Rolling Beta, Self-Weight Bias, and a Leave-One-Out Fix

quant research

factor models

Why a stock’s own weight in an index mechanically inflates its measured beta — and how to isolate the effect with a leave-one-out benchmark.

Published

July 13, 2026

The question

Beta is the standard measure of how much a stock moves relative to a benchmark:

\[\beta_t = \frac{\text{Cov}(R_{i,t}, R_{bench,t})}{\text{Var}(R_{bench,t})}\]

computed over a trailing window so it can be tracked over time rather than collapsed into one number for all history. A rolling beta immediately shows something a static beta hides: the relationship between a stock and its benchmark drifts, sometimes a lot.

def rolling_beta(asset_ret, bench_ret, window=60):
    cov = asset_ret.rolling(window).cov(bench_ret)
    var = bench_ret.rolling(window).var()
    return cov / var

Below is AAPL’s rolling beta to SPY over roughly a decade, computed at two window lengths. The dashed line marks beta = 1 (moving exactly like the index).

AAPL rolling 60-day and 120-day beta to SPY. Beta swings between roughly 0.5 and 2.0 — never stable for long.

The 60-day line reacts faster and is noisier; the 120-day line is smoother but lags. That’s the standard bias–variance trade-off of any rolling window, wearing a time-series costume.

A question worth asking: does index weight inflate beta?

Here’s a subtlety that’s easy to miss. For a cap-weighted index, the index return is literally a weighted sum of its constituents:

\[R_{index} = \sum_i w_i R_i\]

Substituting that into the covariance in the beta formula:

\[\text{Cov}(R_i, R_{index}) = w_i \cdot \text{Var}(R_i) + \sum_{j \ne i} w_j \cdot \text{Cov}(R_i, R_j)\]

That first term means a stock’s own weight mechanically inflates its own measured beta — independent of any real economic sensitivity to the market. A mega-cap holding 7% of an ETF is partly “correlated” with that ETF simply because it is 7% of it, by construction.

Isolating the effect: leave-one-out beta

Comparing beta against the real benchmark is a noisy way to test this — differences could come from genuine market sensitivity, from the benchmark’s hundreds of other holdings, or from the self-weight effect, all tangled together. To isolate just the mechanism, build a synthetic index from a small set of live-weighted holdings, then compute each stock’s beta against that synthetic index two ways: including itself, and excluding itself (with the remaining weights renormalized to sum to 1).

def leave_one_out_beta_report(index_key, window=60, start="2018-01-01"):
    live_weights = get_live_top_holdings(benchmark_ticker)   # real, current ETF weights
    tickers = list(live_weights.index)

    results = {}
    for i in tickers:
        others = [t for t in tickers if t != i]

        w_incl = weights / weights.sum()
        idx_incl = (returns[tickers] * w_incl).sum(axis=1)          # synthetic index, including i

        w_excl = weights[others] / weights[others].sum()
        idx_excl = (returns[others] * w_excl).sum(axis=1)           # synthetic index, excluding i

        beta_incl = rolling_beta(returns[i], idx_incl, window=window).mean()
        beta_excl = rolling_beta(returns[i], idx_excl, window=window).mean()
        results[i] = {"weight": weights[i], "beta_incl_self": beta_incl, "beta_excl_self": beta_excl}

    return pd.DataFrame(results).T

Everything about the calculation is held constant between the two versions — the only thing that changes is whether the stock sits inside or outside the benchmark it’s compared against. Any remaining difference is purely the mechanical effect.

Every SOXX constituent shows a higher beta when included in its own benchmark than when excluded — the self-weight effect predicted by the math.

Result

Run on the semiconductor sector (SOXX and its top-10 weighted constituents):

Every single stock showed a positive self-weight effect (beta_incl_self > beta_excl_self) — no exceptions.
Correlation of index weight with the size of the effect: 0.65 — heavier names get inflated more, exactly as the math predicts.
This is a materially cleaner signal than the raw weight-vs-beta scatter, which only showed a correlation of 0.18 — because that comparison was also picking up noise from the benchmark’s other, non-synthetic holdings.

Why this matters for a model

If you’re training a cross-sectional model — ranking many stocks by predicted beta or predicted alpha — naive beta systematically biases those rankings toward heavily-weighted names, for reasons that have nothing to do with genuine predictive signal. Leave-one-out beta removes that bias and gives a fairer comparison across the universe.

Data quality caveat. The synthetic index here is built only from an ETF’s live top-10 holdings (the maximum Yahoo Finance exposes via its public fund-data endpoint) — not the true full index. Treat this as a demonstration of the mechanism, not an exact real-world beta figure. Live weights come from Ticker.funds_data.top_holdings; there is no free source for full, point-in-time historical index constituents, which is a genuine limitation for anyone trying to build a fully survivorship-bias-free backtest universe on a budget.

