22
One of my favorites is a generalization of correlation: 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, covariance and standard deviation, so these identities ...
18
Nick Higham's specialty is algorithms to find the nearest correlation matrix. His older work involved increased performance (in order-of-convergence terms) of techniques that successively projected a nearly-positive-semi-definite matrix onto the positive semidefinite space.
Perhaps even more interesting, from the practitioner point of view, is his ...
16
You can use changepoint analysis to identify regime change.
You can also look at large angle differences in the eigenvectors between your most up-to-date/recent covariance matrix and the covariance matrix from the prior window.
Another way to identify regime change is using a factor model. If the returns on a particular set of factors is X standard ...
12
Yes it is a better way.
Just take a look to figure 3, from Buss and Vilkov (2012, RFS):
10
Here is a structured list of your bullet points:
covariance,
correlation,
PCA,
factor analysis,
Are similar. They are based on Gaussian assumptions (i.e. correlations means dependencies) and try to identify common factors (i.e. a variable in small dimension) explaining the observed relationships.
co-integration
is more specific in the sense that you ...
10
This is indeed an interesting question.
According to this website, a paper by Goldman Sachs [Tierens and Anadu (2004)] proposes three alternative methods for estimating average stock correlations:
Calculate a full correlation matrix, weighting its elements in line
with the weight of the corresponding stocks in the portfolio/index,
and excluding ...
9
Two ways:
Model the returns using an Ornstein-Uhlenbeck process
You can control the variance of the residual noise in the process to your desired level of correlation. Conceptually you inject gaussian noise into the synthetic OU process to satisfy your requirement.
For example, let's say you have time-series A which is what you are modelling. Time-series ...
9
Your formula looks like cointegration (between the price time series) rather than correlation (between the returns).
To simulate "correlated random walks", i.e., random walks built from correlated innovations, you can just build the desired covariance matrix (for instance, put ones on the diagonal and $\rho$ everywhere else), take multivariate gaussian ...
9
The problem of the selecting the best portfolio (according to some risk measure) with a limited number of assets can be formulated as a mixed integer linear or quadratic program and is reviewed in the recent paper "Portfolio selection problems in practice: a comparison between linear and quadratic optimization models". It can be solved for reasonable sizes ...
8
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 ...
8
This is a misunderstanding of how to apply RMT theory.
The point of the MP distribution is to describe the expected distribution of eigenvalues assuming a symmetric matrix whose elements are drawn from a normal distribution of mean zero and some sigma. So if you observe eigenvalues beyond the level predicted by MP this means you have found factors that are ...
8
Here is the general approach you can follow to generate two correlated random variables. Let's suppose, X and Y are two random variable, such that:
$$X \sim N(\mu_1, \sigma_1^2)$$
$$Y \sim N(\mu_2, \sigma_2^2)$$
and $$cor(X,Y)=\rho$$
Now consider: $y=bx + e_i$, where $x$ $(=\frac{X-\mu_1}{\sigma_1}$) and $y$ $(=\frac{Y-\mu_2}{\sigma_2}$) both follow ...
7
Regarding the optimization question: I haven't compared random matrix estimates to shrinkage estimates, but shrinkage seems to beat (statistical) factor models -- see a series of blog posts at http://www.portfolioprobe.com/tag/ledoit-wolf-shrinkage/
However, my guess is that random matrix estimates behave a lot like factor models, and hence that shrinkage ...
7
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 ...
7
I would suggest a multivariate garch model as a possibility. We aren't exactly overrun with wonderful software for that, but with just bivariate data I would think that the in-sample correlation estimates would be reasonably robust over models and estimation.
It would be good to try two or three ways of doing it to make sure I'm right about that.
You may ...
7
well, it is absolutely in agreement with theory. the correlation as measured by Pearson's coefficient $\rho$ is linear measure in the sense that the bounds [-1,1] are obtained only when transformations of our variables are linear, so if we have variables $X$ and $Y$ then something like $aX+bY+c$ where $a,b\in\mathbb{R^*}$, $c\in\mathbb{R}$ will have ...
