Given:
Formula: \[t = \frac{\hat{\beta} - \beta_0}{\text{SE}(\hat{\beta})}\]
beta_hat <- 0.98
se_beta <- 0.17
beta_null <- 0 # H0: beta = 0
t_crit <- 1.98
# t-statistic
t_beta <- (beta_hat - beta_null) / se_beta
t_beta <- round(t_beta, 4)
cat("t-statistic for H0: beta = 0:", t_beta, "\n")## t-statistic for H0: beta = 0: 5.7647## Critical |t|: 1.98## Reject H0? TRUEResult: \(t = \frac{0.98 - 0}{0.17} = 5.7647\)
Since \(|t| = 5.7647\) \(>\) \(1.98\), we reject \(H_0: \beta = 0\) at the 5% significance level.
Economic interpretation: A \(\hat{\beta} = 0.98\) means the fund moves almost in lockstep with the market. For every 1% increase in the market excess return, the fund’s excess return increases by approximately 0.98%, indicating near-market-level systematic risk.
beta_null1 <- 1 # H0: beta = 1
t_beta1 <- (beta_hat - beta_null1) / se_beta
t_beta1 <- round(t_beta1, 4)
cat("t-statistic for H0: beta = 1:", t_beta1, "\n")## t-statistic for H0: beta = 1: -0.1176## Critical |t|: 1.98## Reject H0? FALSEResult: \(t = \frac{0.98 - 1}{0.17} = -0.1176\)
Since \(|t| = 0.1176\) \(<\) \(1.98\), we fail to reject \(H_0: \beta = 1\) at the 5% significance level.
Interpretation: The fund’s systematic risk is statistically indistinguishable from the market (\(\beta = 1\)). The fund does not exhibit significantly higher or lower market sensitivity than a passive index, suggesting it provides no meaningful leverage or de-risking relative to the market portfolio.
alpha_hat <- 0.0017
se_alpha <- 0.0020
alpha_null <- 0 # H0: alpha = 0
t_alpha <- (alpha_hat - alpha_null) / se_alpha
t_alpha <- round(t_alpha, 4)
cat("t-statistic for Jensen's alpha:", t_alpha, "\n")## t-statistic for Jensen's alpha: 0.85## Critical |t|: 1.98## Reject H0? FALSEResult: \(t = \frac{0.0017 - 0}{0.0020} = 0.85\)
Since \(|t| = 0.85\) \(<\) \(1.98\), we fail to reject \(H_0: \alpha = 0\).
Conclusion: The marketing team’s claim of “positive risk-adjusted performance” is not statistically justified. While the point estimate \(\hat{\alpha} = 0.0017\) is positive, it is statistically indistinguishable from zero at the 5% level. The observed alpha could easily be due to sampling variability rather than genuine manager skill.
## R-squared: 0.5## Systematic variation (%): 50## Idiosyncratic / diversifiable variation (%): 50Interpretation: \(R^2 = 0.50\) means that 50% of the fund’s return variation is explained by systematic (market) risk, and the remaining 50% is idiosyncratic (diversifiable) risk. This relatively low \(R^2\) suggests the fund holds significant unsystematic exposures — perhaps concentrated positions or sector bets — that are not captured by the single market factor.
Formula: \[E[R_i - R_f] = \hat{\beta} \times E[R_m - R_f]\]
E_market_premium <- 0.0070 # 0.70% = 0.0070
E_excess_return <- beta_hat * E_market_premium
E_excess_return_pct <- round(E_excess_return * 100, 4)
cat("CAPM-implied expected monthly excess return:", E_excess_return_pct, "%\n")## CAPM-implied expected monthly excess return: 0.686 %Result: \(E[R_i - R_f] = 0.98 \times 0.70\% = 0.686\%\)
Given:
| Term | Estimate | Std. Error |
|---|---|---|
| \(\hat{\alpha}\) | 0.0029 | 0.0018 |
| \(\hat{b}\) (MKT) | 0.97 | 0.08 |
| \(\hat{s}\) (SMB) | 0.75 | 0.11 |
| \(\hat{h}\) (HML) | −0.13 | 0.13 |
Formula: \(t_k = \frac{\hat{\theta}_k}{\text{SE}(\hat{\theta}_k)}\)
# Estimates
coefs <- c(alpha = 0.0029, MKT = 0.97, SMB = 0.75, HML = -0.13)
ses <- c(alpha = 0.0018, MKT = 0.08, SMB = 0.11, HML = 0.13)
t_stats <- round(coefs / ses, 4)
significant <- abs(t_stats) > t_crit
results_q2f <- data.frame(
Term = names(coefs),
Estimate = coefs,
Std_Error = ses,
t_stat = t_stats,
Significant = significant
)
rownames(results_q2f) <- NULL
print(results_q2f)## Term Estimate Std_Error t_stat Significant
## 1 alpha 0.0029 0.0018 1.6111 FALSE
## 2 MKT 0.9700 0.0800 12.1250 TRUE
## 3 SMB 0.7500 0.1100 6.8182 TRUE
## 4 HML -0.1300 0.1300 -1.0000 FALSESummary:
## SMB loading (s): 0.75## HML loading (h): -0.13##
## Size tilt: Positive and large SMB => SMALL-CAP tilt## Value/Growth tilt: Negative HML => GROWTH tiltInterpretation:
Overall style: The fund is best characterized as a small-cap growth fund, though the growth dimension is not statistically confirmed.
