6B.7 Is the manager any good: performance and luck

6B.7.1Measures#

Let $R_{p,t}$ be the fund's return, $R_{b,t}$ the benchmark TRI's, $r_{f,t}$ the risk-free rate, and $R_{m,t}$ the market's, per period.

• Sharpe ratio (Sharpe 1966): excess return per unit of total risk,
  $$S = \frac{\bar R_p - \bar r_f}{\sigma(R_p)}. \tag{6B.17}$$
  Annualise by $\sqrt{12}$ from monthly data. M04 · 4.5 derives its sampling error, which is large: with i.i.d. returns the standard error of an annualised Sharpe ratio $S$ estimated from $T$ years is about $\sqrt{(1 + S^2/2)/T}$, so 0.47 for $S = 0.5$ over five years.
• Jensen's alpha (Jensen 1968): the intercept of
  $$R_{p,t} - r_{f,t} = \alpha + \beta\,(R_{m,t} - r_{f,t}) + \varepsilon_t, \tag{6B.18}$$
  estimated by OLS. It is the return not explained by market exposure. With more factors (size, value, momentum: Carhart 1997, M06A · 6A.12) it is the return not explained by any of them.
• Tracking error and information ratio: with active return $R_{p,t} - R_{b,t}$,
  $$TE = \sigma(R_p - R_b)\sqrt{m}, \qquad IR = \frac{(\bar R_p - \bar R_b)\,m}{TE}, \tag{6B.19}$$
  for $m$ periods a year. Some authors use the regression alpha and residual volatility instead: $IR = \alpha/\sigma_\varepsilon$.
• Active share (Cremers and Petajisto 2009): with portfolio weights $w_{p,i}$ and benchmark weights $w_{b,i}$ over all securities,
  $$AS = \tfrac12\sum_i |w_{p,i} - w_{b,i}|. \tag{6B.20}$$
  $AS = 0$ is the index; $AS = 1$ shares nothing with it. A fund with active share below about 0.6 that charges an active fee is partly an expensive index fund (6B.13).

Survivorship bias Funds that do badly are merged or closed, and their histories drop out of today's list. Averaging the returns of today's funds over ten years overstates the average fund's performance. The SPIVA scorecards count the dead funds in the denominator for this reason.

6B.7.2The evidence#

SPIVA India S&P Dow Jones Indices' scorecards compare active funds with category benchmarks, after costs and adjusting for survivorship. M06 · 6.11 gives the mid-year 2025 figures (73% of large-cap funds behind their benchmark over 10 years). For the mid-year 2026 scorecard (data to 30 Jun 2026), the S&P website blocked automated access; search extracts of the S&P page give about 85% of large-cap funds behind the S&P India LargeMidCap over 6 months and about 74% over 10 years, more than 70% of funds behind their benchmark over 10 years in every equity category, and about 74% of funds surviving the 10 years. These figures were not read in the scorecard itself; verify before quoting. Since 2024 SPIVA has benchmarked Indian large-cap funds against a large-and-mid-cap index; M06 · 6.11 explains why this inflates measured underperformance when mid-caps lead.

Persistence Carhart (1997) found that the persistence of US fund returns over one year is explained almost entirely by the momentum factor and by expenses: winners' holdings had happened to be momentum stocks, and the only strong persistence was among the worst funds, which stayed bad because their costs stayed high. Past returns are a poor guide to future alpha; past costs are a good guide to future costs.

Luck and skill Suppose $N$ funds have zero true alpha and their alpha $t$-statistics are approximately independent standard normal. The number that clear a one-sided threshold $z$ is binomial with mean

so 400 funds produce about 10 "significant" alphas at $z = 1.96$ with no skill at all, and at least one with probability 0.9999. If instead a fraction of funds is truly skilled, the proportion of false discoveries among the significant funds is $\text{FDR} = \text{FP}/(\text{FP} + \text{TP})$. With 40 skilled funds whose expected $t$-statistic is 2.5 and 360 unskilled ones, the expected numbers are $\text{TP} = 40\,\Phi(0.54) = 28.2$ and $\text{FP} = 360 \times 0.025 = 9.0$: one in four of the "star" funds is a false discovery. M04 · 4.6 develops the multiple-testing corrections (Bonferroni, Holm, Benjamini–Hochberg).

Fama and French (2010) applied this logic to US equity funds from 1984 to 2006. They bootstrapped the cross-section of alpha $t$-statistics under the null of zero true alpha and compared it with the actual cross-section. Net of costs, the actual distribution lay to the left of the simulated one almost everywhere: few funds produced benchmark-adjusted returns sufficient to cover their costs. Gross of costs there was some evidence of skill in both tails (some managers with true positive alpha, some with negative), but the right tail was thin.

6B.7.3Why skill does not reach the investor: Berk and Green (2004)#

Proof

(a) Expected net alpha per rupee at size $q$ is $\phi_t - bq - f$, decreasing in $q$. If it were positive, investors would add money; if negative, they would withdraw it. Competition among investors therefore drives it to zero, which gives $q^*_t$, and with it $\mathbb E_t[\text{net alpha}_{t+1}] = \phi_t - bq_t^* - f = 0$. (b) Adding back the known $bq^*_t$, investors observe $x = a + \varepsilon_{t+1}$, a normal signal of $a$ with noise variance $\sigma^2$. The normal–normal Bayesian update gives the posterior mean $\phi_t + \kappa_t(x - \phi_t)$ with $\kappa_t = s_t^2/(s_t^2 + \sigma^2)$, and $x - \phi_t = (a - bq^*_t + \varepsilon_{t+1}) - (\phi_t - bq^*_t) = \eta_{t+1}$. Apply (a) at $t + 1$: $q^*_{t+1} - q^*_t = (\phi_{t+1} - \phi_t)/b$. (c) $f(\phi_t - f)/b$ is a concave quadratic in $f$ with maximum at $f = \phi_t/2$.∎

The model explains three facts at once. Flows chase past performance (6B.23). Past performance does not predict future net performance, because size adjusts until the expected net alpha is zero (6B.22). And the average active fund does not beat its benchmark after fees even if many managers are skilled: the skill is real, but it is paid to the manager through fees on a larger fund. Net returns are the wrong place to look for skill; fee revenue and size are where it shows. The Indian corollary is capacity: in small-cap categories, where $b$ is large because the stocks are illiquid, inflows erode alpha quickly, which is why some small-cap funds restrict lump-sum inflows after strong years.

Exercisem06b_sip_skill.pyOpen in editor →