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Michael Brenndoerfer

Statistical Data Analysis and Inference › Confidence Intervals

Michael BrenndoerferOctober 22, 20257 min read

Quantitative FinanceAll chapters

Part 1 of 15Mathematical Foundations

Chapter 4Statistical Data Analysis and Inference › Confidence Intervals

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Hypothesis tests give binary decisions, while confidence intervals additionally communicate the precision of an estimate by providing a range of parameter values compatible with the procedure and assumptions.

For i.i.d. normal observations, the following interval has exact Student-t coverage. For a nonnormal sample, it is a large-sample approximation only when the standard error and regularity conditions justify it; dependent returns require an appropriate long-run-variance estimate. The interval is:

\bar{r} \pm t_{1-\alpha/2, n-1} \times \widehat{SE}(\bar{r})

where:

$\bar{r}$ : sample mean (center of the interval)
$t_{1-\alpha/2, n-1}$ : positive upper critical value from the t-distribution with $n-1$ degrees of freedom (for 95% confidence with large $n$ , this is approximately 1.96)
$n$ : sample size
$\alpha$ : significance level (for 95% confidence, $\alpha = 0.05$ )
$\widehat{SE}(\bar{r})$ : estimated standard error of the mean, valid for the stated sampling model
$t_{1-\alpha/2, n-1} \times \widehat{SE}(\bar{r})$ : margin of error

The construction follows from the sampling distribution of the standardized mean under those conditions. Across repeated samples, the procedure uses the positive $t_{1-\alpha/2,n-1}$ critical magnitude so that 95% of the resulting intervals cover the fixed population mean when $\alpha=0.05$ .

In[31]:

Code

# Calculate 95% confidence interval for mean return
# Uses mean_daily, standard_error, and degrees_freedom computed in earlier sections
confidence_level = 0.95
alpha = 1 - confidence_level

# Critical value from t-distribution
t_critical = stats.t.ppf(1 - alpha / 2, degrees_freedom)

# Confidence interval
margin_of_error = t_critical * standard_error
ci_lower = mean_daily - margin_of_error
ci_upper = mean_daily + margin_of_error

# Annualized confidence interval
ci_lower_annual = ci_lower * 252
ci_upper_annual = ci_upper * 252

Out[32]:

Console

95% Confidence Interval for Mean Log Return
--------------------------------------------------
Daily: [-0.002203, 0.002600]
Annualized: [-55.51%, 65.52%]

Point estimate (annualized): 5.00%
Width of CI (annualized): 121.04%

Out[33]:

Visualization

Horizontal confidence-interval bar for annualized mean log return with a point estimate marker and a vertical line at zero return. — Visual representation of the 95% confidence interval for annualized mean log return. The point estimate (orange dot) is surrounded by the confidence interval (blue bar). For the matching two-sided test of H0: μ = 0 at α = 0.05, using the same model and standard error, excluding zero corresponds to rejection and including zero to non-rejection.

Interpreting Confidence Intervals

A 95% confidence interval means: if we repeated this sampling procedure many times, 95% of the constructed intervals would contain the true population mean. It does not mean there is a 95% probability the true mean lies in this specific interval. The true mean is fixed; our interval either contains it or it doesn't.

This distinction matters conceptually. The frequentist interpretation views probability as a long-run frequency. Across many repetitions of the sampling procedure, 95% of intervals would capture the true parameter. The Bayesian interpretation, which allows probability statements about parameters, requires additional assumptions about prior beliefs.

Confidence Intervals and Risk-Adjusted ReturnsLink Copied

Confidence intervals are particularly useful for Sharpe ratios, which are notoriously imprecise. A strategy might show a Sharpe ratio of 1.0, but the confidence interval could span from 0.2 to 1.8, which would change our assessment of the strategy's quality.

The uncertainty of a ratio depends on both its estimated mean and its estimated dispersion. A normal-return model gives the familiar single-period standard-error approximation $\sqrt{(1+\theta^2/2)/n}$ , where $\theta=\mu_d/\sigma_d$ is the population differential-return ratio. Our worked return mixture is not normal, so we use a more general approximation.

For i.i.d. differential returns with positive variance and a finite fourth moment, define $Z=(d-\mu_d)/\sigma_d$ , population skewness $\gamma_1=E[Z^3]$ , and raw kurtosis $K=E[Z^4]$ , where $E[\cdot]$ denotes expectation under the population return law. The delta method uses a linear expression to approximate how small estimation errors in the mean and standard deviation affect their ratio. Applying it jointly to the mean and variance gives:

A=1-\gamma_1\theta+\frac{K-1}{4}\theta^2, \qquad SE(\widehat\theta)\approx\sqrt{\frac{A}{n}}.

