Returns-based style analysis: methodology

Advanced6 min read · updated 24 October 2026 · 2 sources

This page describes the methodology of the style analysis in Portfolio Lab: how style weights are derived from a fund's monthly returns and a selection of style indices, how uncertain they are, how the rolling analysis reveals style drift and how the manager's selection return is measured. The background is explained in the article Style analysis.

The model

The returns-based style analysis of Sharpe (1992) explains a fund's monthly returns by a fixed mix of style indices, such as money market, government bonds, equity regions or value and growth stocks. The mix chosen is the one for which the variance of the difference between the fund and the mix is smallest. The mix is called the style benchmark, and the difference is the manager's selection return.

Portfolio Lab offers three variants: classic Sharpe (weights not negative, summing to 100%), summing to 100% but with negative weights allowed, and entirely without constraints (a regression of the fund return on the indices). The classic variant is a quadratic problem; the code solves it exactly with an active-set method, the second variant via the system of equations of the Lagrange conditions, and the third as an ordinary regression. Because it is the variance that is minimized and not the sum of squares, a constant difference in returns plays no role. It ends up in the selection.

The R² is 1 − Var(residuals) / Var(fund): the share of the variation explained by the style. For index funds it is close to 1; for concentrated active funds it is considerably lower.

Data and defaults

• Templates: asset classes in euros (money market, short-term and ten-year German Bunds, US government bonds, equities USA, Europe, Japan and emerging markets, gold), equity regions in euros, bonds in euros (maturities, euro government bonds, US Treasuries) and US equity styles in dollars (large/small × value/growth like a style box, with short ETF histories extended by index funds). Your own selection is also possible.
• Currency: euro, dollar, Swiss franc or pound. Funds and indices are converted beforehand.
• Months: completed calendar months only; the current month is only an interim figure and is dropped.
• Limits: two to 15 style indices; at least max(12, number of indices + 3) common months. With fewer than three times as many months as indices, the tool warns that the weights are uncertain.

How precise are the style weights?

Style weights are estimates. Portfolio Lab shows standard errors using the approximation of Lobosco and DiBartolomeo (1997):

The more similar the style indices are to one another (high R²_i), the more uncertain the weights become. That is why a few clearly different indices help more than many similar ones. For weights that sit at the zero bound, there is no meaningful standard error; the table leaves them empty.

Rolling and out of sample

The rolling analysis estimates the weights over a moving window of 24 to 120 months in steps of six (default 36). The weights over time reveal style drift; the metric used is the average monthly reallocation ½ Σ |Δw|, annualized, plus the minimum, maximum and range for each style.

More important is measuring selection out of sample, as Sharpe recommends: the weights from the months up to t − 1 are applied to month t. This creates a style benchmark without look-ahead bias: the manager could have known it at the start of each month. The tool shows metrics for this from twelve months out of sample onwards, i.e. from a history of the window length plus twelve months.

Selection and attribution

• Selection p.a.: return of the fund minus return of the style benchmark, both geometric per year.
• Avg. selection (arithmetic): monthly mean of the difference × 12, plus the t-value (mean / (standard deviation / √n)).
• Tracking error and information ratio: standard deviation of the difference × √12, and avg. selection / tracking error. If the difference is practically zero (the fund is the mix), both stay empty.
• Attribution: avg. fund return p.a. = Σ w_i · avg. return_i p.a. + avg. selection p.a. This is arithmetically exact because the weights sum to 100%.
• Calendar years: fund, style benchmark and selection per year, with and without hindsight.

Style map

With the US equity styles template, the tool places the four indices in the corners of a map (value on the left, growth on the right, small at the bottom, large at the top). A fund's position is the weighted average of the corners; negative weights count as zero for this purpose. Over time, this traces a path through the style box.

Limitations

• Style analysis only sees returns, not holdings. If a fund holds something that none of the indices represents (such as high-yield bonds in a pure equity selection), it ends up in the selection or distorts the weights.
• If the fund itself is among the indices, the result is trivial (R² close to 1); the tool points this out.
• For the actual composition based on holdings lists, there is the Look-through.

This page describes how Portfolio Lab calculates; past results are not a forecast. All terms in the glossary.