In the next few minutes:
We can measure returns on three overlapping groups:
An asset earns money in two ways: its price changes, and it may pay cash income.
The sum is the total return, also called the holding period return.
Stock S starts the year at 50, pays a dividend of 1 and ends at 54.
We'll follow stock S through the rest of this reading.
The second split matters for tax: price gains are usually taxed only when realized by a sale.
Check yourself
Stock S returned 10% in year 1, −20% in year 2 and 25% in year 3. What was its average annual return?
The arithmetic mean return is the simple average:
For stock S: (10% − 20% + 25%) ÷ 3 = 5%.
But 100 invested at 5% a year would grow to 115.76. Stock S's path, 100 → 110 → 88 → 110, ended at 110. The arithmetic mean overstates what happened.
The geometric mean return (compound average return) is the one constant rate that reproduces the actual ending value:
For stock S: 1.10 × 0.80 × 1.25 = 1.10, and = 3.23%.
Values per 100 invested, rounded to 2 decimals. Only the geometric rate lands on the actual 110.
The arithmetic-geometric mean inequality: the geometric mean is always less than or equal to the arithmetic mean.
Why the arithmetic mean misleads: it assumes the amount invested stays the same each period. After stock S fell 20%, its 25% gain applied to a smaller base, so it recovered only to 110.
Check yourself
Returns are usually quoted per year. If a period's return repeats c times a year:
Stock S gained 2% last quarter. Annualized: = 8.24%, a bit more than 4 × 2% = 8% because each quarter compounds on the last. Try other quarterly returns:
When you know the holding period in days, c = 365 ÷ days:
Annualizing assumes the return repeats all year. It rarely does, so treat it as a yardstick, not a forecast.
Compounding more and more often has a limit: continuous compounding. The matching return is the continuously compounded (logarithmic) return:
Its key property: continuously compounded returns add up across periods.
9.53% − 22.31% + 22.31% = 9.53% = ln(1.10).
Check yourself
An interest rate can be read as:
For a risky asset, the required rate of return is a risk-free rate plus a risk premium.
The real risk-free rate has no expected inflation and no uncertainty. It reflects pure time preference. The nominal risk-free rate adds an inflation premium, which is why T-bill rates are nominal, not real.
Real rate 2%, expected inflation 3%: nominal = 1.02 × 1.03 − 1 = 5.06%. The common approximation, nominal ≈ real + inflation, gives 5%.
Each added risk raises the required return. Examples:
Required return on a bond, built up (approximate, additive)
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A real return adjusts a nominal return for inflation. It measures the gain in purchasing power. Stock S's 10% with 2% inflation:
An excess return is how much more (or less) an investment earns than a benchmark. Against a 4% risk-free rate:
Stock S, year 1: 10% total, 0.3% transaction costs and 1.2% fees give a 9.7% gross and 8.5% net return.
After-tax nominal return taxes each piece at its own rate. With 20% tax on capital gains and 30% on distributions:
A leveraged return is the gain or loss as a percentage of the investor's own cash. Invest equity plus borrowed at rate , earning on the whole portfolio:
Equity 100, borrow 50 at 6%. Move the portfolio return:
At 10% the leveraged return is 12%. At −20% (stock S's year 2) it is −33%.
Check yourself
| Measure | Rule |
|---|---|
| Total return | Price + capital distribution return |
| Geometric mean | ≤ arithmetic; matches actual growth |
| Annualized | , c = 365 ÷ days |
| Continuously compounded | ln(1 + R); additive |
| Nominal rate | (1 + real)(1 + π) − 1 |
| Required return | Risk-free rate + risk premia |
| Leveraged |
That's the reading.
Now test it on exam-style questions while it's fresh.