Compound Interest and Annuity Chapter Notes | Applied Mathematics for Class 11 - Commerce PDF Download

Compound Interest

Definition: Compound interest refers to the interest calculated on both the initial principal and the accumulated interest from previous periods. It is a crucial concept in finance and investments, determining the growth of savings or the cost of borrowing over time.

Formula:

A = P × (1 + r/n)^(nt)
where:

• A is the future value of the investment/loan, including interest.
• P is the principal investment amount (the initial amount of money).
• r is the annual interest rate (in decimal).
• n is the number of times that interest is compounded per unit
• t is the time the money is invested for, in years.

Compound Interest with Different Compounding Periods

If interest is compounded more frequently than annually, you can use the formula:
A = P × (1 + r/n)^(nt)
Where n is the number of times interest is compounded per unit t.

Continuous Compounding

In the case of continuous compounding, the formula becomes:
A = P × e^(rt)
where
e is Euler's number (approximately 2.71828).

Annuities

Definition: An annuity is a financial product that provides a series of payments made at equal intervals. These payments can be received or paid out over a specified period, often used for retirement income planning or other long-term financial goals.

Types of Annuities

• Ordinary Annuity: Payments are made at the end of each period.
• Annuity Due: Payments are made at the beginning of each period.

Formula for the Future Value of an Ordinary Annuity

FV = P × ((1 + r)^n - 1)/r
where:

• FV is the future value of the annuity.
• P is the regular payment (annuity payment).
• r is the interest rate per period.
• n is the number of periods.

Formula for the Future Value of an Annuity Due

FV = P × ((1 + r)^n - 1)/r × (1 + r)
This formula includes an additional (1 + r) factor to account for the payments occurring at the beginning of each period.

Example: If you deposit $500 at the end of each year into an account that earns 6% interest annually, compounded annually, what will be the value of the annuity after 5 years?
Ans: Using the future value of an ordinary annuity formula:
FV = 500 × ((1 + 0.06)^5 - 1)/0.06
FV ≈ 500 × (1.338225 - 1)/0.06
FV ≈ 500 × 0.338225/0.06
FV ≈ 500 × 5.63708
FV ≈ 2818.54