02: Skill, Luck, and the Value of Active Management

Institutional Investors

Author

Affiliation

Calvin J. Chiou (邱健嘉)

Department of Finance · National Chengchi University

Published

October 2, 2026

Where We Left Off

Last week: the Carhart four-factor model

We established that alpha — the regression intercept in

\[R_{i,t} - R_{f,t} = \alpha_i + \beta_i \cdot RMRF_t + s_i \cdot SMB_t + h_i \cdot HML_t + u_i \cdot UMD_t + \varepsilon_{i,t}\]

is the standard academic measure of fund performance.

. . .

We ended with a question:

Even a well-estimated alpha could be luck. With thousands of funds, some will show large positive alphas by chance alone. How do we tell the two apart?

Today’s papers

Paper	Question
Fama & French (JF 2010)	Bootstrap methodology for distinguishing luck from skill in the cross-section
Berk & van Binsbergen (JFE 2015)	Dollar value added as the correct measure of skill
Andrikogiannopoulou & Papakonstantinou (RFS 2019)	Can the standard tests even detect skill when it is there?

Today’s roadmap

The luck-vs.-skill problem — why standard t-tests fail
Equilibrium accounting — what we know before looking at any fund
Bootstrap simulation — the Fama & French (2010) approach
FDR critique — Andrikogiannopoulou & Papakonstantinou (2019)
Value added — why Berk & van Binsbergen (2015) reframe the question
The Berk-Green model — the equilibrium connecting all three
Synthesis and discussion

Part 1 · The Luck-vs.-Skill Problem

The core statistical challenge

Every estimated alpha contains two components:

\[\underbrace{\hat{\alpha}_i}_{\text{what we observe}} = \underbrace{\alpha_i}_{\text{true skill}} + \underbrace{\text{noise}}_{\text{estimation error}}\]

. . .

With thousands of funds, many will have extreme $\hat\alpha$ purely by chance — even if every manager has zero true skill.

Why standard t-tests are insufficient

The average standard error of a monthly alpha estimate: 0.28% per month (≈ 3.4% annualized).

. . .

Suppose a manager has a genuine annual alpha of +2%. With 10 years of monthly data:

\[t = \frac{\hat{\alpha}}{\hat{\sigma}/\sqrt{T}} = \frac{2\%/12}{3.4\%/\sqrt{120}} \approx 0.63\]

. . .

Far below the conventional threshold of 2.0. A skilled manager looks statistically indistinguishable from an unskilled one at realistic sample sizes.

The multiple comparisons problem

With 3,000 funds tested at the 5% level, we expect 150 false positives even if no fund has skill.

. . .

Standard approach: look at the fund’s own t-statistic.

Problem: comparing one fund’s alpha to a normal distribution ignores the fact that we are looking at the best fund in a very large sample.

. . .

We need a method that correctly accounts for how extreme the best t-statistics should be under the null of no skill.

Part 2 · Equilibrium Accounting

Sharpe’s Arithmetic

The aggregate of all investors holds the market portfolio.

\[\underbrace{\text{passive}}_{\alpha = 0} + \underbrace{\text{active}}_{\alpha = ?} = \underbrace{\text{market}}_{\alpha = 0}\]

. . .

If passive investors have zero alpha by construction, then active investors in aggregate must also have zero alpha — before costs.

(Sharpe 1991)

The negative-sum game

Before costs:

\[\alpha_{\text{active, aggregate}} = 0\]

After costs:

\[\alpha_{\text{active, aggregate}}^{\text{net}} = -\underbrace{\text{expense ratio}}_{\approx 1\%/\text{year}} < 0\]

. . .

Active management is a zero-sum game before fees and a negative-sum game after fees.

This is accounting, not an empirical claim.

Aggregate evidence: Fama & French (2010)

Value-weighted portfolio of all active U.S. equity funds, 1984–2006:

Measure	Four-factor α (ann.)	t-stat
Net (investors receive)	−1.00%	−3.02
Gross (before expense ratio)	−0.05%	−0.15

. . .

Net alpha ≈ −expense ratio. Gross alpha ≈ 0.

