Statistical Data Analysis and Inference › Worked Example: Analyzing Simulated Strategy Returns
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Let's walk through an illustrative analysis of a seeded synthetic return series with small gains and occasional larger losses.
# Simulate an illustrative i.i.d. return mixture with a heavy left tail
np.random.seed(42)
n_days = 504 # Two years of trading
# Base simulated returns with slight positive drift
base_returns = np.random.normal(0.0003, 0.012, n_days)
# Inject occasional large losses into the synthetic series
crash_mask = np.random.random(n_days) < 0.02 # 2% chance of crash day
base_returns[crash_mask] = np.random.normal(-0.04, 0.015, crash_mask.sum())
strategy_returns = base_returnsStrategy Performance Analysis ================================================== Count: 504 Mean (daily): -0.000211 Mean (annualized, linear): -0.053113 Std Dev (daily): 0.012655 Std Dev (annualized): 0.200888 Skewness: -0.247155 Excess Kurtosis: 1.341945 Min: -0.060398 Max: 0.046533 Sharpe Ratio (simple returns, zero reference): -0.264389
In this simulation, the deliberately injected loss days produce negative sample skewness and higher sample kurtosis. Those results describe only the specified synthetic data-generating process.
# Test whether the modeled gross mean return differs from zero
t_stat, p_val = stats.ttest_1samp(strategy_returns, 0)
# Confidence intervals
n = len(strategy_returns)
se = np.std(strategy_returns, ddof=1) / np.sqrt(n)
t_crit = stats.t.ppf(0.975, n - 1)
mean_return = np.mean(strategy_returns)
ci_daily = (mean_return - t_crit * se, mean_return + t_crit * se)
ci_annual = (ci_daily[0] * 252, ci_daily[1] * 252)Statistical Inference ================================================== t-statistic: -0.3739 p-value (two-tailed): 0.7086 95% CI for mean daily return: [-0.001318, 0.000897] 95% CI for annualized arithmetic mean (simple returns): [-33.22%, 22.60%] Conclusion: Cannot conclude strategy returns differ from zero (p ≥ 0.05) Sharpe Ratio: -0.264 (approximate 95% IID-moment CI: [-1.648, 1.119])
This raw mean-return test is not a test of net profitability. To evaluate net returns, account for transaction costs, implementation constraints, and the return convention. Assess benchmark or factor exposures separately when interpreting risk-adjusted performance; use out-of-sample evidence to assess persistence and account for selection when testing many candidates.
Key Parameters
The key parameters for statistical analysis of returns:
- n: Sample size (number of observations). Under the independent common-variance model, larger samples reduce theoretical mean standard errors. Realized interval width also depends on estimated dispersion and the critical value.
- α (alpha): Significance level for hypothesis tests (commonly 0.05). Lower values reduce Type I error under the null and, with the design and alternative held fixed, reduce power.
- ddof: Degrees of freedom adjustment for variance estimation. For i.i.d. observations,
ddof=1gives the usual unbiased estimator of population variance; taking its square root does not make sample standard deviation exactly unbiased. - periods_per_year: Annualization factor (252 for daily trading data). The code multiplies the mean by this factor and volatility and the Sharpe ratio by its square root; skewness and kurtosis remain unchanged.
- confidence_level: Probability coverage for confidence intervals (commonly 0.95). Higher levels produce wider intervals.