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4

If $\Sigma$ is the covariance matrix of all assets and $w$ is the column vector of weightings of the asset in a certain portfolio. Then $$ w^T \Sigma w = VAR $$ is the variance of the portfolio. The contribution to volatility of asset $i$ is given by $$ w_i (\Sigma w)_i/\sqrt{VAR}, $$ where $(\Sigma w)_i$ is the $i_{th}$ entry in the vector $\Sigma w$. Note ...


3

The formula is $$ \mu = \lambda CX $$ in your notation. You find it in many places, e.g. here. The assumption is that you know $\lambda$ which is a strong assumption. Furthermore it only holds if investors are unconstrained (long/short not long only). It is intuitive as it says that given the weighting the return expectation increases with risk aversion ...


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I am not sure if I understood your question correctly but I will try to answer it anyway. If you have a standard normal random vector $z \sim N(\mathbb{0},I_n)$ (where $z,0 \in \mathbb{R}^{n\times1}$ and $I_n \in \mathbb{R}^{n\times n}$ is the identity matrix) and you want to transform it into a multivariate normal $x \sim N(\mu,\Sigma)$ you do it the ...


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One really nice book that comes to my mind is Little, Rubin, Statistical Analysis with Missing Data I read part of it but probably it is too much information in your case. For your application, i think you can categorize the problem into two possible subproblems: First, time series that have unequal starting points (when some stocks' history is ...


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@vanguard2k and @Theja provide useful information. In my experience, unequal starting points is most common, so I'll try to focus on that. The technique that @vanguard2k mentioned for unequal starting points can be thought of like a regression. You start with the longest available data and get the covariance matrix of that. For the next set of available ...


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The clearest and most intuitive article I have seen so far is Kritzman et al., Regime Shifts: Implications for Dynamic Strategies in FAJ (May / June 2012) It not only shows how you can use HMM for financial modelling but it also goes through the actual estimation algorithm (Baum-Welch) step-by-step and even gives full Matlab-code. From the abstract: ...


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One common "model" is to assume the correlation to be constant, such as in a CCC-MVGARCH model. If you want a review of different multivariate GARCH models, you could look at: Silvennoinen and Täräsvirta 2009, Multivariate Garch models, in Handbook of financial time series.


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Not sure your question is about having a process for covariance or to have multivariate GARCH. The standard viewpoint on a stochastic volatility for covariance is to use a Whishart process. See for instance Philipov, A. and M. E. Glickman (2006, July) Multivariate stochastic volatility via wishart processes. Journal of Business & Economic Statistics 24 ...


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I think you're looking for multivariate GARCH models of which this is an overview paper. Multivariate GARCH models have one big drawback: they are pretty hard to estimate due to the number of correlations. This paper by Caporin and McAleer might be of interest in that regard.


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Yes. Correlations max out at 1. However if the correlation is near 1 and the volatility of the spot is significantly larger than the volatility of the future the hedge ratio will be greater than 1. The intuition is if that vol of the future is much smaller than the vol of the spot you might need a lot more futures to minimize the high spot variance.


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Say that you did the calculations in the classic regression way. If you stick the returns of your 4 asset returns in a $(T\times 4)$ matrix $Y$, and your 3 factor returns in a $(T\times 3)$ matrix $X$, then your betas would solve the multiple regressions, collected in a $(3\times 4)$ matrix $$Y = X\cdot \beta + \epsilon$$ You could also add a column of ones ...


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With this solution you have to split your covariance matrix somewhat, but it should give you a vector with betas based on you conditional covariances. Example with two indexes, $x1$ and $x2$, and one asset $y$. $$[\sigma_{y,x1}, \sigma_{y, x2}]\begin{bmatrix} \sigma_{x1}^2 & \sigma_{x1,x2} \\ \sigma_{x1,x2} & \sigma_{x2}^2 \end{bmatrix}^{-1}$$


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A simpler question would be the following: suppose you want to find the covaraince between the returns of two stocks and each of their time series has missing values at different places. What is the best way to compute covariance here? One very sensible way to approach this is to throw away the observations where ony one of the stocks has a return value. Of ...


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Your question is formulated in a very general way, this is why any answer will need to be general as well. In a nutshell and in full generality you need to estimate the joint distribution from your historical data since in most cases correlations alone are not sufficient to define the joint distribution. In a second step you can calculate the distribution ...



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