Hello, dear reader. This is the actual first episode of the series. As I said in the Road to: FIRE – 00.Introduction episode, I think math is the most straightforward way to start. My Italian university background forces me to start every new topic by studying its theory. So, given that I want to reach Financial Independence, I must learn the basics of Mathematical Finance. Now, I won’t take an exam, so I won’t go that deep into the theory, but I will do my best to know at least what I am talking about.

This episode — and the following ones — will follow the structure of the first few lectures from Prof. Coletti’s course “Financial Trading and Algorithms”. I first discovered Paolo Coletti’s existence from this video from Mr. RIP. The course is about a practice-oriented approach to trading in terms of algorithms and computer science. However, the first few lectures on YouTube are more theoretical and look relatively easy for beginners.

Let’s begin!

Time is Money

“Time is Money!”. This is the first sentence of Prof. Coletti in his first video [1]. But let’s clarify this right now:

Time multiples money.

Now it’s good. So let’s dive into it more deeply in the following paragraphs.

Simple and Compound Interest

0 is “now”. The positive numbers are from the following years, while the negative ones are from the past.

Does there exist a (mathematical) method to know how much my capital will be the next year? Sure thing:

C_1 = C_0 \cdot (1 + i)

Where C_1 is the capital the next year, C_0 is the current capital and i is the interest.

Now, let’s imagine we want to know how much the capital will value after 3 years in the simple interest scenario:

C_3 = C_0 \cdot (1 + i) \cdot 3

Let’s generalize it:

C_n = C_0 \cdot (1 + i) \cdot n

where n is the duration (in years, for example) of the investment. This formula is called Simple Interest.

Now, generally speaking, simple interest is not so attractive. In fact, simple interest works well if we are not capitalizing interests, too. But what if we are, instead? The formula changes a bit:

C_3 = C_0 \cdot (1 + i) \cdot (1 + i) \cdot (1 + i)

Let’s generalize it and simplify it a bit:

C_n = C_0 \cdot (1 + i)^n

This is called Compound Interest.

Nice! Let’s complicate things a bit.

Present Net Value

Of course, what we just saw can be used to get the so-called present value, i.e., to know how much the capital obtained in the past values today:

C_{present} = \frac{C}{(1 + i)^{-n}}

Where n will be positive if referring to past years and negative otherwise. Let’s make an example. How much is valuing a 100€ capital obtained three years ago with an interest rate (when computing present-value is called discount rate) of 10% (0.1)?

\frac{100}{(1 + 0.1)^{-(+3)}} = 133.1

Let’s imagine getting more than one capital distributed among different years, as you can see here below

the value is computed as:

C_{present} = \frac{C_1}{(1 + i)^{-3}} + \frac{C_2}{(1 + i)^{0}} + \frac{C_3}{(1 + i)^{1}} + + \frac{C_4}{(1 + i)^{3}}

In a more mathematical way it’s like this:

C_{present} = \sum_{j = 1}^{n}\frac{C_j}{(1 + i)^{-t_j}}

This is called Net Present Value.

Internal Rate of Return

Now, imagine that I want to know if an investment is profitable. To do so, we want to see the value of i such that the NPV equals 0. This will tell us the rate at which the investment will break even. Of course, if we choose an asset and someone gives us an investment rate higher than the IRR, we are happy; otherwise, we will not be satisfied. Moreover, if we are dealing with a compound interest, the solution may have more than one IRR. Here you can find an example made with Google Docs.

Simple Return on Investment

The return is computed as

R = \frac{C_f - C_i}{C_i}

that, after a few math operations become

C_i\cdot(1+R) = C_f

Look who is back! It is very similar to the Simple Interest formula we have seen before. This is the Simple Return. This is the simplest scenario: you buy a stock and sell it afterward. How much did you get in return?

Suppose that R is my x-daily return, and I want to get the yearly. This is done to compare returns from different return rates over different periods of time. The formula to get the yearly return comes from:

(1+R_x)^{\frac{365}{x}} = (1 + R_y)

thanks to which we can get

R_y = (1+R_x)^{\frac{365}{x}}-1

of course if x < 365 then R_t > R_x, otherwise if x > 365 then R_y < R.

Note that if we only have two capital flows, we can apply the IRR formula and get the same value.

Of course, computing such an equation is really hard as it involves solving exponential polynomials. But we can work it out.

Log Return on Investment

This approach is relatively similar to the one we have seen before but is more elegant:

\ln{\frac{C_f}{C_i}} = \ln{C_f} - \ln{C_i}

This is the Log(aritmic) Return on Investment. While this formula looks really good and has some beautiful properties, it doesn’t actually make sense to individual money-savers. This is because, most of the time, investments don’t follow the exponential law. But, anyway, let’s take a look at the properties that are really time-saving and convenient:

1. Simple return is not normally distributed, while the Log return is.
2. Getting the annual return from the daily is way more straightforward than the Simple one: LR_y = LG_d \cdot 365

Conclusions

Let’s recap real quick what we’ve seen in this episode:

1. It’s not true that “time is money”, while instead “time multiplies money”.
2. How interests work in the two main scenarios: simple and compound.
3. Understanding profits with different capital flows over time with the Net Present Value.
4. Understanding if an interest rate is profitable or not with a specific capital flow.
5. How to compute the return of an investment and get the yearly return given a daily one.
6. How to take advantage of logarithms to convert from daily to annual return rate in an easy way.

If something written here is wrong, please don’t hesitate to comment below.

See you around!

Resources

[1] Matematica Finanziaria: https://www.youtube.com/watch?v=0QHbeGTgCJM