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9

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 ...


7

Yes. Check out the Lower Partial Moments literature. In my view the best introduction to this is Narwrocki - A Brief History of Downside Risk Measures. Uryasev established equivalence between CVaR approach and low partial moments. If Markowitz had the tools at the time time, LPM utility functions would be the introductory optimization model as opposed to ...


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

"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 ...


4

There's a huge literature on this topic going back at least 30 years, and I am unfortunately not familiar enough with this literature to give you a great answer to your specific question. However, I will in this answer at least try to point you in some useful directions according to what I've found thus far. Kurtosis, by the way, seems like it is not ...


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 ...


2

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 ...


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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