How are financial formulas such as Effective Annual Rate derived? How can I gain more insight into proving and deriving them myself?

Question

So, I'm starting to learn about finance and I know that there is a large component involved when it comes to differential equations, integration, derivation and series.

For instance, I know that this definition is true, because it fits the definition of a geometric series:

$PV = \frac{P_0}{(1+r)^0} +\frac{P_1}{(1+r)^1} +\frac{P_2}{(1+r)^2} + \frac{P_3}{(1+r)^3}+ ... + \frac{P_n}{(1+r)^n} = \sum_{i=0}^n \frac{P_i}{(1+r)^i} = P_0(1+r)^n$

I believe many of these financial formulas are based on annuities and perpetuities. I discovered these things by reading chapters of Principles of Corporate Finance by Brealey, Myers and Allen. I don't remember it that well but I distinctly remember that they don't go a lot into detail about how these formulas work. Many finance books and courses don't talk about that either.

There are other formulas like EAR(Effective Annual Rate) which I'm not quite sure how to "reverse engineer" so to speak.

EAR goes this way:

$EAR = (1 + \frac{r}{m})^m - 1$

I figured out present value, but only because someone mentioned it was a geometric series. How were the rest of these formulas created? Do they come from some differentiation of a more complicated formula?

I know that some would tell that this doesn't matter, that at the end of the day it is irrelevant and that I just need to know how they are applied. That doesn't work for me very well, I don't really get things unless I figure out why they work the way that they do. I also don't like using formulas without understanding how they really work. I don't suppose someone could help me how to trace the origins of this formulas like I did with Present Value. I don't suppose there's a textbook out there that covers these things as well?

I would really appreciate it.

Answer 1

For EAR, I'll show you with an example which trivially generalizes by replacing numbers by letters.

Suppose an account pays $r = 0.04$ (i.e. $4$%) annual rate compounded $m=4$ times per year.

This Means that each $1/4$ year, an interest payment equal to $(r/m)*(balance) = 0.01b$ is added to the balance, so that after the payment the NEW balance is $b + 0.01b = 1.01b$

At the start, balance = $B$. After $1$ quarter, the new balance is $1.01B$.

After $2$ quarters, the new balance is $(1.01)$(old balance) = $(1.01)(1.01B) = (1.01)^2B$

Etc.... at the end of the year, the balance is $(1.01)^4B$.

The Effective Annual Rate is the interest rate which, if only compounded ONCE a year, would give the same return as the given compounded rate. In other words, by what portion of the original balance has the account grown at the end of a year?

In this example the total change in the account over a year (i.e. total interest paid) is $(1.01)^4B-B =B(1.01^4-1)$ and hence the EAR is that amount divided by the original balance $B$, i.e.

$EAR = 1.01^4-1$ which of course is $(1+r/m)^m - 1$

Answer 2

The EAR is used to compare different investments and decide which is more profitable. Say you have a capital $C$ you want to invest in one of this two investments : one with $k$ periods of time per year and with $i$ interest rate compounded every period and another with $p$ periods and $j$ interest rate. Now a way to compare them is to see which one is more profitable after the same amount of years $r$, then :

$$x = C(1+i/k)^{rk}$$ $$y = C(1+j/p)^{rp}$$

with $x,y$ representing the amount you will obtain for the first and the second investment. But in comparison $r$ and $C$ simplify, so you can omit them. The $-1$ is introduced just for convention.