# Monthly

A **monthly payment** is the amount of money paid each month by the borrower to the bank on a loan from a bank. Conversely, when investing in the bank, the monthly payment is the amount of money paid each month by the bank to the investor.

## Calculation of the monthly installment of a depreciable loan

In a repayable loan, the borrower repays a portion of the borrowed capital each month and pays interest to the bank on the borrowed capital, the whole being the monthly payment.

### Formal calculation

It is assumed here that the 1 ^{st} installment is paid a month after the capital has been paid.

The sum of the monthly installments that will be paid in the future must have a present value equal to the amount borrowed.

To calculate the present value of a monthly payment, we use the interest rate .

Thus, the present value of a monthly payment of *M* € paid in 1 month is{\ displaystyle M / (1 + i)}, where *i* is the monthly interest rate used by the bank, ie{\ displaystyle i = {\ sqrt [{12}] {1 + t}} – 1}, where t is the annual interest rate. The value of the same monthly payment paid in two months is{\ displaystyle M / (1 + i) ^ {2}}, in three months {\ displaystyle M / (1 + i) ^ {3}}etc.

The sum of the current values of the constant monthly payments *M* , equal to the capital borrowed, can be written as follows:

{\ displaystyle C = \ sum _ {k = 1} ^ {k = 12 \ cdot t} {\ frac {M} {{(1 + i)} ^ {k}}}}, *t* being the duration in years of the loan and *C* the capital borrowed.

This sum can be calculated formally:

{\ displaystyle C = M \ cdot {\ frac {1 – {\ frac {1} {{(1 + i)} ^ {12 \ cdot t}}}} {i}}}

The value of *M* is thus calculated, according to the borrowed capital ( *C* ), the monthly interest rate ( *i* ), and the duration of the loan ( *t* , in years):

{\ displaystyle M = {\ frac {C} {\ frac {1 – {\ frac {1} {{(1 + i)} ^ {12 \ cdot t}}}} {i}}} = {\ frac {C \ cdot i} {1 – {\ frac {1} {{(1 + i)} ^ {12 \ cdot}}}}}}

An interesting link makes it possible to assimilate the determination of M:

http://images.math.cnrs.fr/Emprunts-mensualites-interet-taux.html [ archive ]

### Example

A loan of € 100,000 ( *C* = 100,000) over t = 15 years (12 · *t* = 180), with an annual interest rate of 5% ( *i* = 0.05 / 12) results in monthly payments of:

{\ displaystyle M = {\ frac {100,000} cdot 0,05 / 12} {1 – {\ frac {1} {{(1 + 0,05 / 12)} ^ {180}}}}} = 790, 79 \; euro}

Note: The above example uses an approximation of the monthly rate (0.05 / 12). This constant practice of bankers leads to an increased annual effective rate (5.12% in the present case), while the monthly rate{\ displaystyle j} exact is worth {\ displaystyle (1 + j) ^ {12} = 1.05}. So, the monthly rate is calculated exactly by:{\ displaystyle j = (1 + i) ^ {1/12} -1}, and not {\ displaystyle j = i / 12} which is simpler but slightly overestimated.

## Calculation of the monthly installment of a loan in fine

In a loan in fine, the borrower only reimburses the capital at maturity of the loan. Thus, in this case, when the borrower obtains a loan of 100,000 euros for fifteen years, he reimburses this sum after fifteen years in the bank by paying interest meanwhile.

It is easy to calculate the monthly payment with this loan formula. Since the borrower does not repay the capital by paying the monthly installment, it is therefore the price paid each month to the bank for the enjoyment of the capital borrowed, namely, with the usual notations,{\ displaystyle i \ cdot C}. For example, for a loan of 100,000 euros over fifteen years at 5%, this gives 417 euros . The monthly payment does not depend, by definition, on the duration of the loan.

## Comparison of monthly payments in the two loan formulas

In the repayable loan, the monthly payment is constant (with a fixed interest rate of course), but the remaining capital (CRD) decreases as the monthly payments are made. Thus, at the beginning of the loan, the monthly payment consists essentially of interest, whereas towards the end of the loan, it is essentially made up of capital (the share of interest becomes smaller and smaller since it is proportional at the CRD).

In the end loan, on the contrary, the interest is by definition constant (with, once again, a fixed interest rate).

It is interesting to calculate the interest paid ( *Int* ) in both loans:

- depreciable loan: this is the sum of the monthly payments less the capital due, namely

{\ displaystyle Int = 12 \ cdot t \ cdot MC}

- loan in fine: the interest paid is simply the sum of the monthly payments, namely {\ displaystyle Int = 12 \ cdot t \ cdot C \ cdot i}

### Numeric example

- amortizing loan of 100,000 euros over fifteen years, interest rate at 5%:{\ displaystyle Int = 12 \ cdot 15 \ cdot 791-100000 = 42380 \; euros}
- loan in fine of 100,000 euros over fifteen years:{\ displaystyle Int = 12 \ cdot 15 \ cdot 100000 \ cdot 0,05 / 12 = 75000 \; euros}

Paying larger interest can be a paid calculation for high income taxpayers.

## See also

Calculation formula for the remaining capital *C* after monthly investment of a sum *M* for a period in year *t* at a monthly interest rate of *Int _{m}* (in%):

{\ displaystyle C = {\ frac {M \ cdot {\ bigr (} 1 + Int_ {m} {\ bigr)} ^ {12 \ cdot t} -1} {Int_ {m}}}}

The monthly interest rate can be calculated according to the annual rate *Int _{a}* (actuarial rate), according to:{\ displaystyle Int_ {m} = {\ sqrt [{12}] {Int_ {a} +1}} – 1}

Digital application: 200 euros placed every month for 5 years (ie a total investment of 12,000 euros in the absence of interest) at an annual interest rate of 3%. After 5 years, the capital will be € 12,916.19 .