Consistent with equilibrium accounting: in aggregate, active managers neither add nor destroy value before fees. But they definitely do after fees.

What the aggregate tells us

The aggregate result does not imply every manager lacks skill
It is consistent with: skilled and unskilled managers coexist, and roughly cancel each other out in aggregate
The interesting question is in the cross-section: how much do individual managers differ?
That requires a different methodology

Part 3 · Bootstrap Simulation

The Fama & French (2010) approach

Goal: construct the distribution of t(α) that we would observe if no fund had any skill, then compare it to the actual distribution.

. . .

If the actual right tail is fatter than the simulated right tail, that is evidence of genuine skill.

Step 1 — Impose zero true alpha

For each fund $i$, subtract its estimated alpha from its returns:

\[r_{i,t}^* = r_{i,t} - \hat{\alpha}_i\]

The population $\{r_{i,t}^*\}$ has the same time-series and cross-sectional properties as actual fund returns — but true alpha is zero by construction.

Step 2 — Joint monthly resampling

Draw a random sample of months (with replacement). Apply the same draw to all funds simultaneously.

. . .

Why jointly? This preserves:

The cross-correlation of fund returns (funds hold similar stocks)
The joint distribution of factor returns and fund residuals

This is the key methodological departure from Kosowski et al. (2006), who simulate each fund independently.

Steps 3–5 — Estimate, repeat, compare

Step 3: Run the four-factor regression on each fund using the simulated months. Record t(α).

Step 4: Repeat 10,000 times to build the full null distribution.

Step 5: Compare actual t(α) percentiles to the simulated percentiles.

. . .

If actual right tail > simulated	→ evidence of skill
If actual left tail < simulated	→ evidence of unskilled managers

Results: net returns (investors’ perspective)

Below the 80th percentile: actual t(α) beats simulated in < 1% of simulation runs
At the 90th percentile: 90% of simulations beat the actual t(α)
Only at the 97th–99th percentile: results approach the simulated null

Interpretation: most funds do not have sufficient skill to cover costs. A fund at the 90th percentile of historical t(α) is consistent with luck, not skill.

Results: gross returns (before fees)

When expense ratios are added back:

Left tail more extreme than simulated → truly unskilled managers reducing expected returns
Right tail fatter than simulated above the ~60th percentile → genuinely skilled managers

. . .

Fama & French estimate the cross-sectional dispersion of true alpha:

\[\sigma(\alpha^{\text{true}}) \approx 1.25\% \text{ per year}\]

. . .

Only ~16% of funds have true gross alpha > 1.25%/year. Only ~2.4% exceed 2.5%/year.

The CDF picture

Cumulative % of funds

100 |          Actual (gross)
    |       ╱────────────────────
 80 |    ╱╱    ← skill in right tail
    |  ╱╱  Simulated (null)
 60 |╱╱─────────────────────────
    |╲
 40 |  ╲╲  ← poor performance in
    |    ╲╲   left tail (actual
 20 |      ╲╲ more extreme)
    |
  0 +─────────────────────────────
       -4   -2    0    2    4
            t(α)

Left tail: actual < simulated (negative skill). Right tail: actual > simulated (positive skill). Crossover near the 60th–70th percentile.

Why joint sampling matters

Fama & French replicate the Kosowski et al. (2006) period (1975–2002) using their own joint-sampling approach:

Test	% of simulations where right tail exceeds actual
Kosowski et al. (independent sampling)	> 99% (strong skill evidence)
Fama & French (joint sampling)	68–82% (weak to moderate)

. . .

Failure to account for cross-fund correlation inflates the apparent evidence of skill.

Part 4 · The Power Problem

A prior question: can the tests detect skill?

Andrikogiannopoulou & Papakonstantinou (2019) ask a different question:

Suppose skill genuinely exists — and is as large as Fama & French estimate. Would the standard tests be able to detect it?

. . .

Their answer is deeply uncomfortable: often no.

The False Discovery Rate (FDR) methodology

A common alternative to the bootstrap: the FDR estimator (Barras, Scaillet & Wermers 2010).

The key quantity is the proportion of funds with zero true alpha, denoted $\pi^0$.

