Brownian Bridge™

1. You have 500 assets, 20 features, and 10 years of daily data (2,520 observations per asset). You want to predict 1-month (21-day) forward returns using a ridge regression. Why is standard $k$ -fold cross-validation invalid here, and what is the minimum purge gap needed?

Standard

k

-fold is invalid because return distributions are non-normal; purge gap = 21 daysStandard

k

-fold is invalid because: (1) test folds can occur before training folds (training on the future to predict the past), and (2) consecutive observations share 20 days of their 21-day forward return label. The minimum purge gap is 21 days — enough to ensure no overlap between the training label window and the test periodStandard

k

-fold is invalid only for tree models; for ridge regression it is acceptable with

k=10

Standard

k

-fold is invalid because of heteroskedasticity; purge gap = 1 day

2. Ridge regression with penalty $\lambda$ has closed-form solution $\hat{\beta}^{\text{ridge}} = (X^\top X + \lambda I)^{-1} X^\top y$ . As $\lambda \to \infty$ , $\hat{\beta}^{\text{ridge}} \to 0$ . What is the bias and variance of $\hat{\beta}^{\text{ridge}}$ relative to OLS $\hat{\beta}^{\text{OLS}}$ , and under what condition does ridge dominate OLS in MSE?

Ridge has zero bias and lower variance than OLS; it always dominatesRidge introduces bias

-(\lambda(X^\top X + \lambda I)^{-1})\beta

, reduces variance, and dominates OLS in MSE when the squared bias reduction is smaller than the variance reduction — specifically when

\lambda < 2\sigma^2 / \|\beta\|^2

Ridge has the same bias as OLS but higher variance due to the added penaltyRidge dominates OLS only when the design matrix

X^\top X

is singular

3. A quant researcher reports a gradient boosted tree model with in-sample IC = 0.12 and out-of-sample IC = 0.02 over a purged walk-forward evaluation. What does this IC ratio indicate, and what hyperparameter adjustments would you recommend?

IC ratio of 0.02/0.12 = 0.17 — severe overfitting. The model memorised training noise. Reduce model complexity: decrease max_depth (try 2–3), increase min_samples_leaf, decrease n_estimators, or increase the regularisation via subsampling rateIC ratio of 0.17 is acceptable for tree models; increase n_estimators to improve OOS performanceIC ratio of 0.17 means the signal is robust; the low OOS IC is due to transaction costs that should be subtracted from IS ICIC ratio of 0.17 means the model needs more features to generalise

4. Impurity-based feature importance in gradient boosted trees (total reduction in objective function across all splits using each feature) is an unbiased estimator of each feature's true predictive contribution, even when features have different cardinalities or are correlated with each other.

truefalse

5. You construct a cross-sectional equity momentum feature: 12-month return excluding the last month (12-1 momentum). Before training, you winsorise at $\pm3\sigma$ and cross-sectionally standardise. A colleague suggests also using the full 12-month return (including the last month). Why is the 12-1 formulation standard, and what is the danger of using the 12-month return including the last month?

The 12-1 formulation is standard for computational convenience; both are equally valid signalsThe 12-1 formulation excludes the most recent month to avoid short-term reversal contamination: over 1-month horizons, equity returns exhibit negative autocorrelation (bid-ask bounce and microstructure noise create short-term reversal). Including the last month mixes a momentum signal (positive autocorrelation at 2–12 months) with a reversal signal (negative at 1 month), reducing ICThe 12-1 formulation is regulatory — using the last month creates a look-ahead biasThe 12-1 formulation reduces transaction costs by avoiding high-turnover short-term signals

6. The Fundamental Law of Active Management (Grinold 1989) states that the annualised information ratio satisfies $\text{IR} \approx \text{IC} \cdot \sqrt{N}$ , where $N$ is the number of independent bets per year. A strategy has IC = 0.05 and trades monthly across 200 stocks ( $N = 2400$ ). What is the theoretical IR, and why might the realised IR be much lower?

Theoretical IR

\approx 2.45

; realised IR lower because transaction costs reduce net return and because stocks are not independent (factor correlations reduce effective

N

)Theoretical IR

\approx 0.05 \times 200 = 10

; realised IR lower because of margin requirementsTheoretical IR

= \text{IC} = 0.05

;

\sqrt{N}

factor only applies to binary betsTheoretical IR

\approx 2.45

; realised IR will equal this if the IC estimate is unbiased

Quiz: ML in Signal Generation

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