library(knitr)
library(kableExtra)
library(ggplot2)
library(dplyr)

Question 1: Single-Factor (Market) Model

Setup and Data

The single-factor (market) model is:

\[R_i - R_f = \alpha + \beta(R_m - R_f) + \varepsilon\]

# Given values
alpha_est   <- 0.0017
alpha_se    <- 0.0020
beta_est    <- 0.98
beta_se     <- 0.17
R2_q1       <- 0.50
E_mkt_prem  <- 0.0070   # 0.70% expressed as decimal
n_q1        <- 96       # months
t_crit      <- 1.98     # critical |t| at 5% level

Part (a): t-statistic for β — Test H₀: β = 0

Formula:

\[t_{\hat{\beta}} = \frac{\hat{\beta} - \beta_0}{SE(\hat{\beta})}\]

Under \(H_0: \beta = 0\), the null value \(\beta_0 = 0\).

t_beta_zero <- (beta_est - 0) / beta_se
cat("t-statistic for H0: beta = 0:", round(t_beta_zero, 4), "\n")
## t-statistic for H0: beta = 0: 5.765
cat("Critical |t|:", t_crit, "\n")
## Critical |t|: 1.98
cat("Reject H0?", abs(t_beta_zero) > t_crit, "\n")
## Reject H0? TRUE

Decision: Since \(|t| = 5.7647| > 1.98\), we reject \(H_0: \beta = 0\) at the 5% significance level.

Economic Interpretation of β:
\(\hat{\beta} = 0.98\) means that for every 1% increase in the market excess return, the fund’s excess return is expected to increase by approximately 0.98%. The fund has slightly less systematic (market) risk than the market portfolio itself. A \(\beta < 1\) indicates a defensive fund — it amplifies market moves by a factor of 0.98, meaning it underperforms the market in bull markets but loses slightly less in bear markets.

Part (b): Test H₀: β = 1

Formula:

\[t_{\hat{\beta}=1} = \frac{\hat{\beta} - 1}{SE(\hat{\beta})}\]

t_beta_one <- (beta_est - 1) / beta_se
cat("t-statistic for H0: beta = 1:", round(t_beta_one, 4), "\n")
## t-statistic for H0: beta = 1: -0.1176
cat("Critical |t|:", t_crit, "\n")
## Critical |t|: 1.98
cat("Reject H0?", abs(t_beta_one) > t_crit, "\n")
## Reject H0? FALSE

Decision: Since \(|t| = 0.1176 < 1.98\), we fail to reject \(H_0: \beta = 1\) at the 5% level.

Interpretation: Although the point estimate of \(\hat{\beta} = 0.98\) is slightly below 1, it is not statistically distinguishable from 1. The fund’s systematic risk is statistically consistent with having the same market exposure as the market portfolio itself (i.e., consistent with a passive market index fund in terms of systematic risk).

Part (c): t-statistic for α (Jensen’s Alpha)

Formula:

\[t_{\hat{\alpha}} = \frac{\hat{\alpha}}{SE(\hat{\alpha})}\]

t_alpha <- alpha_est / alpha_se
cat("t-statistic for alpha:", round(t_alpha, 4), "\n")
## t-statistic for alpha: 0.85
cat("Critical |t|:", t_crit, "\n")
## Critical |t|: 1.98
cat("Reject H0 (alpha = 0)?", abs(t_alpha) > t_crit, "\n")
## Reject H0 (alpha = 0)? FALSE

Decision: Since \(|t| = 0.85 < 1.98\), we fail to reject \(H_0: \alpha = 0\) at the 5% level.

Assessment of Marketing Claim: The marketing team’s claim of “positive risk-adjusted performance” is not statistically justified. While \(\hat{\alpha} = 0.0017\) (i.e., +0.17% per month above CAPM expectation) is positive in sign, it is indistinguishable from zero at the 5% significance level. The result could easily be due to sampling variation. Advertising positive alpha on this basis would be misleading.

Part (d): Interpreting R²

R2_q1_val        <- R2_q1
systematic_frac  <- R2_q1_val
diversifiable_frac <- 1 - R2_q1_val

cat("R-squared:", R2_q1_val, "\n")
## R-squared: 0.5
cat("Systematic (market) fraction:", round(systematic_frac, 4), "=", 
    round(systematic_frac * 100, 2), "%\n")
## Systematic (market) fraction: 0.5 = 50 %
cat("Diversifiable (idiosyncratic) fraction:", round(diversifiable_frac, 4), "=", 
    round(diversifiable_frac * 100, 2), "%\n")
## Diversifiable (idiosyncratic) fraction: 0.5 = 50 %

Interpretation:
\(R^2 = 0.50\) means that 50% of the variance in the fund’s excess returns is explained by movements in the market excess return — this is the systematic component.

