Ahmer Nadeem Khan
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Statistical Alpha Signal Combination & Robust Backtesting in Long/Short Equity

Thirteen cross-sectional equity factors, run through a statistical gate on the S&P 100 (2020–2026), then combined into one signal. Every information coefficient is corrected for the autocorrelation that overlapping forward returns inject and carries a bootstrap confidence interval, so a factor is kept only on robust evidence. The result is a market-neutral signal (beta ≈ 0, Sharpe 0.99) that is profitable across the stable bull and bear sub-periods and takes its main drawdown at the 2022–2023 regime transition.

Signal in-sample tear sheet

Sharpe
0.99
Ann. return
+13.2%
Max drawdown
−16.5%
Beta to SPY
−0.02
Win rate
55.9%
Profit factor
1.18
Ann. alpha
+13.5%
Ann. volatility
13.5%
Sortino
0.94
Calmar
0.80
VaR 95%
−1.4%
CVaR 95%
−1.9%
Cumulative growth of $1
transition Alpha +73% SPY +69%
0.75×1.00×1.25×1.50×1.75× 20222023202420252026
Turnover per rebalance
0

Gross fraction of the book traded at each 31-day rebalance.

Drawdown
0−5%−10%−15%
Monthly returns (%)
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
2021
-1.1
2022
+5.2
+1.1
-0.0
+4.9
+1.9
+2.5
-5.9
+2.4
-0.4
+0.4
-1.4
+3.3
2023
-10.5
-0.1
+0.4
+1.2
-3.4
-0.7
+2.4
-1.0
+0.4
-1.2
+3.4
-0.9
2024
+1.5
+6.6
-2.0
-1.7
+3.3
+2.4
-5.4
+0.3
+1.7
+2.2
+3.6
+3.4
2025
+0.6
-3.4
+0.9
+8.9
+3.8
-1.1
+4.7
-0.4
+3.2
+2.5
-3.5
+3.7
2026
+2.7
-1.2
+1.8
+9.3
+10.7
-3.5

SignalSigned IC-weighting (EWM, 126-day half-life) of Sector-Neutral Momentum + Low Volatility

ExecutionRebalanced every 31 days, 10 bps cost, dollar-neutral unit-exposure long/short on S&P 100 constituents, benchmarked against SPY over Dec 2021–Jun 2026

The sample holds one bear regime (about 8 independent 31-day windows), so the bear Sharpe (0.89) is suggestive but not statistically significant. The universe is current S&P 100 membership, so survivorship bias inflates the absolute figures. And factor selection is in-sample; a stricter study would split signal generation over train and test.

The two factors

Sector-Neutral Momentum. Each stock's twelve-month return, skipping the most recent month ($P_{t-21}/P_{t-252} - 1$), minus the average return of its GICS sector. A positive score means the stock is outrunning its sector peers. Subtracting the sector average keeps the factor from turning into a bet on whichever sector led. (Moskowitz and Grinblatt, 1999.) Note: Moskowitz and Grinblatt in fact argue against the residual. They find that industry movement explains much of momentum, so the within-sector component left after sector-neutralizing is substantially weaker. Neutralizing here is a deliberate trade of raw strength for diversification against a concentrated sector bet.
Low Volatility. The negative of each stock's trailing 21-day return volatility ($-\operatorname{std}$ of daily returns). A positive score means calmer recent price action, so the book leans toward the steadiest names, the documented tendency of low-volatility stocks to earn higher risk-adjusted returns. (Ang, Hodrick, Xing and Zhang, 2006.)

Combining the factors

Equal-weight. Z-score each factor across the cross-section and average them with fixed weights $1/k$.
IC-weighted. Weight each factor by its trailing information coefficient (IC), the correlation between the factor's scores and the next period's returns, a running measure of how well it has ranked winners from losers. Signs are kept and the weights are scaled to unit gross each day ($\sum_j |w_j| = 1$), so a factor whose recent IC has turned negative is flipped and held short rather than applied in its losing direction. The trailing average can be a flat window or exponentially weighted; the selected signal uses an EWM with a 126-day half-life, letting the weights re-orient toward recent behaviour. The weights are lagged one day.
Lasso. Regress the next period's returns on the z-scored factor scores over a trailing window, refit each day, and use the fitted coefficients as weights. The L1 penalty drives redundant factors to exactly zero, pricing in the correlation between factors that independent IC-weighting ignores. Each training window does not contain the current date $t$.

