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1) To be honest, any horizon is problematic in this respect. Simple sampling statistics 101 will tell you that the standard error around any estimate of true mean returns is the root time * variance. So for eg stocks at 20 vol, that's a +/-40% 1y 95% confidence interval around your sample mean ;-) With 100 years of data, that's still +/-4%! Which is in-line ...


5

You're not going to get an analytic formula except in special cases of function $\rho(x)$. And you're probably going to want $\rho$ convex. If $\rho$ is convex, the problem is a convex optimization problem and can be efficiently solved numerically. If $\rho$ isn't convex, the optimization problem may be difficult to solve. If $\rho(x) = |x|$ you basically ...


4

The problem lies in the definition of risk. It seems that in the cited paper, the authors treat risk as a concept connected with the uncertainty of the out-of-sample performance of the portfolio. In that way portfolios constructed using the proposed robust estimators would be what they call minimum-risk portfolios. Contrasted with minimum-variance ...


3

Assume the weights of the two assets are $w$,$1-w$ respectively;the expected returns and standard deviations are denoted by $\mu$,$\sigma$ with subscripts 1,2,p(for portfolio),i.e,we have $\mu_1$,$\mu_2$,$\mu_p$,$\sigma_1$,$\sigma_2$,$\sigma_p$.The correlation coefficent is $\rho$ Then $$\sigma_p^2=w^2\sigma_1^2+(1-w)^2\sigma_2^2+2w(1-w)\sigma_1\sigma_2\rho ...


3

Risk is a broader concept than variance. That paper is specifically focused on robust estimators (i.e., estimators that are less sensitive to outliers) of dispersion. A robust estimator of dispersion is not the same thing as variance (which may be a dispersion parameter for some classes of distributions). Nevertheless, these robust estimators could be used ...


3

For academic references, you will likely have to look in the very early optimization literature. Uniqueness of the MV portfolio follows immediately from the lemma that a strictly convex function on a convex set has no local minima. The standard textbook reference is Convex Optimization by Boyd and Vandenberghe. See section 4.2.2 in particular. A free ...


2

Answering "No" to the title question, I'll mention that variance is a rather poor measure of risk, even if convinient and nicely behaving. Variance is not even a risk measure, with the standard deviation eventully being a deviation risk measure, while not necessarily for downside risk (see David Nawrocki-"A Brief History of Downside Risk Measures" for ...


2

This article by Eric Falkenstein is exactly what you are looking for: Early Low Vol Literature Now Everywhere EDIT Falkenstein has a new post out on the academic origins of the approach: Here


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Only certain aspects of the risks that you bear in power markets given exposure to variable quantity swaps can be hedged. To your point, you have to have some expectation of what the load will look like. Even if you immediately go out and buy power against this expected qty you are subject to the risk that the load will deviate from said qty. There is no ...


2

The general formula for the global minimum variance portfolio is $w=\frac{C^{-1} 1}{1^T C^{-1} 1}$ where C is the covariance matrix and 1 is a vector of 1's. In this case the covariance matrix is diagonal with $\sigma_i^2$ in the ith diagonal element. Its inverse is also diagonal and has $\frac{1}{\sigma_i^2}$ in the ith diagonal element. Evaluating the ...


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yes u is the unit vector of all ones


1

It doesn't make sense to use the (co)variance(s) of asset values; if you did, by cutting an investment's share of the allocation by half, you would also cut its variance by a factor of 4. In a meaningful portfolio design, the volatility (variance) of an asset, by itself, is the same no matter how much or how little of your portfolio you put in it. Why doesn'...


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Just to save you some time, there is a non-existence proof for this class of problems. The models assume perfect information, what has been missed is that there are no estimators that converge to the population parameter under incomplete information. Consider the static model equation $\tilde{w}=R\bar{w}+\epsilon,R>1$. The maximum likelihood estimator ...


1

That part of the paper is showing why the efficient frontier is the same regardless of whether you are maximizing utility, maximizing returns given variance, or minimizing variance given returns. Inequality constraints tend to be a bit more work to deal with analytically, so that might be a reason why they use the equality constraint on one of them. ...


1

Intuitively speaking this statement should be clear, as in case the risk-free rate is equal to the expected return of the global minimum variance portfolio you can just assume that the minimum variance portfolio is just an investment into the risk-free rate. Therefore the intersection between the efficient frontier and the tangent line between $r_f$ and the ...


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Yes, the assumption in MPT is normal distribution for returns. You can programme yourself in R or Excel, following elementary linear algebra. Eric Zivot (U Wash) has a spreadsheet solution here: https://faculty.washington.edu/ezivot/econ424/solverex.pdf https://faculty.washington.edu/ezivot/econ424/Efficient%20Portfolios%20in%20Excel%20Using%20the%...


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If the stock you'd like to hedge with is the same as the option's underlying obviously just find the net delta and hedge with that amount of stock. If you have different types of stocks and would like to hedge with an index you can multiply the delta with the beta of each stock versus the index. Beta is analogous to delta in a way. With delta we describe ...


1

If short sell is allowed, I remember there's a unique analytical solution, otherwise it has to be solved numerically. Is your approache different? IMHO the issue of min variance approach is really not how to solve this constrained optimization problem, but how to estimate asset return and var/covar matrix accurately.


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