The core idea of finance isn't about picking winning stocks; it's about a single, powerful principle: a dollar today is worth more than a dollar tomorrow. This isn't just because of inflation. It's because a dollar today can be invested to earn a return, becoming more than a dollar in the future. This is the Time Value of Money (TVM), and mastering its mechanics is the key to valuing almost any financial asset, from a simple bond to a complex corporation.

We move value forward in time through compounding and bring it back to the present through discounting. Let's look at the formulas. The future value (FV) of a single sum is:

To find out what a future cash flow is worth today, we just rearrange the formula to solve for the present value (PV). This process is called discounting.

These formulas work for a single lump sum. But finance usually deals with streams of cash flows. For example, a company might expect profits of $100,000 this year, $120,000 next year, and $110,000 the year after. To find the total present value, you'd discount each of these uneven cash flows back to today and add them up.

When a stream of cash flows is constant and occurs at regular intervals, we have special cases called annuities and perpetuities. An annuity is a series of equal payments for a fixed number of periods. Think of a car loan or a mortgage payment.

A perpetuity is a series of equal payments that continues forever. While seemingly theoretical, it's a critical tool for valuing assets with long or indefinite lifespans, like the stock of a stable company.

The formula for a perpetuity is much simpler. Since the payments never end, we just divide the cash payment by the interest rate.

Interest rates are usually quoted as a nominal annual rate. However, the real earning power of your money depends on how often interest is compounded. A 12% annual rate compounded monthly is better than the same rate compounded annually.

To compare different compounding frequencies, we calculate the Effective Annual Rate (EAR).

What happens if we compound more and more frequently? What if we compounded every second, or every millisecond? We approach a theoretical limit known as continuous compounding an important concept in financial modeling, especially for derivatives pricing.

These TVM concepts come together in loan amortization. When you take out a loan, each payment you make is split between interest and principal. Early on, most of your payment goes toward interest. Over time, more and more of it goes toward paying down the principal balance.

An amortization schedule is a table that details this breakdown for every payment over the life of the loan. It's built using the annuity formula to first calculate the fixed payment amount, and then tracks the balance payment by payment.

Above is a sample for a $100,000 loan over 15 years (180 months) at a 6% annual rate (0.5% per month). Notice how the interest portion of the payment shrinks while the principal portion grows with each payment.

Time to review these core financial mechanics.

Let's test your understanding of these calculations.

Understanding these formulas is the first step. The real skill is applying them to untangle complex financial situations and make informed decisions.