alpha_ff <- 0.0029
se_alpha_ff <- 0.0018
t_alpha_ff <- round(alpha_ff / se_alpha_ff, 4)
cat("FF3 Alpha:", alpha_ff, "\n")## FF3 Alpha: 0.0029## t-statistic: 1.6111## Significant at 5%? FALSEResult: \(t_\alpha = \frac{0.0029}{0.0018} = 1.6111\)
Since \(|t| = 1.6111\) \(<\) \(1.98\), the FF3 alpha is not statistically significant at the 5% level.
Interpretation: After controlling for market, size, and value exposures, the fund earns a monthly alpha of 0.29%, but this is indistinguishable from zero statistically. We cannot conclude that the manager adds value beyond the three Fama–French factor exposures. The positive point estimate may reflect sampling variation rather than genuine stock-selection or timing skill.
## CAPM R2: 0.75## FF3 R2: 0.92## FF3 Adjusted R2: 0.918## Improvement in R2: 0.17Interpretation of the rise from 0.75 to 0.92: Adding the SMB and HML factors explains an additional 17% of the fund’s return variation. This indicates that the fund’s returns are materially driven by size and value premiums, not just broad market movements. The single-factor model was leaving substantial systematic variation unexplained.
Why adjusted \(R^2\) is appropriate for model comparison: Raw \(R^2\) mechanically increases with every added predictor, even irrelevant ones. The adjusted \(R^2\) penalizes for the number of predictors: \[\bar{R}^2 = 1 - \frac{(1-R^2)(n-1)}{n-k-1}\] where \(k\) is the number of regressors. Here, adjusted \(R^2 = 0.918\) (vs raw \(R^2 = 0.920\)) — very close, confirming that the added factors (SMB, HML) provide genuine explanatory power and are not inflating fit spuriously.
Model: \[\text{logit}\, P(\text{Up}) = \beta_0 + \beta_1 r_{t-1} + \beta_2 \Delta VIX_{t-1}\]
Coefficients: \(\beta_0 = -0.02\), \(\beta_1 = 5.4\), \(\beta_2 = -0.38\)
Today’s inputs: \(r_{t-1} = 0.010\), \(\Delta VIX = 1.5\)
Formulas: \[\text{logit} = \beta_0 + \beta_1 r_{t-1} + \beta_2 \Delta VIX\] \[P(\text{Up}) = \frac{e^{\text{logit}}}{1 + e^{\text{logit}}} = \frac{1}{1 + e^{-\text{logit}}}\]
b0 <- -0.02
b1 <- 5.4
b2 <- -0.38
r_lag <- 0.010
dVIX <- 1.5
logit_val <- b0 + b1 * r_lag + b2 * dVIX
prob_up <- 1 / (1 + exp(-logit_val))
pred_class <- ifelse(prob_up >= 0.5, "Up", "Down")
cat("Logit value:", round(logit_val, 4), "\n")## Logit value: -0.536## P(Up) : 0.3691## Predicted class (threshold = 0.5): DownStep-by-step:
\[\text{logit} = -0.02 + 5.4(0.010) + (-0.38)(1.5) = -0.02 + 0.054 - 0.57 = -0.536\]
\[P(\text{Up}) = \frac{1}{1 + e^{-(-0.536)}} = 0.3691\]
Predicted class: Down (probability \(0.3691 < 0.5\))
\(\beta_1 = 5.4 > 0\) (lagged return): A positive lagged return increases the log-odds of an “Up” day tomorrow. This reflects short-term momentum — a day of positive returns tends to be followed by another positive day. This is consistent with return autocorrelation documented at short horizons.
