This document answers all four questions of the final examination of Application of Financial Software Package. Every coefficient test uses the two-sided rule reject \(H_0\) when \(|t| > t_{crit} \approx 1.98\) (the large-sample 5% two-tailed critical value). All numerical results are computed in R so the analysis is fully reproducible, and figures are rounded to four decimals unless stated otherwise.
The general single-coefficient \(t\)-statistic for a hypothesised value \(\theta_0\) is
\[ t \;=\; \frac{\hat{\theta} - \theta_0}{\widehat{\mathrm{SE}}(\hat{\theta})}. \]
The market model regresses the fund’s excess return on the market excess return over \(n = 96\) months:
\[ R_i - R_f \;=\; \alpha + \beta\,(R_m - R_f) + \varepsilon . \]
# --- Reported regression output -------------------------------------------
alpha <- 0.0017; se_alpha <- 0.0020 # Jensen's alpha & its SE
beta <- 0.98; se_beta <- 0.17 # market beta & its SE
R2_1 <- 0.50 # coefficient of determination
mkt_prem <- 0.0070 # E[R_m - R_f] = 0.70% per month
n1 <- 96 # sample size
tcrit <- 1.98 # critical |t| at 5%\[ t_{\beta=0} = \frac{\hat\beta - 0}{\mathrm{SE}(\hat\beta)} = \frac{0.98}{0.17}. \]
t_beta0 <- (beta - 0) / se_beta
data.frame(statistic = "t (beta = 0)",
value = r4(t_beta0),
crit = tcrit,
decision = ttest_decision(t_beta0, tcrit))Result. \(t_{\beta=0} = 5.7647\), which far exceeds \(1.98\), so we reject \(H_0\) at the 5% level: the market beta is highly significant.
Economic interpretation. \(\hat\beta = 0.98\) means that for every \(1\%\) change in the market excess return, the fund’s excess return moves by about \(0.98\%\). The fund carries essentially the same systematic (non-diversifiable) exposure to market risk as the market portfolio itself. Beta is the slope of the security characteristic line and measures the fund’s contribution to a diversified investor’s portfolio risk.
\[ t_{\beta=1} = \frac{\hat\beta - 1}{\mathrm{SE}(\hat\beta)} = \frac{0.98 - 1}{0.17}. \]
t_beta1 <- (beta - 1) / se_beta
data.frame(statistic = "t (beta = 1)",
value = r4(t_beta1),
crit = tcrit,
decision = ttest_decision(t_beta1, tcrit))Result. \(t_{\beta=1} = -0.1176\), which is tiny in magnitude, so we fail to reject \(H_0:\beta = 1\).
Interpretation. Although \(\hat\beta = 0.98 \ne 1\) numerically, the difference from one is statistically indistinguishable from sampling noise. The fund’s systematic risk is not statistically different from that of the market: it is neither aggressive (\(\beta>1\)) nor defensive (\(\beta<1\)) in any reliable sense. Note the contrast with part (a): the same coefficient is significantly different from 0 yet not different from 1 — the hypothesis being tested, not just the estimate, determines the conclusion.
\[ t_{\alpha} = \frac{\hat\alpha - 0}{\mathrm{SE}(\hat\alpha)} = \frac{0.0017}{0.0020}. \]
t_alpha <- alpha / se_alpha
data.frame(statistic = "t (alpha = 0)",
value = r4(t_alpha),
crit = tcrit,
decision = ttest_decision(t_alpha, tcrit))Result. \(t_{\alpha} = 0.85 < 1.98\), so we fail to reject \(H_0:\alpha = 0\).
Does the data justify advertising “positive risk-adjusted performance”? No. Jensen’s alpha is the average return in excess of the CAPM benchmark. The point estimate is positive (\(0.17\%\) per month), but with a standard error of \(0.20\%\) it is statistically indistinguishable from zero. The fund’s apparent outperformance is well within the range expected from luck. Advertising “statistically positive risk-adjusted performance” would be misleading: the honest statement is that there is no reliable evidence of skill (positive alpha).
\(R^2 = 0.50\) means that 50% of the variation in the fund’s excess returns is explained by the market factor — this is the systematic portion of risk that cannot be diversified away. The remaining 50% is idiosyncratic (firm-specific, diversifiable) variation captured by the residual \(\varepsilon\). In a market model, \(R^2\) is exactly the fraction of total variance attributable to systematic market movements.
Under CAPM the intercept is zero in equilibrium, so the expected excess return is
\[ E[R_i - R_f] = \beta \cdot E[R_m - R_f] = 0.98 \times 0.70\%. \]
capm_er <- beta * mkt_prem
cat("CAPM-implied monthly excess return:",
r4(capm_er), "=", r4(capm_er * 100), "% per month\n")## CAPM-implied monthly excess return: 0.0069 = 0.686 % per monthThe CAPM-implied expected excess return is 0.686% per month (\(0.0069\)). This is the return the fund should earn purely as compensation for bearing market risk; any genuine skill would appear as a positive, significant alpha on top of this — which, from part (c), we do not observe.