The estimator:

\[\hat\pi^0 = \frac{\#\{p_i > \lambda\}}{N(1 - \lambda)} \tag{2}\]

where $p_i$ is the two-sided p-value for fund $i$’s alpha and $\lambda$ is a tuning parameter (typically 0.5).

. . .

BSW (2010) report $\hat\pi^0 \approx 75\%$ — three-quarters of funds have zero true alpha.

The logic behind the estimator

If most fund p-values are large (near 1), that suggests most funds are consistent with zero alpha.

\[\hat\pi^0 = \frac{\#\{p_i > \lambda\}}{N(1-\lambda)}\]

. . .

The denominator $N(1-\lambda)$ is the expected number of large p-values if every fund has zero alpha.

The numerator counts actual large p-values.

If the ratio is high, most funds look like they have zero alpha.

The problem: this confuses zero alpha with low power

A fund with genuine alpha of ±1% will still have a large p-value most of the time when $T$ is short.

. . .

Therefore $\hat\pi^0$ does not distinguish between:

The fund truly has zero alpha, or
The fund has non-zero alpha but the test lacks power to detect it

Both cases produce large p-values. The estimator counts both as zero-alpha funds.

The magnitude of the bias

Andrikogiannopoulou & Papakonstantinou simulate fund returns with known true alpha distributions — calibrated to Fama & French’s own estimates — and ask how well the FDR recovers $\pi^0$.

. . .

True $\|\alpha\|$	% misclassified as zero-alpha	$T$ needed to avoid
±1% per year	~90%	Implausibly long
±2% per year	~65%	> 200 years
±3.5% per year	~30%	~380 months

A striking calibration

Under a data-generating process where only 1% of funds truly have zero alpha (consistent with Fama & French’s estimate of $\sigma = 1.25\%$):

. . .

The FDR estimator reports $\hat\pi^0 \approx 70\%$.

. . .

An estimator that says “70% of funds have zero alpha” regardless of whether 1% or 75% actually do is not measuring what it claims to measure.

Why the bias is always upward

The factor $\frac{1}{1-\lambda}$ amplifies every underpowered test result.

. . .

Whenever a fund has non-zero alpha but a large p-value (which happens most of the time when $T$ is small), it inflates the numerator of $\hat\pi^0$.

The bias cannot go downward: underpowered tests always make $\hat\pi^0$ too large.

Power comes from $T$, not $N$

Adding more funds $N$ does not help at all — the bias operates fund-by-fund, and averages to the same wrong answer.

. . .

Only lengthening $T$ (each fund’s return history) helps — and convergence is extremely slow.

Even 200 years of monthly data is insufficient to correctly identify funds with ±1% annual alpha.

. . .

Implication: Any study that reports FDR estimates with typical CRSP mutual fund sample lengths (80–120 months) should be read with extreme caution.

The methodological warning

“Low power” is not a caveat to be noted and set aside
It can reverse the economic conclusion: a finding of “75% of funds have zero alpha” may be a finding about our instruments, not our managers
This applies beyond mutual funds: any setting with short panels and small true effects
The Fama-French bootstrap is not immune, but is substantially more robust — it tests the entire cross-section rather than fund-by-fund p-values

Part 5 · Value Added — A Different Framing

Berk & van Binsbergen (2015): the question

Rather than ask “does alpha exist?”, they ask:

How much dollar value does the average mutual fund extract from financial markets?

. . .

This turns out to be a fundamentally different question — and leads to a different answer.

Why alpha is the wrong measure of skill

Under standard neoclassical assumptions (rational investors, competitive markets, managers optimizing):

. . .

Net alpha → driven to zero by investor competition. Every fund that attracts capital until expected net alpha = 0. \[\alpha^{\text{net}} = 0 \quad \text{in equilibrium}\]

. . .

Gross alpha → equals the fee the manager charges. \[\alpha^{\text{gross}} = \alpha^{\text{net}} + f_i = 0 + f_i = f_i\]

Gross alpha is a choice variable, not a skill measure.

