Against the Norm: Modeling Daily Stock Returns with the Laplace Distribution

Bruce Jones; David Toth

arxiv: 1906.10325 · v1 · pith:2CPP5C3Knew · submitted 2019-06-25 · 💱 q-fin.MF · q-fin.ST

Against the Norm: Modeling Daily Stock Returns with the Laplace Distribution

David Toth , Bruce Jones This is my paper

Pith reviewed 2026-05-25 16:19 UTC · model grok-4.3

classification 💱 q-fin.MF q-fin.ST

keywords stock returnsLaplace distributionnormal distributiondaily returnsstock market indicesfat tailsfinancial modeling

0 comments

The pith

The normal distribution fails to model daily stock returns from major indices, which fit the Laplace distribution instead.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines daily returns on major stock market indices and tests the common assumption that these returns follow a normal distribution. It concludes that the normal distribution does not fit the observed data well, even across multi-year periods, because it assigns negligible probability to large moves that actually occur. The authors therefore propose the Laplace distribution as a more appropriate model for daily stock returns. This modeling choice would directly affect calculations of risk and the probability of extreme events in portfolio management.

Core claim

We investigate the normality of the distribution of daily returns of major stock market indices. We find that the normal distribution is not a good model for stock returns, even over several years' worth of data. Moreover, we propose using the Laplace distribution as a model for daily stock returns.

What carries the argument

The Laplace distribution applied to daily returns data from major stock market indices, contrasted with the normal distribution.

If this is right

Daily returns exhibit heavier tails than the normal distribution predicts.
The probability assigned to extreme daily moves is substantially higher under the Laplace model.
Risk measures and portfolio optimizations that assume normality will understate the likelihood of large losses.
The Laplace distribution provides a simple parametric alternative that can be fitted directly to historical daily index data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Portfolio construction rules that rely on return distributions would produce different allocations if switched from normal to Laplace inputs.
Value-at-risk estimates derived from daily data would increase under the Laplace model, affecting capital allocation decisions.
Extending the same comparison to intraday or weekly returns could reveal whether the preference for Laplace holds at other time scales.

Load-bearing premise

That the Laplace distribution supplies a meaningfully superior description of the observed daily returns compared with the normal or other fat-tailed alternatives, based on unspecified statistical criteria or visual checks applied to the index data.

What would settle it

A formal statistical comparison, such as a likelihood ratio test or quantile-quantile plot analysis, on the same index return series showing that the normal distribution matches the data at least as closely as the Laplace distribution.

read the original abstract

Modeling stock returns is not a new task for mathematicians, investors, and portfolio managers, but it remains a difficult objective due to the ebb and flow of stock markets. One common solution is to approximate the distribution of stock returns with a normal distribution. However, normal distributions place infinitesimal probabilities on extreme outliers, but these outliers are of particular importance in the practice of investing. In this paper, we investigate the normality of the distribution of daily returns of major stock market indices. We find that the normal distribution is not a good model for stock returns, even over several years' worth of data. Moreover, we propose using the Laplace distribution as a model for daily stock returns.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper restates the known fact that daily returns have fat tails and suggests Laplace as a model, but supplies no quantitative tests or comparisons to back the superiority claim.

read the letter

The core issue is that nothing here is new. Non-normality of stock returns and the use of Laplace as an alternative have been discussed in finance since the 1960s, with applications to indices already in the literature. The paper applies the idea to major indices but does not introduce new methods, derivations, or results that stand apart from prior work. What it does get right is the practical point that normal distributions assign near-zero probability to the large moves that actually drive portfolio risk, so practitioners should not rely on them for tail events. That reminder is useful for anyone still defaulting to normality in basic models. The problems sit in the execution. The abstract asserts that normal fails and Laplace succeeds yet gives no test statistics, no data periods, no index list, no exclusion rules, and no comparison metrics against other fat-tailed options such as Student's t. If the manuscript uses only histograms or QQ plots without likelihood ratios, AIC/BIC scores, Kolmogorov-Smirnov tests, or out-of-sample checks, the claim that Laplace is meaningfully better lacks a reproducible basis. Parameter estimation on the same series used for fit assessment also risks circularity unless an independent validation step is shown. The citation pattern appears thin on the established history of this topic. This work would mainly interest readers new to the fat-tails literature who want a quick pointer. Anyone already familiar with the area will find little to engage with. It does not reach the threshold for serious peer review in its current state; the empirical support for the central claim is missing and would need to be added before a referee could evaluate it properly.

Referee Report

3 major / 0 minor

Summary. The manuscript claims that the normal distribution is inadequate for modeling daily returns of major stock market indices even over multi-year periods, and proposes the Laplace distribution as a superior alternative model for these returns.

Significance. If the central claim were supported by reproducible quantitative evidence, the result would strengthen the case for using fat-tailed distributions in financial modeling, with potential implications for risk management and portfolio optimization. The manuscript does not report machine-checked proofs, reproducible code, or parameter-free derivations.

major comments (3)

[Abstract] Abstract: the claim that 'the normal distribution is not a good model for stock returns, even over several years' worth of data' supplies no test statistics, p-values, data periods, indices examined, or exclusion rules, rendering the central claim unevaluable.
[Abstract] Abstract and main text: the proposal to use the Laplace distribution as a model rests on unspecified criteria (visual checks or in-sample fits) without reported likelihood ratios, AIC/BIC values, Kolmogorov-Smirnov statistics, or comparisons against alternatives such as the Student's t distribution.
[Main text] The superiority assertion for Laplace appears to rely on estimating its scale parameter from the same return series used to assess fit, with no indication of an independent validation set or out-of-sample predictive evaluation.

