Mathematics

Financial Mathematics

Applied financial mathematics: calculation of percentages, simple and compound interest, capitalization and amortization annuities, APR calculation and index numbers.

A percentage expresses a number as a fraction of 100. For example, a percentage r=12%r = 12\% means 12 out of 100.
Percentage Multiplier: is the factor by which an amount is multiplied to apply a percentage rr (increase or decrease).
Iv=(1±r100)I_v = \left( 1 \pm \dfrac{r}{100} \right)
Percentage Increase: (Multiplier >> 1) To increase an initial amount C0C_0 by a percentage rr, the final amount is:
Cf=C0(1+r100)C_f = C_0 \left( 1 + \dfrac{r}{100} \right)
Percentage Decrease: (Multiplier << 1) To decrease an initial amount C0C_0 by a percentage rr, the final amount is:
Cf=C0(1−r100)C_f = C_0 \left( 1 - \dfrac{r}{100} \right)
Successive Percentages: To calculate multiple percentage increases or decreases in a row, multiply the respective percentage multipliers.

Example: 15% increase, 8% decrease, and 25% increase.
Cf=C0(1+15100)⋅(1−8100)⋅(1+25100)C_f = C_0 \left( 1 + \dfrac{15}{100} \right) \cdot \left( 1 - \dfrac{8}{100} \right) \cdot \left( 1 + \dfrac{25}{100} \right)

I is the profit (interest) generated by an amount of money, called principal C0C_0, when deposited in an entity that offers an annual interest rate rr (in %), over a period of tt (years).

I=C0(r100t)I = C_0 \left( \dfrac{r}{100} t \right)
The final accumulated amount at the end of the period Cf=C0+IC_f = C_0 + I is:
Cf=C0(1+r100t)C_f = C_0 \left( 1 + \dfrac{r}{100} t \right)

If the interest is paid in periods other than a year, the time variable tt must be adjusted to the corresponding period.

Semiannual
r2⋅100⋅s\displaystyle \frac{r}{2 \cdot 100} \cdot s
Quarterly
r4⋅100⋅tr\displaystyle \frac{r}{4 \cdot 100} \cdot tr
Monthly
r12⋅100⋅m\displaystyle \frac{r}{12 \cdot 100} \cdot m
Daily
r365⋅100⋅d\displaystyle \frac{r}{365 \cdot 100} \cdot d
If the interest is added to the initial principal account, the amount generating interest increases each period.
The final accumulated amount at the end of the period is CfC_f.
Cf=C0(1+r100)t\begin{aligned} C_f &= C_0 \left( 1 + \dfrac{r}{100} \right)^t \end{aligned}
I=Cf−C0\begin{aligned} I &= C_f - C_0 \end{aligned}

If the interest is compounded in periods other than a year, the variable term must be adjusted to the period.

Semiannual
(1+r2⋅100)s\displaystyle \left(1 + \frac{r}{2 \cdot 100}\right)^s
Quarterly
(1+r4⋅100)tr\displaystyle \left(1 + \frac{r}{4 \cdot 100}\right)^{tr}
Monthly
(1+r12⋅100)m\displaystyle \left(1 + \frac{r}{12 \cdot 100}\right)^m
Daily
(1+r365⋅100)d\displaystyle \left(1 + \frac{r}{365 \cdot 100}\right)^d
An annuity is a fixed amount of money called a periodic payment, aa, deposited at regular intervals, earning an interest rate rr in %, to obtain a final accumulated amount CfC_f after a certain time, tt.
Cf=a(1+i)(1+i)t−1i\begin{aligned} C_f &= a(1+i) \dfrac{(1+i)^t - 1}{i} \end{aligned}
withi=r100\begin{aligned} \text{with} \quad i &= \dfrac{r}{100} \end{aligned}
It is the fixed amount of money or periodic payment, aa, paid at regular intervals to pay off a loan amount CC, at a certain annual interest rate rr in %, over a certain time, tt in years.
a=C⋅(1+i)t⋅i(1+i)t−1\begin{aligned} a &= C \cdot \dfrac{(1+i)^t \cdot i}{(1+i)^t - 1} \end{aligned}
withi=r100\begin{aligned} \text{with} \quad i &= \dfrac{r}{100} \end{aligned}
If we want to calculate the loan amount based on the number of payments and the fixed payment amount:
C=a⋅(1+i)t−1(1+i)t⋅iC = a \cdot \dfrac{(1+i)^t - 1}{(1+i)^t \cdot i}

As in other financial math formulas, if the payment period is a different interval, the term ii and the time variable tt must be adjusted accordingly.

Semiannual
t=st=s
i=r2⋅100\displaystyle i = \frac{r}{2 \cdot 100}
Quarterly
t=trt=tr
i=r4⋅100\displaystyle i = \frac{r}{4 \cdot 100}
Monthly
t=mt=m
i=r12⋅100\displaystyle i = \frac{r}{12 \cdot 100}
Daily
t=dt=d
i=r365⋅100\displaystyle i = \frac{r}{365 \cdot 100}

EAR (Effective Annual Rate) or APY (Annual Percentage Yield): (% of capital growth in one year) It is an indicative reference of the effective annual cost or yield of a financial product regardless of its term:

EAR=[(1+ip)p−1]⋅100withi=r100\text{EAR} = \left[ \left( 1 + \dfrac{i}{p} \right)^p - 1 \right] \cdot 100 \qquad \text{with} \qquad i = \dfrac{r}{100}
Where pp is the number of compounding periods or payments per year.
They are used to quickly compare data, in the form of a percentage.
Example: Water consumption per inhabitant per day table
2018201920202021
Caceres163185221203
Asturias161172195185
Valencia138143168162
To create the index numbers table, we set the consumption in 2018 as the baseline, which will correspond to 100%, and the rest of the data will be a percentage of that baseline data.
Water Consumption Index Numbers Table
2018201920202021
Caceres100113135124
Asturias100106121115
Valencia100103121117