Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1808.04233

Connecting Sharpe ratio and Student t-statistic, and beyond

Eric Benhamou
View original source ↗Read source document ↗Export metadata ↓

Abstract

Sharpe ratio is widely used in asset management to compare and benchmark funds and asset managers. It computes the ratio of the excess return over the strategy standard deviation. However, the elements to compute the Sharpe ratio, namely, the expected returns and the volatilities are unknown numbers and need to be estimated statistically. This means that the Sharpe ratio used by funds is subject to be error prone because of statistical estimation error. Lo (2002), Mertens (2002) derive explicit expressions for the statistical distribution of the Sharpe ratio using standard asymptotic theory under several sets of assumptions (independent normally distributed - and identically distributed returns). In this paper, we provide the exact distribution of the Sharpe ratio for independent normally distributed return. In this case, the Sharpe ratio statistic is up to a rescaling factor a non centered Student distribution whose characteristics have been widely studied by statisticians. The asymptotic behavior of our distribution provide the result of Lo (2002). We also illustrate the fact that the empirical Sharpe ratio is asymptotically optimal in the sense that it achieves the Cramer Rao bound. We then study the empirical SR under AR(1) assumptions and investigate the effect of compounding period on the Sharpe (computing the annual Sharpe with monthly data for instance). We finally provide general formula in this case of heteroscedasticity and autocorrelation.

Explore related subjects

q-fin.STstat.ME

Keep this discovery

BibTeXRIS

Eric Benhamou. 2018-08-02. Connecting Sharpe ratio and Student t-statistic, and beyond. https://arxiv.org/abs/1808.04233

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Log S-fBM model: Statistical analysis

The Log S-fBM model, introduced by Wu et al., is a stochastic volatility model whose log volatility is a stationary fractional Brownian motion (S-fBM): a stationary Gaussian process with power-decaying autocovariance driven by the Hurst exponent $H$, and variance scaled by an intermittency coefficient. A key property is that it reconciles rough volatility, where $H$ is typically near $0.1$ (see Gatheral et al.), with multifractal volatility, where $H$ is close to $0$ as in Bacry, Muzy et al.: the model's volatility measure converges to a multifractal random measure as $H\to0$. Numerical findings in Wu et al. show intermittency of order $0.02$ across financial assets, motivating a small intermittency approximation of log volatility moments for calibration via the general method of moments (GMM). In this work, we conduct a statistical analysis of the Log S-fBM model. We derive scaling properties of the S-fBM process and the Log S-fBM integrated volatility measure, present deviation inequalities with tail distributions sensitive to $H$ and intermittency, and develop a hypothesis test for the null Hurst exponent, i.e.\ rough versus multifractal dynamics. Finally, we revisit scale invariance of the log volatility increment process via explicit small-intermittency formulas, reproducing analogous properties in both regimes.

q-fin.ST↗
arXiv2026

Asymmetric Long-Memory GARCH: Sign-Dependent Kernel Injection in a Two-Dimensional Markov Chain

We introduce ALM-GARCH, an asymmetric long-memory GARCH model in which positive and negative innovations enter conditional variance with different injection amplitudes and kernel offsets. These departures define testable level and memory channels relative to a nested symmetric benchmark. Positive Harris recurrence holds for interior configurations under a Foster-Lyapunov condition. Across five equity indices and Bitcoin, joint symmetry is rejected throughout, driven primarily by the level channel. The memory channel is supported for the Nikkei 225, KOSPI, and Bitcoin but is weakly identified when the positive branch is nearly inactive. Out-of-sample performance is broadly comparable to standard benchmarks.

q-fin.ST↗
arXiv2026

Modeling Trade Durations under Temporal Granularity Effects in Forex Markets

Trade durations in high-frequency foreign exchange data exhibit increased occurrence near integer values. To address this empirical phenomenon, we propose the granularity-adjusted autoregressive conditional duration (GA-ACD) model. It is based on a novel two-component mixture distribution consisting of a standard generalized gamma component for regular durations and a second component that locally redistributes probability mass around integer values to capture heaping. Conditional dynamics are modeled within a score-driven framework, allowing the scale parameter to vary over time in response to past durations, and enabling maximum likelihood estimation of all model parameters. A simulation study shows that ignoring heaping leads to biased parameter estimates and distorted inference regarding both the distribution and the dynamics of durations. An empirical analysis demonstrates that integer-duration clustering is pervasive across major currency pairs and that the GA-ACD model outperforms the standard generalized gamma ACD model.

q-fin.ST↗