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I use the 'implied correlation' defined as $$\rho = \frac{V^2_P-\sum V^2_j}{(\sum V_j)^2-\sum V^2_j}$$ for $V_p$ the VaR (or volatility) of the portfolio, and $V_j$ the VaRs (or volatilities) of the individual components. Essentially it shows what would be the common correlation that I would need to use in order to aggregate the stand-alone risks to the ...

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In 2006 Choueifaty proposed a measure of portfolio diversification, called the Diversification Ratio (DR), which he defined as the ratio of the weighted average of the volatilities of the assets in the portfolio, to the portfolios overall volatility. The DR of a long only portfolio is greater than or equal to one, and equals unity for a ...

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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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You can also use the Herfindahl-Hirschman-Index (HCI) as a measure for concentration. In portfolio analysis, you can calculate it as $\frac{1}{N} \leq HCI(x) = \sum_{i=1}^N x_i^2 \leq 1$ where $x$ is a vector of $N$ portfolio asset weights. One can easily see that $HCI(x) = 1$ if 100% is invested in a single asset, and $HCI(x) = 1/N$ if the portfolio is ...

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Alex C's and Kiwiakos' answers are definitely the most realistic approaches. If you are open to consider also other kinds of risk measures, further alternatives might be thought of. Variance / correlation based approaches interprete "diversification" as how much your assets are heterogeneous from the point of view of deviations from the historical mean. In ...

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You can not account or skewness in the mean-variance framework as skewness is the third central moment. Thus what I would do is formulate the skewness in terms of the asset returns. I.e. for each time-step you have $$r_t = \sum_{i=1}^5 w_i r^i_t,$$ where $r_t^i$ is the return of asset $i$ at time $t$, $w_i$ is the weight and $r_t$ the portfolio return ...

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The state of the art is Asymptotic Lower Bounds for Optimal Tracking: a Linear Programming Approach by Jiatu Cai, Mathieu Rosenbaum, Peter Tankov. Hence the references of this paper are the ones to read. In the paper, they explain how you have to consider the prefactors (your $\lambda$ and $\Theta$) to be able to stay close to a target portfolio trajectory. ...

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One of the more prominent proponents has a current paper on the reasons - and why this anomaly "will persist": Why the Low Volatility Anomaly Will Persistby Eric G. Falkenstein, March 2015 Abstract Common explanations of the low volatility anomaly involve biases or frictions that cause investors to overpay for high volatility assets, giving them a ...

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