Full analysis

Download this post (PDF) Download the notebook (.ipynb)

--- title: "Rolling Beta, Self-Weight Bias, and a Leave-One-Out Fix" description: "Why a stock's own weight in an index mechanically inflates its measured beta — and how to isolate the effect with a leave-one-out benchmark." date: 2026-07-13 categories: [quant research, factor models] image: ../../images/posts/leave-one-out-beta.png format: html: default pdf: documentclass: article geometry: margin=1in toc: true --- ## The question Beta is the standard measure of how much a stock moves relative to a benchmark: $$\beta_t = \frac{\text{Cov}(R_{i,t}, R_{bench,t})}{\text{Var}(R_{bench,t})}$$ computed over a trailing window so it can be tracked over time rather than collapsed into one number for all history. A **rolling** beta immediately shows something a static beta hides: the relationship between a stock and its benchmark drifts, sometimes a lot. ```python def rolling_beta(asset_ret, bench_ret, window=60): cov = asset_ret.rolling(window).cov(bench_ret) var = bench_ret.rolling(window).var() return cov / var ``` Below is AAPL's rolling beta to SPY over roughly a decade, computed at two window lengths. The dashed line marks beta = 1 (moving exactly like the index). ![AAPL rolling 60-day and 120-day beta to SPY. Beta swings between roughly 0.5 and 2.0 — never stable for long.](../../images/posts/aapl-rolling-beta.png) The 60-day line reacts faster and is noisier; the 120-day line is smoother but lags. That's the standard bias–variance trade-off of any rolling window, wearing a time-series costume. ## A question worth asking: does index weight inflate beta? Here's a subtlety that's easy to miss. For a cap-weighted index, the index return is *literally* a weighted sum of its constituents: $$R_{index} = \sum_i w_i R_i$$ Substituting that into the covariance in the beta formula: $$\text{Cov}(R_i, R_{index}) = w_i \cdot \text{Var}(R_i) + \sum_{j \ne i} w_j \cdot \text{Cov}(R_i, R_j)$$ That first term means a stock's **own weight mechanically inflates its own measured beta** — independent of any real economic sensitivity to the market. A mega-cap holding 7% of an ETF is partly "correlated" with that ETF simply because it *is* 7% of it, by construction. ## Isolating the effect: leave-one-out beta Comparing beta against the real benchmark is a noisy way to test this — differences could come from genuine market sensitivity, from the benchmark's hundreds of other holdings, or from the self-weight effect, all tangled together. To isolate just the mechanism, build a **synthetic index** from a small set of live-weighted holdings, then compute each stock's beta against that synthetic index two ways: including itself, and excluding itself (with the remaining weights renormalized to sum to 1). ```python def leave_one_out_beta_report(index_key, window=60, start="2018-01-01"): live_weights = get_live_top_holdings(benchmark_ticker) # real, current ETF weights tickers = list(live_weights.index) results = {} for i in tickers: others = [t for t in tickers if t != i] w_incl = weights / weights.sum() idx_incl = (returns[tickers] * w_incl).sum(axis=1) # synthetic index, including i w_excl = weights[others] / weights[others].sum() idx_excl = (returns[others] * w_excl).sum(axis=1) # synthetic index, excluding i beta_incl = rolling_beta(returns[i], idx_incl, window=window).mean() beta_excl = rolling_beta(returns[i], idx_excl, window=window).mean() results[i] = {"weight": weights[i], "beta_incl_self": beta_incl, "beta_excl_self": beta_excl} return pd.DataFrame(results).T ``` Everything about the calculation is held constant between the two versions — the only thing that changes is whether the stock sits inside or outside the benchmark it's compared against. Any remaining difference is *purely* the mechanical effect. ![Every SOXX constituent shows a higher beta when included in its own benchmark than when excluded — the self-weight effect predicted by the math.](../../images/posts/leave-one-out-beta.png) ## Result Run on the semiconductor sector (SOXX and its top-10 weighted constituents): - **Every single stock** showed a positive self-weight effect (`beta_incl_self > beta_excl_self`) — no exceptions. - **Correlation of index weight with the size of the effect: 0.65** — heavier names get inflated more, exactly as the math predicts. - This is a materially cleaner signal than the raw weight-vs-beta scatter, which only showed a correlation of 0.18 — because that comparison was also picking up noise from the benchmark's other, non-synthetic holdings. ## Why this matters for a model If you're training a **cross-sectional model** — ranking many stocks by predicted beta or predicted alpha — naive beta systematically biases those rankings toward heavily-weighted names, for reasons that have nothing to do with genuine predictive signal. Leave-one-out beta removes that bias and gives a fairer comparison across the universe. ::: {.note} **Data quality caveat.** The synthetic index here is built only from an ETF's live top-10 holdings (the maximum Yahoo Finance exposes via its public fund-data endpoint) — not the true full index. Treat this as a demonstration of the *mechanism*, not an exact real-world beta figure. Live weights come from `Ticker.funds_data.top_holdings`; there is no free source for full, point-in-time historical index constituents, which is a genuine limitation for anyone trying to build a fully survivorship-bias-free backtest universe on a budget. ::: ::: {.download-block} <div class="dl-title">Full analysis</div> [Download this post (PDF)](index.pdf){.btn .btn-primary} [Download the notebook (.ipynb)](/files/notebooks/index_beta_analysis.ipynb){.btn .btn-outline-secondary} :::