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Show how style analysis can be used to cope with factor exposures that change over time.

FRM® Part II · Risk Management and Investment Management · Alpha (and the Low-Risk Anomaly) · FRM2.IM.3.f

The idea

Style analysis, due to William Sharpe, regresses a fund's returns on a set of asset class or style index returns. The weights are constrained to be non-negative and to sum to one, so the coefficients read as the fund's effective portfolio mix. The residual is the selection return, or alpha, relative to this style benchmark. A single regression over a long sample assumes the exposures are constant, which fails for managers who change style. Style analysis copes by re-estimating the constrained regression over a rolling window, so the weights, and the style benchmark built from them, move with the manager. A short window reacts faster but gives noisier weights. A long window is more precise but slow to detect a change. Judging a drifting manager against a fixed benchmark credits the drift to skill or blames the manager for it. Exam point: rolling constrained regression tracks time-varying exposures.

Formulas

Selection return against a style benchmark

selection return = Rp − (w1 × R1 + w2 × R2 + … + wk × Rk)

Rp
the fund's return
w1, w2, wk
the style weights, non-negative and summing to one
R1, R2, Rk
each style index's return

Use the weights from the latest rolling window; stale or full-sample weights mislabel style drift as skill.

A practice question

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

In Sharpe's style analysis, what does the set of estimated regression weights represent?

  1. The fund's alpha in each asset class, summing to the total selection return.Why not: Alpha is the residual of the regression, not the weights; the weights describe exposure.
  2. The fund's effective portfolio mix across the chosen asset class or style indices, with weights constrained to be non-negative and to sum to one. - correct
  3. The fund's market timing ability, measured as the sensitivity of returns to squared market returns.Why not: Describes a nonlinear market-timing regression, not a style weight.
  4. The standard deviation of the fund's returns attributable to each asset class.Why not: Weights are exposures, not risk contributions.

Style analysis is a constrained regression of fund returns on style indices. The weights (non-negative, summing to one) show the implied portfolio mix; the residual is selection return.

A harder one

A style analysis shows a fund's weights are 60% in a value index, 30% in a growth index and 10% in a bond index. Over the period the indices returned 8%, 12% and 4% respectively, and the fund returned 9.0%. Which is closest to the fund's selection return (alpha) against its style benchmark?

  1. 1.0%Why not: Builds the benchmark with equal weights (8%) instead of the estimated 60/30/10 style weights.
  2. -1.0%Why not: Swaps the value and growth weights (30/60/10) when building the style benchmark.
  3. -3.0%Why not: Compares the fund with the growth index alone rather than with the weighted style benchmark.
  4. 0.2% - correct

Style benchmark return = 0.6 x 8 + 0.3 x 12 + 0.1 x 4 = 4.8 + 3.6 + 0.4 = 8.8%. Selection return = 9.0 - 8.8 = 0.2%.

The hardest tier

A fund returned 9.0% in a quarter in which a value index returned 6.0% and a growth index returned 10.0%. A rolling 36-month style analysis ending at the start of the quarter gave weights of 40% value and 60% growth. The full-sample constant weights are 60% value and 40% growth, and the weights from three years earlier were 80% value and 20% growth. Which is closest to the fund's alpha using the rolling style benchmark?

  1. 1.4%Why not: Uses the full-sample constant weights (60/40), which ignore the manager's recent shift in style.
  2. 0.6% - correct
  3. 2.2%Why not: Uses the stale weights from three years earlier instead of the current rolling estimate.
  4. -0.6%Why not: Subtracts in the wrong order (benchmark minus fund), giving the wrong sign.

Rolling style benchmark = 0.4 x 6.0 + 0.6 x 10.0 = 8.4%. Alpha = 9.0 - 8.4 = 0.6%. Constant or stale weights would give 1.4% or 2.2%, mislabelling style drift as skill.

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