\(\beta_2 = -0.38 < 0\) (\(\Delta VIX\)): A rise in the VIX (fear index) decreases the log-odds of an “Up” day. This captures risk aversion and market stress — when implied volatility spikes, investors become fearful and equity returns tend to be negative or weak the following day.
Confusion Matrix (200-day hold-out):
| Actual Up | Actual Down | Total | |
|---|---|---|---|
| Predicted Up | 67 | 44 | 111 |
| Predicted Down | 33 | 56 | 89 |
| Total | 100 | 100 | 200 |
TP <- 67 # True Positive (Predicted Up, Actual Up)
FP <- 44 # False Positive (Predicted Up, Actual Down)
FN <- 33 # False Negative (Predicted Down, Actual Up)
TN <- 56 # True Negative (Predicted Down, Actual Down)
N <- 200
# Accuracy
accuracy <- (TP + TN) / N
# Sensitivity (True Positive Rate for "Up")
sensitivity <- TP / (TP + FN)
# Specificity (True Negative Rate for "Down")
specificity <- TN / (TN + FP)
# Precision for "Up" predictions
precision <- TP / (TP + FP)
cat("Accuracy :", round(accuracy, 4), "\n")## Accuracy : 0.615## Sensitivity: 0.67## Specificity: 0.56## Precision : 0.6036Results:
\[\text{Accuracy} = \frac{TP + TN}{N} = \frac{67 + 56}{200} = 0.615\]
\[\text{Sensitivity} = \frac{TP}{TP + FN} = \frac{67}{67 + 33} = 0.67\]
\[\text{Specificity} = \frac{TN}{TN + FP} = \frac{56}{56 + 44} = 0.56\]
\[\text{Precision} = \frac{TP}{TP + FP} = \frac{67}{67 + 44} = 0.6036\]
# With balanced classes (100 Up, 100 Down), majority class = either (both 50%)
# Naive rule predicts the same class always
majority_class_count <- max(100, 100) # Both classes are equal
naive_accuracy <- majority_class_count / N
cat("Naive classifier accuracy:", naive_accuracy, "\n")## Naive classifier accuracy: 0.5## Model accuracy: 0.615## Model beats naive rule? TRUENaive rule accuracy: Since both classes are balanced (100 “Up”, 100 “Down”), a naive majority-class predictor achieves \(\frac{100}{200} = 0.5000\) (50%) accuracy.
Model accuracy: \(0.615\) — the model beats the naive rule (\(61.5\% > 50\%\)).
Why accuracy alone is inadequate for a trading system: Accuracy treats false positives and false negatives symmetrically, but the costs of trading errors are asymmetric. Missing a genuine “Up” day (false negative) vs. entering a trade on a “Down” day (false positive) have very different P&L consequences. In a long-only trading context: - False positives (predicting Up when market is Down) cause direct monetary losses. - False negatives (predicting Down when market is Up) result in missed gains (opportunity cost).
A more economically relevant criterion is precision (what fraction of “Up” predictions are correct, directly linked to trade win rate) or F1 score balancing precision and recall — or even better, a direct measure of strategy P&L under the predicted signals.
Given: Sample mean monthly return \(\bar{r} = 0.70\%\), sample standard deviation \(s = 5.50\%\), \(n = 48\) months.
Formula (monthly): \[SR_{\text{monthly}} = \frac{\bar{r}}{s}\]
Annualization: Scale by \(\sqrt{12}\) (12 months per year): \[SR_{\text{annual}} = SR_{\text{monthly}} \times \sqrt{12}\]
r_bar <- 0.0070 # 0.70%
s <- 0.0550 # 5.50%
n_months <- 48
SR_monthly <- r_bar / s
SR_annual <- SR_monthly * sqrt(12)
cat("Monthly Sharpe Ratio :", round(SR_monthly, 4), "\n")## Monthly Sharpe Ratio : 0.1273## Scaling factor : sqrt(12) = 3.4641## Annualized Sharpe : 0.4409Results:
\[SR_{\text{monthly}} = \frac{0.0070}{0.0550} = 0.1273\]
\[SR_{\text{annual}} = 0.1273 \times \sqrt{12} = 0.1273 \times 3.4641 = 0.4409\]
Scaling factor: \(\sqrt{12}\), because returns scale by \(T\) and standard deviations scale by \(\sqrt{T}\), so the Sharpe ratio scales by \(\sqrt{T}\).
