The use of mathematics in financial transactions has many applications in the economy and is sometimes very complex, especially in advanced techniques such as those used for financial markets.
In the field of credit and in spite of sometimes confusing terminology , the interaction between mathematics and finance in general does not pose any particular problem.
We will apply, from some tools used by the financial technique , to compare the proportional rate used by banks to calculate the interest of a mortgage and the equivalent rate that is usually used for investment transactions . The first, associated with the various ancillary costs, will make it possible to determine the TEG, which makes it possible to compare objectively the offers of loan
The different terms used
When you borrow, the proposal from your banker translates into an annual rate, while your repayments are made monthly, hence the notion of proportionality that we will see below that defines the proportional or periodic rate.
On the other hand, the return of a deposit on a regulated booklet that runs for several years makes use of the notion of capitalization (the vested interests in turn produce interest). From a sum obtained at term, called acquired value , it is possible to perform various discounting calculations and find the equivalent rate.
Good to know: be aware that in a year we count 360 days in the field of credit and 365 in the calculations on investment transactions.
The proportional or periodic rate
This is both the simplest form of use and the most common application since it is used to calculate a fixed rate and fixed maturity mortgage . Among other things, it allows you to edit a depreciation schedule and uses simple interest calculations.
It makes it possible to report the departure (nominal) rate to a sub-annual period (less than the reference period). Thus, if we want to reduce it to the month, it suffices to divide it by twelve.
The proportional rate is based on simple interests. The calculations are made on the borrowed capital.
Formula for calculating a simple interest investment interest (generally for a period of less than one year):
Interest = original capital * annual tx * number of days / 365
Proportional rate formula for calculating a monthly loan payment
Nominal Tx * Duration of the desired period / 12
Example quantified for a 4.84% annual loan
(3.84 / 100 * 1/12) = 0.32%
Value acquired over a period of less than one year
Acquired value = nominal value + Interest
Equivalent or actuarial rate
This ratio is based on compound interest. He finds multiple applications in the context of financial investments and perfectly illustrates the old adage: “time is money”. It makes it possible to calculate the overall effective rate by including the cost related to the ancillary costs. We are interested in transactions for which interest is paid in arrears (or after-tax), ie at the end of the annual period (in the case of most financial investments).
We will take a concrete application and check for example that a monthly rate of 0.5% is not equivalent to 6% annual.
tm is the monthly tx and ta is the annual tx:
1 + t = (1 + tm) 12 from which t = (1 + tm) 12 – 1
To know: the converse is also true: an annual rate of 6% will not give an equivalent rate of 0.5% per month.
Value formula acquired over a period of more than one year
with VA = acquired value, VN = nominal value, i = interest, n = period
VA = VN + (1 + i) n