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Characterise what style analysis is.

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

The idea

Style analysis, developed by William Sharpe, infers a fund's effective asset mix from its returns alone. Regress the fund's returns on the returns of a set of asset-class or style indices. The weights are constrained to be non-negative and to sum to one, which mirrors a long-only, fully invested portfolio, and there is no unconstrained intercept. The fitted weights are the fund's style exposures, and the style-weighted return of the indices is a customised benchmark. The R² shows how much of the fund's return variation the style mix explains. The residual is credited to security selection and any timing. Run over rolling windows, the regression reveals style drift. It needs only returns, not holdings. The weights are estimates that carry error, so read them as exposures, not as reported positions.

Formulas

Volatility of return not explained by style

σ residual = σ × √(1 − R²)

σ
the fund's total return volatility
R²
the R² of the style regression

Take the square root of the unexplained variance share. Multiplying σ by 1 − R², or by √R², gives the wrong part.

Selection return against the style benchmark

selection = Rp − Σ wᵢ × Rᵢ

Rp
the fund's return
wᵢ
the fitted style weight on index i
Rᵢ
the return of style index i

Use the fitted style weights, not equal weights or a single index, and subtract the benchmark from the fund.

A practice question

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

In a style analysis regression, the unexplained part of the fund's return (the residual) is attributed to:

  1. the asset allocation implied by the style weights.Why not: The style weights, not the residual, represent the asset mix.
  2. the market-wide return shared by all funds.Why not: Market exposure is part of the style benchmark, not the residual.
  3. the fund's expenses, which are the only difference from the benchmark.Why not: Expenses are one piece of the residual, not the whole of it.
  4. security selection and other decisions that differ from the style mix. - correct

The fitted style-weighted index return is the passive style benchmark. What is left (the residual) is credited to selection and timing.

A harder one

A fund's style regression has an R² of 0.85 and the fund's total return volatility is 16%. The standard deviation of the part of its return NOT explained by style is closest to:

  1. 6.2% - correct
  2. 2.4%Why not: Multiplies 16% by (1 - R²) instead of taking the square root of the unexplained variance share.
  3. 14.8%Why not: The standard deviation explained by style: 16% x sqrt(0.85).
  4. 13.6%Why not: Multiplies 16% by the R², which gives the explained, not unexplained, share.

Unexplained variance share = 1 - 0.85 = 0.15. Residual standard deviation = 16% x sqrt(0.15) = 6.2%.

The hardest tier

A fund's style benchmark weights are 50% large growth, 30% mid-cap value, 15% small-cap and 5% T-bills, with index returns of 12%, 9%, 15% and 2% respectively. The fund returned 11.8%. The selection return relative to the style benchmark is closest to:

  1. -0.75%Why not: Reverses the sign by computing benchmark minus fund instead of fund minus benchmark.
  2. 0.75% - correct
  3. 2.30%Why not: Compares the fund with the equal-weighted average of the four indices (9.5%), not the fitted style weights.
  4. -0.20%Why not: Compares the fund with the large growth index alone (12%).

Style benchmark = 0.5 x 12 + 0.3 x 9 + 0.15 x 15 + 0.05 x 2 = 11.05%. Selection = 11.8% - 11.05% = 0.75%.

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