Estimated on \(n = 144\) monthly observations:
\[ R_i - R_f = \alpha + b\cdot \mathrm{MKT} + s\cdot \mathrm{SMB} + h\cdot \mathrm{HML} + \varepsilon . \]
ff <- data.frame(
term = c("Intercept (alpha)", "MKT (b)", "SMB (s)", "HML (h)"),
est = c(0.0029, 0.97, 0.75, -0.13),
se = c(0.0018, 0.08, 0.11, 0.13)
)
R2_2 <- 0.92
adjR2_2 <- 0.918
n2 <- 144\[ t_j = \frac{\hat\theta_j}{\mathrm{SE}(\hat\theta_j)}, \qquad j \in \{\alpha,b,s,h\}. \]
ff$t <- ff$est / ff$se
ff$sig <- ifelse(abs(ff$t) > tcrit, "Significant", "Not significant")
transform(ff, est = r4(est), se = r4(se), t = r4(t))Conclusions at the 5% level:
data.frame(
loading = c("SMB (size)", "HML (value/growth)"),
estimate = c(ff$est[3], ff$est[4]),
significant = c(abs(ff$t[3]) > tcrit, abs(ff$t[4]) > tcrit)
)Style label: a small-cap fund that is essentially style-neutral on the value/growth dimension (a small-cap “blend,” leaning marginally and insignificantly toward growth).
cat("alpha =", r4(ff$est[1]), "(", r4(ff$est[1]*100), "% per month ),",
"t =", r4(ff$t[1]), "->", ttest_decision(ff$t[1], tcrit), "\n")## alpha = 0.0029 ( 0.29 % per month ), t = 1.6111 -> Fail to reject H0The intercept is the three-factor (FF) alpha: average return after controlling for market, size, and value exposures. The estimate is positive (\(0.29\%\) per month) but \(t = 1.6111 < 1.98\), so it is not statistically significant. The manager therefore does not demonstrably “add value” beyond what is mechanically generated by loading on the market and small-cap factors. Much of what a single-factor model might have mislabeled as skill is, in fact, a size premium the manager harvests passively. Genuine skill requires a significantly positive alpha, which is absent.
data.frame(
model = c("CAPM (1 factor)", "Fama-French (3 factors)"),
R2 = c(0.75, R2_2),
adjusted_R2 = c(NA, adjR2_2)
)Moving from the one-factor CAPM (\(R^2 = 0.75\)) to the three-factor model (\(R^2 = 0.92\)) shows that the size (SMB) and value (HML) factors explain a large block of return variation the market factor alone misses — consistent with the highly significant SMB loading found in (f). Roughly \(17\) percentage points of additional variation are now attributed to systematic factor exposure rather than left in the residual.
Why adjusted \(R^2\) for model comparison. Ordinary \(R^2\) can never decrease when predictors are added, even if those predictors are pure noise, so it mechanically favours larger models and cannot diagnose overfitting. Adjusted \(R^2\) applies a degrees-of-freedom penalty,
\[ R^2_{\text{adj}} = 1 - (1 - R^2)\,\frac{n-1}{\,n-k-1\,}, \]
rewarding added regressors only if they improve fit by more than chance. Here \(R^2_{\text{adj}} = 0.918\) sits essentially on top of \(R^2 = 0.92\), confirming that SMB and HML earn their place — the improvement is genuine, not an artifact of adding parameters.
\[ \operatorname{logit} P(\text{Up}) = \beta_0 + \beta_1\,r_{t-1} + \beta_2\,\Delta\mathrm{VIX}_{t-1}, \qquad P(\text{Up}) = \frac{1}{1 + e^{-\,\text{logit}}}. \]
logit_val <- b0 + b1 * r_lag + b2 * dVIX
p_up <- 1 / (1 + exp(-logit_val))
class_pred <- ifelse(p_up >= 0.5, "Up", "Down")
cat("logit =", r4(logit_val), "\n")## logit = -0.536## P(Up) = 0.3691## Predicted class (0.5 threshold): DownStep by step,
\[ \text{logit} = -0.02 + 5.4(0.010) + (-0.38)(1.5) = -0.536, \]
\[ P(\text{Up}) = \frac{1}{1+e^{-(-0.536)}} = 0.3691. \]
Because \(0.3691 < 0.5\), the predicted class is “Down.” Intuitively, the large negative VIX-change contribution (\(-0.57\) in log-odds) overwhelms the small positive momentum contribution (\(+0.054\)), pushing the probability below one-half.
