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Account for why it is hard to gauge the performance of actively managed portfolios.

FRM® Part II · Risk Management and Investment Management · Portfolio Performance Evaluation · FRM2.IM.7.f

The idea

Judging an active manager is hard for statistical reasons. Abnormal returns are small, often one or two percentage points a year, while annual return volatility is far larger, so the signal is buried in noise. The standard error of an alpha falls only with the square root of the number of observations, so decades of data are needed for significance. A manager also changes the portfolio's risk and composition over time, so beta or volatility estimated over a long window describes no single period. Shortening the window makes the estimate noisier. A wrong benchmark, or omitted style factors, can create apparent alpha. Finally, with thousands of managers, some will beat the market several years running by luck alone. A good record is not proof of skill without a significance test.

Formulas

Observations needed for alpha to reach a target t-statistic

n = (t × σ ÷ α)²

t
the target t-statistic, about 2
σ
the residual standard deviation per period
α
the expected alpha per period

Shows why skill is hard to detect: an alpha one-fifth of residual risk needs about a century of annual data. Forgetting to square, or to include t, understates n.

A practice question

From the question bank, with the reason each wrong option is wrong.

The most fundamental statistical difficulty in evaluating an active manager's skill is that:

  1. alpha cannot be measured when the manager's returns are not normally distributed.Why not: Alpha can be estimated; the problem is its precision, not its measurability.
  2. beta is always unstable, so alpha is undefined.Why not: Beta can change but alpha is still estimable; the central problem is the noise level.
  3. annual abnormal returns are small relative to volatility, so very long samples are needed. - correct
  4. managers' returns are always perfectly correlated with the benchmark.Why not: Correlation is high for many funds, but that is not the fundamental limit.

A signal of one or two percent per year is hidden in volatility of 15% or more, so skill can only be distinguished from luck over many years.

A harder one

A manager's expected alpha is 1% per year and the annual residual standard deviation is 5%. About how many years of data are needed for the expected t-statistic of alpha to reach 2?

  1. 25Why not: Computes (5/1)² and forgets the critical value of 2.
  2. 100 - correct
  3. 10Why not: Computes 2 x 5/1 and forgets to square the ratio, giving 10 years.
  4. 50Why not: Squares only the volatility term, 2 x 5²/1, leaving the 2 unsquared.

n = (t x sigma/alpha)² = (2 x 5/1)² = 100 years. This illustrates why skill is almost impossible to detect from a single manager's track record.

The hardest tier

Suppose 1,000 managers have no skill and each has an independent 50% chance of beating the benchmark in a given year. The expected number that beat the benchmark in each of four consecutive years is closest to:

  1. 125Why not: Counts only three years of wins: 1,000 x 0.5³ rather than 0.5^4.
  2. 63 - correct
  3. 250Why not: Divides 1,000 by 4 years instead of taking 0.5 to the fourth power.
  4. 500Why not: Applies the one-year probability and ignores the other three years.

Probability of four wins in a row = 0.5^4 = 6.25%. Expected count = 1,000 x 0.0625 = 62.5, about 63. A long winning streak among many managers is partly luck.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

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