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You've found parameterizations where fantastically long samples are required for sample 4th moments to converge on population 4th moments. Quick evidence of imprecise estimation Let $k_i$ denote your estimated kurtosis in simulation $i$. Looking across your $i = (1,\ldots, 1000)$ simulations, your $k_i$ estimates are all over the place. What's your ...


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If we are talking about risk management (Hence, the risk neutral world), normality allows us to get closed form solutions. For instance, the Black and Scholes equation assumes Gaussian returns (Equivalently, the stock follows a geometric Brownian motion). Your thought is correct, although you can not simply adjust for kurtosis. You need to define properly ...


6

It probably does not have a mean or a variance. Ratio variables often don't. As these are accounting ratios, there are several candidate distributions and their ratios wouldn't have a first moment. There are a couple of things to remember. If none of the variables in a ratio can be negative, then you likely have a truncated distribution. In practice, this ...


3

The skewness and kurtosis values you obtain appear to be of realistic magnitude. In general higher frequencies are more non-normal, i.e. have higher skewness and kurtosis. If non-normal returns are aggregated the central limit theorem starts working and the return distribution coverges to a normal. Convergence can be quite slow under fat tails. You can try ...


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Since there are upvotes on Alex C's answer, it seems that people on this site are inclined to believe it. It is a bad idea to let bad memes go unchecked. Science, after all, should be self-correcting. So here is the mathematical logic correcting Alex C's answer. Alex C's comment: So "peakedness in the centre" and "fat in the tails" describe exactly the ...


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Quailtatively a (zero skewness) Leptokurtic distribution, after being standardized to have zero mean and unit variance shows three features when you plot the density and compare it to a standard normal N(0,1) distribution: higher peak, higher (fatter) tails, and lower mid-range(*). All three properties go together, even though people sometimes mention only ...


2

I'm not sure if this would qualify as "empirical work" but you should definitely read Dynamic Hedging if you haven't already. Taleb talks a lot about this. To be more specific, have a look at page 264, the section called "Higher Moment Bets". I hope this is helpful, although I am not sure if this is what you are looking for.


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The only thing weird is skewness not being lower for the weekly vs daily. In any case, take a look at table 1.1 from Campbell, Lo and Mackinlay, and check that your values are not far off the ballpark. Actually, with annual data, you should have nearly zero skewness and zero excess kurtosis (on the market). However, asset allocation might lead to severe ...


1

You haven't provided any parameters to the simulateBEKK method. If you want to simulate fake returns data that are realistic you will need to provide parameters that provide realistic results. In the following code I randomly generated some parameters which seem to simulate a more realistic process than you were doing in the question. By running some ...


1

Mean-variance efficiency generally only considers the first two moments. Skew and kurtosis are outside of it, and Sharpe is commonly criticized for not accounting for skew and kurtosis (leading to metrics like Omega as a replacement). Regarding your specific question, it depends entirely on what risks a given investor/trader wants to take. One person ...


1

Let me begin with the fact that economists and those in finance have been struggling with the source of kurtosis since Mandelbrot published “On the Variation of Certain Speculative Prices,” in 1963. There was a serious attempt to properly found finance on solid mathematical ground in the 1960s, but it failed. I believe it failed for three reasons and this ...


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I suggest you have a look at the paper: Schloegel, Erik (2010) "Option Pricing Where the Underlying Assets Follow a Gram/Charlier Density of Arbitrary Order", Journal of Economic Dynamics and Control, Vol. 37, No. 3, pp. 611-631 available on SSRN. A random variable $Y$ that follows a Gram/Charlier Type A series distribution has the probability density ...


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VaR for GED in R package(fGarch) qged(p, mean = 0, sd = 1, nu = 2) #Example qged(.01, mean=1000, sd=2000) [1] -3652.696 where, $1-p$ is confidence level.


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