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## Hot answers tagged minimum-variance

6

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

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 $$\sigma_p^2=w^2\sigma_1^2+(1-w)^2\sigma_2^2+2w(1-w)\sigma_1\sigma_2\rho ... 4 The problem lies in the definition of risk. It seems that in the cited paper, the authors treat risk as a concept connected with the uncertainty of the out-of-sample performance of the portfolio. In that way portfolios constructed using the proposed robust estimators would be what they call minimum-risk portfolios. Contrasted with minimum-variance ... 4 This has already been dealt with multiple times. As @Dom explains, the purpose is to simplify partial derivatives. For exposition's sake, assume there are only two assets with weight vector \omega=(\omega_1,\omega_2), then we seek to minimize a function of the form:$$f(\omega_1,\omega_2)=\frac{1}{2}\left(\omega_1^2\sigma_2^2+\omega_2^2\sigma_2^2+2\omega_1\...

3

The objective of hedging is to reduce the variance of the (position+hedge) portfolio. So which of these two solutions gives a smaller variance? You could calculate it numerically and compare the variances. However, in general ... the answer is going to be: whichever of commodity 1 or commodity 2 has higher correlation ($\rho$) with jet fuel. The percent of ...

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 ...

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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 ...

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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

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There can be zero weights, and for that matter there can be (and often will be) negative weights as well unless you specifically have a constraint saying there can't be. Consider the case where you have 2 risky assets with different variance that are perfectly positively correlated with each other. The minimum variance portfolio will then consist only of the ...

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Identifying your variables: You will need a weight for each of the 26 assets in each of the 9 portfolios. Suppose you take each portfolio in turn and create a stacked vector: $\mathbf{w} = [w_{1,1} \; .. \;w_{1,26} \; w_{2,1} \; .. \; w_{2,26} \; .. \;w_{9,26}]$ Equality constraints: Each weight of an asset cross section has to sum to the holding, $W_j$: ...

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The model you were assigned comes from the following paper: de Miguel et al (2009) A Generalized Approach to Portfolio Optimization: Improving Performance by Constraining Portfolio Norms Instead of using an additive penalty term, the ridge shrinkage of the portfolio weight vector should, or works best, as a separate constraint: {\underset{w}{\arg\min}} \...

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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

Let us fix the asset universe with $N$ assets whose returns are multivariate normally distributed with covariance matrix $\Sigma$. You are already invested in $K<N$ assets (your portfolio) and you wish to add other assets from that universe to your portfolio to form a hedge(d) portfolio. Let us assume that the hedge should be self-financing. Let us ...

1

A lot of things we use in economics and financial economics in particular are inconsequential, but practical. If you have a quadratic program, include this fraction conveniently gets rid of pesky constants in the first order conditions. Specifically, $\nabla_\omega \omega' \Sigma \omega = (\Sigma + \Sigma') \omega$, using the convention that vectors are ...

1

It does not matter whether you measure covariance of two portfolios or two securities, the formula is the same. Simply instead of returns and expected values for securities, put those for portfolios.

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The intuition that "if I have an N stock portfolio and an (N+1)th stock becomes available, buying some of it will lower portfolio variance" is not correct. It is true if all stocks are uncorrelated, or if stock correlations are low. But it can fail in general, as the example given in your book demonstrates. Suppose you initially invest in stocks 2,3,4. 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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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 ...

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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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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 ...

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