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The minimum variance solution loads up on securities that have low variances and co-variances. Theoretically you are correct that this should have a low expected return profile. However, it turns out - in contradiction to modern portfolio theory - that securities that have low-volatility or low-beta experience higher returns than high-volatility or high-...

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I might be misunderstanding your question. My thoughts: being short gamma is being long volatility your comment re gamma increasing regardless of direction only holds for ATM options. For ITM options, being short gamma is being long the underlying. For OTM options, being short gamma is being short the underlying. Some graphs: Below, except as ...

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Actually, co-skewness is represented by a rank 3 tensor, rather than a matrix. I'm going to reproduce the formulation from Bhandari and Das, Options on portfolios with higher-order moments, but I'll add and omit some details. The co-skewness tensor is $$S_{ijk} = E \left[ r_i \times r_j \times r_k \right] = \frac{1}{T} \sum_{t=1}^T r_i(t) \times r_j(t) \... 11 The following papers may help. A New Look at Minimum Variance Investing by Bernd Scherer Minimum Variance Portfolio Composition by Clarke, De Silva & Thorley Under a multifactor risk-based model, if the global minimum variance portfolio dominates the market portfolio, the implication is that the market portfolio is not multifactor efficient and that ... 10 Being short gamma simply means that you are short options regardless of whether they are puts or calls. The most common type of investor that is willing to be short gamma is someone who sells options, also known as a premium collector. These investors commonly use strategies such as short puts, covered calls, iron condors, vertical credit spreads, and a ... 10 I'll answer by way of example. Suppose I want to buy a stock that is relatively cheap. Firstly, I need to define what is meant by cheap, so I might choose to look at the price-to-earnings ratio. Then I need to define what is meant by relative, so I might compare stocks only within a given sector. This may work well at first, but then I notice that as I try ... 10 Unlike the tangency portfolio on the efficient frontier (which represents the most efficient portfolio in terms of max expected sharp ratio), min var portfolios have no ex ante theory that suggests it should outperform a cap weighted market portfolio. The same can be said about other risk-weighted portfolio construction schemes, including equal risk ... 9 Yes Strategic Asset Allocation: Determining the Optimal Portfolio with Ten Asset Classes Strategic Asset Allocation and Commodities The Case for Commodities An Asset Class for All Seasons: The Benefits of a Strategic Allocation to Commodities No Should Investors Include Commodities in Their Portfolios After All? New Evidence My Take Although there ... 9 Van Tharp, in his book Definitive Guide to Position Sizing, identifies 31 separate models for money management. In said book he specifically warns against using both the Kelly Criterion and Optimal f. In addition to the models identified by Van Tharp there is Ralph Vince's Leverage Space Portfolio Model. 9 +1 for asking an excellent question. I agree with the answers of @Owen and @chrisaycock - I'm late to the party but perhaps this will shed some light. How practitioners or academics answer this question will tell you a lot about their view on the nature and sources of returns and risk. For example, the Fama-French "equilibrium" school of thought would argue ... 9 Perform a returns analysis by regressing the returns of your composite strategy on the returns of the component strategies. Constrain the beta coefficients to sum to 100% and bound them from 0 to 1. You will then have the % explained by each component. 8 The PortfolioAnalytics package will create weights without reference to current weights, if that's what you want. It should also have much of the reporting that you like from Rmetrics fPortfolio. There is a longer seminar presentation on Portfolioanalytics from 2010's R/Finance conference here: Complex Portfolio Optimization with Generalized Business ... 7 This is the website to the R/Finance conference this year. Tons of great links. http://www.rinfinance.com/agenda/ Brian Peterson's slide (Building and Testing Quantitative Strategy Models in R) mentions Portfolio-Analytics (which I think is based on R/Metrics). And here is a paper based on Portfolio-Analytics. http://cran.r-project.org/web/packages/... 7 You are absolutely right to point out that most proactive participants in options markets prefer to be long gamma, and it is typically reactive market makers who take the opposite side of their trades. While the typical options trader (I find it difficult to call anyone trading options an "investor") does not hedge his position, market makers will attempt ... 