1. Time value of money
One dollar today is worth more than one dollar later. Everything in this section discounts or compounds along that idea, where r is the periodic rate and n the number of periods.
| Quantity | Formula | Reads as |
|---|---|---|
| Future value | FV = PV × (1 + r)ⁿ | Grow PV forward n periods |
| Present value | PV = FV ÷ (1 + r)ⁿ | Discount FV back to today |
| FV — ordinary annuity | A × [((1 + r)ⁿ − 1) ÷ r] | Level payments, end of period |
| PV — ordinary annuity | A × [(1 − (1 + r)⁻ⁿ) ÷ r] | Present worth of a payment stream |
| Annuity due | (ordinary annuity) × (1 + r) | Payments at start of period |
| Perpetuity | PV = A ÷ r | Level payment forever |
Watch the period
Always match r and n to the same period. For monthly cash flows, use the monthly rate and the number of months — not the annual figures.
2. Stated vs effective rates
| Quantity | Formula |
|---|---|
| Effective annual rate (EAR) | EAR = (1 + stated ÷ m)ᵐ − 1 (m = compounding periods/yr) |
| EAR — continuous compounding | EAR = eʳ − 1 |
| Real vs nominal | (1 + nominal) = (1 + real)(1 + inflation) |
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3. Discounted cash flow
| Measure | Definition | Decision rule |
|---|---|---|
| NPV | Σ CFₜ ÷ (1 + r)ᵗ − initial outlay | Accept if NPV > 0 |
| IRR | The rate r that makes NPV = 0 | Accept if IRR > required return |
4. The three return measures
- Holding period return (HPR):
(P₁ − P₀ + income) ÷ P₀. - Time-weighted return (TWR): compounds the HPR of each sub-period —
[(1+R₁)(1+R₂)…(1+Rₙ)] − 1. Removes the effect of cash flow timing, so it measures the manager's performance. - Money-weighted return (MWR): the IRR of all the investor's cash flows. Reflects the size and timing of deposits/withdrawals, so it measures the investor's experience.
- Geometric mean return:
[Π(1 + Rᵢ)]^(1/n) − 1— the correct way to average returns over time. Always ≤ the arithmetic mean.
5. Dispersion & risk-adjusted return
| Statistic | Formula | Tells you |
|---|---|---|
| Variance (σ²) | Σ(xᵢ − x̄)² ÷ N | Spread of outcomes |
| Standard deviation (σ) | √variance | Spread, in original units |
| Coefficient of variation | σ ÷ mean | Risk per unit of return — lower is better |
| Sharpe ratio | (Rₚ − R_f) ÷ σₚ | Excess return per unit of total risk |
Sample variance divides by N − 1 (not N) — a frequent exam trap.
Common mistakes
- Mismatching rate and period in any TVM problem — the single biggest source of wrong answers.
- Averaging returns arithmetically across time instead of geometrically.
- Confusing TWR and MWR — TWR judges the manager, MWR judges the investor's timing.
- Using N instead of N−1 for a sample standard deviation.
Sources
- CFA Institute — cfainstitute.org.
- CFA Program Curriculum, Quantitative Methods (Level I).
Study summary only. "CFA" and "Chartered Financial Analyst" are trademarks of CFA Institute, which does not endorse this material.