The remaining 50% is diversifiable (idiosyncratic) risk — variance attributable to factors specific to this fund (stock selection, sector bets, manager decisions) that are not explained by broad market movements. For a single-factor model, \(R^2\) is also equal to the squared correlation between the fund return and the market return.

Part (e): CAPM-Implied Expected Monthly Excess Return

Formula:

\[E[R_i - R_f] = \hat{\beta} \times E[R_m - R_f]\]

capm_expected_excess <- beta_est * E_mkt_prem
cat("CAPM-implied expected monthly excess return:", 
    round(capm_expected_excess, 4), "\n")
## CAPM-implied expected monthly excess return: 0.0069
cat("In percentage terms:", round(capm_expected_excess * 100, 4), "%\n")
## In percentage terms: 0.686 %

Result: The CAPM predicts the fund should earn a monthly excess return of:

\[E[R_i - R_f] = 0.98 \times 0.70\% = 0.686\%\]

This is the return the fund must achieve to be fairly compensated for its level of systematic risk. Any realized return above this benchmark would represent alpha.

Question 2: Fama–French Three-Factor Model

Setup and Data

The Fama–French three-factor model is:

\[R_i - R_f = \alpha + b \cdot MKT + s \cdot SMB + h \cdot HML + \varepsilon\]

# Coefficient estimates
alpha_ff  <- 0.0029;  alpha_ff_se  <- 0.0018
b_mkt     <- 0.97;    b_mkt_se     <- 0.08
s_smb     <- 0.75;    s_smb_se     <- 0.11
h_hml     <- -0.13;   h_hml_se     <- 0.13

R2_ff     <- 0.92
adjR2_ff  <- 0.918
n_ff      <- 144
t_crit_ff <- 1.98

Part (f): t-statistics for All Four Coefficients

Formula for each coefficient:

\[t_{\hat{\theta}} = \frac{\hat{\theta}}{SE(\hat{\theta})}\]

t_alpha_ff <- alpha_ff / alpha_ff_se
t_b_mkt    <- b_mkt   / b_mkt_se
t_s_smb    <- s_smb   / s_smb_se
t_h_hml    <- h_hml   / h_hml_se

results_ff <- data.frame(
  Coefficient = c("Intercept (α)", "MKT (b)", "SMB (s)", "HML (h)"),
  Estimate    = c(alpha_ff, b_mkt, s_smb, h_hml),
  Std_Error   = c(alpha_ff_se, b_mkt_se, s_smb_se, h_hml_se),
  t_statistic = round(c(t_alpha_ff, t_b_mkt, t_s_smb, t_h_hml), 4),
  Significant = abs(c(t_alpha_ff, t_b_mkt, t_s_smb, t_h_hml)) > t_crit_ff
)

kable(results_ff, caption = "Fama-French Three-Factor Model: t-statistics") %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"),
                full_width = FALSE)
Fama-French Three-Factor Model: t-statistics
Coefficient Estimate Std_Error t_statistic Significant
Intercept (α) 0.0029 0.0018 1.611 FALSE
MKT (b) 0.9700 0.0800 12.125 TRUE
SMB (s) 0.7500 0.1100 6.818 TRUE
HML (h) -0.1300 0.1300 -1.000 FALSE

Summary of significance at 5% level (|t| > 1.98):

for (i in 1:nrow(results_ff)) {
  sig_label <- ifelse(results_ff$Significant[i], "SIGNIFICANT", "NOT significant")
  cat(sprintf("%-20s t = %7.4f  -->  %s\n",
              results_ff$Coefficient[i],
              results_ff$t_statistic[i],
              sig_label))
}
## Intercept (α)       t =  1.6111  -->  NOT significant
## MKT (b)              t = 12.1250  -->  SIGNIFICANT
## SMB (s)              t =  6.8182  -->  SIGNIFICANT
## HML (h)              t = -1.0000  -->  NOT significant
  • α: \(t = 1.6111\) → borderline; NOT significant at 5% (just below threshold)
  • MKT (b): \(t = 12.125\) → Significant
  • SMB (s): \(t = 6.8182\) → Significant
  • HML (h): \(t = -1\) → NOT significant

Part (g): Investment Style Classification

cat("SMB loading (s):", s_smb, " -- Sign: POSITIVE, Magnitude: LARGE\n")
## SMB loading (s): 0.75  -- Sign: POSITIVE, Magnitude: LARGE
cat("HML loading (h):", h_hml, " -- Sign: NEGATIVE, Magnitude: SMALL\n")
## HML loading (h): -0.13  -- Sign: NEGATIVE, Magnitude: SMALL

Size Tilt — SMB loading:
\(\hat{s} = 0.75 > 0\) and statistically significant. A positive SMB loading means the fund co-moves with small-cap stocks relative to large-cap stocks. The fund has a clear small-cap bias.