The value-added measure

The correct measure: multiply the benchmark-adjusted gross return by AUM.

\[\boxed{V_{i,t} = q_{i,t-1} \cdot \bigl(R_{i,t}^{\text{gross}} - R_{i,t}^{\text{benchmark}}\bigr)}\]

Managerial skill: $S_i = \mathbb{E}[V_{i,t}]$

. . .

Peter Lynch illustration:

Period	Monthly gross α	Average AUM	Monthly value added
First 5 years (Magellan)	2.00%	$40M	$800K
Last 5 years	0.20%	$10B	$20M

Alpha fell by 90%; value added rose 25-fold.

Why not gross alpha?

Gross alpha fails unless all funds have the same AUM.

Under decreasing returns to scale, at the optimum: \[\alpha^{\text{gross}}_i = \frac{a_i^2}{4 q_i b_i}\]

Cross-sectional correlation between gross alpha and value added: 23% ($R^2 = 5\%$).

Value added always measures skill: \[q_i \cdot \alpha^{\text{gross}}_i = \frac{a_i^2}{4 b_i} = S_i \checkmark\]

AUM varies enormously; fees vary little. Therefore AUM, not gross alpha, is where skill shows up.

The right benchmark: Vanguard, not FFC

FFC factors are not tradable passive alternatives for most of the sample:

Factor	α vs. Vanguard	t-stat
Market	2 bp/month	0.83
SMB	22 bp/month	2.80
HML	35 bp/month	3.37
UMD	70 bp/month	3.38

. . .

Using FFC as the benchmark overstates the hurdle rate — especially in the 1970s and 80s when small-cap and value index funds didn’t exist.

Berk & van Binsbergen use actual Vanguard index funds available to investors at each date.

Main result: average fund adds positive value

Statistic	Value (Y2000 $/month)
Time-weighted mean	+$270,000
t-statistic	5.74
Annual equivalent	+$3.2 million
Median fund	−$20,000
% with positive value added	43%

. . .

Most funds destroy value. The average is positive because skilled managers control most of the capital.

The distribution of skill

         % of funds
    30 |    ■
       |   ■■■
    20 |  ■■■■■
       | ■■■■■■■
    10 |■■■■■■■■■■■
       |■■■■■■■■■■■■■■
     0 +──────────────────────
         -3M   0   +1M  +7M
           Value Added ($/month)

Median: -$20K     Mean: +$270K

Right-skewed: a minority of large, skilled funds drives the positive average.

Persistence of value added: up to 10 years

Fama & French found alpha persistence largely unpredictable beyond 1 year.

. . .

Berk & van Binsbergen sort funds on the Skill Ratio (t-statistic of past value added) and track future value added:

Top decile consistently beats bottom decile at every horizon 3–10 years out
Null rejected at 95% confidence at most horizons (non-parametric binomial test)
Top decile controls ~25% of total AUM — investors are rewarding skill

Investors can identify skill

A surprising finding:

Current compensation (AUM × fee) predicts future value added better than past value added does.

. . .

Since fee rates vary little across funds, investors — through their capital allocation — determine compensation.

This means investors are correctly inferring managerial quality and directing capital accordingly.

This contradicts the “dumb money” narrative of investors chasing past returns.

Where does the value go?

\[V_{i,t} = \underbrace{q_{i,t-1} \cdot f_{i,t-1}}_{\text{fees (managers)}} + \underbrace{q_{i,t-1} \cdot \varepsilon_{i,t}}_{\text{return to investors}}\]

. . .

Net alpha to investors ≈ zero — consistent with Fama & French.

Managers capture all economic rents through fees. Skilled managers charge higher effective fees (through larger AUM, not higher fee rates) and investors pay willingly because expected net alpha is zero.

. . .

About half of total value added comes from diversification services; the other half from stock selection and market timing.

