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17

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 ...


17

I would consider a motion chart that plots the eigenvalues of the covariance matrix over time. For a static view you can create a table: rows represent dates, and columns represent eigenvectors. The entries of the table represent changes in the angle of the eigenvector from the previous row. This will show how stable your covariance structure is. You can ...


11

The PCA analysis does not really tell you what the bonds do but it tells you how the rates move together. The variations of $n$ rates (i.e. 1 y, 2y, ...) are split up in (at first) abstract factors like $$ \Delta R_i = \sum_{j=1}^n e_{i,j} f_j $$ where $\Delta R_i$ is the change in the rate $i$ and $f_j$ is factor $j$ and $e_{i,j}$ is the (factor loading=) ...


9

I am implementing a method in Java to calculate the variance, covariance, and value at risk for a portfolio, which should be flexible for use with any number of assets in a portfolio. I am struggling with how to calculate the covariance of the assets as I can only find formulae to do so for two or three sets of values. Are you sure you ...


9

Here's an interesting possibility: correlation network analysis + motion chart. Thanks to the hot research efforts in social network analysis (SNA), network analysis and graphics libraries such as R and Gephis are now easily accessible. I am well-versed in correlation analysis, and have a feeling that SNA can be effectively adapted for it. After all, the ...


9

Nick Higham happens to have given a talk on this very subject this summer; he continues to actively work to improve nearest correlation matrix algorithms. You can see his talk and notes here: http://mxm.mxmfb.com/rsps/ct/c/629/r/90368/l/48110


8

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 ...


7

If $X \sim N(\mu, V)$ is multivariate gaussian, you can write $X = \mu + C Y$ where $ Y \sim N(0,1) $ is a standard Gaussian and $C$ is the lower-triangular Choleski matrix of $V$. You can then express $ v = \sum_{i=1}^n (X_i - S/n)^2 $, where $ S = \sum_{i=1}^n X_i $, in terms of $Y$ and $C$. (I do not reproduce the computations: they are straightforward.) ...


7

You are correct: evaluating volatility forecasts is quite different from evaluating forecasts in general, and it is a very active area of research. Methods can be classified in several ways. One criterion is to consider evaluation methods for single forecasts (e.g., for the time series of returns of a specific portfolio) vs multiple simultaneous forecasts ...


6

The short answer is that I don't know, but your question gives some hints about how to find out. The key thing for me is that you want a minimum variance portfolio. I don't think you should be thinking about some abstract mathematical operation that is "best", but rather look over a few mathematical operations and see which seems to work best for your ...


6

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 ...


5

The standard estimator of the covariance matrix is: $$\widehat{ \mathrm{cov}}(X) = \frac 1 {n-1} \sum_{i=1}^n (X_i-\bar X)(X_i-\bar X)^T,$$ where $X_i$ is the column vector containing the $i$th observation of all the observables. Each summand is an outer product of a vector with itself, i.e., a square matrix having rank at most one. Therefore ...


5

In Oracle Crystal Ball, we use an old algorithm, that works pretty well and converges fast. It is from Iman-Conovar. Here is the reference: Iman, R.L., Conover, W.J. 1982. A distribution-free approach to inducing rank correlation among input variables. Commun. Statist.-Simula. Computa. 11, 311-334. That said, Prof. Higham's method based on optimization ...


5

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 ...


4

I think an extremely interesting strand of research on this topic is represented by extensions of vine copulas with time-varying parameters. For vine copulas in general have a look at this site from the Technische Universität München: Vine Copula Models One of their research projects, which is the most relevant in this context, is:Time varying vine copula ...


4

I'd look at the evolution of a heat map based on the correlation structure (literally the lower triangle). I'd probably write a script in R or python that writes out the heat map per t to disk, then use a command line program like imagemagick to stitch images together into an animated gif, for example. I'm sure you could do it entirely in Processing too, ...


4

You probably want to take it back to how one evaluates forecast models in general: using some metrics over one- or many-step forecasts, see e.g. here for a Wikipedia discussion. But instead of forecasting first moments, it would now be second moments. This can still use (root) mean squared error, or mean absolute percentage error, or related measures; ...


