Mattos System

Mattos System Essay New methods of interest calculus and capital amortization

Mattos System by Erick Mattos <[email protected]> is licensed under Creative Commons Attribution-No Derivative Works 3.0 Unported License.

CAS Table and Mattos Tables analysis Herein it will be used the variables: P = $ 10.000,00; i = 1% monthly.; n = 10 installments; and the arithmetic progression formulas: a n=a 1 n−1⋅r e S n = Formula 1 Calculation of nth term

n⋅a 1a n  2

Formula 2 Terms sum

Constant Amortization System On regular calculation of CAS Table the interest of each payment is calculated monthly over the debit balance. See following table: INSTALMENT

0 1 2 3 4 5 6 7 8 9 10 TOTALS

INTEREST 0 $100.00 $90.00 $80.00 $70.00 $60.00 $50.00 $40.00 $30.00 $20.00 $10.00 $550.00

CAS TABLE AMORTIZATION 0 $1,000.00 $1,000.00 $1,000.00 $1,000.00 $1,000.00 $1,000.00 $1,000.00 $1,000.00 $1,000.00 $1,000.00 $10,000.00

PAYMENT 0 $1,100.00 $1,090.00 $1,080.00 $1,070.00 $1,060.00 $1,050.00 $1,040.00 $1,030.00 $1,020.00 $1,010.00 $10,550.00

BALANCE $10,000.00 $9,000.00 $8,000.00 $7,000.00 $6,000.00 $5,000.00 $4,000.00 $3,000.00 $2,000.00 $1,000.00 $0.00 $0.00

Remaking the table under another point-of-view, having the capital divided into independent equal parts as if it were diverse contracts acquired on the same period with different and consecutive due dates and remunerating each part by simple interest, we have: INSTALMENT

0 1 2 3 4 5 6 7 8 9 10 TOTALS

INTEREST 0 $10.00 $20.00 $30.00 $40.00 $50.00 $60.00 $70.00 $80.00 $90.00 $100.00 $550.00

MATTOS TABLE AMORTIZATION PAYMENT 0 0 $1,000.00 $1,010.00 $1,000.00 $1,020.00 $1,000.00 $1,030.00 $1,000.00 $1,040.00 $1,000.00 $1,050.00 $1,000.00 $1,060.00 $1,000.00 $1,070.00 $1,000.00 $1,080.00 $1,000.00 $1,090.00 $1,000.00 $1,100.00 $10,000.00 $10,550.00

BALANCE $10,000.00 $9,000.00 $8,000.00 $7,000.00 $6,000.00 $5,000.00 $4,000.00 $3,000.00 $2,000.00 $1,000.00 $0.00 $0.00

We observe that the payments and the interests of the two tables are the same but in an inverse order. We have the usual table in a decreasing order of interest charging and this one in an increasing order. Mattos System by Erick Mattos <[email protected]>

Mathematically, the interests are in an arithmetic progression. On exposed models, the initial term is equal to one on the first table (a1=1) and to ten on the second one (a1=10), the period is equal to ten (n=10) and the factor is equal to one negative on the first table (r=-1) and to one positive on the second one (r=1). The total interest calculation is then done by: J=

n⋅{1[1n−1⋅1]} P ⋅i⋅ ⇒ 2 n

1n J = P⋅ ⋅i 2

Formula 3 Total interest

We can observe that remuneration of the parts is given by simple interest and that in no moment it is calculated interest on interest. The CAS and Mattos System are linked to an arithmetic progression and there is not any exponentiation as we usually see on compound interest or interest on interest.

Mattos System Interest linked to Amortization The analysis of this document informations make us to have as a suggestion to interest calculation under the premise of remuneration by simple interest that it be transacted over the amortization realized, as on the Mattos Table. Whatever the amortization is, one should calculate simple interest over it as if it were an independent contract. This way one has the proper remuneration, transparency and ease of calculations. Even in the case of payment delay one easily verify the amount of interest to be paid, keeping on mind the amortization, the period and the interest rate which will be always the effective. One can by this methodology even loan at interest without previous payments calculation. This way, the borrower would make a payment when possible or following contract of which would be deduced the amortization and the interest at the exact moment of the event, by the following formulas:

PMT =A A⋅i⋅t ⇒

PMT = A⋅1 i⋅t 

⇔

Formula 4 Calculation of the amortization of a payment

PMT A= 1i⋅t 

or

A=

P n

Formula 5 Calculation of the amortization under fix payments

Mattos Fix System This system intends to remunerate correctly the contracts by simple interest as on the previous tables without the inconvenient of variable payments, by means of fix payments. The calculation of fix payments of Mattos Fix System is given by: Mattos System by Erick Mattos <[email protected]>

1n P⋅ ⋅i J P 2 P PMT =  ⇒  ⇒ n n n n

P 1n PMT = ⋅[ ⋅i1] n 2

Formula 6 Calculation of fix payments

We have then the table created with the goal of having fix payments and correct interest charging over the total period: MATTOS FIX TABLE PAYMENT 0 0 $1,000.00 $1,055.00 $1,000.00 $1,055.00 $1,000.00 $1,055.00 $1,000.00 $1,055.00 $1,000.00 $1,055.00 $1,000.00 $1,055.00 $1,000.00 $1,055.00 $1,000.00 $1,055.00 $1,000.00 $1,055.00 $1,000.00 $1,055.00 $10,000.00 $10,550.00

INSTALMENT INTEREST AMORTIZATION

0 1 2 3 4 5 6 7 8 9 10 TOTALS

0 $55.00 $55.00 $55.00 $55.00 $55.00 $55.00 $55.00 $55.00 $55.00 $55.00 $550.00

BALANCE $10,000.00 $9,000.00 $8,000.00 $7,000.00 $6,000.00 $5,000.00 $4,000.00 $3,000.00 $2,000.00 $1,000.00 $0.00 $0.00

Observe that the remuneration is paid by installments of total system interest which is not linked to the actual installment, then it is very important to point out the need of recalculation in case of any variable alteration. However in case of delayed installments it is enough to add the amortization complementary interest of the period to the payment. I point out that this system is in conformance to the law, that it does not use interest on interest and that is less expensive to the borrower, being this way more fair to him and to the society which has on credit an important tool for wealth production. Politically it is much interesting to a society that the best investments be the production and not the financial market, being this system a valorous adjustment to the importance between them. Unfortunately, among financial theoretician, it will have argumentation if those systems fit on molds and established classifications. However the presented systems have as main goal to bring financial justice and calculation easiness, burdening less and reducing mathematical operations complexity, difficulty of immense majority which are not able to comprehend and to calculate the systems actually employed. The use of the presented systems should be analyzed over the political prism of social justice and common wealth and it must be a political decision. No more to say, to anyone interested,

Erick Pereira Mattos [email protected] Mattos System by Erick Mattos <[email protected]>