Value/Growth Tilt — HML loading:
\(\hat{h} = -0.13 < 0\) (negative). A negative HML loading means the fund co-moves with growth stocks (low book-to-market) rather than value stocks (high book-to-market). However, \(h\) is not statistically significant (\(|t| = 1 < 1.98\)), so this growth tilt is weak and uncertain.

Conclusion: The fund is best classified as a small-cap, mild growth fund — strongly tilted toward small-capitalization stocks, with a weak (statistically insignificant) inclination toward growth stocks.

Part (h): Intercept Interpretation — Does the Manager Add Value?

cat("Alpha estimate:", alpha_ff, "(i.e.,", alpha_ff * 100, "% per month)\n")
## Alpha estimate: 0.0029 (i.e., 0.29 % per month)
cat("t-statistic for alpha:", round(t_alpha_ff, 4), "\n")
## t-statistic for alpha: 1.611
cat("Critical |t|:", t_crit_ff, "\n")
## Critical |t|: 1.98
cat("Significant?", abs(t_alpha_ff) > t_crit_ff, "\n")
## Significant? FALSE
annualized_alpha <- (1 + alpha_ff)^12 - 1
cat("Annualized alpha (approx):", round(annualized_alpha * 100, 4), "%\n")
## Annualized alpha (approx): 3.536 %

Interpretation:
\(\hat{\alpha} = 0.0029\) implies approximately +0.29% per month (roughly 3.54% annualized) of return above what the three Fama–French factors predict.

However, with \(t = 1.6111\), this alpha is not statistically significant at the 5% level (just below the threshold of 1.98). The evidence for genuine managerial skill beyond the three factor exposures is suggestive but not conclusive. We cannot reject the hypothesis that the true alpha is zero. The manager does not convincingly add value beyond the factor exposures at conventional significance levels.

Part (i): R² Rise from 0.75 to 0.92; Adjusted R² as Comparison Metric

R2_single  <- 0.75
R2_ff_val  <- 0.92
adjR2_ff_val <- 0.918

improvement_R2 <- R2_ff_val - R2_single
cat("Single-factor R²:", R2_single, "\n")
## Single-factor R²: 0.75
cat("FF Three-factor R²:", R2_ff_val, "\n")
## FF Three-factor R²: 0.92
cat("Improvement in R²:", round(improvement_R2, 4), "\n")
## Improvement in R²: 0.17
cat("FF Adjusted R²:", adjR2_ff_val, "\n")
## FF Adjusted R²: 0.918

Rise from 0.75 to 0.92:
Adding SMB and HML as additional factors increases explained variance by 17 percentage points. This means the SMB and HML factors collectively capture an additional 17% of the variation in the fund’s returns that the market factor alone could not explain — reflecting the fund’s substantial small-cap exposure. The three-factor model is a materially better description of this fund’s return-generating process.

Why Adjusted R² is the Appropriate Metric:
\(R^2\) can only increase (or stay the same) when additional predictors are added, even if those predictors have no true explanatory power. This mechanical inflation makes raw \(R^2\) unsuitable for comparing models with different numbers of predictors.

The adjusted \(R^2\) penalizes for the number of predictors:

\[\bar{R}^2 = 1 - \frac{(1-R^2)(n-1)}{n-k-1}\]

where \(n\) is the number of observations and \(k\) is the number of predictors. It increases only if the new predictor improves fit by more than chance would predict. With \(\bar{R}^2 = 0.918\) for the three-factor model (versus \(R^2 = 0.92\)), the small penalty for two additional factors confirms the improvement is genuine, not mechanical.

