Hot answers tagged covariance
9
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=) ...
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
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
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
...
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
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 ...
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
The answer of user27915816 led me into the right direction, yet I think I found an even better generalization:
Distance Correlation (dCor)
There are several reasons for that:
It generalizes classical (i.e. linear) correlation in the sense that linearity is a special case. It gives identical readings for linear dependence.
There are analogs for variance, ...
2
Mutual information measures how much knowing one variable reduces uncertainty about another variable. It considers any type of dependency (linear or non-linear), it's measured in bits, and it is widely used in machine learning, computer vision NLP and other fields.
1
@pyCthon's comment hit home. So I did some tests.
I compared a parametric computation to a Monte Carlo computation of IR Vol for a small set of fixed income securities. I was particularly concerned whether I could identify factors that would indicate that the difference would exceed 10% of the MC result.
Here's are my summary findings:
Vanilla IR ...
1
If you assume that your monthly returns are independent from each other, then the annualized variance of each series, and the covariance can be annualized. This assumption allows you to use V(x1+X2+...+x12) = V(x1) + V(x2) + ... + V(x12) where xi is the return for the month "i". Actually, for this to happen you only need a weaker assumption: that is that ...
1
Wikipedia gives:
$\sigma(x,y) = E[xy] - E[x]E[y]$
and
$\sigma(ax+by,cz) = ac\, \sigma(x,z) + bc\, \sigma(y,z)$
(paraphrasing the $\sigma(ax+by,cW+dV)$ rule).
So
$\sigma(I,A) = \sigma([aA+bB+cC+dD],A)$
$\sigma(I,A) = a\,\sigma(A,A) + b\,\sigma(B,A) + c\,\sigma(C,A) + d\,\sigma(D,A)$
$\sigma(I,A) = a\,\sigma^2(A) + b\,\sigma(B,A) + c\,\sigma(C,A) + ...
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