Daniel asks: What's up with the finance questions?

The following is a summary of a conversation I had with Daniel Liu ,who asked about the relevance of the finance problems that I had posted recently. As this is a chat, you should read it and play the part of Daniel, and fill in the blanks accordingly. In particular, where it says [Daniel spends time thinking], you should do just that before reading on, because not all the answers are given.

Disclaimer:
This is only "one" of the many reasons why I'm posting finance questions.
Lots of creative license was used, and the conversation was paraphrased / edited, so are not necessarily our real responses.
I made fictitious Daniel smarter or dumber as needed to help the flow.

Calvin: These finance questions are meant to help you think about what they mean, and how we can apply our mathematical knowledge to situations in the “real world”, instead of just abstract problem solving. Of course, for the questions which require more in-depth knowledge, that will be out of the reach of some people, similar to how some of the questions which require (say) Law of Quadratic Reciprocity will not be easily solved. For the sake of making this discussion understandable, I will choose basic simple examples, but extend them out from there.

Calvin: Let’s start with the example of Rule of 72. This is technically a business / finance question, but at it’s heart it’s a question about approximation, specifically the Taylor series. By understanding where the approximation comes from, we have a much better understanding of the model, and hence also of where the model breaks down

[Daniel spends time thinking]

Daniel: K, I got it after looking at wikipedia.

Calvin: So, why does the rule of 72 work?

Daniel: No idea. I just saw that’s what the rule said to do, and I did it.

Calvin: Alright, set up the equations and solve for t properly.

[Daniel spends time thinking]

Daniel: The answer is [fill in the blank].

Calvin: Right, so this is has logarithms in it, which are ugly without a calculator. Why can we apply the rule of 72 in our head? Is it even true?

[Daniel spends time thinking]

Daniel: I guess we want to show that $\frac{ \ln 2 } { \ln (1+ r \% ) } \approx \frac{ 72 } { r }$. I don’t see why that is true.

Calvin: Hint: What is the taylor series of $1 + r$?

[Daniel spends time thinking]

Daniel: I see that the Taylor series is of $\ln (1+r)$ is [fill in the blank], hence the approximation would be $\frac{ 0.69 } { r \% - \frac{ r \% ^2 } { 2} }$

Calvin: Great, so we get the approximation $\frac{ 69}{r}$. So why do you think it’s “Rule of 72” instead of “Rule of 69”?

Daniel: Well, we approximated $\ln (1+r)$ and since we took $\frac{ 1}{\ln (1+r) }$ and increased the denominator, i guess we should increase the numerator too?

Calvin: Great observation! But not the real reason. The true reason is that 72 is much more friendly to divide....

Daniel: Wow! People are lazy!

Calvin: What are the shortcomings of the Rule of 72?

[Daniel spends time thinking]

Daniel: If $r > 1$, then clearly this will not work. Also, when $r = 100\%$, it should take 1 year to double, instead of $\frac{ 72}{100}$ years, which is like 30% off.

Calvin: Great! So, how small must r be? At what value of r does this rule not work? This goes into error analysis in physics. We introduced an error of $+ 2-3 \%$ going from 69 to 72. We introduced a small negative error going from $r - \frac{ r^2}{2}$ to $r$ ...

[Daniel spends time thinking]

Daniel: The latter error becomes larger as r gets large. What is the amount of error in the second one?

[Daniel spends more time thinking]

Daniel: Ahhhh the percentage error is [fill in the blank]

Calvin: So, if we accept an error range of 5%, what is the approximate largest r we should use?

Daniel: We start with (say) - 3 %, and it can go up to +5%, so we have a +8% margin, which means we can use r up to 16%

Calvin: Great that you remembered the initial error. Now you know much more about the rule of 72 than most finance people. Congrats! So, now you know how to make quick estimates of doubling time in your head / without a calculator.

Daniel: Typically interest doesn’t go up beyond 16% right?

Calvin: It could, for example: credit card, payday loans, loansharks, etc.

[Skipping a entire tangential portion about the above question]

Calvin: The idea is that
1. We start off with a model of the world as we see it
2. We get some complicated equation that governs it
3. We start to make simplifying assumptions and approximations
4. We get a much simpler equation, that works, but only in a local neighborhood.
5. We sacrifice accuracy for speed, and so we need to know when things are no longer accurate.
6. By understanding the shortcomings, we can patch the model. A model with many patches is almost as bad, since we will then need to do alot of calculations

Calvin: So, let’s say we want to look at the scenario when interest is 100% to 120%. What do we do?

Daniel: I would just use the original formula?

Calvin: Well, that goes against what we just talked about, so humor me.

Daniel: We’re stuck. for $r > 1$, the $r^2 , r^3$ terms will just explode right

Calvin: Hint: Consider $\ln (2.1 \pm A )$. What are the error terms?

[Daniel spends time thinking]

Daniel: Hm, if we have $A = r - 1$, then we can take the maclaurin expansion about $\ln 2.1$, and the error terms are managable since $|A| < 0.1$.

Calvin: Right! SO what we’re essentially doing, is that we are just taking local approximations applied to a complicated formula. We must understand the limits of the approximation, which tells us the neighborhood that it works in. When we move out of the neighbourhood, we simply pick another point and plunk ourselves down there and calculate again. This way, our calculations are going to be much faster.

Daniel: What’s the time difference between solving an exponential equation and a linear one?

Calvin: Significant. Linear equations can be solved via Crammer’s Rule. Exponential / Logarithm equations require looking up log tables.

(An entire portion about logarithms, see the teaser here.)