# How does the time value of money work?

Learn from Mathematical Finance

The time value of money (TVM) is a fundamental financial concept that highlights the importance of time in the valuation of money. It asserts that a dollar today is worth more than a dollar in the future due to its potential earning capacity. This principle is crucial for making informed financial decisions, whether you're investing, borrowing, or saving.

Key Concepts of Time Value of Money

1. Present Value (PV):

- Definition: Present value represents the current worth of a sum of money that will be received or paid in the future, discounted at a specific interest rate.

- Formula: PV = FV / (1 + r)^n

- Where:

- PV = Present Value

- FV = Future Value

- r = Interest Rate

- n = Number of Periods

2. Future Value (FV):

- Definition: Future value refers to the amount of money an investment will grow to over time at a given interest rate.

- Formula: FV = PV × (1 + r)^n

- Where:

- FV = Future Value

- PV = Present Value

- r = Interest Rate

- n = Number of Periods

3. Interest Rate:

- Definition: The interest rate is the percentage at which money grows over a specified period. It can be expressed as an annual percentage rate (APR) or an effective annual rate (EAR).

4. Compounding:

- Definition: Compounding is the process of earning interest on both the initial principal and the accumulated interest from previous periods. This leads to exponential growth of the investment.

5. Discounting:

- Definition: Discounting is the reverse process of compounding, used to determine the present value of a future sum of money. It involves applying a discount rate to account for the time value of money.

6. Annuities:

- Definition: An annuity is a series of equal payments made at regular intervals. The value of an annuity can be calculated in terms of its present value or future value.

- Types: Ordinary annuities (payments at the end of each period) and annuities due (payments at the beginning of each period).

7. Loan Amortization:

- Definition: Loan amortization involves spreading loan payments over time, with each payment covering both interest and principal. Understanding TVM helps in calculating monthly payments and the total interest paid over the life of the loan.

Practical Applications of TVM

- Investment Decisions: Evaluating potential investments by comparing their present values to determine which offers the highest return.

- Retirement Planning: Calculating how much to save now to achieve a desired amount in retirement.

- Loan Analysis: Determining the cost of borrowing and planning repayment schedules based on the time value of money.

Understanding TVM is essential for effective financial planning and decision-making, as it helps in evaluating the worth of future cash flows in today's terms. By applying TVM principles, individuals and businesses can make more informed choices regarding investments, savings, and financing.