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I wrote this paper a couple of years ago where we discuss this kind of topic. On page 6, you see a formula that comes from a paper from Acerbi available in Szego's book: $$\sigma^2(ES^{(N)}_\alpha(X)) \overset{N>>1}{=} \frac{1}{N(1-\alpha)^2} \int_0^{F^{-1}(1-\alpha)} dx \int_0^{F^{-1}(1-\alpha)} dy \{ \min( F(x), F(y) ) - F(x)F(y) \}$$ This should ...


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More often than not, I prefer to work with a scenario representation. That is, I will simulate from the distribution and calculate the VaR and CVaR as appropriate. This is especially the case for forward-looking analysis of portfolios' CVaR, rather than in evaluating the historical returns of some portfolio. If for some reason I can't do the scenario ...


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Values of VaR are just the inverses of the cumulative distributions. CVaR is not a very commonly used term, its more frequently used synonym is Expected Shortfall. See http://www.maths.manchester.ac.uk/~saralees/chap17.pdf for the list of Expected Shortfall values for more than 20 distributions.


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I don't think you can say anything general on this type of setup, certainly not from an empirical point of view. Assume the market conditions change between the two periods, then $ES$ could be higher or lower. If you assume some distribution for you returns, then they should probably be the same if the two periods have the same length.


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Sure a lot of traditional (mutual) buy side funds use MPT. They also mostly subscribe to the efficient market hypotheses. And they also do not hide the fact that they have no interest to lobby many retirement investment and savings schemes to allow for long/short investments but hold on to long-only. And finally, most of them underperform simple benchmark ...


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Lots of wealth management firms still use MPT; in my experience regulators like it because they understand it. If asset returns are normally distributed, the standard deviation of the portfolio is a coherent risk measure (this can be seen by noting that the normal distribution's CVaR, which is a coherent risk measure, can be written as $$\mu+c \sigma$$ ...


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I am engineer studying Finance, therefore Im not an expert in Math/Stat, but not noob. I disagree with the previous answer. In fact, I know portfolio managers and hedge fund assesors that usses MPT. It must be said that you need to know what that represents, and also not only focus your investment in MPT, but consider other methods. Like in every other ...


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MPT should be called Medieval Portfolio Theory, it is a theory from 50 years ago with huge theoretical flaws (mean-variance utility, use of Pearson's correlation that is not coherent, based on historical data). Come on, it is an error maximizer. The least one could do is Michoud resampling, but it is patented. Or a bayesian Black-Litterman would be more ...


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a) The formula for Beta is: $$\beta_i=\frac{\sigma_{i,M}^2}{\sigma_M^2}=\frac{0.165^2}{0.11^2}=2.25$$ b) So by the CAPM equation, the expected return for the asset is: $$E(R_i)=r_f+\beta(R_M-r_f)=0.04+2.25(0.12-0.04)=0.22=22\%$$ c) If the variance of the stock is $0.22^2$, since this variance was multiplied by $\beta=2.25$, we get: ...



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