Instructions. Answer all questions and show every formula and calculation. Round to four decimals unless told otherwise. The critical value used for all two-sided 5% t-tests is \(|t| \approx 1.98\).
All numbers below are computed live in R so that every figure is reproducible and verifiable from the code chunks.
A fund’s monthly excess returns are regressed on the market excess return over \(n = 96\) months:
\[ R_i - R_f \;=\; \alpha \;+\; \beta\,(R_m - R_f) \;+\; \varepsilon \]
| Term | Estimate | Std. Error |
|---|---|---|
| Intercept (\(\alpha\)) | 0.0017 | 0.0020 |
| Market premium (\(\beta\)) | 0.98 | 0.17 |
Also reported: \(R^2 = 0.50\), average market risk premium \(E[R_m - R_f] = 0.70\% = 0.0070\), critical \(|t| \approx 1.98\).
n_obs <- 96
alpha <- 0.0017; se_alpha <- 0.0020
beta <- 0.98; se_beta <- 0.17
R2_1 <- 0.50
mkt_prem <- 0.0070 # E[Rm - Rf]
tcrit <- 1.98The t-statistic for a coefficient against a null value \(\beta_0\) is
\[ t \;=\; \frac{\hat\beta - \beta_0}{\mathrm{SE}(\hat\beta)}, \qquad H_0:\beta = 0 \;\Rightarrow\; t = \frac{\hat\beta}{\mathrm{SE}(\hat\beta)}. \]
t_beta0 <- beta / se_beta
r4(t_beta0)## [1] 5.7647\(t = 0.98 / 0.17 = 5.7647\). Since \(|t| = 5.7647 > 1.98\) we reject \(H_0:\beta = 0\) at the 5% level — \(\beta\) is highly significant.
Economic interpretation. \(\beta = 0.98\) means the fund moves almost one-for-one with the market: a 1% rise in the market excess return is associated with about a 0.98% rise in the fund’s excess return. The fund carries essentially market-level systematic (non-diversifiable) risk, marginally defensive since \(\beta < 1\).
\[ t \;=\; \frac{\hat\beta - 1}{\mathrm{SE}(\hat\beta)} \]
t_beta1 <- (beta - 1) / se_beta
r4(t_beta1)## [1] -0.1176\(t = (0.98 - 1)/0.17 = -0.1176\). Since \(|t| = 0.1176 < 1.98\) we fail to reject \(H_0:\beta = 1\).
Conclusion. The fund’s beta is statistically indistinguishable from 1. Its systematic risk is not significantly different from that of the market — we cannot claim it is either more aggressive or more defensive than the market.
\[ t \;=\; \frac{\hat\alpha}{\mathrm{SE}(\hat\alpha)} \]
t_alpha <- alpha / se_alpha
r4(t_alpha)## [1] 0.85\(t = 0.0017 / 0.0020 = 0.85\). Since \(|t| = 0.85 < 1.98\) we fail to reject \(H_0:\alpha = 0\).
On the marketing claim. Although the point estimate of alpha is positive (\(0.17\%\) per month), it is not statistically significant. The data do not justify advertising “positive risk-adjusted performance” — the positive alpha is well within sampling noise and could plausibly be zero (or negative).
\(R^2 = 0.50\) means the market factor explains 50% of the variance of the fund’s excess return. Decomposition of total risk:
c(systematic = r4(R2_1), diversifiable = r4(1 - R2_1))## systematic diversifiable
## 0.5 0.5CAPM sets \(\alpha = 0\), so the expected excess return is driven entirely by beta:
\[ E[R_i - R_f] \;=\; \beta \cdot E[R_m - R_f] \]
capm_er <- beta * mkt_prem
r4(capm_er) # decimal## [1] 0.0069r4(capm_er * 100) # percent## [1] 0.686\(E[R_i - R_f] = 0.98 \times 0.0070 = 0.0069\), i.e. about 0.686% per month.
