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1

Unfortunately, there exist no closed form for this. The Lagrangean reads $$L(w,\lambda)=w^TM_3(w\otimes w)-\lambda(w^T\mathbf{1}-1)$$ with first order conditions \begin{align} \frac{\partial L }{\partial w_i}&=3w^TM_{3,i}w-\lambda \quad \forall i \\ \frac{\partial L }{\partial \lambda}&=w^T\mathbf{1}-1 \end{align} where $M_{3,i}$ is the $i$th ...

3

Diversification is key. The clear cut answer is diversification. A weighted combination of assets will more often than not show a lower return variance than even the asset with the lowest variance across the asset universe. The setup Without loss of generality, let us assume there exist two assets $a$ and $b$ with variance $\sigma_a^2=\alpha^2<\sigma_b^2=... 1 If you start out with a matrix specifying the covariance of every pair of assets, and an alpha for every asset (because people usually do), and define an objective function that maximizes the alpha and minimizes the variance of the portfolio, and run a quadratic optimizer, but don't specify a lot of constraints, then you may well end up with 100% of a long-... 1 I want to add two comments to this. 1. Empirical utility-based-optimization and moments I would argue that comparing different degrees of a Taylor approximated utility optimization (a.k.a. a moment based model with two, three, four, ..., infinite moments) adds additional assumptions to your model when working with an asset universe whose statistics are not ... 1 If utility is your measure of performance, then it will still be your measure of performance out of sample, since it is what you care about. You can see utility as a measure of the balance between profit and risk, where risk is some combination of variance, skew, kurtosis... Your wealth is a random variable$X$that can be described by its moments. First ... 2 When$x_i$is the return of the$i$th asset, the returns of portfolio$\vec{w}$are$\sum_i w_i x_i$. The covariance of the returns of two portfolios,$\vec{w}$and$\vec{v}$are then $$\sum_i \sum_j w_i v_j \operatorname{cov}\left(x_i, x_j\right).$$ Now note that$\Sigma_{i,j} = \operatorname{cov}\left(x_i,x_j\right)$. The rest is confirming that this ... 1 I agree with @Kermittfrog's comment, that this only works if you do not impose any budget constraint (in the sense that your weights sum up to one). Other than that, I am sorry that I can not precisely answer your question where it was first derived (tbh: I am not even sure if it was explicity derived at all somewhere because it simply follows from the very ... 1 Just divide covariance by the square roots of the two variances. In this case you would want $$\frac{1/a}{\sqrt{\frac{1}{a}\frac{c}{b^2}}},$$ which takes value $$\frac{|1^{\top}\Sigma^{-1}\mu|}{\sqrt{(1^{\top}\Sigma^{-1}1)(\mu^{\top}\Sigma^{-1}\mu)}}.$$ 5 You are asking two questions: whether MVO works in real life and portfolio managers actually use it? whether defined benefit schemes use this tool? Concerning 1. the answer is generally no, although it kind of works with 2 assets. The elegant Markowitz solution showing the theoretical Sharpe and minimal variance optimal portfolio are numerically unstable. ... 3 The Lagrangian 'solution' can yield negative contributions to portfolio risk, which is a bad look. An alternative definition is via the symmetric square root of the covariance,$\Sigma^{1/2}$. For portfolio$\vec{w}$define $$\vec{r} = \Sigma^{1/2}\vec{w}.$$ The norm of$\vec{r}$is the volatility of the portfolio. Moreover, this definition is equivariant ... 6 In this answer, I am assuming that you want to keep correlations constant. To begin with, note that the$N\times N$covariance matrix$\Sigma$with element$\Sigma_{i,j}=Cov(x_i,x_j)$can be written as $$\Sigma = \mathbf{SRS}$$ where$\mathbf{S}$is a diagonal matrix of the simple volatilties$\sigma_i$, and$\mathbf{R}$is the correlation matrix. Thus in ... 3 Let's stick with the nomenclature in the literature and let$\gamma$denote the decision maker's risk aversion coefficient. The optimization problem is $$\max_{\mathrm{w}} \mathrm{w}^T\mathrm{\mu}-\frac{1}{2}\gamma \mathrm{w}^T\mathrm{\Sigma}\mathrm{w} \quad s.t. \mathrm{w}^T\mathrm{e}=1$$ where$e$denotes a vector of ones. The corresponding Lagrangean ... 7 A few more steps beyond your last equation gives the answer. With$C = \mathbf{1}^T\mathbf{\Sigma}^{-1}\mathbf{1}$, we have $$\sigma_P^2 = [C^{-1} \mathbf{\Sigma}^{-1}\mathbf{1}]^T \mathbf{\Sigma} [C^{-1}\mathbf{\Sigma}^{-1}\mathbf{1}] = C^{-2}\mathbf{1}^T(\mathbf{\Sigma}^{-1})^T\mathbf{\Sigma} \mathbf{\Sigma}^{-1}\mathbf{1}$$ Since$[(\mathbf{\Sigma}^{-1})^...

4

Let \begin{align} a&\equiv \mathbf{1}^T\mathbf{\Sigma}^{-1}\mathbf{1}\\ b&\equiv \mathbf{1}^T\mathbf{\Sigma}^{-1}\boldsymbol{\mu}\\ c&\equiv \boldsymbol{\mu}^T\mathbf{\Sigma}^{-1}\boldsymbol{\mu} \end{align} Then \begin{align} \mathrm{E(minVarPortfolio)}& = \frac{b}{a}\\ \mathrm{V(minVarPortfolio)}& = \frac{1}{a}\\ \mathrm{E(...

1

There is no closed-form analytical solution for the long-only minimum-variance portfolio. Only the the unconstrained (short-sales allowed) portfolio. See here. Modifying the unconstrained portfolio to become the constrained portfolio in the manner you described is not going to be equal to the true constrained portfolio solution, which must be obtained by ...

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