Step-by-step bootstrap procedure:
set.seed(42)
# Simulate monthly returns for illustration
r_sim <- rnorm(48, mean = 0.0070, sd = 0.0550)
B <- 10000
SR_boot <- numeric(B)
for (b in 1:B) {
r_b <- sample(r_sim, size = 48, replace = TRUE) # i.i.d. resample
SR_boot[b] <- mean(r_b) / sd(r_b)
}
SR_monthly_original <- mean(r_sim) / sd(r_sim)
SE_boot <- sd(SR_boot)
cat("Original monthly SR (simulated):", round(SR_monthly_original, 4), "\n")## Original monthly SR (simulated): 0.073## Bootstrap SE of monthly SR: 0.149cat("95% Bootstrap CI: [",
round(quantile(SR_boot, 0.025), 4), ",",
round(quantile(SR_boot, 0.975), 4), "]\n")## 95% Bootstrap CI: [ -0.2097 , 0.3765 ]Steps:
Why ordinary i.i.d. bootstrap is inappropriate: Monthly financial returns exhibit serial correlation (autocorrelation, volatility clustering, GARCH effects). The i.i.d. bootstrap destroys the time-series dependence structure by resampling observations independently, leading to incorrect standard errors.
Fix — Block Bootstrap: Use a stationary block bootstrap (or overlapping block bootstrap). This resamples contiguous blocks of consecutive observations (e.g., blocks of 3–6 months), preserving local time-series dependence while still generating diverse bootstrap samples.
lambda_min_cv <- 0.030
lambda_1se <- 0.065
factors_min <- 14
factors_1se <- 7
cat("Lambda (min-CV): ", lambda_min_cv, "-> retains", factors_min, "factors\n")## Lambda (min-CV): 0.03 -> retains 14 factors## Lambda (1-SE rule): 0.065 -> retains 7 factorsRecommended choice: \(\lambda = 0.065\) (the 1-standard-error rule)
Rationale:
Parsimony and interpretability: Retaining 7 factors instead of 14 produces a simpler, more transparent model that is easier to maintain and monitor in production.
Overfitting protection: The 1-SE rule selects the most regularized model whose CV error is within one standard error of the minimum. The difference in in-sample CV performance between \(\lambda = 0.030\) and \(\lambda = 0.065\) is statistically negligible, while \(\lambda = 0.065\) provides substantially stronger regularization.
Out-of-sample degradation: In financial applications, in-sample factor signals are prone to data-snooping and overfitting. With 60 candidate factors and only 48–144 months of data, many factors are likely spurious. The sparser model (7 factors) is more likely to generalize to new data.
Transaction costs: Fewer factors typically means fewer positions, lower turnover, and reduced transaction costs in live trading.
# Illustrate walk-forward scheme schematically
T_total <- 60 # total months (illustrative)
train_window <- 36
test_window <- 6
folds <- data.frame(
Fold = integer(),
Train_Start = integer(),
Train_End = integer(),
Test_Start = integer(),
Test_End = integer()
)
fold <- 1
train_end <- train_window
while ((train_end + test_window) <= T_total) {
folds <- rbind(folds, data.frame(
Fold = fold,
Train_Start = 1,
Train_End = train_end,
Test_Start = train_end + 1,
Test_End = train_end + test_window
))
train_end <- train_end + test_window
fold <- fold + 1
}
print(folds)## Fold Train_Start Train_End Test_Start Test_End
## 1 1 1 36 37 42
## 2 2 1 42 43 48
## 3 3 1 48 49 54
## 4 4 1 54 55 60Walk-Forward Scheme (Step-by-Step):
Initial training window: Use the first \(W\) months (e.g., \(W = 36\)) as the training set. Fit the LASSO with the chosen \(\lambda\) and record the selected factors and coefficients.
Out-of-sample test step: Apply the fitted model to the next \(h\) months (e.g., \(h = 6\)) without refitting. Record predicted signals and actual returns.
Roll forward: Expand (or roll) the training window by \(h\) months and re-estimate the model. Repeat until the end of the data.
Aggregate OOS performance: Collect all out-of-sample predictions. Compute strategy returns, Sharpe ratio, maximum drawdown, and other metrics solely from the held-out periods.
Why standard random \(k\)-fold cross-validation is unsafe for this problem:
Random \(k\)-fold CV shuffles observations randomly and assigns them to folds without regard to time. This means the training set for any fold will contain future data relative to some of its test observations — a form of look-ahead bias (data leakage). For instance, if month 48’s return is used to train a model that is then tested on month 12, the model has been “told” information from the future. This produces optimistically biased CV estimates that will not generalize in real trading, where the model can only use information available at the time of the decision.
End of Assignment