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 return is useful for one-period expectation, while geometric mean return captures compound growth over multiple periods.
- Effective annual rates convert different compounding conventions into comparable annual returns.
- Time value of money problems require consistent cash-flow timing, sign convention, and compounding frequency.
Rates, Returns, and Time Value of Money
Quantitative Methods is a 6-9% Level I topic, and return measurement is the entry point. A return translates a cash investment into a rate that can be compared across assets, managers, and periods. Always identify the beginning value, ending value, income received, and timing of external cash flows before choosing a formula.
Holding period return is the total return earned over one holding period. The core formula is HPR = (ending value - beginning value + income) / beginning value. If a share is bought for 50, pays a 2 dividend, and is sold for 55, HPR is (55 - 50 + 2) / 50 = 14%.
Arithmetic mean return is the simple average: sum of periodic returns / N. It is best for estimating a single-period expected return when each period is equally likely. Geometric mean return is the compound rate: [(1+r1)(1+r2)...(1+rN)]^(1/N) - 1. It is lower than the arithmetic mean when returns vary.
Money-weighted return is the internal rate of return on a portfolio's actual cash flows. It gives more weight to periods when more money was invested. Time-weighted return removes the effect of external deposits and withdrawals by compounding subperiod returns. It is better for evaluating manager skill when clients control cash flows.
Interest-rate quotes also require precision. A stated annual rate is incomplete until the compounding frequency is known. If the stated rate is 12% compounded monthly, the monthly rate is 1%, and the effective annual rate is (1.01)^12 - 1 = 12.68%. The formula is EAR = (1 + stated rate / m)^m - 1.
Time value of money rests on the idea that a dollar today can be invested, so it is worth more than a dollar later. Future value compounds present money forward: FV = PV(1+r)^N. Present value discounts future money back: PV = FV / (1+r)^N. Use the same periodic rate and number of periods.
An annuity is a series of equal cash flows. For an ordinary annuity, payments occur at the end of each period. For an annuity due, payments occur at the beginning of each period, so the value is one period of interest higher than the comparable ordinary annuity. A perpetuity is a level cash flow that continues forever: PV = PMT / r.
Uneven cash-flow problems use the cash-flow keys or a timeline. Net present value is the sum of discounted cash flows minus the initial outlay: NPV = sum[CF_t / (1+r)^t] - initial investment. The internal rate of return is the discount rate that sets NPV to zero.
| Return or TVM tool | Best use | Exam trap |
|---|---|---|
| HPR | One actual holding period | Leaving out income |
| Arithmetic mean | One-period expectation | Using it for compound growth |
| Geometric mean | Multi-period growth | Averaging returns first |
| Money-weighted return | Investor's actual IRR | Cash-flow timing dominates |
| Time-weighted return | Manager performance | Missing subperiod breaks |
| EAR | Comparing rate quotes | Ignoring compounding frequency |
Calculator discipline matters. Cash outflows and inflows need opposite signs. The number of periods must match the periodic rate. A 10-year loan with monthly payments has N = 120, not 10, and the monthly rate is the annual nominal rate divided by 12 when monthly compounding is stated.
For Level I, most TVM errors are setup errors. Draw a short timeline, label cash flows, convert the quoted rate, and decide whether cash flows are beginning-of-period or end-of-period. The formula is usually simple once timing is clean.
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:
For evaluating an investment manager when clients control external deposits and withdrawals, the most appropriate return measure is:
A stated annual rate of 8% compounded quarterly has an effective annual rate closest to: