Formula for converting simple interest to compound?

Question

Is there a formula for calculating the compound interest where the simple annual interest rate is known and also how often during the year the interest is marked up? I think it may also be known as the effective rate.
Ta
Wazza

Answer 1

Let say principal = A
simple annual interest rate = i
no of year = n

Principal after n year for compound interest
= A(1 + i)^n

So, effective rate (compound interest rate) for n years
= [A(1 + i)^n - A]/A
= (1 + i)^n - 1

effective rate (compound interest rate) per year
= [(1 + i)^n - 1] / n

Answer 2

Simple Annual Interest Formula

Answer 3

Simple Interest Formula Definition

Answer 4

For the best answers, search on this site https://shorturl.im/axFbk

Compound interest is R=((1+r)^n)-1. Simple interest is R=r*n. If you know r and n then you can calculate either one. If you know n and R then you can reverse the formulas to find one or the other. For example, 1% per month is 12% simple and 12.68% compounded. 12% p.a. compounded monthly is 0.949% per month, which is 11.39% simple.

Answer 5

"(Compound Interest) A pattern of interest accrual such that the rate of interest earned in each investment period is constant is called compound interest. If 25 is the initial investment and 26 is the constant interest rate, the accumulated value at the end of 27investment periods is given by 28

Under a pattern of simple interest, interest is not reinvested at the end of each period to earn additional interest. Therefore the constant amount of interest earned in each investment period can be thought of as a rate of interest, called the simple interest rate, applied to the principal amount only. In contrast, compound interest assumes that the interest earned in one period is reinvested in the next period (at the same rate as the initial investment) to earn additional interest.
The formulas given in the definitions of simple and compound interest are easily derived from the definitions. First, suppose that the simple interest amount earned in each period is 29 At the end of the first period, the accumulated amount is 30At the end of the second period, the accumulated amount is 31 32At the end of 33periods, the accumulated amount is 34 35
Next, if the compound interest rate is 36then 37is the amount of interest earned in the first period. At the end of the first period, the accumulated amount is 38 39The accumulated amount at the end of the first period can be thought of as the principal amount at the beginning of the second period. The amount of interest earned in the second period is thus 40and the accumulated amount at the end of the second period is

41

42

43

44

At the end of 45periods the accumulated amount is 46 47
Note that with simple interest, the effective rate of interest is different for each time period and is in fact decreasing. This can be seen by finding the effective rate for the 48period, as the ratio of the interest earned in the 49period to the accumulated value at the beginning of the 50period. Since the interest earned in any period is 51we have 52If 53is thought of as the simple interest rate 54, this equation simplifies to

55

which is a decreasing function of 56 57
In contrast, the effective rate of interest for a compound interest pattern is constant and is just the compound interest rate, as can be seen from

58 59

which is constant by the definition of compound interest.

Example (Simple and Compound Interest) (a) If 1050 is invested for three years at 4%, find the accumulated value and total interest earned at the end of the period under both simple and compound interest.
Under simple interest, the 4% simple interest rate determines the constant interest amount by 60 61The accumulated value after 3 years is 62 63The total interst earned in the three year period is 64Note that 65
Under compound interest, the 4% compound interest rate is the same as the annual effective rate of interest. The accumulated value after 3 years is 66 67The total interest earned in the three year period is 68The extra 5.11 is due to interest compounding, or interest earned on interest.
(b) If 100 is invested at 3% simple interest, find the accumulated value after 3 years and the effective rates of interest in the third and fourth year.
The accumulated value after three years is given by the formula 69where 70and 71Therefore,

72 73

The effective rate of interest in the third year is the amount of interest in the third year divided by the accumulated amount at the end of the second year.

74 75 76

The effective rate of interest in the fourth year is the amount of interest in the fourth year divided by the accumulated amount at the end of the third year.

77 78

(c) If 1115 is invested to earn a constant 44.6 each year, how many years will it take for the account value to double?
The situation described is simple interest, since the interest amount is constant. We want to solve for 79 given that the accumulated value is 2230.
80

81

(d) If 2130 is invested to earn a constant rate of 4%, what is the accumulated value after 4 years? How many whole years will it take for the account value to become greater than 3000?
The accumulated value after 4 years is given by the formula 82 where 83Substituting, we get

84

To find the number of years it will take for the account value to grow to over 3000, we must solve the equation

85

86 87

therefore, the number of whole years for the account value to grow to over 3000 is 9"