4

You can use the Exponentially Weighted Average directly aswell, finding the covariances and then normalizing back to the correlations: $ \sigma_{t+1,jk} = (1-\lambda) \sum_{n=0}^\infty \lambda^{n} r_{j,t-n} r_{k,t-n} $ (this assumes average returns 0 etc etc. More general versions can be derived)


4

The given matrix can not represent a covariance matrix since it would imply that asset 1 is negatively correlated to asset 2 and asset 3. But asset 2 is negatively correlated to asset 3 which contradicts the first statement. In general a covariance matrix has to be positive semi-definite and symmetric, and conversely every positive semi-definite symmetric ...


3

Is the covariance of the raw return forecasts a good forecaster of the covariance of market returns? As you suggest, the covariance of the raw return forecasts is a lousy forecast of the covariance of market returns. Grinold & Kahn explain why quite eloquently in Active Portfolio Management, 2nd edition (pg. 275). It might be tempting to augment the ...


3

For the stationary multivariate normal case, the expected returns vector does not matter. This is because the cross-sectional mean is subtracted out before calculating the standard deviation. The cross-sectional mean can be more conveniently thought of as like the return on an equally weighted portfolio. Similarly, I would argue that the expected ...


3

1) Calculate exponential averages (EMA) for time series A & B. 2) Calculate exponential standard deviations for A & B. My little hack for this is to calculate an EMA of squared returns, then subtract the squared EMA of simple returns, then take the square root of this. sqrt( ema(return^2) - ema(return)^2 ) 3) Apply the same concept to calculating ...


3

Any explanations? Yes. Within each asset category we find that stocks may be: Unattractively underperforming the category norm Attractive as they meet the expected norm Unsustainable as their returns exceed the category norm and may suffer mean reversion By focusing on low variance, we exclude type (3) stocks that damage portfolio performance through ...


3

The Newey-West procedure is meant to adjust the covariance matrix of the parameters to account for autocorrelation and heteroskedasticity. It is typically used in financial applications when one estimates the alpha (a parameter in a regression model) of a portfolio or strategy. One would adjust the standard errors using the Newey-West procedure in order to ...


3

There are many techniques, but I would begin with Stambaugh Analyzing Investments Whose Histories Differ in Lengths. The full information maximum likelihood approach he describes basically involves regressing the short history series against the long history series to obtain the covariance with the longer history securities and adding back the covariance of ...


3

There are several measures discussed in the literature, the classical approach is Markowitz mean-variance portfolio optimization. The formula for portfolio return variance is $$\sigma_p^2 = \sum_i w_i^2 \sigma_{i}^2 + \sum_i \sum_{j \neq i} w_i w_j \sigma_i \sigma_j \rho_{ij}$$ where $\rho_{ij}$ are the correlations betweent the assets. Others suggeste ...


2

In mean-variance portfolio work, the elements of the covariance matrices are highly volatile and infused with error, so how to obtain forecasts that are usable ? A simple idea is to use a Stein-equal covar shrinkage estimator which, in practice, is easy to calculate and produces superior portfolios when evaluated on out-of-sample data ( see Continuous Time ...


2

I think a good approach is to compare your two covariance matrices on a set of random portfolios (see for instance http://www.portfolioprobe.com/about/applications-of-random-portfolios/assess-risk-models/). What you want is a high correlation (across the portfolios) between the predicted and realized portfolio volatility. We're never going to estimate the ...


2

Once we start building time-varying copulas like Lopes suggests in that paper, I think we are better off venturing into the world of state space models. When viewed in a bayesian context, the similarities between the approaches are striking to me. The advantage of the copula, as I understand it, is that it is a quick and dirty way to understand the ...


2

Maybe I don't get your question well, but what it appears that your goal is to buy securities in order to reduce the correlation between your portfolio constituents. So firstly you need a metric of diversification. Something simple you can use, is calculate the correlation matrix, and the weights of each position. A simple metric would be the sum of ...



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