Question 3: Logistic Regression for Market Direction

Setup

The logistic regression model is:

\[\text{logit}\, P(\text{Up}) = \beta_0 + \beta_1 r_{t-1} + \beta_2 \Delta VIX_{t-1}\]

\[P(\text{Up}) = \frac{1}{1 + e^{-(\beta_0 + \beta_1 r_{t-1} + \beta_2 \Delta VIX_{t-1})}}\]

beta0   <- -0.02
beta1   <-  5.4
beta2   <- -0.38
r_lag   <-  0.010
dvix    <-  1.5
thresh  <-  0.5

Part (j): Predicted Probability and Class

Step 1 — Compute the linear predictor (log-odds):

\[\eta = \beta_0 + \beta_1 \cdot r_{t-1} + \beta_2 \cdot \Delta VIX\]

eta <- beta0 + beta1 * r_lag + beta2 * dvix
cat("Linear predictor (log-odds), eta:", round(eta, 4), "\n")
## Linear predictor (log-odds), eta: -0.536
# Sigmoid function
p_up <- 1 / (1 + exp(-eta))
cat("Predicted probability P(Up):", round(p_up, 4), "\n")
## Predicted probability P(Up): 0.3691
cat("Predicted class (threshold 0.5):", ifelse(p_up >= thresh, "Up", "Down"), "\n")
## Predicted class (threshold 0.5): Down

Calculation:

\[\eta = -0.02 + 5.4 \times 0.010 + (-0.38) \times 1.5 = -0.536\]

\[P(\text{Up}) = \frac{1}{1 + e^{-(-0.536)}} = 0.3691\]

Predicted class: Since \(P(\text{Up}) = 0.3691 < 0.5\), the model predicts “Down” for tomorrow.

Part (k): Economic Interpretation of β₁ and β₂

cat("beta1 (lagged return):", beta1, " -- POSITIVE\n")
## beta1 (lagged return): 5.4  -- POSITIVE
cat("beta2 (delta VIX):", beta2, " -- NEGATIVE\n")
## beta2 (delta VIX): -0.38  -- NEGATIVE

β₁ = 5.4 (Positive — Lagged Return):
A positive \(\beta_1\) means that a higher lagged return increases the log-odds of an “Up” day tomorrow. This captures a momentum effect: days following positive returns are more likely to be positive themselves. The large magnitude (5.4) indicates that even a 1% positive lagged return substantially increases the probability of a subsequent up day.

β₂ = −0.38 (Negative — ΔVIX):
A negative \(\beta_2\) means that an increase in VIX decreases the probability of an “Up” day. This captures the fear gauge effect: rising VIX signals increasing market fear/uncertainty, which is associated with negative or volatile market returns. When investors become more fearful (VIX rises), the market is more likely to decline or be turbulent the next day.

Part (l): Confusion Matrix Metrics

# Confusion matrix values
TP <- 67  # Predicted Up, Actual Up
FP <- 44  # Predicted Up, Actual Down
FN <- 33  # Predicted Down, Actual Up
TN <- 56  # Predicted Down, Actual Down
N  <- TP + FP + FN + TN

cat("Confusion Matrix:\n")
## Confusion Matrix:
cm_df <- data.frame(
  " "           = c("Predicted Up", "Predicted Down", "Total"),
  Actual_Up     = c(TP, FN, TP + FN),
  Actual_Down   = c(FP, TN, FP + TN),
  Total         = c(TP + FP, FN + TN, N),
  check.names = FALSE
)
kable(cm_df, caption = "Confusion Matrix (200-day holdout)") %>%
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)
Confusion Matrix (200-day holdout)
Actual_Up Actual_Down Total
Predicted Up 67 44 111
Predicted Down 33 56 89
Total 100 100 200

Formulas and Calculations:

\[\text{Accuracy} = \frac{TP + TN}{N}\]

\[\text{Sensitivity (Recall)} = \frac{TP}{TP + FN}\]

\[\text{Specificity} = \frac{TN}{TN + FP}\]

\[\text{Precision} = \frac{TP}{TP + FP}\]

accuracy    <- (TP + TN) / N
sensitivity <- TP / (TP + FN)
specificity <- TN / (TN + FP)
precision   <- TP / (TP + FP)

metrics_df <- data.frame(
  Metric    = c("Accuracy", "Sensitivity (Recall for Up)", 
                "Specificity", "Precision (for Up)"),
  Formula   = c("(TP+TN)/N", "TP/(TP+FN)", "TN/(TN+FP)", "TP/(TP+FP)"),
  Value     = round(c(accuracy, sensitivity, specificity, precision), 4)
)

kable(metrics_df, caption = "Performance Metrics") %>%
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)
Performance Metrics
Metric Formula Value
Accuracy (TP+TN)/N 0.6150
Sensitivity (Recall for Up) TP/(TP+FN) 0.6700
Specificity TN/(TN+FP) 0.5600
Precision (for Up) TP/(TP+FP) 0.6036
  • Accuracy = \((67 + 56)/200 = 0.615\) → The model correctly classifies 61.5% of all days.
  • Sensitivity = \(67/100 = 0.67\) → Of actual “Up” days, 67% are correctly identified.
  • Specificity = \(56/100 = 0.56\) → Of actual “Down” days, 56% are correctly identified.
  • Precision = \(67/111 = 0.6036\) → Of all “Up” predictions, 60.36% are correct.

