Financial Calculations: Compound Interest, Amortization, and NPV

Compound Interest

Compound interest earns interest on both the principal and previously accumulated interest. The formula A = P(1 + r/n)^(nt) calculates the future value, where P is principal, r is annual rate, n is compounding frequency, and t is time in years. At 7% annual return, $10,000 becomes $19,672 in 10 years with annual compounding and $20,097 with monthly compounding.

The Rule of 72

Divide 72 by the annual interest rate to estimate doubling time. At 8% interest, money doubles in approximately 9 years (72/8). This quick mental math helps evaluate investment opportunities and understand the cost of debt.

Loan Amortization

Monthly payment = P * [r(1+r)^n] / [(1+r)^n - 1], where P is the loan amount, r is the monthly interest rate, and n is the total number of payments. In the early years, most of each payment goes to interest; by the end of the term, most goes to principal. An amortization schedule shows this breakdown for each payment.

Net Present Value (NPV)

NPV calculates the present value of future cash flows minus the initial investment. A positive NPV means the investment creates value at the chosen discount rate. NPV = -Initial_Investment + Sum(Cash_Flow_t / (1+r)^t) for each period t. The discount rate typically reflects the cost of capital or the required rate of return.

Internal Rate of Return (IRR)

IRR is the discount rate that makes NPV equal to zero. It represents the annualized return of an investment. Compare IRR to your hurdle rate (minimum acceptable return): if IRR exceeds the hurdle rate, the investment is potentially worthwhile. Be cautious with IRR for projects with non-conventional cash flows (multiple sign changes), which can produce multiple IRR values.

Practical Applications

Mortgage comparison: Calculate total interest paid over the life of different loan offers, not just monthly payments. Investment evaluation: Use NPV when comparing projects of different sizes and durations. Savings planning: Use compound interest calculations to determine how much to save monthly to reach a retirement goal.