Finance interest

Interest is the rate that a financing-issuer collects revenue, overtop of the financed amount.
The interest rate does not necessarily reflect the total payback amount, just the allocation of revenue.

Interest rates can be decided by the financing issuer, but are normally set relative to the country's overnight-rate (the cost the federal-reserve, or bank-of-canada charges to borrow money).

In some countries this rate must be below a certain APR, meaning it can be illegal to collect revenue to quickly on a financing.

Tutorials

khan academy: interest/compound interest

https://www.youtube.com/watch?v=GtaoP0skPWc&list=PL9ECA8AEB409B3E4F

Terminology

yield:
return:

principal:           amount borrowed
compounding-period:  interval that interest is added to the principal (ex. quarterly)
periodic-return:     percentage paid per compounding-period

Overview of Financing/Interest

Simple Interest

In simple interest, the interest-rate is multiplied against the initial principal each compounding period.

${\texttt {principal}}\times (1+({\texttt {period}}\times {\texttt {interest-rate}}))$

Example:
If we borrowed 1000$
with an interest rate of 10%
and our interest accrues weekly

week	calculation	payback-amount
week 0	1000$ * (1 + (0*0.1))	1000$
week 1	1000$ * (1 + (1*0.1))	1100$
week 2	1000$ * (1 + (2*0.1))	1200$
week 3	1000$ * (1 + (2*0.1))	1300$

Compound Interest

In compound interest, the interest is calculated on the initital principal and the interest collected so far.
It accrues interest exponentially.

${\texttt {principal}}\times ({\texttt {interest-rate}})^{\texttt {Nth-compounding-period}}$

Example:
If we borrowed 1000$
with an interest rate of 10%
and our interest accrues weekly

week	calculation	payback-amount
week 0	1000$ * (1 + 0.1) ** 0	1000$
week 1	1000$ * (1 + 0.1) ** 1	1100$
week 2	1000$ * (1 + 0.1) ** 2	1210$
week 3	1000$ * (1 + 0.1) ** 3	1331$

Prime-Rate/Overnight Rate

NOTE:

TL;DR the overnight-rate is what the central bank charges to lend money.
the prime-rate is the rate a bank will charge people most likely to pay back money (which is influenced by the overnight-rate)

The Bank of Canada sets their cost of borrwing nightly based on the economy.
If country's inflation or GDP are low, the overnight-rate will be decreased to encourage borrowing/spending, and stimulate the economy.
Banks borrow money from the Bank of Canada, and are charged the overnight-rate for borrowing.
They in turn finance businesses and individuals at an interest-rate relative to the overnight-rate they borrwed at.

The prime-rate is set at the bank's discretion, and is the rate the bank would charge their most dependable customers to borrow money.
Financings interest rates are generally expressed as prime + N%

Rate Types

Nominal Interest Rate

Does not account for inflation.

$(1+{\texttt {periodic-return}})^{\texttt {compounding-periods}}-1$

if nominal_rate == 5%:
    5$ interest on each 100$ paid

https://www.educba.com/real-interest-rate-formula/

Real Interest Rate

Accounts for inflation.

${\frac {1+{\texttt {nominal-interest-rate}}}{1+{\texttt {inflation-rate}}}}-1$

https://www.educba.com/real-interest-rate-formula/

Effective Interest Rate

Accounts for compounding interest (but not inflation).

${\bigg (}1+{\dfrac {\texttt {nominal-interest-rate}}{\texttt {compounding-periods}}}{\bigg )}^{\texttt {compounding-periods}}-1$

TODO:

could you sub in real-interest-rate to account for inflation here?

https://corporatefinanceinstitute.com/resources/knowledge/finance/effective-annual-interest-rate-ear/

https://www.calculatorsoup.com/calculators/financial/effective-interest-rate-calculator.php

Annual Percentage Rate

https://corporatefinanceinstitute.com/resources/knowledge/finance/annual-percentage-rate-apr/

Other

Present Value

The present value determines what the value a future amount of money would be today if we had it.
You can use the present value to compare an investment opportunity today to the promise of money in the future.

If we had the choice of investing 100$ for a 5% interest rate,
or receiving 110$ a year from now, which would be the best choice?
you can compare them by comparing the present value of the 110$ against today's 100$.

year-1  100$        # convert to year-3 with: 100$ * (1 + 0.05%) ** 2
year-2
year-3        110$  # convert to year-1 with: 110$ / (1 + 0.05%) ** 2

${\texttt {present-value}}={\dfrac {\texttt {amount}}{({\texttt {interest-rate}})^{\texttt {periods}}}}$

Assuming our interest period is yearly, we could convert the 110$ to year-1 using

${\dfrac {110\$}{(1+0.05\%)^{2yrs}}}$