Part (m): Naive Baseline Comparison and Limitations of Accuracy

# Naive majority class: predict "Up" always (100 Up, 100 Down -- balanced)
# With balanced classes, majority = either class at 50%
actual_up   <- 100
actual_down <- 100
naive_accuracy <- max(actual_up, actual_down) / N
cat("Naive majority-class accuracy:", round(naive_accuracy, 4), 
    "(", round(naive_accuracy*100, 2), "%)\n")
## Naive majority-class accuracy: 0.5 ( 50 %)
cat("Model accuracy:", round(accuracy, 4), "(", round(accuracy*100, 2), "%)\n")
## Model accuracy: 0.615 ( 61.5 %)
cat("Model beats naive?", accuracy > naive_accuracy, "\n")
## Model beats naive? TRUE
cat("Improvement:", round((accuracy - naive_accuracy)*100, 2), "percentage points\n")
## Improvement: 11.5 percentage points

Naive Accuracy:
The classes are balanced (100 Up, 100 Down). A naive classifier that always predicts the majority class (either “Up” or “Down”) achieves:

\[\text{Naive Accuracy} = \frac{\max(100, 100)}{200} = 0.5 = 50\%\]

Does the model beat it? Yes — the model achieves 61.5% accuracy vs. the naive 50.0%, a gain of 11.5 percentage points.

Why Accuracy Alone is Inadequate for Trading:

Accuracy treats all errors equally (a missed “Up” day = a missed “Down” day), but in a trading system the costs of errors are asymmetric:

  • A false positive (“Up” predicted, market goes Down) causes the trader to enter a losing long position.
  • A false negative (“Down” predicted, market goes Up) causes the trader to miss a profitable day.

These two errors have very different P&L consequences depending on position sizes, transaction costs, and leverage. Accuracy also ignores the magnitude of returns — correctly predicting a +3% day is far more valuable than correctly predicting a +0.1% day.

More Economically Relevant Criterion:
A better metric for a trading system is the Sharpe ratio of the strategy derived from the model’s signals — this directly measures risk-adjusted profitability. Alternatively, precision (of “Up” predictions) is more relevant when the cost of false positives (bad trades) is high. For directional trading, the F1 score (harmonic mean of precision and recall) or Matthews Correlation Coefficient (MCC) are more balanced than raw accuracy.

Question 4: Resampling and Regularization in a Backtest

Setup

mu_monthly  <- 0.0070   # 0.70% mean monthly return
sigma_monthly <- 0.0550  # 5.50% std dev
n_months    <- 48

Part (n): Monthly Sharpe Ratio and Annualization

Monthly Sharpe Ratio formula (assuming risk-free rate is embedded in excess returns):

\[SR_{monthly} = \frac{\bar{r}}{\hat{\sigma}}\]

Annualization: Monthly Sharpe ratios are annualized by multiplying by \(\sqrt{12}\), since returns scale linearly with time while standard deviations scale with the square root of time.

\[SR_{annual} = SR_{monthly} \times \sqrt{12}\]

SR_monthly  <- mu_monthly / sigma_monthly
scaling     <- sqrt(12)
SR_annual   <- SR_monthly * scaling

cat("Mean monthly return:", mu_monthly, "(", mu_monthly*100, "%)\n")
## Mean monthly return: 0.007 ( 0.7 %)
cat("Monthly std deviation:", sigma_monthly, "(", sigma_monthly*100, "%)\n")
## Monthly std deviation: 0.055 ( 5.5 %)
cat("Monthly Sharpe Ratio:", round(SR_monthly, 4), "\n")
## Monthly Sharpe Ratio: 0.1273
cat("Scaling factor: sqrt(12) =", round(scaling, 4), "\n")
## Scaling factor: sqrt(12) = 3.464
cat("Annualized Sharpe Ratio:", round(SR_annual, 4), "\n")
## Annualized Sharpe Ratio: 0.4409

Results:

\[SR_{monthly} = \frac{0.0070}{0.0550} = 0.1273\]

\[SR_{annual} = 0.1273 \times \sqrt{12} = 0.1273 \times 3.4641 = 0.4409\]

Scaling factor used: \(\sqrt{12}\), based on the i.i.d. returns assumption under which variance scales as \(12\sigma^2\) per year, so standard deviation scales as \(\sqrt{12}\,\sigma\).

