What is Compounding?

Compound interest refers to the method of calculating interest in addition to the interest generated by the principal in the next interest-bearing cycle. ^[1]

[fù lì]

The calculation of compound interest is correct

Compound interest final value

The Commercial Press's "English-Chinese Dictionary of Securities Investment" explains: compound interest compound rate; compound interest; interest on interest. Interest arising from the principal and interest accrued in the previous interest period. That is to say, the new interest earned from the undrawn interest at the principal's interest rate is often referred to as interest, interest, and profit rolling. Not only does the principal generate interest, but also interest. The formula for compound interest is:

Where: P = principal; i = interest rate; n = holding period

Ordinary annuity final value

Ordinary annuity final value: refers to the sum of the principal and interest of the same amount of income or expenditure at the end of each period within a certain period, that is, the amount of each period is converted to the final value of the last period at the compound interest, and then added up to the end of the annuity value.

For example: a deposit of 1 yuan per year, an annual interest rate of 10%, after 5 years, the final value of each year and the final value of the annuity, the formula is: F = A [(1 + i) ^ n-1] / i, recorded as F = A (F / A, i, n).

Derived as follows:

1 yuan at the end of each year

Final value at the end of 2 years = 1 * (1 + 10%) = (1 + 10%)

Deposit $ 1 at the end of 2 years

3 years end value = 1 * (1 + 10%) ^ 2 + 1 * (1 + 10%) = (1 + 10%) ^ 2+ (1 + 10%)

Deposit $ 1 at the end of 3 years

Final value at the end of 4 years = 1 * (1 + 10%) ^ 3 + 1 * (1 + 10%) ^ 2 + 1 * (1 + 10%) = (1 + 10%) ^ 3+ (1+ 10%) ^ 2+ (1 + 10%)

Deposit $ 1 at the end of 4 years

5 years end value = 1 * (1 + 10%) ^ 4 + 1 * (1 + 10%) ^ 3 + 1 * (1 + 10%) ^ 2 + 1 * (1 + 10%) = ( 1 + 10%) ^ 4+ (1 + 10%) ^ 3+ (1 + 10%) ^ 2+ (1 + 10%)

The final value of the one-year annuity deposited at the end of 5 years F = (1 + 10%) ^ 4+ (1 + 10%) ^ 3+ (1 + 10%) ^ 2+ (1 + 10%) + 1

If there are many annuities, calculating the final value using the above method is obviously quite cumbersome. Since the annual payment is equal, and the coefficient for converting the final value is regular, a simple calculation method can be found.

Assuming that the annual payment amount is A, the interest rate is i, and the number of periods is n, the final value of the annuity F calculated based on compound interest is:

F = A + A × (1 + i) ^ 1 + ... + A × (1 + i) ^ (n-1),

Sum formula for proportional series

F = A [1- (1 + i) ^ n] / [1- (1 + i)]

F = A [1- (1 + i) ^ n] / [1-1-i]

F = A [(1 + i) ^ n-1] / i where [(1 + i) ^ n-1] / i is the ordinary annuity final value coefficient, or the postpaid annuity final value coefficient, and the interest rate is i The final value of the annuity after n periods is recorded as (F / A, i, n). You can check the table of the general value of the annuity.

For example: An investor invests 5,000 yuan (A) in the first year and receives a 3% (i) return each year. After that, he invests the sum of these principals and profits together with the 5,000 yuan to be paid each year. Then, after 30 years (n), his total assets will become: F = 5000 × [(1 + 3%) ^ 30-1] / 3% = 237877.08. Among them, the investor invested a total of 5000X30 = 150,000 yuan, and received a total of 87877.08 yuan in interest.