Concept: an active fund doesn't need to be bad to lose to an index fund. It only needs to charge more than the skill it delivers. This test shows the hurdle an active fund must clear, the odds of clearing it over your holding period, and the dollar cost of being wrong.
What This Resource Does
This test converts an active fund's extra fees into three numbers:
- The hurdle: the extra gross return the manager must earn every year just to tie the index fund.
- The odds: the probability the fund beats its benchmark over your holding period, for a manager with no skill and for one with real skill.
- The break-even skill rate: how likely it must be that your chosen manager is skilled before paying the higher fee makes sense.
It also adjusts published "percentage of funds that beat the index" figures for survivorship bias.
Inputs You Need
| Input | Where to find it |
|---|---|
| Active fund all-in annual cost (expense ratio, platform or advice fee, trading costs) | Fund fact sheet, broker fee schedule |
| Index fund all-in annual cost | Use the Tracking Difference True-Cost Estimator |
| Benchmark gross return assumption (r) | Your own conservative assumption |
| Tracking error of the active fund | Standard deviation of monthly (fund − benchmark) returns × √12, from provider data |
| Holding period (n years) | Your plan |
| Amount invested | Your plan |
| Number of funds at the start and end of a study period, and how many beat the benchmark | Scorecard report, with its methodology notes |
The Formulas
Step 1: The hurdle
Hurdle = Active all-in cost − Index all-in cost
This is the gross alpha (return above the benchmark before costs) the manager must produce every year to tie.
Step 2: Net excess return
Net excess return = Gross alpha − Hurdle
A manager with no skill has gross alpha of zero, so their net excess return equals the negative of the hurdle.
Step 3: Probability of beating the benchmark
Probability ≈ Φ( Net excess return × √n ÷ Tracking error )
Φ is the standard normal cumulative probability, available in any spreadsheet as NORM.S.DIST(z, TRUE). It assumes annual excess returns are independent and roughly normal, so treat the result as an estimate.
Step 4: Dollar cost of a no-skill active fund
Cost = Amount × [(1 + r − Index cost)ⁿ − (1 + r − Active cost)ⁿ]
Step 5: Break-even probability of skill
Break-even probability of skill = Hurdle ÷ Gross alpha if skilled
This assumes a fund is either unskilled (alpha of zero) or skilled (the alpha above). If you believe the chance your pick is skilled is below this figure, the index fund is the better expected choice.
Step 6: Survivorship-adjusted beat rate
Reported beat rate = Funds beating the benchmark ÷ Funds surviving
Adjusted beat rate = Funds beating the benchmark ÷ Funds at the start
Funds that closed or merged after poor results disappear from the survivors, which flatters the reported figure.
Step 7: Required skill for a target probability
Gross alpha needed for a target probability = Hurdle + z × Tracking error ÷ √n
Here z is the standard normal value for your target (z = 0.84 for 80%, z = 0 for 50%).
Worked Example
Illustrative figures.
| Assumption | Value |
|---|---|
| Active fund all-in cost | 0.85% |
| Index fund all-in cost | 0.05% |
| Gross benchmark return (r) | 7% |
| Tracking error | 4% |
| Amount invested | $50,000 |
Step 1: The Hurdle
Hurdle = 0.85% − 0.05% = 0.80% a year
Steps 2 and 3: Odds for a Manager With No Skill
Net excess return is −0.80%, so the test statistic is −0.80% ÷ 4% × √n = −0.20 × √n.
| Holding period | Working | Probability of beating the index |
|---|---|---|
| 1 year | Φ(−0.200) | 42% |
| 5 years | Φ(−0.447) | 33% |
| 10 years | Φ(−0.632) | 26% |
| 20 years | Φ(−0.894) | 19% |
| 30 years | Φ(−1.095) | 14% |
The longer you hold, the less likely the fund is to win, because the fee gap accumulates while luck averages out.
Odds for a Manager With Real Skill
Assume the manager has a true gross alpha of +1.0%, so net excess return is +0.20% and the statistic is +0.05 × √n.
| Holding period | Working | Probability of beating the index |
|---|---|---|
| 1 year | Φ(0.050) | 52% |
| 10 years | Φ(0.158) | 56% |
| 30 years | Φ(0.274) | 61% |
Even a genuinely skilled manager beats the index only a little more than half the time, because the net edge (0.20%) is small against 4% of yearly noise.
Step 4: Dollar Cost of a No-Skill Active Fund (20 Years)
| Fund | Net return | Growth factor | Ending value |
|---|---|---|---|
| Index fund | 6.95% | 1.0695²⁰ = 3.834 | $191,700 |
| Active fund, no skill | 6.15% | 1.0615²⁰ = 3.299 | $164,950 |
| Cost of choosing active | about $26,750 |
Step 5: Break-Even Probability of Skill
Suppose a skilled manager delivers +1.5% gross alpha.
Break-even probability of skill = 0.80% ÷ 1.5% = 53%
You would need to be more than 53% sure your chosen manager is skilled. If you believe only 5% of managers have that skill, the expected gross alpha is 5% × 1.5% = 0.075%. Against a 0.80% hurdle, the expected net result is about −0.725% a year.
Step 6: Survivorship Adjustment
A scorecard covers a 10-year period.
| Item | Count |
|---|---|
| Funds at the start | 100 |
| Funds surviving to the end | 70 |
| Survivors that beat the benchmark | 35 |
Reported beat rate = 35 ÷ 70 = 50%
Adjusted beat rate = 35 ÷ 100 = 35%
The 30 closed funds are missing from the reported figure. Counting them, only about a third of the original funds beat the benchmark.