Part (o): Bootstrap Procedure for Sharpe SE

Bootstrap Procedure — Step by Step:

steps_df <- data.frame(
  Step = 1:6,
  Description = c(
    "Collect the original sample of n = 48 monthly returns: {r1, r2, ..., r48}",
    "Draw B = 1,000 (or 10,000) bootstrap samples, each of size n = 48, by resampling WITH replacement from the original data",
    "For each bootstrap sample b, compute the sample mean (mu*_b) and sample std dev (sigma*_b)",
    "Compute the bootstrap Sharpe ratio: SR*_b = mu*_b / sigma*_b",
    "Repeat steps 2-4 B times to obtain the distribution {SR*_1, SR*_2, ..., SR*_B}",
    "The bootstrap standard error of the Sharpe ratio is: SE_boot = std({SR*_1, ..., SR*_B})"
  )
)

kable(steps_df, caption = "Bootstrap Procedure for Sharpe Ratio Standard Error") %>%
  kable_styling(bootstrap_options = c("striped", "hover")) %>%
  column_spec(1, bold = TRUE, width = "1cm") %>%
  column_spec(2, width = "14cm")
Bootstrap Procedure for Sharpe Ratio Standard Error
Step Description
1 Collect the original sample of n = 48 monthly returns: {r1, r2, …, r48}
2 Draw B = 1,000 (or 10,000) bootstrap samples, each of size n = 48, by resampling WITH replacement from the original data
3 For each bootstrap sample b, compute the sample mean (mu*_b) and sample std dev (sigma*_b)
4 Compute the bootstrap Sharpe ratio: SR*_b = mu*_b / sigma*_b
5 Repeat steps 2-4 B times to obtain the distribution {SR*_1, SR*_2, …, SR*_B}
6 The bootstrap standard error of the Sharpe ratio is: SE_boot = std({SR*_1, …, SR*_B}) 
set.seed(42)
# Simulate 48 monthly returns consistent with the problem parameters
returns_sim <- rnorm(n_months, mean = mu_monthly, sd = sigma_monthly)

B <- 5000  # number of bootstrap replications
SR_boot <- numeric(B)

for (b in 1:B) {
  boot_sample <- sample(returns_sim, size = n_months, replace = TRUE)
  SR_boot[b]  <- mean(boot_sample) / sd(boot_sample)
}

SE_boot_monthly <- sd(SR_boot)
cat("Bootstrap SE of monthly Sharpe ratio:", round(SE_boot_monthly, 4), "\n")
## Bootstrap SE of monthly Sharpe ratio: 0.1495
cat("Bootstrap SE of annualized Sharpe:", round(SE_boot_monthly * sqrt(12), 4), "\n")
## Bootstrap SE of annualized Sharpe: 0.5178
# Quick visualization
hist(SR_boot * sqrt(12), breaks = 40, col = "steelblue", border = "white",
     main = "Bootstrap Distribution of Annualized Sharpe Ratio",
     xlab = "Annualized Sharpe Ratio",
     sub = paste0("SE = ", round(SE_boot_monthly * sqrt(12), 4)))
abline(v = SR_annual, col = "red", lwd = 2, lty = 2)
legend("topright", legend = c("Sample Sharpe"), col = "red", lty = 2, lwd = 2)

Why i.i.d. Bootstrap is Inappropriate for Monthly Returns:

Monthly financial returns exhibit serial dependence — specifically:

  1. Volatility clustering (ARCH effects): High-volatility months tend to cluster together, and low-volatility months do too. Standard i.i.d. resampling destroys this temporal structure.
  2. Autocorrelation: Returns may exhibit momentum (positive autocorrelation) or mean reversion, both of which are broken when observations are shuffled randomly.

When these dependencies are present, i.i.d. resampling underestimates the true variability of the Sharpe ratio estimator, producing artificially narrow confidence intervals.

The fix: Use the block bootstrap (specifically, the stationary block bootstrap by Politis & Romano, or the circular block bootstrap). Instead of resampling individual observations, contiguous blocks of consecutive months are resampled. The block length is chosen to be long enough to preserve the serial dependence structure. This maintains the temporal correlation within blocks while still achieving variance estimation consistency.

