Institutional Investors
Department of Finance · National Chengchi University
2026-10-02
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?
| 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? |
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.
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.
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.
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)
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.
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.
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.
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.
Draw a random sample of months (with replacement). Apply the same draw to all funds simultaneously.
Why jointly? This preserves:
This is the key methodological departure from Kosowski et al. (2006), who simulate each fund independently.
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 |
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.
When expense ratios are added back:
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.
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.
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.
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.
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.
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.
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:
Both cases produce large p-values. The estimator counts both as zero-alpha funds.
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 |
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.
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.
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.
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.
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 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.
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.
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.
| Statistic | Value (Y2000 $/month) |
|---|---|
| Time-weighted mean | +$270,000 |
| t-statistic | 5.74 |
| Annual equivalent | +$3.2 million |
| Median fund | −$20,000 |
| % with positive value added | 43% |
Most funds destroy value. The average is positive because skilled managers control most of the capital.
% of funds
30 | ■
| ■■■
20 | ■■■■■
| ■■■■■■■
10 |■■■■■■■■■■■
|■■■■■■■■■■■■■■
0 +──────────────────────
-3M 0 +1M +7M
Value Added ($/month)
Median: -$20K Mean: +$270KRight-skewed: a minority of large, skilled funds drives the positive average.
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:
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.
\[V_{i,t} = \underbrace{q_{i,t-1} \cdot f_{i,t-1}}_{\text{fees (managers)}} + \underbrace{q_{i,t-1} \cdot \varepsilon_{i,t}}_{\text{return to investors}}\]
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.
A competitive model of the mutual fund industry with:
| 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 |
Both are right. They measure different things.
Fama & French
Berk & van Binsbergen
The resolution: Skill exists and is measurable in dollars. Competition drives the return to investors to zero. Rents flow to managers, not investors.
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 investorsDo 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.
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.
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?
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?
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.
Readings for Week 4:
Calvin J. Chiou
jjchiou@nccu.edu.tw
Office: Commerce Building 261233
See you next Thursday.
Institutional Investors · Slides 02 · NCCU Finance