version of a problem because the original problem is intractable isn't solving the original problem analytically. That's precisely the situation where you would be interested in taking a numerical approach.

I think we're agreeing but we're just posing the distinction in slightly different terms.

None-the-less, I like the air resistance example. That's the same one I was leaning towards and noticed it's also a good example of where numerical methods shade into 'models' and models shade into 'simulations'. [scare quotes are obligatory here since there seems to be some disagreement about what properly constitutes a model and what properly constitutes a simulaiton -- I've never been able to get a straight answer on that.]

One more comment and I'll leave it to you guys.  Jamalov was discussing algebraic solutions as related to the fields of finance and economics.  In my opinion, it's one thing to use an algebraic solution to track the motion of planets where one has reasonably exact and observable data.  It's another to use algebra to solve "fuzzy" problems like economics where the data is not always so clearly defined.

Certainly. That presumably goes a long way to explaining why the financial sector prefers physicists and mathematicians. They're already familiar with the necessary techniques from their previous experience with quantum mechanics where states of a system are expressed in terms of probabilities.

I've not had cause to do the advanced financial stuff yet but I'm told it's very easy if you're already used to quantum mechanics. I've heard that from enough varied credible sources that I'm inclined to believe that I could turn my hand to it should the need ever arise. (It probably will one day.)

The paper compares numerical and algebraic methods to calculate the standard deviation of a combination of two variables. Both methods are correct in maths world. But they are, at the same time, both wrong in the real world as they are only valid if the two variables are independent which, as we all learned to our cost, is not the case with investments!

You are right of course juanitoz. Covariance is an important consideration. In the portfolio model we actually take the covariance into account. In the capital budgeting model we assume, as you point out, that sales in successive years are not correlated. I did look at the effect of correlation on NPV and found that there was an effect but this effect is rather weak. If you are interested i can post the download link for that excel file.

Yes, you are also right Jamalov! My point was not really about the topic under discussion in paper, but a much more general comment about about the field.

True, covariance is a critical as measure of correlation in the model (in both alegraic and numerical cases). But my point was not about the model, which is fine as a model, but about the modelling. ie whether the model accurately reflects the real world. Even if two variables appear to have a low correlation, there can be a underlying dependency in the real world which has not been exhibited so is not modelled. Then, at another time, the dependency shows itself and the model is no longer accurate.

Going back to the laws of motion as an example (though I'm stretching the analogy a bit now). Imagine I have two cannons. One fires further but with higher variance. The other shorter but more accurately. I distribute my limited number of cannon balls a certain way to maximise effectiveness in preparation for an attack. But on the day of the battle, there is a strong headwind. Both cannons are fire less far than expected and my cannon balls are not in the best places.

Even in non-mathematic dependant jobs, algebra, or more specifically the concepts it teaches, can help one deduce answers that might otherwise allude them. It is helpful to at least have a basic understanding of it, if for no other reason than to see something with a missing component and be able to implement the missing link and complete the figurative equation.

Thank you nomadic knight. You may have a point there.

That's the kind of thing I'd personally really like to believe because it's always satisfying for us to be able to tell people about the various advantages of the things we like. But unfortunately I don't think it's true.

I'm not aware of what might have been found where that has been studied directly but I do know that psychologists have directly investigated whether studying formal logic, Latin, or chess makes you any more logical in your approach to issues outside of the domain itself (i.e. Do formal logic, Latin, chess confer the student with any transferrable skills?). The answer is a very resounding "no".

The only exception to the general finding that technical skills offer no advantage outside of the technical domain in which they have been cultivated appears to be with statistics. Studying statistics has been found to aid people in thinking about matters far from the formal discipline itself.

There is of course the classic statistics job interview question "How many piano tuners are there in the country?" There is no right answer to the question but it's supposed to be an opportunity for the candidate to demonstrate that they are able to transfer their statistical reasoning skills to unfamiliar situations.

However, a version of your sentiment which does ring true for me is that if you don't know any algebra then you're ill prepared to take advantage of it wherever an algebraic approach is possible. That's a natural situation but it can still be compared to the historically contingent situation in Europe for the better part of two millenia where not knowing Latin meant you would be unable to read and benefit from the most learned texts of the age.