Amount
Amount is the initial capital added to the interest in the period.
Interest
Consists in the payment of an investment, borrowed capital or in the rent paid or collected for the use of the money. The difference between an investment redeemed and initial amount is also called interest.
At any Monetarist Economy, the cost to lend or borrow any amount should be measured applying an index between the credit and its value for a certain period of time. This index is called interest rate. Said rate is used to evaluate both payment rate of a capital for people who has resources, and for those who hasn't, and, will have to borrow capital. The first group should consider the risk factors, expenses, inflation and the gain expected when investing said rate. The higher, the better. For the second group, the smaller the better.
Simple Interest
Interest rate always refers to the initial capital. Therefore, the rate is called proportional since it presents a stable variation along time. Example: Example: 1% a day reaches 30% a month, which represents 360% a year and so on.
Consider the initial capital P applied at simple interest of rate i per period, for n periods.
Taking into account that simple interest refers to initial capital, we have the following formula:
J = P.i.n |
J = Interest after n periods, of capital P applied at an interest rate per period equal to i.
At the end of n periods, capital is equal to initial capital plus interest from the period.
Initial capital added to interest from the period is called AMOUNT (M). So:
M = P + J
J = P + P.i.n
M = P + P.i.n
M = P(1 + i.n) Therefore,
M = P(1+i.n) |
Example:
In the amount of $3,000.00 is applied an interest rate of 5% a month, for five years. Calculate the amount and interest by the end of the period.
Solution:
We get: P = 3,000.00, i = 5% = 5/100 = 0.05 and n = 5 years = 5,12 = 60 months.
J = 3,000.00 x 0.05 x 60 = 9,000,00
M = 3000(1 + 0,05x60) = 3,000(1+3) = $12,000.00
Compound Interest
Interest rate refers to initial capital plus interest accrued up to the previous period. The rate exponentially varies in the period. In this interest system, 1% a day does not reach 30% a month, which does not represent 360% a year.
Compound interest is more common in the financial system, being more useful for daily problems. Interest from each period is consolidated to the main to calculate interest for the upcoming period.
Capitalization is when interest is consolidated to the main. After three months of capitalization, we have:
1st month: M =P.(1 + i)
2nd month: the main is equivalent to the previous month: M = P x (1 + i) x (1 + i)
3rd month: The main one is equal to the amount of previous month: M = P x (1 + i) x (1 + i) x (1 + i)
So, we have the formula:
M = P(1 + i) n |
The rate i must be expressed in the same time measurement of n , which means that both must be in the same unit, that is, interest rate per year for n years. |
In the system, the rate entered is processed as annual rate. Thus, n must be converted into years, that is, 1 month is equal to 1/12 or 30/360, so time is in the same unit as interest rate. |
In order to calculate interest, exclude the main from the amount at the end of the period:
J = M - P |
Example:
In the amount of $6,000.00 is applied an interest rate of 3.5% a month, for one year.
Solution:
P = R$6,000.00
n = 1 year = 12 months
i = 3.5 % a.m. = 0.035
M = ?
Applying the formula: M = P (1 + i)n , we get:
M = 6,000 (1 + 0.035) 12
M = 6,000 x 1.511 = 9,066.41
So, the amount isR$9,066.41
Relation between interest and progression
Simple interest: M( n ) = P + P.i.n ==> P.A. starting by P and ratio P.i.n
Compound interest: M( n ) = P . ( 1 + i ) n ==> P.G. starting by P and ratio ( 1 + i ) n
Hence:
In capitalization of simple interest, balance grows in arithmetic progression
In capitalization of compound interest, balance grows in geometric progression
Consider the initial balance of R$1,000.00 and an interest rate of 50% in the period.
Simple Interest | |
Period | Balance |
1 | 1,500.00 |
2 | 2,000.00 |
3 | 2,500.00 |
4 | 3,000.00 |
5 | 3,500.00 |
6 | 4,000.00 |
7 | 4,500.00 |
8 | 5,000.00 |
9 | 5,500.00 |
10 | 6,000.00 |
Compound Interest | |
Period | Balance |
1 | 1,500.00 |
2 | 2,250.00 |
3 | 3,375.00 |
4 | 5,062.50 |
5 | 7,593.75 |
6 | 11,390.63 |
7 | 17,085.94 |
8 | 25,628.91 |
9 | 38,443.36 |
10 | 57,665.04 |
When calculating compound interest, it is not possible to divide an annual rate to obtain the daily rate, since compound interest presents a geometric progression, which means, interest levies on interest, so, we use:
Equivalent Rate
i q = (1 + i t)q/t -1
i q = rate for the desired period
i t = rate for the given period
q = desired period
t = given period
a.a. = a year
a.d. = a day.
Example:
9.7% a.a. equivalent to:
i q = (1 + 0.097) 1/360 - 1 = 0,000257197
iq= 0.02572% (with 5 significant digits) which means:
9.7% a.a. equivalent to 0.02572% a.d.
For compound interest:
FV= Future Value
PV = Current Value
i= Rate (*)
n= Period (*)
(*) both variables must be in the same time period. |
So:
FV = 14,500 (1 + 0.097) 1/360 = 14,503.73
J = 14,503.73 - 14,500.00 = 3.73
* 1/360 = 1day in one year
Or, using the daily equivalent rate:
FV = 14,500 (1 + 0.000257197) 1 = 14,503.73
J = 14,503.73 - 14,500.00 = 3.73
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