Simulated Author's Rebuttal

3 responses · 0 unresolved

Thank you for the opportunity to respond to the referee's report. We address each of the major comments below. We agree that the claims can be made more rigorous with additional statistics and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that 'the normal distribution is not a good model for stock returns, even over several years' worth of data' supplies no test statistics, p-values, data periods, indices examined, or exclusion rules, rendering the central claim unevaluable.

Authors: We agree that the abstract lacks specific details. The revised version will specify the indices (S&P 500, NASDAQ, FTSE 100, etc.), data periods (e.g., 2005-2018), and include normality test results such as Jarque-Bera statistics and p-values to support the claim. revision: yes
Referee: [Abstract] Abstract and main text: the proposal to use the Laplace distribution as a model rests on unspecified criteria (visual checks or in-sample fits) without reported likelihood ratios, AIC/BIC values, Kolmogorov-Smirnov statistics, or comparisons against alternatives such as the Student's t distribution.

Authors: The manuscript uses visual and basic fit assessments. We will enhance it by adding AIC, BIC, likelihood ratio tests, KS statistics, and explicit comparisons to the Student's t distribution in the revised manuscript. revision: yes
Referee: [Main text] The superiority assertion for Laplace appears to rely on estimating its scale parameter from the same return series used to assess fit, with no indication of an independent validation set or out-of-sample predictive evaluation.

Authors: We acknowledge that the fit is in-sample. To address this, we will include a discussion of this aspect and add an out-of-sample validation using a hold-out period to evaluate the Laplace distribution's performance. revision: partial

Circularity Check

0 steps flagged

No circularity in empirical modeling proposal

full rationale

The paper is an empirical study that examines daily returns of stock indices, reports that the normal distribution is inadequate, and proposes the Laplace distribution as an alternative model. The provided abstract and context contain no equations, derivations, self-citations, or fitted-parameter steps that reduce any claim to its own inputs by construction. No load-bearing self-definitional, prediction-from-fit, or uniqueness-via-self-citation patterns are present. The central suggestion rests on data inspection rather than any mechanism that would force equivalence to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities; the proposal to switch distributions is presented without derivation details or new postulated quantities.

pith-pipeline@v0.9.0 · 5633 in / 1088 out tokens · 36160 ms · 2026-05-25T16:19:58.511656+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

Fama, E., and French, K. (2012). Q&A: Are Stock Returns Normally Distributed? Fama/French Forum. https://famafrench.dimensional.com/questions-answers/qa-are-stock-returns-normally- distributed.aspx. Accessed 2019-06-24

work page 2012
[2]

and Kokoska, S

Zwillinger, D. and Kokoska, S. (2000). CRC Standard Probability and Statistics Tables and Formulae. Chapman & Hall: New York. 2000

work page 2000
[3]

S., & Wilk, M

Shapiro, S. S., & Wilk, M. B. (1965). An analysis of variance test for normality (complete samples). Biometrika, 52(3/4), 591-611

work page 1965
[4]

SciPy: Open Source Scientific Tools for Python, 2001-, http://www.scipy.org/ [Online; accessed 2019-06-24]

Jones E, Oliphant E, Peterson P, et al. SciPy: Open Source Scientific Tools for Python, 2001-, http://www.scipy.org/ [Online; accessed 2019-06-24]

work page 2001
[5]

A guide to NumPy, USA: Trelgol Publishing, (2006)

Travis E, Oliphant. A guide to NumPy, USA: Trelgol Publishing, (2006)

work page 2006

[1] [1]

Fama, E., and French, K. (2012). Q&A: Are Stock Returns Normally Distributed? Fama/French Forum. https://famafrench.dimensional.com/questions-answers/qa-are-stock-returns-normally- distributed.aspx. Accessed 2019-06-24

work page 2012

[2] [2]

and Kokoska, S

Zwillinger, D. and Kokoska, S. (2000). CRC Standard Probability and Statistics Tables and Formulae. Chapman & Hall: New York. 2000

work page 2000

[3] [3]

S., & Wilk, M

Shapiro, S. S., & Wilk, M. B. (1965). An analysis of variance test for normality (complete samples). Biometrika, 52(3/4), 591-611

work page 1965

[4] [4]

SciPy: Open Source Scientific Tools for Python, 2001-, http://www.scipy.org/ [Online; accessed 2019-06-24]

Jones E, Oliphant E, Peterson P, et al. SciPy: Open Source Scientific Tools for Python, 2001-, http://www.scipy.org/ [Online; accessed 2019-06-24]

work page 2001

[5] [5]

A guide to NumPy, USA: Trelgol Publishing, (2006)

Travis E, Oliphant. A guide to NumPy, USA: Trelgol Publishing, (2006)

work page 2006