Style exposure: resemblance in returns

Style exposure asks which mix of broad market styles moved most like the portfolio over the last three months. The answer is a statistical resemblance, not a list of what is owned. Different holdings can move together because they respond to similar market forces. A fitted mix makes that shared movement visible, while the unexplained part records what the broad styles did not capture in this particular window.

Sharpe's returns-based style analysis compares sequences of returns, rather than scoring each stock's characteristics. Per-stock factor models need information about the underlying companies. Funds, wealth products and parts of Asian markets do not always have that information in a comparable form. Returns provide a shared observable series. The calculation finds non-negative weights that minimize the squared differences between the portfolio's returns and the style mix's returns, with a total weight of 100%.

The basis contains 13 styles, listed below with their ETF tickers. Growth, value and small-cap distinguish parts of the US stock market; other rows cover markets and asset classes outside it. Quality, low-volatility and momentum are omitted because their correlations with US large-cap exceed 0.9. Highly similar series make their individual weights difficult to distinguish. Long-term Treasuries, represented by TLT, remain separate from aggregate bonds because their interest-rate exposure differs. BIL represents cash and short-term bills in USD. REIT means real estate investment trust: a listed company owning income-producing property.

The table shows the fitted example beside a separate fit of SPY over the same dates. Each colored bar begins at the left edge and uses its own weight as a fraction of the cell width. Unlike the stress contribution chart, there is no central zero line and no scaling to the longest bar. The numbers and bar lengths express the same weights.

The example returned 1.24% over the window, while the style mix returned 1.12%. The example line uses unchanged quantities. The fitted curve starts with the estimated weights and follows each basis ETF's cumulative price change, also without rebalancing. The solver fits overlapping returns; it does not force every cumulative point, or the last point, to match. A small gap in the cumulative chart is therefore compatible with a close statistical fit.

R-squared (R²), the coefficient of determination, measures the fraction of return variation explained by the fitted styles. It is one minus the variance of the residuals divided by the variance of the example's returns. Residuals are the differences left after fitting. R² is displayed rounded down to two decimals: 0.99 for the example and 0.99 for SPY. Unexplained volatility is the residual standard deviation, annualized using the three-day horizon: 0.29% and 0.51%. An amber note appears below 0.60; no low-fit example is shown here.

The fixed three-month window describes recent behavior, independently of the replay span. Markets close at different times, so a single day's Asian and US prices can reflect different information. Returns over overlapping 3-day intervals soften that misalignment. This example has 63 returns from 66 price levels. Neighboring intervals share days and are not independent observations. With fewer than 42 samples the card displays insufficient data, rather than a fitted mixture.

The solver enforces non-negative weights and a total of 100%, and solves that constrained problem directly. It does not allow a negative weight to offset another style. No standard errors are supplied. The weights depend on the window and on the chosen basis, so a precise percentage is a numerical solution for these samples, not a statement that the exposure is known with that precision.

A holding represented by a proxy already follows an index ETF. That ETF can itself closely resemble one of the basis styles, mechanically making the fitted result closer. The card displays a note whenever any proxied share is positive. A high fit can therefore partly reflect how the replay was constructed, rather than independent information about the original holding. Proxy disclosure remains relevant even when both cumulative curves look almost identical.

The questionnaire is not used. There is no rolling drift view or per-holding breakdown. The result describes one recent historical window for the portfolio as a whole. It does not assign a judgement to a style, measure whether the mixture fits a person's circumstances, or forecast how these exposures behave in another window.

Sharpe, W. F., Asset Allocation: Management Style and Performance Measurement, Journal of Portfolio Management 18(2), 1992.

Fama & French, Common Risk Factors in the Returns on Stocks and Bonds, Journal of Financial Economics 33(1), 1993.

Carhart, M., On Persistence in Mutual Fund Performance, Journal of Finance 52(1), 1997.