7 The minimum variance optimization framework does not guarantee positive return whatsoever. As a matter of fact what you are trying to do is something close to the following:$$\underset{w}{\arg \min} \quad w' Q w \quad \text{s.t} \quad Aw \leq b,\quad \sum_i w_i=1$$The fact that you get positive return is a nice result that you get from your backtest (i.... 7 In my experience, a VaR or CVaR portfolio optimization problem is usually best specified as minimizing the VaR or CVaR and then using a constraint for the expected return. As noted by Alexey, it is much better to use CVaR than VaR. The main benefit of a CVaR optimization is that it can be implemented as a linear programming problem. Another option I have ... 7 Bernd Scherer has done exactly this test in his text "Portfolio Construction and Risk Budgeting 4th Edition". There is an SSRN paper by Scherer called "Resampled Efficiency and Portfolio Choice (2004)" you can take a look at as well. I would suggest you skip re-sampling (especially if you have a long-only portfolio) and take a look at Meucci's Robot ... 7 It appears that you are re-running the regression with each new data point. Instead, you should use an update/online formula (see an excellent answer by the famous Dr. Huber at stats.se). You can find an implementation in the R package biglm. If it doesn't have all the features you need (no windowing out of old data) you can at least adapt it and use it ... 7 If you measure risk by the standard deviation of the portfolio return$$ \sigma = \sqrt{w^T \Sigma w}, $$then it is usual to define risk contributions for each asset by$$ \sigma_i = w_i (\Sigma w)_i/\sigma, $$then diversified could mean that these \sigma_i are evenly spread over the assets in the portfolio. You find this approach and more in this paper ... 6 The Kelly criterion is a very popular bet-sizing method. Edward Thorp has written a great deal on this topic. You can try googling for more, or start with his review of the concept, or a recent paper, Medium Term Simulations of The Full Kelly and Fractional Kelly Investment Strategies. This is not specific to futures, but I'm not sure why you would need ... 6 In statistical arbitrage, quant traders attempt to build a neutral portfolio by balancing various assets against each other. Each asset's size within the portfolio isn't determine necessarily by how much money it's expected to generate, but by how correlated it is against other assets. A simple approach is sector neutrality, in which sector/industry ... 6 Short gamma is a bet on volatility (expressed as hedging costs) not getting too large. The key concept here is that you get paid to be short gamma. Consider that any option is sold for a bit more more than its intrinsic value (the extra bit is often called volatility value.). If nothing moves, then the option ultimately expires precisely at intrinsic ... 6 Hull - Options, Futures and other Derivatives. 6 There's a strong theoretical argument that makes the case for active management that is also supported by empirical research. First, check out Jonathan Berk's paper "Five Myths of Active Management". The paper reads like a clever Gedankenexperiment. Starting with a theoretical approach is better than starting with an empirical approach because as Berk ... 6 The VaR constraint is convex and quadratic and can be handled with any solver supports quadratic constraints, like Guribi, cplex (from IBM) or xpress (from FICO). The CVaR can be formulated as a linear program if you are able to perform monte-carlo simulations on the returns. Briefly, the LP model is \begin{eqnarray*} c &\ge& \alpha + {1 \over (... 6 Strictly speaking, this is a proxy hedging problem. You have to hedge one currency with another. The one period covariance matrix is assumed to be$$\Sigma_{1}=\left[\begin{array}{cc} 0.03 & 0\\ 0 & 0.025 \end{array}\right]\left[\begin{array}{cc} 1 & 0.8\\ 0.8 & 1 \end{array}\right]\left[\begin{array}{cc} 0.03 & 0\\ 0 & 0.025 \end{...

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This is indeed a subtle point. What is generally meant with this statement is that correlation is going up in bear markets, so it is not so much the "turmoil" part (i.e. volatility per se) but the "trend" (i.e. negative in this case) part. Putting it another way is that when you control for volatility not the correlation but the covariance (which is the part ...

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The estimation of a covariance matrix is unstable unless the number of historical observations $T$ is greater than the number of securities $N$ (5000 in your example). Consider that 10 years of data represents only 120 monthly observations and about 2500 daily observations. Depending on the application, using data dating farther back than 10 years may be ...

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After having done a lot of research on the topic I found the following excellent research piece on ETF.com: Wealthfront modifies historic asset-class returns with current market implied expected returns (Black-Litterman) as well as with the in-house views of Chief Investment Officer Burton Malkiel’s team. In addition, Wealthfront sets minimum and ...

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Since $dW_A$ and $dW_B$ are already correlated as per the way you construct it, your portfolio being the sum of the two is already correlated. If you want it very explicitity written out, then you could rewrite $dW_B = \rho dW_A + \sqrt{1-\rho^2}dW_Z$ where $dW_Z$ is independent of $dW_A$. More generally (higher dimensions) you can use Cholesky. Now with ...

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