A dollar today is worth more than a dollar next year — because today's dollar can be invested, and because next year's dollar is uncertain. The time value of money is the arithmetic that makes every capital decision comparable across time. This is the single most important quantitative idea in the school.
- 01State the formulas for future value and present value.
- 02Compute both by hand for common household cases.
- 03Explain discounting and compounding as inverses of one another.
- 04Use the arithmetic to compare two cash flows that occur at different times.
Future value
Future value asks: if I invest P today at rate r, what will it be worth in n years? The answer, with annual compounding, is F = P × (1 + r)^n. The exponent is where the magic sits. At 7% real returns, money doubles roughly every ten years — the Rule of 72 (72 / 7 ≈ 10). Compounded over a working life, that turns a modest saver into a wealthy retiree, and it is why the discipline of Course 102 matters more than the tactics of Course 202.
Present value
Present value is the inverse operation. Given a promised cash flow F in n years, at a discount rate r, its present value is P = F / (1 + r)^n. Every capital decision in the rest of the Academy — buy a house, take a job, invest in a business — reduces to a comparison of present values. Learn to do it in your head for round numbers.
Discounting
Discounting is the act of translating a future dollar into today's dollars. The rate you discount at expresses two things: your alternative use of money (the opportunity cost) and your uncertainty about the promise being kept (the risk premium). A higher discount rate makes future dollars worth less today; a lower discount rate makes them worth more.
Compounding
Compounding is discounting in reverse. Frequency matters: $10,000 at 7% for one year is $10,700 compounded once, $10,722.90 compounded monthly, and $10,725.08 compounded continuously. The differences are small at one year, meaningful at ten, and dominant at fifty.
Build a small time-value calculator. Given a present value P, an annual rate r, and a number of years n, it should produce the future value F = P × (1 + r)^n. Given a future value F, r, and n, it should produce the present value P = F / (1 + r)^n. Compare compounding annually, monthly, and continuously.
Parameters
- — P (present value, dollars)
- — F (future value, dollars)
- — r (annual rate, decimal — e.g. 0.07 for 7%)
- — n (years, integer or fractional)
- — k (compounding periods per year — 1, 12, or ∞)
Expected
For P = $10,000, r = 7%, n = 30, annual compounding: F ≈ $76,123. Present value of $76,123 in 30 years at 7% ≈ $10,000. The two operations are strict inverses.
- F = P × (1 + r)^n. P = F / (1 + r)^n.
- Compounding is exponential. Small rates over long horizons produce large sums.
- The Rule of 72 gives the doubling time: years ≈ 72 / r%.
- Every capital decision reduces to a comparison of present values.