3.1 Rates, Returns, and Time Value of Money
Key Takeaways
- Holding period return combines income and price change over the investor's actual holding period.
- Arithmetic mean answers a one-period expectation question; geometric mean captures realized compound growth and is always lower when returns vary.
- Money-weighted return is the portfolio IRR (driven by cash-flow timing); time-weighted return isolates manager skill.
- Effective annual rate converts any compounding convention to a comparable annual figure, and TVM problems demand consistent periodicity and sign convention.
Rates, Returns, and Time Value of Money
Quantitative Methods is a 6-9% weight on the 2026 CFA Level I exam, meaning roughly 11 to 16 of the 180 multiple-choice items. The exam is computer-based, split into two 90-question sessions totaling 4 hours 30 minutes, and Quant questions are heavily computational. Return measurement is the entry module: a return restates a cash investment as a rate that can be compared across assets, managers, and periods. Before choosing any formula, identify four things: beginning value, ending value, income received, and the timing of external cash flows.
Single-period and average returns
Holding period return (HPR) is the total return over one holding period: HPR = (P1 - P0 + income) / P0. Buy a share for 50, collect a 2 dividend, sell at 55, and HPR = (55 - 50 + 2)/50 = 14%. The arithmetic mean is sum of returns / N and is the best estimate of a single future period's return. The geometric mean is the realized compound rate: [(1+r1)(1+r2)...(1+rN)]^(1/N) - 1. When returns vary at all, the geometric mean is strictly below the arithmetic mean; the gap widens with volatility.
Returns of +50% then -50% average to 0% arithmetically but to (1.5 x 0.5)^(1/2) - 1 = -13.4% geometrically, which matches the actual loss of value.
The harmonic mean appears in cost-averaging: buying a fixed dollar amount each period yields an average purchase price equal to the harmonic mean of prices, which is below both other means. A trimmed mean discards the most extreme observations and a winsorized mean replaces them, both to reduce outlier influence.
Money-weighted vs. time-weighted
Money-weighted return (MWR) is the internal rate of return on the portfolio's actual cash flows; it overweights periods when more money was invested. Time-weighted return (TWR) compounds subperiod returns and removes the distorting effect of client deposits and withdrawals, so the CFA Institute and GIPS standards prefer it for judging manager skill when the client controls cash flows.
Converting rate quotes
A stated (nominal) annual rate is meaningless until you know the compounding frequency m. The effective annual rate (EAR) is EAR = (1 + stated/m)^m - 1. At 12% compounded monthly, the periodic rate is 1% and EAR = 1.01^12 - 1 = 12.68%. At 8% compounded quarterly, EAR = 1.02^4 - 1 = 8.24%. With continuous compounding, EAR = e^(stated) - 1. Always convert competing quotes to EAR before comparing.
Time value of money mechanics
A dollar today can be invested, so it exceeds a dollar later. Future value compounds forward: FV = PV(1+r)^N. Present value discounts back: PV = FV/(1+r)^N. An ordinary annuity pays at period-end; an annuity due pays at period-start and is worth exactly (1+r) times the ordinary annuity (on the BA II Plus, set BGN mode). A perpetuity is a level forever cash flow: PV = PMT/r. Uneven cash flows use the cash-flow worksheet; net present value (NPV) is sum[CF_t/(1+r)^t] - outlay, and the internal rate of return (IRR) is the rate that sets NPV to zero.
| Return or TVM tool | Best use | Common exam trap |
|---|---|---|
| HPR | One actual holding period | Omitting income (dividend/coupon) |
| Arithmetic mean | One-period expectation | Using it for multi-period growth |
| Geometric mean | Realized compound growth | Averaging returns before compounding |
| Money-weighted return | Investor's actual IRR | Forgetting timing dominates result |
| Time-weighted return | Manager performance, GIPS | Failing to break at cash-flow dates |
| EAR | Comparing rate quotes | Ignoring compounding frequency |
Worked annuity example
Suppose you save 5,000 at the end of each year for 20 years earning 6% annually. The future value of an ordinary annuity is FV = PMT x [((1+r)^N - 1)/r] = 5,000 x [(1.06^20 - 1)/0.06] = 5,000 x 36.786 = 183,928. If instead each deposit were made at the start of the year (an annuity due), the entire balance earns one extra year of interest, so the answer is 183,928 x 1.06 = 194,964. To value a stream of unequal cash flows, fall back to the cash-flow worksheet and discount each amount at the periodic rate.
The present value of a 4% perpetuity paying 8 forever is 8/0.04 = 200; a growing perpetuity paying 8 next year and growing 2% is 8/(0.04 - 0.02) = 400.
Calculator discipline and traps
The only calculators allowed are the Texas Instruments BA II Plus (or Professional) and the HP 12C. Outflows and inflows must carry opposite signs or the calculator returns an error or a nonsensical result. Periodicity must be consistent: a 10-year loan with monthly payments has N = 120 and a monthly rate equal to the annual nominal rate divided by 12, not N = 10. Most Level I TVM mistakes are setup errors rather than arithmetic errors, so draw a short timeline, label each cash flow with a sign, convert the quoted rate to the correct periodic rate, and decide beginning-of-period versus end-of-period before pressing CPT.
Clear the time-value worksheet (2nd CLR TVM) between problems to avoid carrying a stale P/Y, I/Y, or PMT into the next question. A final reminder candidates forget under time pressure: the rate I/Y is entered as a whole-number percentage on the BA II Plus, not as a decimal, and N must be expressed in the same units as the periodic rate.
An investor buys a stock for 40, receives a 1.20 dividend, and sells the stock for 43.60 after one year. The holding period return is closest to:
Annual returns of +50% followed by -50% produce a geometric mean return closest to:
A stated annual rate of 8% compounded quarterly has an effective annual rate closest to: