7

To supplement the other answer, yes there are optimization reasons for the covariance matrix being symmetric positive definite (SPD). All positive definite matrices are invertible and its inverse is also positive definite. This guarantees a unique global minimum in a quadratic optimization problem (MVO). Lots of material available on the topic: https://www....


6

Suppose the covariance matrix is $V$ (which is n by n) and the weights are $w$ (of length n). Then the Portfolio Variance is $V_p = w^T V w$ and the Risk Contribution (in terms of variance) of asset $k$ is $RC_k=w_k \sum_j V[k,j]w_j$ in words this is "the weight of asset k times the inner product of the k-th row of $V$ and the weight vector". (Sometimes ...


6

What does 'simulate a covariance matrix' mean? If the question means, generate an arbitrary correlation matrix for 1000 stocks, then we can choose any symmetric matrix with all 1s down the diagonal, so long as every element is between -1 and 1 and the matrix is positive semi-definite. The large size of the matrix means that putting random values in every ...


6

Positive definite matrix $A$ is defined as $x^TAx > 0$ for all vectors $x$. Since a term $w^T\Sigma w$ in Markowitz (and other models as well) expresses variance in returns, it is a measure of dispersion. Any measure of dispersion has to be positive (or maybe zero but it is a case where there is no uncertainty and hence no risk). Negative dispersion is ...


5

This is the result of the Sherman-Morrison inversion for the sum of an invertible matrix and an outer product. You will find this (and many other helpful methods) in the Matrix Cookbook. Specifically, this is equation 160 on p 18: $$ \left(\boldsymbol{A}+\boldsymbol{bc}^T\right)^{-1}=\boldsymbol{A}^{-1}-\frac{\boldsymbol{A}^{-1}\boldsymbol{bc}^T\boldsymbol{A}...


5

here is how to get covariance matrix from correlations:


5

If you assume that a financial asset price has a change that is a wiener process then you can view the future value of that asset as the initial value plus the sum of the independent daily changes (for equity or returns based then you would need log version of this): $$ S_t = S_0 + \sum \Delta S_i $$ where $\Delta S_i = S_i - S_{i-1} $ is a wiener process. ...


4

Just to expand on Alex answer. Empirically it is simply not true. Focusing on the diagonal of the variance-covariance matrix, we know that there is a large variance risk premium. Take a look at table 3 from Carr and Wu (2009). Regarding covariances we do not have much evidence, because there are no options on every single pair of stocks. However, we do know ...


3

OK, so think of it this way... Your standard (Markowitz) covariance matrix is a sample observation. That may or not be close to the population sigmas and correlations of your sampled markets. Even if close, the sample-vs-population errors will create asset allocation errors. The identity matrix here is the "complete strategic ignorance" covariance ...


3

You can think of it in Bayesian terms. To start with, knowing nothing at all about stocks, you might assume that stock returns are i.i.d with unit variance. This would be your prior. It is very simple and is well behaved because the identity is invertible. Then you would gather some empirical data on stock returns and measure the actual variances and ...


3

I will just clarify Point 2 in StackG excellent answer. (It's really a comment, but it's too long and has too much math symbols to fit in the comment field.) Suppose you're given a covariance matrix $C$ for the returns of $n$ assets. (1000 $\times$ 1000 is 1 million entries - should not be too large for modern computers to work with, but do be mindful of ...


3

There is nothing wrong mathematically (nor ethically) with this objective function. However, this objective is weird in a couple of ways. First, there is no weighting on these which implies you prefer to minimize these terms in accordance with their orders of magnitude. As has been pointed out, the correlation term is likely much larger so your optimization ...


3

I think what you are effectively looking at is $$\ \begin{align} \log(S_{AUDCAD})&=\log(S_{AUDUSD})\pm\log(S_{USDCAD})\\ \Rightarrow z&=x\pm y \end{align} $$ Thus, $$ \sigma_z^2=\mathrm{E}\left(\left(x\pm y\right)^2\right)- [\mathrm{E}(x\pm y)]^2 =\sigma_x^2+\sigma_y^2\pm 2\sigma_{xy} $$ Hence, $$ \tag{1} \sigma_{xy}=\frac{\sigma_z^2-\sigma_x^2-\...


2

As an addition to the already rich answers, I would suggest you to read the following paper by Marcos L. De Prado on the computation of Forward-Looking Correlation Matrices. Estimation of Theory-Implied Correlation Matrices https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3484152


2

The Ledoit-Wolf estimate cited by @develarist can be quite good, but as you say you already knew about "shrinking". It takes the population of correlations observed as an effective Bayesian prior for any given correlation, so it sort of inherently assumes that all pairs are similar an some sense. It would not work well, say, with known block sets of highly ...


2

Quantile regression is considered a robust procedure but lacks the quality of being fully differentiable. There are also regularized regression models like ridge regression, lasso regression and elastic net regression that implicitly consider the covariance of the data like OLS, but additionally reduce volatility in estimates through the introduction of bias....


2

In theory, the Ledoit and Wolf shrinkage estimator is supposed to guarantee a positive-definite matrix, given that it adds a positive-definite matrix (the target) to a semi-positive one (the sample covariance). I can see four reasons why you didn't get a positive-definite matrix: Your true covariance is effectively not full rank, i..e you have perfect ...


2

For the same reason you can't meaningfully measure covariance/correlation using price of individual assets...correlation (covariance by extension) represents the comovement in deviations from individual means. You can't represent that if the mean continues to change (ie, series considered aren't stationary). Same goes for multiple assets as is represented ...


2

The relationship between covariance, standard deviation and correlation is: $$ corr(x,y) = \frac{cov(x,y)}{\sigma_x \sigma_y}$$ So to construct your matrix you will have the variances in the diagonal: $$ cov(x,x) = corr(x,x) \times \sigma_x \times \sigma_x = 1 \times \sigma_x^2 = \sigma_x^2 $$ And for the covariances: $$ cov(x,y) = corr(x,y) \times \...


2

The interpretation and units problem, ie the lack of an easily intuitive answer, is precisely why quants/econometricians etc. tend to shy away from talking too much about covariances [even if they are absolutely necessary; and frequently used]. Thus if anything involving covariances has to interpreted, let alone explained, the default is usually to express ...


2

As a first step it is helpful to draw a 'time circle' from 0000 to 2359 GMT/UTC and plot on this circle the opening and closing times of the major markets/exchanges. This gives an overview of the situation and makes it easy to answer questions like "at xxxx GMT what markets are open and what markets are closed". (Actually you may need to have ...


1

While the close vote might be reasonable, there is mathematical arguments that show there is a limit to how negatively correlated a set of assets can be. It is even a classic quant interview question: Let the correlation matrix be $$\Omega = \rho \mathbf{1} + (1-\rho)I_d,$$ where $\mathbf 1$ is the $d \times d$ matrix with 1's everywhere. What is the range ...


1

After having tried this with randomly generated vectors, I am consistently seeing the correlation matrix of randomly generated numbers, regardless of which distribution they are sampled from, are always more well-conditioned than the covariance matrix. Which is strange because the covariance matrix exists before the correlation matrix: the correlation ...


1

Yes, you can compare matrix condition numbers if evaluating them for the same problem, for example taking the matrix's inverse. For L2: For the additional mathematical characterization of conditioning and its impact, check out the first half of these lecture notes from a class I took: https://github.com/mandli/intro-numerical-methods/blob/master/...


1

Usually, when one talks about exponential smoothing, they talk about it's halflife. So, for example, suppose we exponentially smooth some quantity ( argument carries over to covariance matrix but I'd rather just rather consider the scalar quantity case ) and call the exponentially smoothed estimate $\hat{smth_t}.$ So, this means that we have: $\hat{...


1

I don’t see how just calculation of Portfolio variance would need an invertible var-covar matrix, I mean you don’t even have to use the matrix notation to calculate it. It may be so that lower time frames would output unstable values of the individual variances and pairwise correlations. However there are certain methods to achieve a stable covariance ...


1

The paper you presented gave a thorough description. Although it actually didn't refer to Schweinler-Wignler Orthogonalisation, it referred to Lowdin Orthogonalisation, Symmetric Orthogonalisation and Canonical Orthogonalisation and the determination of the Schweinler-Wigner Matrix. The paper can be summarised as: You assume an initial set of linearly ...


1

I have actually considered the problem that you are working on, though configured somewhat differently. There isn't going to be a universal answer to your question. See, in particular, Holland, Paul W. Covariance Stabilizing Transformations. Ann. Statist. 1 (1973), no. 1, 84--92. Nonetheless, there are answers, some already mentioned. I would argue ...


1

This is not a complete answer, more a different perspective to the answers already given. If you have some a-priori knowledge about the covariance structure and about the factors influencing it, you should try to reflect this in your statistical model. Three ideas: Divide your sample into subpopulations with identical factor values and estimate separately. ...


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