Why premia may persist or vanish: risk-based and behavioural explanations and tracking-error regret
Distinguish risk-based from behavioural explanations of factor premia, calculate how often a tilt underperforms over different horizons, and understand why regret drives investors out.
What you will learn
- The two families of explanation for factor premia and what each implies for the future.
- How to estimate the probability that a tilt underperforms the market over 1, 5, 10 and 20 years from a premium and a tracking error.
- What tracking-error regret is and why it is the main practical risk of factor investing.
- A test you can apply to any proposed factor before accepting it.
Explanation
Risk-based explanations
Under a risk-based explanation the premium is compensation for bearing a risk others want to avoid. Value stocks tend to have low profitability, high leverage or distressed operations whose earnings fall most in recessions; small caps are illiquid and fail more often. If this is right, the premium should persist because it is a price for pain, and it can be large and negative exactly when you need money most.
Behavioural explanations
Under a behavioural explanation the premium comes from systematic mistakes of other investors (over-extrapolation, under-reaction to news, lottery preferences as in lesson 6) combined with limits to arbitrage: short-selling constraints, career risk of fund managers, investors who withdraw after bad years. Then the premium persists only as long as the mistakes and the limits persist; publication and cheap ETFs weaken the limits, one reading of the decay evidence in lesson 10. Many researchers hold a mixed view: value and size partly risk, momentum and low volatility mostly behavioural with institutional frictions.
A test for any new factor
Before accepting a factor, ask four questions: is it persistent across decades, pervasive across countries and asset classes, robust to small changes in definition, and is there an economic reason, risk-based or behavioural, why it should continue? A pattern that fails two of these is probably a data-mining artefact. Of hundreds of published predictors only a handful pass; those are in lessons 2–6.
Tracking-error regret
Tracking-error regret is the discomfort of lagging the index everyone around you owns, even when your absolute return is fine. It may be why premia survive: most investors cannot sit through five or ten years of lagging, sell at the bottom and hand the premium to those who stay. If a tilt has expected excess return μ and tracking error σ per year, and years are roughly independent, the probability of trailing the market over N years is approximately Φ(−μ√N / σ), where Φ is the standard normal distribution function.
Worked example
Hypothetical example: a diversified multi-factor tilt with an assumed net premium of 2 % a year and a tracking error of 5 % a year against a world market-cap ETF. Returns are assumed normal and independent across years; this is a simplification that understates the length of real drawdowns. Fees are already deducted in the 2 %; taxes are ignored.
Step 1 – the ratio μ / σ = 2 / 5 = 0.4 (this is the information ratio from lesson 8).
Step 2 – for each horizon compute z = 0.4 × √N and look up Φ(−z):
| Horizon | z = 0.4 × √N | Probability of trailing the market |
|---|---|---|
| 1 year | 0.40 | 34 % |
| 3 years | 0.69 | 24 % |
| 5 years | 0.89 | 19 % |
| 10 years | 1.26 | 10 % |
| 20 years | 1.79 | 4 % |
Step 3 – size of a bad outcome: a two-standard-deviation shortfall over 5 years is 2 × 5 % × √5 = 22 percentage points below the expected 10 % cumulative gain, about 12 points behind the market; on a €100,000 tilt roughly €12,000 less than the index fund.
Step 4 – sensitivity: with a net premium of only 1 % (the pessimistic case of lesson 10), the probability of trailing after 10 years rises to 26 % and after 20 years to 19 %.
Reading: one in ten investors who do everything right will still be behind the market after a decade. If you cannot accept that in advance, a factor tilt is not for you, and that honest answer is a valid outcome of this course.
Common mistake
Practice activity
On paper: a friend proposes a tilt with an expected net premium of 1.5 % and a tracking error of 3 %. (a) Compute the information ratio. (b) Estimate the probability of trailing the market over 5 and 10 years, using Φ(−1.12) ≈ 0.13 and Φ(−1.58) ≈ 0.06. (c) Your friend says the premium is behavioural and the strategy is now available in a 0.10 % fee ETF with large inflows. Which of the four tests should worry you most and why?
Show solution
(a) IR = 1.5 / 3 = 0.5. (b) 5 years: z = 0.5 × √5 = 1.12, about 13 %. 10 years: z = 0.5 × √10 = 1.58, about 6 %. (c) The economic-reason test. A purely behavioural premium depends on limits to arbitrage; a cheap, popular ETF removes them, so the premium can shrink despite a persistent past. Plan with a lower premium, which raises every probability in (b).
Limitations: the normal, independent-years model ignores fat tails and the tendency of factor returns to cluster in multi-year regimes, so real worst cases are longer and deeper.
Key takeaways
- Risk-based premia should persist but hurt at bad times; behavioural premia depend on limits to arbitrage and can fade with popularity.
- Test a factor for persistence, pervasiveness, robustness and an economic reason before using it.
- Probability of trailing the market ≈ Φ(−IR × √N); with IR 0.4 about 10 % of disciplined investors are still behind after 10 years.
- Tracking-error regret, not volatility, is the risk that ends most factor plans; decide your tolerance before you start.
Next step
In the final lesson you write a factor-tilt policy that fixes the allocation, the underperformance you will tolerate in years and euros, and the rules for reviewing it.
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