Here $A$ combines mean uncertainty, dispersion uncertainty, and their covariance. Under a normal distribution, $\gamma_1=0$ and $K=3$ , so $A=1+\theta^2/2$ . A Wald interval is the estimate plus or minus a normal critical value times its estimated standard error. A nondegenerate first-order interval also requires $A>0$ .

The code estimates the same coefficient as the mean square of $z_i-\widehat\theta(z_i^2-1)/2$ , where $z_i$ uses sample centering and denominator- $n$ dispersion. These influence values approximate each observation's contribution to the ratio's first-order estimation error. Squaring avoids numerical cancellation in the expanded expression for $A$ . The point estimate retains the denominator- $(n-1)$ sample standard deviation; this changes only lower-order terms in the asymptotic calculation.

This interval is approximate, not an exact finite-sample 95% guarantee. Finite sample moments do not establish finite population moments. Serial dependence requires the long-run variance of the ratio's influence function rather than this i.i.d. coefficient.

In[34]:

Code

def sharpe_confidence_interval(
    returns, reference_return=0.0, confidence=0.95, periods_per_year=252
):
    """
    Compute an approximate annualized Sharpe-ratio interval relative to a
    matching-period reference return.

    returns and reference_return must use the same period and return
    convention. The default reference_return=0.0 makes the zero-reference
    assumption explicit. Reference may be a finite scalar or a vector matching
    returns exactly. The IID approximation requires positive population
    variance, finite fourth moment, and nonzero first-order influence variance.
    Annualize both the estimate and its moment-based standard error by
    sqrt(periods_per_year). This is not an exact finite-sample interval;
    serial dependence requires an influence-function long-run variance.
    """
    returns = validate_return_sample(returns)
    periods_per_year = validate_periods_per_year(periods_per_year)
    confidence = np.asarray(confidence, dtype=float)
    if confidence.ndim != 0 or not np.isfinite(confidence) or not 0 < confidence < 1:
        raise ValueError(
            "confidence must be a finite scalar strictly between zero and one"
        )
    confidence = float(confidence)
    reference = np.asarray(reference_return, dtype=float)
    if reference.ndim != 0 and reference.shape != returns.shape:
        raise ValueError(
            "reference_return must be a scalar or match the one-dimensional returns exactly"
        )
    if not np.all(np.isfinite(reference)):
        raise ValueError("reference_return must contain only finite values")
    excess_returns = validate_return_sample(returns - reference)
    n = len(excess_returns)
    mean_excess = np.mean(excess_returns)
    std_return = np.std(excess_returns, ddof=1)
    sharpe_period = mean_excess / std_return
    sharpe = sharpe_period * np.sqrt(periods_per_year)

    # Denominator-n moments and a nonnegative influence-square coefficient
    z = (excess_returns - mean_excess) / np.std(excess_returns, ddof=0)
    influence = z - 0.5 * sharpe_period * (z**2 - 1)
    variance_coefficient = np.mean(influence**2)
    if (
        not np.isfinite(sharpe)
        or not np.isfinite(variance_coefficient)
        or variance_coefficient <= 0
    ):
        raise ValueError(
            "a finite ratio and positive finite influence variance are required"
        )
    se_sharpe_period = np.sqrt(variance_coefficient / n)
    se_sharpe = se_sharpe_period * np.sqrt(periods_per_year)

    z_critical = stats.norm.isf((1 - confidence) / 2)
    if not np.isfinite(z_critical) or z_critical <= 0:
        raise ValueError("confidence must produce a positive finite critical value")

    ci_lower = sharpe - z_critical * se_sharpe
    ci_upper = sharpe + z_critical * se_sharpe

    if not np.all(np.isfinite([se_sharpe, ci_lower, ci_upper])) or ci_lower >= ci_upper:
        raise ValueError(
            "the interval must be finite and numerically distinguishable for these inputs"
        )
    return sharpe, ci_lower, ci_upper, se_sharpe


sharpe, sr_lower, sr_upper, se_sr = sharpe_confidence_interval(
    log_returns, reference_return=0.0
)

Out[35]:

Console

Approximate 95% IID-Moment Interval for Log-Return Ratio
--------------------------------------------------
Sharpe Ratio: 0.1632
Standard Error: 1.0008
Approximate 95% IID-moment CI: [-1.7984, 2.1248]

Under the IID finite-fourth-moment approximation, the CI includes zero.