Statistical Gating and Robustness

Newey–West (HAC) t-statistic. Overlapping multi-day forward returns make adjacent IC observations serially correlated, which deflates the naive standard error and inflates the t-stat. The HAC correction widens the error to absorb that autocorrelation: $$t_{\text{NW}} = \frac{\bar{\text{IC}}}{\hat{\sigma}_{\text{NW}} / \sqrt{T}}, \qquad \hat{\sigma}^2_{\text{NW}} = \hat{\gamma}_0 + 2\sum_{j=1}^{L}\left(1 - \frac{j}{L+1}\right)\hat{\gamma}_j$$ with $L = h-1$ (horizon minus one) and $\hat{\gamma}_j$ the autocovariances. Without it, overlapping 21-day returns inflate t-statistics several-fold.
Block bootstrap 95% CI. Resamples the IC series in blocks of length $h$ to preserve serial dependence, takes the mean IC of each of $B = 10{,}000$ resamples, and reports the 2.5th and 97.5th percentiles. A factor is retained only when this interval excludes zero.
IC decay. Mean IC recomputed against the $h$-day-forward return across horizons, $\overline{\text{IC}}(h) = \operatorname{mean}_t \operatorname{corr}\big(S_{i,t},\, r_{i,\,t \to t+h}\big)$ for $h = 1 \ldots 20$ days. The horizon where predictive power is strongest before it fades marks the signal's natural holding period, which informs the rebalance cadence.

The thirteen factors. Momentum: Momentum (12–1), Cross-Sectional Momentum, Sector-Neutral Momentum, Residual Momentum. Mean-reversion: Short-Term Reversal, RSI, Bollinger z-score. Risk: Low Volatility, Rolling Sharpe, Quality proxy. Tail: Rolling CVaR, Rolling Max Drawdown, Rolling Skewness.

Execution

Daily cross-section
$S_{i,t}$
Dollar-neutral
$w_{i,t} = S_{i,t} - \bar{S}_t$
Unit gross exposure
$\hat{w}_{i,t} = \dfrac{w_{i,t}}{\sum_j |w_{j,t}|}$
Holding period
$p_{i,t} = \hat{w}_{i,\,\tau(t)}$
Cost model
$$\begin{gathered} R_t \\ = \textstyle\sum_i p_{i,t}\, r_{i,t} - c\,\text{TO}_t \end{gathered}$$

No lookahead. The forward return $r_{i,t} = P_{i,t+1}/P_{i,t} - 1$ pays today's positions the move from $t$ to $t{+}1$.

Costs. Turnover $\text{TO}_t = \sum_i |p_{i,t} - p_{i,t-1}|$ is billed at $c = 10$ bps, with the first day charged the full build $\text{TO}_0 = \sum_i |p_{i,0}|$.

Residual beta. Dollar-neutrality ($\sum_i w_i = 0$) does not imply beta-neutrality. Writing each stock's beta as $\beta_i = \bar{\beta} + \tilde{\beta}_i$, the book's beta is $\beta_p = \sum_i w_i \beta_i = \bar{\beta}\sum_i w_i + \sum_i w_i \tilde{\beta}_i = \sum_i w_i \tilde{\beta}_i$: the average term cancels for free, but a residual survives when the long and short baskets carry unequal average beta. So beta sits near zero only for a balanced book, not by construction. The realized beta in the chosen signal displayed above is an outcome of dollar-neutrality, but beta-neutrality would require additional constraints (orthogonality to the estimated beta vector) which complicates the backtesting and execution.

Held weights. The holding period is $H = 31$ trading days, and $\tau(t) = H\lfloor t/H \rfloor$ is the most recent rebalance date. The target vector is scaled to unit gross exposure ($\sum_i |\hat{w}_{i,t}| = 1$) so the portfolio is normalized in terms of total wealth and daily returns stay comparable regardless of the signal's magnitude. Between rebalances the book carries the target weight vector $\hat{w}_{i,\tau(t)}$ rather than a fixed number of shares, so it is re-set to those weights every day. A buy-and-hold book instead fixes share counts at the rebalance and lets the weights drift as prices move. Holding the weights fixed is a small execution simplification; since the vector is set at the rebalance from past data, it adds no lookahead.

Simplified cost. The 10 bps rate is linear in turnover and identical for every name, so it stands in for spread and commissions but ignores market impact (which grows with trade size), borrow fees on the short leg, and per-name liquidity. It is a screening-grade estimate: adequate for ranking candidates on equal footing, and optimistic for a large book.