## About calculating the overall effective rate

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### The basic formula

A priori, nothing is simple and our goal is to make this formula more readable.

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It is an equality between two sums of flows (SYGMA) of the same duration, starting from the last to the first.

Ak expresses the loan flow, and Ak 'the repayment flow.

(1 + i) ^ tK or ^ tK 'expresses that this equality is the result of an actuarial calculation, ensured by the rise of each flow to the power of its rank.

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These are the details given by Decree No. 2002-928 of June 10, 2002 taken pursuant to Article 1 of Decree No. 2002-927 of June 10, 2002 relating to the calculation of the overall effective rate applicable to consumer credit and amending the consumer code.

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"K is the serial number of a loan ;

K 'is the serial number of a reimbursement or payment of charges;

AK is the amount of loan number K;

A'K 'is the amount of reimbursement or payment of charges n ° K';

m is the serial number of the last loan ;

m 'is the serial number of the last reimbursement or the last payment of charges;

tK is the interval, expressed in years and fractions of years, between the date of loan n ° 1 and that of subsequent loans n ° 2 to m ;

tK 'is the interval, expressed in years and fractions of years, between the date of loan n ° 1 and those of repayments or payments of charges n ° 1 to m';

i is the overall effective rate which can be calculated (either by algebra, or by successive approximations, or by a computer program) when the other terms of the equation are known, by contract or otherwise ".

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This formula essentially teaches that there are as many loans as there are repayments.

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To be complete, article R.314-3 of the Consumer Code specifies, and this is essential, that the desired rate (t) is obtained "according to the compound interest method" . In other words, this succession of constraints to the calculation makes it possible to write the Basic Formula in a different way, converted into an equation of the type:

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K - F (costs) x 1- (1 + Tp) ^ -m / Tp = 0

## In the Basic Formula, Ak is always equal to Ak '

Taking into account the basic postulates, (t) must be calculated from a sequence of terms in geometric progression, obtained according to the structural formula:

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(K - F) - Ak ^ m - Ak ^ m-1 - Ak ^ m-2 .... - Ak

## In the Basic Formula, Ak is invariant

These apparently complex formulas amount to calculating the sum of a sequence of terms in geometric progression, term (k) obtained from:

K (or K ') = C (the amount borrowed) x Tp / 1- (1 + Tp) ^ -Ak'

The value of K is therefore always the same.

We commented on the section dedicated to the case law of the firm (at the foot of the home page) , the very last judgment obtained corresponding to this hypothesis (Nancy, Tribunal d'instance, October 11, 2019).

## Deviations from the Formula

Research in actuarial mathematics has developed various "optimization" methods widely used by banks.

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It should be understood that, in the appendix to article L.313-1 (now L.314-2 of the Consumer Code), the Basic Formula is a residual formula, which should only be used when examples 1 to 6 cannot be applied. In reality, these examples concern extremely marginal hypotheses of atypical credits.

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By inventing deferred amortization (total or partial) or amortization stages (see these words in the lexicon ) lenders have recourse to atypical credit assumptions to lower the value of Ak and Ak '.

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So that the TEG obtained with a deferral would be lower than the TEG without deferral, while the credit costs the borrower more. The counterintuitive result given to the borrower pushes him to accept the repayment formulas which, in reality, are those which cost him the most.

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