For a managed equity fund, estimated on \(n = 144\) monthly observations:
\[ R_i - R_f \;=\; \alpha \;+\; b\cdot \text{MKT} \;+\; s\cdot \text{SMB} \;+\; h\cdot \text{HML} \;+\; \varepsilon \]
| Term | Estimate | Std. Error |
|---|---|---|
| Intercept (\(\alpha\)) | 0.0029 | 0.0018 |
| MKT (\(b\)) | 0.97 | 0.08 |
| SMB (\(s\)) | 0.75 | 0.11 |
| HML (\(h\)) | -0.13 | 0.13 |
Also reported: \(R^2 = 0.92\), Adjusted \(R^2 = 0.918\), critical \(|t| \approx 1.98\).
ff <- data.frame(
term = c("alpha", "MKT (b)", "SMB (s)", "HML (h)"),
estimate = c(0.0029, 0.97, 0.75, -0.13),
se = c(0.0018, 0.08, 0.11, 0.13)
)
n_ff <- 144; k_ff <- 3; R2_ff <- 0.92\[ t_j \;=\; \frac{\hat\theta_j}{\mathrm{SE}(\hat\theta_j)} \]
ff$t_stat <- r4(ff$estimate / ff$se)
ff$significant_5pct <- ifelse(abs(ff$t_stat) > tcrit, "YES", "no")
ff| Coefficient | \(t\) | Significant (5%)? |
|---|---|---|
| \(\alpha\) | 1.6111 | no |
| MKT (\(b\)) | 12.125 | YES |
| SMB (\(s\)) | 6.8182 | YES |
| HML (\(h\)) | -1 | no |
Significant at 5%: the market loading \(b\) and the size loading \(s\). Not significant: the intercept \(\alpha\) and the value loading \(h\).
Classification: a small-cap fund with at most a weak (statistically insignificant) growth lean — best described as small-cap, style-neutral on the value–growth dimension.
alpha_ff <- ff$estimate[1]
t_alpha_ff <- ff$t_stat[1]
c(alpha_monthly = r4(alpha_ff), alpha_annual_approx = r4(alpha_ff*12),
t = r4(t_alpha_ff))## alpha_monthly alpha_annual_approx t
## 0.0029 0.0348 1.6111The intercept \(\alpha = 0.0029\) (\(\approx 0.29\%\)/month, roughly \(3.48\%\)/year) is the factor-adjusted (abnormal) return — performance beyond what the three factor exposures explain. Its t-statistic is \(1.6111 < 1.98\), so it is not significant.
Does the manager add value? The point estimate hints at positive skill, but we cannot statistically conclude the manager adds value beyond passive factor exposure: the alpha is within sampling noise of zero. The fund’s returns are largely a repackaging of market + small-cap exposure.
The single-factor CAPM achieved \(R^2 = 0.75\); adding SMB and HML raised it to \(0.92\). This +17 pp jump shows that a large share of what the CAPM treated as idiosyncratic noise is in fact systematic exposure to the size factor (and, to a lesser extent, value). The factor model captures the fund’s behavior far better.
Why adjusted \(R^2\) for model comparison. Ordinary \(R^2\) can never decrease when predictors are added — even useless ones — so it mechanically favors larger models. Adjusted \(R^2\) penalizes the number of predictors \(k\):
\[ \bar R^2 \;=\; 1 - (1 - R^2)\,\frac{n - 1}{\,n - k - 1\,} \]
It rises only when a new predictor improves fit more than expected by chance, making it the fair criterion across models of different dimension. Verifying the reported value:
adj_R2 <- 1 - (1 - R2_ff) * (n_ff - 1) / (n_ff - k_ff - 1)
r4(adj_R2)## [1] 0.9183\(\bar R^2 = 1 - (1-0.92)\frac{143}{140} = 0.9183\), which matches the reported \(0.918\) up to rounding (they agree to three decimals). It is barely below \(R^2 = 0.92\), confirming the extra factors add genuine explanatory power rather than overfitting.