Step 7: How Much Skill Would Make Active a Good Bet?
You hold for 10 years and want an 80% chance of beating the index.
Gross alpha needed = 0.80% + 0.84 × 4% ÷ √10 = 0.80% + 1.06% = about 1.86% a year
That means sustained outperformance of nearly 2 percentage points a year before fees, for a decade.
Verdict
- Over 20 years, a no-skill active fund has about a 19% chance of beating the index and costs about $26,750 on a $50,000 stake.
- To justify the fee, you would need more than a 53% chance of having picked a skilled manager, in a field where skill is rare and hard to spot beforehand.
- Skill, if it exists, shows up in a small edge. Fees are certain.
How to Read It
- The fee gap is the starting handicap. Before costs, the average active dollar earns the average market return. After costs, active investors as a group must earn less than index investors by the cost difference.
- Odds fall with time. Short horizons are closer to a coin flip. Over decades, the fee gap dominates.
- Tracking error cuts both ways. A "closet indexer" with a tracking error of 1% has a tiny chance of earning the hurdle. A concentrated fund with a tracking error of 8% has more upside but also a deeper downside.
- Past outperformance is a weak guide. A fund that beat the index for five years may simply have been lucky. Use Step 7 to check how much skill that record would require.
- Look at the same category. Compare funds against the right benchmark, such as small-cap against a small-cap index, or the result is meaningless.
- There may be reasons to hold active funds. Areas with less efficient pricing, or a specific objective such as income or a values screen, may justify a higher fee. Price it using this test first.
- The normal-distribution shortcut is rough. Real excess returns have fat tails and drift. Use the numbers to compare, not to predict.
Step-by-Step Method
- Find the all-in annual cost of the active fund and the index alternative.
- Calculate the hurdle (Step 1).
- Find the active fund's tracking error from monthly data.
- Calculate the probability of beating the index at your holding period, for zero skill and for plausible skill (Steps 2 and 3).
- Calculate the dollar cost of choosing a no-skill active fund (Step 4).
- Calculate the break-even probability of skill and compare it with your honest belief (Step 5).
- If you are using a scorecard, adjust for survivorship (Step 6).
- Check how much skill your target odds would require (Step 7).
- Choose active only if you can state, in writing, why this fund clears the hurdle.
- Review annually, and switch if the reason no longer holds.
Active Fund Break-Even Worksheet (Copy This)
| Line | Item | Your figure |
|---|---|---|
| A | Active fund all-in annual cost | |
| B | Index fund all-in annual cost | |
| C | Hurdle (A − B) | |
| D | Holding period, n | |
| E | Tracking error | |
| F | Gross alpha if skilled (my honest estimate) | |
| G | Net excess return if skilled (F − C) | |
| H | Probability of beating the index, no skill: Φ(−C × √D ÷ E) | |
| I | Probability of beating the index, skilled: Φ(G × √D ÷ E) | |
| J | Amount invested | |
| K | Index ending value | |
| L | Active ending value, no skill | |
| M | Dollar cost of no skill (K − L) | |
| N | Break-even probability of skill (C ÷ F) | |
| O | My honest probability this manager is skilled | |
| P | Gross alpha needed for 80% odds (C + 0.84 × E ÷ √D) |
Survivorship Table
| Item | Count |
|---|---|
| Funds at the start | |
| Funds surviving | |
| Survivors beating the benchmark | |
| Reported beat rate | |
| Adjusted beat rate (÷ funds at start) |
Active vs Index Checklist (Copy This)
| # | Check | Done |
|---|---|---|
| 1 | Calculated the hurdle using all-in costs on both sides | ☐ |
| 2 | Compared against the right benchmark for the fund's category | ☐ |
| 3 | Obtained tracking error from monthly data, not just the stated figure | ☐ |
| 4 | Calculated the odds at my realistic holding period | ☐ |
| 5 | Calculated the dollar cost of no skill | ☐ |
| 6 | Wrote down why this manager should be skilled, with evidence beyond past returns | ☐ |
| 7 | Compared my belief in skill against the break-even probability | ☐ |
| 8 | Adjusted any scorecard figures for survivorship | ☐ |
| 9 | Checked whether the fund's returns are net of fees | ☐ |
| 10 | Checked for a manager change, since past records belong to the old team | ☐ |
| 11 | Checked turnover and tax cost if held in a taxable account | ☐ |
| 12 | Set a review date and a rule for when I would switch | ☐ |
Common Mistakes This Test Catches
- Looking at fees only as a percentage of assets. The hurdle shows what the manager must earn every year just to tie.
- Choosing a fund for its past outperformance. A strong record can be luck, and Step 7 shows how much skill it would take to make it real.
- Reading scorecards that only count survivors. Closed funds vanish from the sample, which makes active funds look better.
- Comparing against the wrong benchmark. A fund that beats a broad index may only be taking more of a risk it was supposed to avoid.
- Ignoring tracking error. A fund can have a high chance of tying and no chance of winning.
- Judging over one year. One-year odds are close to a coin flip, which hides the fee gap.
- Forgetting manager turnover. A track record belongs to the people who earned it.
- Ignoring taxes. High turnover in a taxable account adds a tax drag the fee comparison misses.
- Treating the model as a forecast. The odds are an estimate that depends on assumptions.
Important Note
Fund performance data, scorecard methodology, and fee disclosure rules differ by country and provider, and they change over time. The fees, tracking error, alpha, and probabilities above are illustrative, not forecasts, and the probability formula assumes independent, roughly normal excess returns, which real markets only approximate. Readers should check current fund documents or consult a qualified financial adviser.
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