Present / Future Value

Present value, future value, and the time value of money

Two formulas tie a cash flow today to one in the future: FV = PV · (1 + i)^n and PV = FV / (1 + i)^n, where i is the interest (or discount) rate per period and n is the number of periods. Behind them sits the principle of the time value of money — R$ 100 today is worth more than R$ 100 tomorrow because the cash on hand can be invested and earn interest. The discount rate captures both the opportunity cost and the risk of the future flow.

A concrete example: at 1% per month, R$ 1,000 today grows to R$ 1,616.07 in 5 years (60 months) — that is the future value. Conversely, R$ 1,616.07 due in 5 years is worth exactly R$ 1,000 today at the same discount rate — that is the present value. With multiple cash flows, the present value generalizes to PV = Σ CF_t / (1 + i)^t, which is the basis of NPV (net present value). Fixed annuities have a closed-form factor for PMT-style flows; the Gordon perpetuity formula PV = D / (r − g) values a perpetual stream growing at rate g.

Where PV and FV are used

Project evaluation (NPV, IRR), corporate valuation (DCF), the classic buy-now-vs-pay-in-installments decision, retirement planning, bond pricing (a bond's price is the PV of its coupons plus principal), pension actuarial calculations, MBA and CFA exam questions on time value of money.

FAQ

Which discount rate should I use? For projects, the cost of capital (WACC) or the required return that reflects the project's risk. For personal decisions, your investment opportunity rate (Tesouro Selic, CDB) is a reasonable floor. The higher the rate, the more aggressively future flows are discounted.

How do I evaluate cash vs installments? Bring all installments to present value at your investment rate and compare with the cash price. If cash < PV of installments, paying upfront is better; if cash > PV, financing wins (assuming you actually invest the difference).

What about inflation? If flows are nominal, the rate must be nominal too; if they are real, the rate must be real. Mixing the two distorts the result. Use the Fisher equation (1 + nominal) = (1 + real) · (1 + inflation) to convert.

Why is the rate squared, cubed, etc.? Because each period interest is applied on the previous accumulated balance — it is the same compound logic. PV and FV are the two sides of the same compounding coin: FV looks forward, PV looks backward.