A model predicts the probability that tomorrow’s market return is positive (“Up”):
\[ \operatorname{logit} P(\text{Up}) \;=\; \beta_0 \;+\; \beta_1\,(\text{lagged return } r_{t-1}) \;+\; \beta_2\,(\Delta \text{VIX}_{t-1}) \]
Estimated coefficients: \(\beta_0 = -0.02\), \(\beta_1 = 5.4\), \(\beta_2 = -0.38\). Today’s inputs: \(r_{t-1} = 0.010\), \(\Delta\text{VIX} = 1.5\).
b0 <- -0.02; b1 <- 5.4; b2 <- -0.38
r_lag <- 0.010; d_vix <- 1.5The logistic link and its inverse:
\[ z = \operatorname{logit}P(\text{Up}) = \beta_0 + \beta_1 r_{t-1} + \beta_2 \Delta\text{VIX}, \qquad P(\text{Up}) = \frac{1}{1 + e^{-z}}. \]
z <- b0 + b1*r_lag + b2*d_vix
p_up <- 1 / (1 + exp(-z))
c(logit_z = r4(z), P_Up = r4(p_up))## logit_z P_Up
## -0.5360 0.3691predicted_class <- ifelse(p_up >= 0.5, "Up", "Down")
predicted_class## [1] "Down"\(z = -0.02 + 5.4(0.010) + (-0.38)(1.5) = -0.02 + 0.054 - 0.57 = -0.536\).
\[ P(\text{Up}) = \frac{1}{1 + e^{0.536}} = 0.3691. \]
Since \(P(\text{Up}) = 0.3691 < 0.5\), the predicted class is “Down.”
On a 200-day hold-out test set (0.5 threshold):
| Actual Up | Actual Down | Total | |
|---|---|---|---|
| Predicted Up | 67 | 44 | 111 |
| Predicted Down | 33 | 56 | 89 |
| Total | 100 | 100 | 200 |
TP <- 67; FP <- 44; FN <- 33; TN <- 56
N <- TP + FP + FN + TN
accuracy <- (TP + TN) / N
sensitivity <- TP / (TP + FN) # true-positive rate for "Up" (recall)
specificity <- TN / (TN + FP) # true-negative rate for "Down"
precision <- TP / (TP + FP) # precision for "Up" predictions
data.frame(
metric = c("Accuracy", "Sensitivity (TPR, Up)", "Specificity", "Precision (Up)"),
formula = c("(TP+TN)/N", "TP/(TP+FN)", "TN/(TN+FP)", "TP/(TP+FP)"),
value = r4(c(accuracy, sensitivity, specificity, precision))
)\[ \begin{aligned} \text{Accuracy} &= \frac{TP+TN}{N} = \frac{67+56}{200} = 0.615,\\[4pt] \text{Sensitivity} &= \frac{TP}{TP+FN} = \frac{67}{100} = 0.67,\\[4pt] \text{Specificity} &= \frac{TN}{TN+FP} = \frac{56}{100} = 0.56,\\[4pt] \text{Precision} &= \frac{TP}{TP+FP} = \frac{67}{111} = 0.6036. \end{aligned} \]
actual_up <- TP + FN # 100
actual_down <- FP + TN # 100
naive_acc <- max(actual_up, actual_down) / N
c(actual_up = actual_up, actual_down = actual_down,
naive_accuracy = r4(naive_acc), model_accuracy = r4(accuracy))## actual_up actual_down naive_accuracy model_accuracy
## 100.000 100.000 0.500 0.615Since the two classes are tied (100 Up vs 100 Down), there is no strict majority class. A constant naive classifier that always predicts “Up” (or always predicts “Down”) is therefore correct \(100/200 = 0.5000\) of the time. The model’s accuracy of 0.615 beats this benchmark by \(11.5\) percentage points.
Why accuracy alone is inadequate for a trading system.
A more economically relevant criterion: the risk-adjusted profitability of the resulting strategy — its (out-of-sample) Sharpe ratio, or equivalently the realized P&L / economic value of trading on the signal, which weights each decision by the money actually made or lost.
