Financial Mathematics

Compound interest grows faster because interest is earned not only on the original principal but also on previously accumulated interest - this creates exponential rather than linear growth. The practical implication is enormous for long-term savings: starting to save early, even with small amounts, can result in dramatically larger balances than starting later with larger amounts. For example, someone who invests $200/month starting at age 25 will likely have more at retirement than someone who invests $400/month starting at age 35, despite contributing less total money. This is why financial advisors emphasize "time in the market" and early saving habits.

APR (Annual Percentage Rate) is the stated interest rate without accounting for compounding, while APY (Annual Percentage Yield) shows the effective annual rate including compounding effects. The difference matters in several situations: (1) When comparing savings accounts, a higher APY means more earnings - two accounts with the same APR but different compounding frequencies will have different APYs. (2) When borrowing, lenders often advertise APR while APY shows the true cost - a credit card with 18% APR compounded daily has an APY of about 19.7%. (3) For informed decision-making, consumers should compare APYs for savings and understand that advertised APRs on loans understate the actual interest cost when compounding occurs more than annually.

In an amortized loan, each payment covers both interest and principal, but the proportion changes over time. Interest is calculated on the remaining balance, so when the balance is high (early years), the interest portion is large. As you pay down the principal, less interest accrues each month, so more of your fixed payment goes toward principal. For example, on a $200,000 mortgage at 6%, your first payment might be $1,199, with $1,000 going to interest and only $199 to principal. Years later, the same $1,199 payment might have only $200 going to interest and $999 to principal. This is why making extra principal payments early in a loan can save substantial interest over time - it reduces the balance that interest is calculated on for all remaining payments.

Present value allows us to compare money received or paid at different times by expressing all values in today's dollars, accounting for the time value of money (the fact that money available now is worth more than the same amount later due to earning potential). Examples of useful applications: (1) Lottery winnings - choosing between a $1 million lump sum today or $50,000 per year for 25 years requires calculating the present value of the annuity to make a fair comparison. (2) Business investments - evaluating whether to spend $100,000 now on equipment that will generate $30,000 per year in savings for 5 years. (3) Retirement planning - determining how much a pension promising $40,000/year is worth compared to a lump sum offer. (4) Legal settlements - calculating fair compensation for future lost wages in today's dollars. Present value is essential for any decision involving money flows at different times.