Part 6 · The Berk-Green Equilibrium

Berk & Green (2004): the theoretical backbone

A competitive model of the mutual fund industry with:

Fund managers endowed with permanent skill (positive gross alpha on first dollar)
Decreasing returns to scale: alpha falls as AUM grows
Rational investors: flow capital to funds with positive expected alpha
Equilibrium: capital flows until every fund’s expected net alpha = 0

The model’s predictions

Prediction	FF (2010)	BvB (2015)
Zero net alpha in aggregate	Violated: aggregate ≈ −1%/year	Not directly tested
Zero net alpha per fund	Largely confirmed	Confirmed: investors earn ≈ 0
Investors identify skill	Not tested	Confirmed: compensation predicts VA
Skill is rewarded with AUM	Not tested	Confirmed: top decile controls 25% of AUM
Persistence in alpha	Consistent with zero	Skill ratio persists 10 years

Reconciling Fama-French and Berk-van Binsbergen

Both are right. They measure different things.

Fama & French

Return-based measure
Net alpha ≈ −expense ratio
Few funds cover costs
Skill rare or undetectable

Berk & van Binsbergen

Dollar-value measure
Average value added = +$3.2M/year
Skill widespread and persistent
Investors identify it

. . .

The resolution: Skill exists and is measurable in dollars. Competition drives the return to investors to zero. Rents flow to managers, not investors.

Part 7 · Synthesis

Three papers, one coherent story

Alpha-based tests (FF 2010)
  → Net alpha ≈ −fees; skill detectable only in extreme tails
  → But: bootstrap needs joint sampling to avoid false positives

  ↓ challenged by

Power critique (AP 2019)
  → FDR overstates π⁰ because it confuses low power with zero skill
  → Even FF's bootstrap has limited power at realistic T

  ↓ reframed by

Dollar value-added (BvB 2015)
  → Switch from return to dollars: avg fund adds $3.2M/year
  → Skill is widespread; investors allocate rationally
  → Net return ≈ 0: rents captured by managers, not investors

Implications for performance evaluation

Do not rely on alpha alone. Only the extreme right tail (≥97th percentile) of the t(α) distribution provides reliable signal.
Mind the benchmark. FFC factors embed hindsight bias. Traded alternatives (Vanguard index funds) are more appropriate.
Think in dollars, not percentages. A $10B fund with 0.2% alpha adds more value than a $1M fund with 10% alpha.
Persistence in value added lasts a decade. It is short in alpha because rational flows compete away the alpha — not because skill disappears.
Net alpha ≈ 0 for the best funds. There is no free lunch. Skilled managers capture their own rents through AUM.

Implications for research methodology

Report power simulations before reporting FDR estimates. A table showing $\hat\pi^0$ at varying $T$ is not optional.
Joint resampling is mandatory in bootstrap studies of fund performance. Independent sampling inflates evidence of skill.
Low power is not just a caveat. In the AP (2019) calibration, an estimator tuned to detect skill reports “75% zero-alpha funds” when only 1% truly have zero alpha. That is a structural failure.

Discussion

Discussion questions

Fama & French find VW net alpha ≈ −1%/year. Is this evidence that managers lack skill, or that investors rationally drive net alpha to zero? How would you distinguish these empirically?
Berk & van Binsbergen argue gross alpha equals the fee the manager charges. Does this mean a fund charging 2% is more skilled than one charging 0.5%? What does the theory actually predict?
The FF bootstrap jointly samples months for all funds. Why does this matter? What happens if months are sampled independently?
FFC factors have large positive alpha relative to Vanguard funds (especially HML and UMD). What does this imply about alpha estimates from the 1970s and 80s?
Suppose a manager has true annual alpha of 2% on $100M. Under Berk-Green, what happens to alpha and AUM over time? Is this consistent with BvB’s 10-year persistence finding?

Discussion questions (continued)

Write down the assumption behind the FDR equation (2) in words. When does it fail? Why is the bias in $\hat\pi^0$ always upward?
AP (2019) show that adding more funds $N$ does not help, but increasing $T$ does. Explain the asymmetry. Does the Fama-French bootstrap suffer from the same problem?
You are refereeing a paper that applies FDR to 3,000 hedge funds with an average of 48 monthly observations each. What simulation would you ask the authors to run before believing their estimated proportions?
The median fund has negative value added. The average is positive. What does this asymmetry tell you about the structure of the industry and the distribution of managerial talent?