A candidate strategy earns a sample mean monthly return of \(0.70\%\) with a sample standard deviation of \(5.50\%\) over \(48\) months.
mu_m <- 0.0070 # mean monthly return
sd_m <- 0.0550 # monthly standard deviation
T_m <- 48\[ \text{SR}_{\text{monthly}} = \frac{\bar r}{s_r}, \qquad \text{SR}_{\text{annual}} = \sqrt{12}\;\text{SR}_{\text{monthly}}. \]
SR_m <- mu_m / sd_m
scale <- sqrt(12)
SR_ann <- SR_m * scale
c(monthly_Sharpe = r4(SR_m), scaling_factor = r4(scale),
annualized_Sharpe = r4(SR_ann))## monthly_Sharpe scaling_factor annualized_Sharpe
## 0.1273 3.4641 0.4409\(\text{SR}_{\text{monthly}} = 0.0070/0.0550 = 0.1273\).
Scaling factor \(= \sqrt{12} = 3.4641\). It is \(\sqrt{12}\) (not \(12\)) because, under i.i.d. returns, the mean scales with the horizon \(T\) while the standard deviation scales with \(\sqrt{T}\); their ratio therefore scales with \(\sqrt{T}\).
\[ \text{SR}_{\text{annual}} = \sqrt{12}\times 0.1273 = 0.4409. \]
Because monthly returns are serially dependent, use a block bootstrap (its i.i.d. special case is just block length = 1). This makes no normality assumption about the return distribution.
Procedure (block bootstrap), step by step:
(No SE is computed here because the raw 48-month return series is not provided — the question asks for the procedure, which is described above.)
Why the ordinary i.i.d. bootstrap is inappropriate here. It would draw individual months independently (step 2 with \(\ell = 1\)). But monthly financial returns are not independent: they exhibit serial dependence — autocorrelation in the mean and, especially, volatility clustering (GARCH-type effects). Resampling single months destroys this time-series structure and biases the standard error, typically understating it.
The fix is the block bootstrap used above, which resamples contiguous blocks of consecutive months and so preserves the short-run autocorrelation and volatility clustering. The stationary (Politis–Romano) bootstrap, which randomizes the block length, is the standard refinement.
Deploy \(\lambda = 0.065\) (the 1-SE, 7-factor model). The 1-SE rule chooses the most parsimonious model whose CV error is within one standard error of the minimum. The CV curve is noisy and its minimum typically sits at a \(\lambda\) that is too small, retaining factors that will not generalize. With only \(48\) months but \(60\) candidate factors — a high-dimensional, low-sample, multiple- testing setting highly prone to overfitting and data-snooping — the simpler 7-factor model is more robust out-of-sample, more stable, and more interpretable. The negligible loss in in-sample CV fit buys substantial protection against fitting noise.
Time-respecting (walk-forward) scheme:
Why standard random k-fold CV is unsafe. Random k-fold shuffles observations, so the training folds routinely contain future months used to predict past ones. This is look-ahead bias / data leakage: it exploits information unavailable in real time and yields over-optimistic, unrealizable performance. It also breaks the temporal autocorrelation structure and lets information bleed between adjacent (correlated) months. Walk-forward respects the arrow of time — only the past is ever used to predict the future — giving an honest estimate of live performance.
| Item | Value |
|---|---|
| Q1a t(beta=0) | 5.7647 |
| Q1b t(beta=1) | -0.1176 |
| Q1c t(alpha) | 0.8500 |
| Q1e CAPM E[Ri-Rf] | 0.0069 |
| Q2 t(alpha) | 1.6111 |
| Q2 t(MKT) | 12.1250 |
| Q2 t(SMB) | 6.8182 |
| Q2 t(HML) | -1.0000 |
| Q2 adj R^2 | 0.9183 |
| Q3j P(Up) | 0.3691 |
| Q3l accuracy | 0.6150 |
| Q3l sensitivity | 0.6700 |
| Q3l specificity | 0.5600 |
| Q3l precision | 0.6036 |
| Q3m naive acc | 0.5000 |
| Q4n monthly Sharpe | 0.1273 |
| Q4n annualized Sharpe | 0.4409 |
Document generated with R Markdown; all figures rounded to four decimals.