Present Value and Future Value

  • PV is the value today; FV is the value at a future time.
  • Core relation (compound growth): FV = PV \cdot (1 + i)^n
  • With periodic rate: i = \frac{r}{m},\quad n = t \cdot m
  • Present value (discounting): PV = \frac{FV}{(1 + i)^n}
  • Default assumption: annual compounding unless stated (i.e., m = 1,\; i = r,\; n = t)

Simple vs Compound Interest; Growth Over Time

  • Simple interest: FV_{simple} = PV \cdot (1 + r t)
  • Compound interest: FV_{compound} = PV \cdot (1 + i)^n
  • Key idea: compound interest earns interest on previously earned interest; the effect grows with time
  • Example intuition: over long horizons, compound growth far exceeds simple interest

Compounding Frequency and Time Horizon

  • More frequent compounding (higher m) increases FV for the same nominal rate
  • Monthly compounding: i = \frac{r}{12},\quad n = 12t
  • If nothing is said, assume yearly compounding; adjust m and i when given a different frequency

Present Value, Discounting, and Applications

  • PV of a future amount: PV = \frac{FV}{(1 + i)^n}
  • Discounting = removing the effect of interest to find what a future amount is worth today
  • Common uses: bonds, stocks, and evaluating projects by the value of future cash flows today
  • When we say “value” in finance, we usually mean present value

Four Known Variables and Quick Takeaways

  • If you know PV, i, n, you can find FV with FV = PV(1+i)^n
  • If you know FV, i, n, you can find PV with PV = \frac{FV}{(1+i)^n}
  • If there are payments (an annuity), formulas incorporate Pmt (not shown here); timing (beginning vs end of period) matters (annuity due vs ordinary annuity)
  • Default is annual compounding unless specified; for other frequencies, convert to i and n accordingly

Illustrative Example (conceptual)

  • A present value of $10 invested at 5.5% for 200 years grows to about FV \approx 10 \cdot (1 + 0.055)^{200} \approx 4.47 \times 10^{5}
  • This demonstrates how compounding dramatically affects long horizons
  • To find present value of a future target, use PV = \frac{FV}{(1+i)^n} with the appropriate i and n