Comparing Laplace and Normal Tails in Compounded Return Simulations

Summary

The document raises a distribution-choice question: why a simulated Laplace process appears to have slightly lower probabilities of crossing selected cumulative-return thresholds than a normal process, despite the Laplace distribution’s reputation for heavier tails. The example simulates many paths of daily simple returns over a fixed horizon, compounds each path by multiplying one plus each return, and compares the resulting frequency of crossing upper and lower thresholds.

The reported frequencies are close for the two distributions, but they do not by themselves establish which distribution has heavier tails. Tail behavior depends on parameterization and scale matching, and compounding many small returns changes the shape of the cumulative-return distribution. The document provides no answer explaining the discrepancy and no checks of the random generator’s parameter conventions, variance matching, or simulation uncertainty. Its main lesson is to verify the distributions and compare equivalent moments and tail probabilities before drawing conclusions from a simulation.

Key ideas

• A distribution’s tail properties depend on its parameterization and the scale used for comparison.
• Compounding daily returns transforms the distribution of outcomes over a multi-period horizon.
• Similar simulated threshold frequencies do not establish that two distributions have the same tail behavior.
• A fair comparison should verify the random generator’s parameters and match relevant moments such as variance.
• Simulation uncertainty and threshold choice can affect estimated probabilities of rare outcomes.

Tags

# Probability of outlier events for laplace distribution


# Probability of outlier events for laplace distribution












I've read that the laplace distribution is better for forecasting purposes than the normal distribution due to it better accounting for fat tails. However, when I run the numbers in matlab, laplace ends up with skinnier tails than a normal distribution. Am I doing anything wrong? Here is the code:

```
%laplace distribution

a = laprnd(755,100000,0,0.0034);
for x = 1:length(a)
   rets(x) = prod(1+a(:,x))-1;
end

[sum(rets>.0419)/length(rets),sum(rets<-.1688)/length(rets)]

ans = 

0.3122    0.0264

%Normal Distribution

j = normrnd(0,0.0034,755,100000);
for y = 1:length(j)
    retsj(y) = prod(1+j(:,y))-1;
end

[sum(retsj>.0419)/length(retsj),sum(retsj<-.1688)/length(retsj)]

ans = 

0.3151    0.0267
```

laprnd code is here: https://www.mathworks.com/matlabcentral/fileexchange/13705-laplacian-random-number-generator

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.