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17

Just to be painfully clear, it only seems to make sense to consider the logarithm of returns, i.e. $X=\log (1+\frac r{100})$ for a simple return of $r\%$ in an arbitrary period because this is what sums when returns are temporally aggregated. A basic property of cumulants is that cumulants of all orders are additive under convolution, for which a proof can ...


8

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 ...


6

"Skewness" quantifies how asymetric a distribution is about the mean. "Kurtosis" quantifies how peaked or flat the distribution is. Skewness is defined as: $E[ (X - mean)^3 ] = \frac{(\sum (x_i - x_{mean})^3 )}{N}$ and Kurtosis as: $E[ (X - mean)^4 ] = \frac{(\sum (x_i - x_{mean})^4 )}{N}$ where X is your distro values (x_1, x_2, ... x_N), mean is the ...


5

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 ...


4

In my opinion you have two choices: You calculate annual returns from the daily returns that you have - I guess it is clear how. Subsequently you calculate your statistics on these $11$ data points. When I look at your comment above, this could be what you want to achieve. Then you have the ex-post statistics on your data. The drawback is that $11$ data ...


4

That can be a somewhat difficult question to answer, given that the context may yield different distributions. Nevertheless, I think that you could try to fit the best distribution algorithmically. For instance, lately I found this package at Matlab file exchange: Finding the best distribution that fits the data Link (...) This is where Mike's allfitdist ...


4

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 ...


3

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 ...


3

Assuming you have return time series $$ r_1(1), r_1(2), \ldots, r_1(T) \qquad \text{and} \qquad r_2(1), r_2(2), \ldots, r_2(T) $$ for the 2 assets and asset weights $w_1$ and $w_2$, we can follow the calculation of the $N$-asset portfolio skewness laid out in another answer for a similar question. To extend it to include portfolio kurtosis, we need the co-...


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.


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 ...


1

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 ...


1

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.


1

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

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 ...


1

What is the data basis that you start from? If you just have the covariance matrix, then you can only calculate portfolio variance or volatility by $$ w^T \Sigma w$$ where $w$ are the portfolio weights and $\Sigma$ is the covariance matrix. If you have the individual asset continuously compounded returns $r^j_t$ where $j$ indexes assets, $j=1,\ldots,N$, and $...


1

I think the best answer is to test different distributions with your specific data set and see which fits the data the best... skewness and kurtosis are just a small piece of information , there is still a good deal of information we don't know and won't have with out having the data set in front of us


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