Part (p): LASSO λ Selection

lambda_min_cv <- 0.030; factors_min <- 14
lambda_1se    <- 0.065; factors_1se <- 7

cat("Min-CV lambda:", lambda_min_cv, "-> factors retained:", factors_min, "\n")
## Min-CV lambda: 0.03 -> factors retained: 14
cat("1-SE rule lambda:", lambda_1se, "-> factors retained:", factors_1se, "\n")
## 1-SE rule lambda: 0.065 -> factors retained: 7
cat("Factor reduction:", factors_min - factors_1se, "fewer factors with 1-SE rule\n")
## Factor reduction: 7 fewer factors with 1-SE rule

Recommendation: Deploy \(\lambda = 0.065\) (the 1-SE rule solution) with 7 factors.

Reasoning:

The 1-SE rule selects the most parsimonious model (highest regularization) whose cross-validation error is within one standard error of the minimum. This is the preferred choice in a financial backtest context for several reasons:

  1. Overfitting / Data Snooping Risk: With 60 candidate factors over a finite backtest, there is a high probability that some of the 14 factors retained by \(\lambda_{min}\) are spurious — they improve in-sample fit through noise-fitting, not true predictive signal. The 1-SE rule’s 7-factor model is less susceptible to this.

  2. Parsimony and Interpretability: Fewer factors are easier to interpret, monitor, and trade. A 7-factor model is more robust to factor collinearity and parameter instability.

  3. Out-of-Sample Generalization: The CV error difference between the two models is not statistically meaningful (within one standard error), yet the simpler model carries substantially lower variance. By the bias-variance tradeoff, the 7-factor model will typically generalize better to new data.

  4. Transaction Costs: More factors often mean more rebalancing signals and higher turnover, increasing trading costs. A sparser model is more practical to implement.

The additional 7 factors in the min-CV model represent marginal in-sample improvement that is likely not recoverable out-of-sample.

Part (q): Walk-Forward (Time-Respecting) Validation

# Illustrative walk-forward scheme
scheme_df <- data.frame(
  Window = paste("Window", 1:5),
  Training_Period   = c("Months 1–36",  "Months 1–42",  "Months 1–48",  
                         "Months 1–54",  "Months 1–60"),
  Test_Period       = c("Months 37–42", "Months 43–48", "Months 49–54", 
                         "Months 55–60", "Months 61–66"),
  Type = rep("Expanding Window", 5)
)

kable(scheme_df, caption = "Illustrative Walk-Forward (Expanding Window) Scheme") %>%
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)
Illustrative Walk-Forward (Expanding Window) Scheme
Window Training_Period Test_Period Type
Window 1 Months 1–36 Months 37–42 Expanding Window
Window 2 Months 1–42 Months 43–48 Expanding Window
Window 3 Months 1–48 Months 49–54 Expanding Window
Window 4 Months 1–54 Months 55–60 Expanding Window
Window 5 Months 1–60 Months 61–66 Expanding Window

Walk-Forward Procedure — Step by Step:

  1. Initial Training Window: Use the first \(T_0\) months (e.g., 36 months) to estimate the LASSO model — select \(\lambda\) via inner cross-validation on this training window and train the final model.
  2. Test on the Next Block: Apply the fitted model to predict returns on the next \(h\) months (e.g., 6 months). Record the out-of-sample predictions and returns.
  3. Expand/Roll Forward: Extend the training window by \(h\) months (expanding window) — or shift it forward by \(h\) months (rolling window of fixed length). Re-estimate the model.
  4. Repeat: Continue until the full dataset is exhausted.
  5. Evaluate: Aggregate all out-of-sample predictions and compute the overall performance metrics (Sharpe ratio, hit rate, etc.) on the collected test-period observations.

Note on inner cross-validation for λ selection: Within each training window, a nested time-series cross-validation (or hold-out) must be used to select \(\lambda\) — never using future data.

Why Random k-Fold CV is Unsafe for This Problem:

Standard random k-fold CV randomly assigns observations to folds, meaning:

  1. Look-ahead bias / data leakage: Future observations end up in the training fold while earlier observations are in the test fold. The model “sees the future” during training, producing optimistically biased performance estimates.
  2. Dependence violation: Financial return data are temporally dependent (due to autocorrelation, momentum, and macro cycles). Random shuffling breaks this structure and violates the i.i.d. assumption underlying standard k-fold CV theory.
  3. Factor selection contamination: With LASSO over 60 factors, random CV would select factors based on future information, dramatically inflating the apparent predictive power in a way that cannot be replicated in live trading.