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
accuracy <- (TP + TN) / N
sensitivity <- TP / (TP + FN) # true-positive rate for "Up"
specificity <- TN / (TN + FP) # true-negative rate
precision <- TP / (TP + FP) # PPV 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))
)actual_up <- TP + FN # 100
actual_down <- FP + TN # 100
naive_acc <- max(actual_up, actual_down) / N
data.frame(
rule = c("Model (0.5 threshold)", "Naive majority-class"),
accuracy = r4(c(accuracy, naive_acc))
)The classes are balanced (\(100\) up, \(100\) down), so a majority-class rule scores \(0.5\). The model’s \(0.615\) beats the naive rule by about 11.5 percentage points — a real but modest edge.
Why accuracy alone is inadequate for a trading system. Accuracy weights every error equally and ignores the economics of the bet:
A more economically relevant criterion is the risk-adjusted return of the strategy itself — the (out-of-sample) Sharpe ratio of the P&L generated by trading on the signal. Closely related useful measures are expected profit/utility per trade (cost-aware), or the AUC/ROC when one wants a threshold-independent ranking quality.
mu_m <- 0.0070 # sample mean monthly return = 0.70%
sd_m <- 0.0550 # sample SD monthly return = 5.50%
n_obs <- 48 # months\[ SR_{\text{monthly}} = \frac{\bar r}{s} = \frac{0.0070}{0.0550}, \qquad SR_{\text{annual}} = \sqrt{12}\; SR_{\text{monthly}}. \]
SR_month <- mu_m / sd_m
scale_factor <- sqrt(12)
SR_annual <- SR_month * scale_factor
cat("Monthly Sharpe :", r4(SR_month), "\n")## Monthly Sharpe : 0.1273## Scaling factor : 3.4641 (= sqrt(12))## Annualized Sharpe: 0.4409The monthly Sharpe is 0.1273 and the annualized Sharpe is 0.4409. The scaling factor is \(\sqrt{12} \approx 3.4641\): under the i.i.d. assumption the annual mean scales by \(12\) while the annual standard deviation scales by \(\sqrt{12}\), so the ratio scales by \(12/\sqrt{12} = \sqrt{12}\). (Implicitly the risk-free rate is treated as \(0\) for the excess return here.)
A nonparametric bootstrap estimates \(\mathrm{SE}(\widehat{SR})\) without a normality assumption:
Why the ordinary i.i.d. bootstrap is inappropriate here. Monthly financial returns are not i.i.d.: they exhibit serial dependence — autocorrelation in the mean and, especially, volatility clustering (GARCH effects). Resampling individual months independently destroys this temporal dependence, which biases the variance estimate (typically understating the true standard error) and yields over-optimistic confidence intervals.
The fix: a block bootstrap. Resample contiguous blocks of consecutive months so that within-block dependence is preserved — the moving-block bootstrap, or the stationary bootstrap of Politis & Romano (random block lengths, which keeps the resampled series stationary). The block length should grow with \(n\) to capture the dependence horizon.
# Illustrative block-bootstrap for a Sharpe-ratio SE.
# `r_series` would be the vector of 48 monthly returns (not provided here).
library(boot)
sharpe <- function(x) mean(x) / sd(x)
# tsboot preserves dependence via fixed-length blocks (moving-block bootstrap)
set.seed(1)
bb <- tsboot(r_series, statistic = sharpe,
R = 10000, l = 6, sim = "fixed") # block length l ~ 6 months
se_SR <- sd(bb$t) # bootstrap standard error
ci_SR <- quantile(bb$t, c(0.025, 0.975)) # percentile 95% CIdata.frame(
rule = c("Minimum-CV-error", "One-standard-error (1-SE)"),
lambda = c(0.030, 0.065),
factors_retained = c(14, 7)
)I would deploy the one-standard-error solution: \(\lambda = 0.065\) with 7 factors. The 1-SE rule selects the most regularized (sparsest) model whose cross-validation error is within one standard error of the minimum — i.e. the simplest model that is statistically indistinguishable in predictive accuracy from the best one. The reasons are sharper in finance:
If the sole objective were raw predictive accuracy on a stationary process with ample data, the minimum-CV \(\lambda\) could be defended — but for a deployable trading strategy, the robustness of the 1-SE choice wins.
A valid out-of-sample scheme must never let future data inform the past.
Why standard random \(k\)-fold CV is unsafe here. Random \(k\)-folding shuffles observations, so a training fold routinely contains months that occur after the test months. This is look-ahead bias / data leakage: the model is allowed to “see the future” when predicting the past, which is impossible in live trading and inflates measured performance. Because returns are temporally ordered and serially dependent, only a forward-chaining (walk-forward) protocol reproduces the real information set available at each decision date and gives an honest estimate of deployable performance.