Wrapping Up

What we covered today

Equilibrium accounting constrains aggregate active performance: zero before costs, negative after
Bootstrap simulation (FF 2010) is the correct tool for cross-sectional inference; joint monthly resampling is essential
FDR critique (AP 2019): the standard alternative confuses zero alpha with low statistical power; the bias is large, upward, and irreducible without implausibly long samples
Value added (BvB 2015): switching from alpha to dollars reveals widespread skill, 10-year persistence, and rational investor behavior
The Berk-Green equilibrium ties everything together: managers are skilled, but competition drives net returns to investors toward zero

The bottom line for investors

. . .

Skill exists. The average fund adds $3.2M/year in dollar terms. Value added persists 10 years. Investors identify it.

. . .

But investors do not capture it. Competition drives net alpha to approximately zero. Managers extract all rents through fees and AUM growth.

. . .

For the typical investor, active management costs about 1% per year and returns approximately zero alpha. The beneficiary is the manager, not you.

Coming up next

Readings for Week 4:

Active Share (Cremers & Petajisto, RFS 2009) — How much does a fund deviate from its benchmark, and does it matter?
R Lab: computing Active Share from holdings data

References

Andrikogiannopoulou, A., & Papakonstantinou, F. (2019). Reassessing false discoveries in mutual fund performance: Skill, luck, or insufficient data? Review of Financial Studies, 32(5), 2095–2137.
Barras, L., Scaillet, O., & Wermers, R. (2010). False discoveries in mutual fund performance: Measuring luck in estimated alphas. Journal of Finance, 65(1), 179–216.
Berk, J. B., & Green, R. C. (2004). Mutual fund flows and performance in rational markets. Journal of Political Economy, 112(6), 1269–1295.
Berk, J. B., & van Binsbergen, J. H. (2015). Measuring skill in the mutual fund industry. Journal of Financial Economics, 118(1), 1–20.
Fama, E. F., & French, K. R. (2010). Luck versus skill in the cross-section of mutual fund returns. Journal of Finance, 65(5), 1915–1947.
Kosowski, R., Timmermann, A., Wermers, R., & White, H. (2006). Can mutual fund “stars” really pick stocks? Journal of Finance, 61(6), 2551–2595.
Sharpe, W. F. (1991). The arithmetic of active management. Financial Analysts Journal, 47(1), 7–9.

Questions?

Calvin J. Chiou
jjchiou@nccu.edu.tw
Office: Commerce Building 261233

See you next Thursday.

Where We Left Off

Last week: the Carhart four-factor model

Today’s papers

Today’s roadmap

Part 1 · The Luck-vs.-Skill Problem

The core statistical challenge

Why standard t-tests are insufficient

The multiple comparisons problem

Part 2 · Equilibrium Accounting

Sharpe’s Arithmetic

The negative-sum game

Aggregate evidence: Fama & French (2010)

What the aggregate tells us

Part 3 · Bootstrap Simulation

The Fama & French (2010) approach

Step 1 — Impose zero true alpha

Step 2 — Joint monthly resampling

Steps 3–5 — Estimate, repeat, compare

Results: net returns (investors’ perspective)

Results: gross returns (before fees)

The CDF picture

Why joint sampling matters

Part 4 · The Power Problem

A prior question: can the tests detect skill?

The False Discovery Rate (FDR) methodology

The logic behind the estimator

The problem: this confuses zero alpha with low power

The magnitude of the bias

A striking calibration

Why the bias is always upward

Power comes from \(T\), not \(N\)

The methodological warning

Part 5 · Value Added — A Different Framing

Berk & van Binsbergen (2015): the question

Why alpha is the wrong measure of skill

The value-added measure

Why not gross alpha?

The right benchmark: Vanguard, not FFC

Main result: average fund adds positive value

The distribution of skill

Persistence of value added: up to 10 years

Investors can identify skill

Where does the value go?

Part 6 · The Berk-Green Equilibrium

Berk & Green (2004): the theoretical backbone

The model’s predictions

Reconciling Fama-French and Berk-van Binsbergen

Part 7 · Synthesis

Three papers, one coherent story

Implications for performance evaluation

Implications for research methodology

Discussion

Discussion questions

Discussion questions (continued)

Wrapping Up

What we covered today

The bottom line for investors

Coming up next

References

Questions?