The walk-forward scheme ensures the model only ever trains on data available at the time of prediction, mirroring actual live trading conditions and producing realistic out-of-sample performance estimates.

Summary

summary_df <- data.frame(
  Question = c(
    "Q1(a): β t-stat (H₀:β=0)",
    "Q1(b): β t-stat (H₀:β=1)",
    "Q1(c): α t-stat",
    "Q1(d): Systematic R²",
    "Q1(e): CAPM Expected Excess Return",
    "Q2(f): Significant Factors",
    "Q2(g): Fund Style",
    "Q2(h): Manager Alpha",
    "Q3(j): P(Up)",
    "Q3(l): Accuracy",
    "Q3(l): Sensitivity",
    "Q3(l): Specificity",
    "Q3(l): Precision",
    "Q3(m): Naive Accuracy",
    "Q4(n): Monthly SR",
    "Q4(n): Annualized SR"
  ),
  Result = c(
    paste0("t = ", round((0.98-0)/0.17, 4), " → Reject H₀"),
    paste0("t = ", round((0.98-1)/0.17, 4), " → Fail to Reject H₀"),
    paste0("t = ", round(0.0017/0.0020, 4), " → NOT significant"),
    "50% systematic, 50% diversifiable",
    paste0(round(0.98*0.0070*100, 4), "% per month"),
    "MKT, SMB significant; α borderline; HML not",
    "Small-cap, mild growth tilt",
    paste0("α = +0.29%/month, NOT significant (t = ", round(0.0029/0.0018, 4), ")"),
    paste0(round(1/(1+exp(-(-0.02+5.4*0.010+(-0.38)*1.5))), 4), " → Predict Down"),
    paste0(round((67+56)/200, 4)),
    paste0(round(67/100, 4)),
    paste0(round(56/100, 4)),
    paste0(round(67/111, 4)),
    "0.5000 (balanced classes)",
    paste0(round(0.0070/0.0550, 4)),
    paste0(round((0.0070/0.0550)*sqrt(12), 4))
  )
)

kable(summary_df, caption = "Complete Summary of Key Results") %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"),
                full_width = TRUE) %>%
  row_spec(0, bold = TRUE)
Complete Summary of Key Results
Question Result
Q1(a): β t-stat (H₀:β=0) t = 5.7647 → Reject H₀
Q1(b): β t-stat (H₀:β=1) t = -0.1176 → Fail to Reject H₀
Q1(c): α t-stat t = 0.85 → NOT significant
Q1(d): Systematic R² 50% systematic, 50% diversifiable
Q1(e): CAPM Expected Excess Return 0.686% per month
Q2(f): Significant Factors MKT, SMB significant; α borderline; HML not
Q2(g): Fund Style Small-cap, mild growth tilt
Q2(h): Manager Alpha α = +0.29%/month, NOT significant (t = 1.6111)
Q3(j): P(Up) 0.3691 → Predict Down
Q3(l): Accuracy 0.615
Q3(l): Sensitivity 0.67
Q3(l): Specificity 0.56
Q3(l): Precision 0.6036
Q3(m): Naive Accuracy 0.5000 (balanced classes)
Q4(n): Monthly SR 0.1273
Q4(n): Annualized SR 0.4409

Key Takeaways

Q1 — Single-Factor Model: The fund has market beta statistically indistinguishable from 1.0, and its Jensen’s alpha (+0.17%/month) is not statistically significant. The single-factor model explains 50% of return variation, leaving substantial idiosyncratic risk.

Q2 — Fama–French: Adding SMB and HML lifts explanatory power from R²=0.75 to R²=0.92. The fund is clearly small-cap tilted (strong, significant SMB loading). Alpha of +0.29%/month is economically meaningful but just misses statistical significance — the manager shows promise but not proven skill.

Q3 — Logistic Regression: The model captures momentum (positive β₁) and the fear premium (negative β₂). It achieves 61.5% accuracy vs. 50% naive — meaningful but not dominant. Precision (60.4%) matters more than accuracy for trading applications.

Q4 — Resampling & LASSO: The annualized Sharpe of 0.4409 is estimated from a short 48-month window — bootstrap SEs would reveal substantial uncertainty. The 1-SE LASSO (7 factors) is preferred over the minimum-CV solution (14 factors) to guard against overfitting. Walk-forward validation is essential to avoid look-ahead bias that would render backtest results meaningless.