Basic, lesson 1 of 3
Measuring return
The average of yearly returns is not what an investor earns. Compounding, volatility, and cash flows in and out all change the number, and interviewers test whether you know which one to quote.
9 min, 3 checks
1Arithmetic and geometric
The arithmetic mean adds the yearly returns and divides by the number of years. The geometric mean asks what single yearly return, compounded, gets you from the starting value to the ending value. The geometric one is what an investor actually earned.
Geometric return = (Product of (1 + r_t))^(1/n) - 1
Worked example
Up 50%, down 50%
- 1Arithmetic mean: (50 - 50) / 2 = 0%.
- 2Wealth: $100 x 1.5 x 0.5 = $75.
- 3Geometric mean: 0.75^(1/2) - 1 = -13.4% a year.
The average says flat. The investor lost a quarter of the money.
Check 1
A fund returns +20%, -10%, and +5% in three years. What is its geometric annual return, in percent, to one decimal?
2Volatility drag
The gap between the two means grows with volatility. A handy approximation links them: the compound return is roughly the average return minus half the variance. Two funds with the same average return do not leave the client equally rich; the calmer one compounds faster.
Geometric return ~ Arithmetic return - Volatility^2 / 2
Use decimals: 20% volatility squared is 0.04, half of it is 0.02, or 2 points.
| Volatility | Drag (s^2 / 2) | Approximate compound return |
|---|---|---|
| 10% | 0.5 points | 7.5% |
| 20% | 2.0 points | 6.0% |
| 30% | 4.5 points | 3.5% |
Check 2
An asset has an average annual return of 10% and volatility of 30%. Using the approximation, what is its compound return, in percent?
3Time weighted and money weighted
A manager does not choose when clients add or withdraw money, so funds report a time weighted return: chain each period's return and ignore the size of the pot. A money weighted return is the IRR of the client's own cash flows, so it rewards or punishes the timing of contributions.
Worked example
The client added at the top
- 1Year 1: $100 grows 20% to $120. The client adds $100, so $220 is invested.
- 2Year 2: the fund falls 10%, to $198.
- 3Time weighted: (1.20 x 0.90)^(1/2) - 1 = 3.9% a year.
- 4Money weighted: the IRR of -100 now, -100 in a year, +198 in two years is about -0.7% a year.
The fund did fine. The client's timing turned it into a small loss.
Check 3
Which return should you use to judge the portfolio manager's skill?
Terms in this lesson: Standard deviation.
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