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My rebellion against ignorance

Photo credit to Torsten Dederichs

Math is for people who like understanding things

What is published by others and what feedback I have received from friends could be distilled to a single postulate: an equation is a torturous device that must be danced around before apologetically presenting it to the reader lest they be offended by having a correct, unambiguous answer. Working from this anti-math postulate leads to incomplete, clumsy and/or incorrect explanations about subjects which obey concise mathematical rules. This destructive postulate undermines a vast literature of financial math for three reasons: (1) it is insulting to the reader from the beginning and the educated reader is often aware of the insult, (2) it inevitably leads to the dumbing down of material into half-truths and incomplete thoughts where real and complete explanations are available and (3) in many cases it leads to incorrect statements - ones which are objectively and indefensibly false. I aim to avoid these offenses at all costs when I write about money.

The world is full of literature about money for the mathematically ignorant and emotionally undisciplined. I am not writing to those people. If this blog is confusing or the equations unclear, the reader does not need me to hold their hand through an abbreviated explanation of calculus, they need a calculus and/or differential equations course which is available for free online in many places. No portion of this text requires more than a cursory understanding of calculus and the ability to follow step by step derivations. Most of the broad conclusions can be understood with algebra skills that all reasonably educated adults possess.

Ubiquitous and terrible ways to avoid writing equations

Perhaps the most common ignorance inducing statement made in the financial math world is the attempt to describe the interest on a loan with a couple of examples.

From Nerdwallet.com retrieved 2017-05-22

a 30-year, fixed-rate $300,000 mortgage with a 6% APR . . . for a total repayment amount of $647,515. If, however, you took out the same mortgage and paid $40,000 in one-time fees upfront, you would have a 4% APR and end up paying . . . a total cost of $555,607.

From DaveRamsey.com retrieved 2017-05-22

A $175,000, 30-year mortgage with a 4% interest rate will cost you $68,000 more over the life of the loan than a 15-year mortgage will.

That is to say, describe the following equation

$$ \text{Interest} = f(B_0,r,t_\text{term}) $$

using only a few example points. \(B_0 =\) loan amount, \(r =\) interest rate, and \(t_\text{term} =\) the loan term. This is fatuously incomplete because if we know nothing else about that equation, we have no idea what kinds of functions we might be dealing with. Is interest a linear or exponential function of \(B_0\)? of \(r\)? logarithmic? sinusoidal? The author imparts almost no information with their examples except illustrations of their "Debt is really costly" or "Low-interest rates are preferable to high ones" thesis.

The real equation is not as complicated as the black box version suggests it might be because it is not an independent function of \(r\) and \(t_\text{term}\) it is a function of their product \(rt_\text{term}\). It also scales linearly with \(B_0\) That information takes the mystery from a function of 3 variables to a scaling relation of 1.

$$ \frac{\text{Interest}}{B_0} = f(rt_\text{term})= \frac{rt_\text{term}}{1-e^{-rt_\text{term}}}-1 $$

This equation is simple to understand, easy to apply, and relevant to every loan with a constant repayment rate. For reasons I cannot understand, it also seems to be a giant secret in the finance guru business. This problem is widespread in the world of financial literature, not limited to describing the interest on a loan (though that is the most common case).

Business Insider thinks \(e\) is too confusing to publish

Consider this pile of garbage from businessinsider.com in an article "11 Personal Finance Equations You Need To Know" which is introduced with the overselling line "We've rounded up 11 math equations that can be used every single day. Write them down, whip out your pencil, and prepare to budget like a genius" despite the fact that only one of their 11 equations has anything to do with budgeting and it's so tautological, it must be a joke. (Variables changed for consistency with this blog.)

$$ P = \frac{B_0 \frac{r}{f}}{1-\left(1+\frac{r}{f}\right)^{-t_\text{term} f}} $$

The equation describes the payments on a loan as a function of the loan size, the payment frequency, the loan term, and the interest rate. It is misleading or insulting for three reasons: it appears to be derived from a finite difference analysis of the differential equation of a loan balance which it is not, no work is shown, and the approximations made are not stated.

If we look at the real equation, the one which is the result of a 1st order ODE, we see that the authors at business insider decided the readers would not know what \(e\) is so they gave it a cumbersome approximation. They assume a sizable portion of their readership would be able to properly multiply and convert units in frequency and interest rate, raise quantities to fractional powers, and apply order of operations to a complex fraction but \(e\) was too foreign. Replacing \(e\) with 2.71 was also not an option for some indefensible, unknown reason.

$$ P = \frac{B_0 \frac{r}{f}}{1-e^{-rt_\text{term}}} $$

From the same article is this ridiculous attempt to compute the net present value of an annuity. \(F\) is the cash flow rate ($ per year), \(r\) is the rate of return, and \(t\) is the length of the annuity.

$$ \text{NPV} = F\left( \frac{1}{r}-\frac{1}{r(1+r)^t}\right) $$

The real equation that any 2nd-year college student in a STEM curriculum should be able to derive is both more correct and concise.

$$ \text{NPV} = \frac{F}{r} \left( 1 - e^{-rt} \right) $$

Again, Business Insider believes its readership is too uneducated to understand \(e\)? Do they think the same thing about \(\pi\)? What is wrong with their editors? Why is their instinct to publish cumbersome, inaccurate formulas instead of combating ignorance with explanation?

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© MC Byington