7
If you look at tick data, you will probably get an even better analysis. However, vix correlation tends to be negative with spx but remember that this is generally more true for when spx tanks. When spx goes up, the correlation isn't as strong. Why? People panic after a drop, therefore leading to people buying options. They don't care about black scholes ...
7
Apart from numerical stability errors, Cholesky and PCA (without dim reduction) shall produce exactly the same distribution, they are two symmetric decomposition of the same covariance matrix and thus are equivalent for transforming a standard normal vector.
Of course when doing different things with PCA components, such as in dim reduction or quasi Monte ...
7
I think it's alive and well. I don't think there's a specific "decoupling" time, but if you look at e.g. Munnix et al. "Statistical causes for the Epps effect in microstructure noise", it seems that the biased correlation is about 60% of the real value for 1 min data and about 90% for 5 min data, so you could say that 5 min is pretty safe, but 1 min is ...
7
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 ...
7
To clarify notation, you have an universe of $n=2000 \space$ stocks and two portfolio vectors $\mathbf{a},\mathbf{b}\in\mathbb{R}^{n}$ with $\left\|\mathbf{a}\right\|_{1}=\left\|\mathbf{b}\right\|_{1}=1$. Further, you have Estimators for the true Variance $\operatorname{Var}\left[\mathbf{a}\right]$ resp. $\operatorname{Var}\left[\mathbf{b}\right]$ and the ...
7
I personally use the simple Garch(1,1) for volatility filtering in the risk management area.
In fact in most cases I don't even estimate the parameters, I stick 0.94 for mean reversion, 0.04 for the squared error and I get the constant by matching the series variance. My experience is that there is no point pretending to finetune parameters when vol is ...
7
First you need to correct the formula to:
$$
W_t^2 = \rho W_t^1 + \sqrt{1-\rho^2} Z_t,
$$
where $Z_t$ is a BM independent of $W_t^1$
If you calculate the variance and the covariance, then you see that it is true:
$$
V[W_t^1] = t
$$
and
$$
V[W_t^2] = \rho^2 V[W_t^1] + (1-\rho^2) V[Z_t] = \rho^2 t + (1-\rho^2) t = t,
$$
which is the desired variance.
For the ...
7
We can obtain a closed-form expression for price correlation given (log) return correlation when the two stocks follow geometric Brownian motion:
$$S_1(t) = S_1(0)e^{(\mu_1- \frac{1}{2} \sigma_1^2)t}e^{\sigma_1Z_1(t)},\\ S_2(t) = S_2(0)e^{(\mu_2- \frac{1}{2} \sigma_2^2)t}e^{\sigma_2Z_2(t)},$$
where $\text{corr}(Z_1(t),Z_2(t)) = E[Z_1(t)Z_2(t)]=\rho t$. ...
6
I would entirely separate the investigation and analysis of volatility and correlation between two asset classes.
Think of it this way: If volatility is extremely high then high fluctuation to both, the up and down side will contribute to the stability of high volatility. However, for correlations it makes a big difference whether the large move has been ...
6
Extra market volatility alone will cause correlations and stock volatilities to spike as you describe, even when overall market structure remains unchanged.
There's a minor variation of the very simple CAPM model that captures precisely this behavior.
To be specific, let's say every security $S_A, S_B, \dots$ (or yield, if you want bonds in this) has a ...
6
It is hard to find a stable non-trivial dependence structure in financial data. Usually when such is found it is hard to rationalize.
One of my favorite (although I am sure there are others) is the so called "Presidential Puzzle". This is an old finding by Santa-Clara and Valkanov (2003) They find that "
Excess return in the stock market is higher under ...
6
I just want to add to vonjd's answer some info on the comparison of the 3 methods. This is too big for a comment so I'm posting as a separate answer but please upvote his answer, not mine.
Do the differences in methodologies matter in practice?
To gauge the practical importance of the biases in methods 2 and 3, we calculate the weighted stock correlation ...
6
Let us consider a basket $B$ with components $S_1,\dots,S_n$ :
$$B(t) = \sum_{i=1}^nw_iS_i(t)$$
At time $t$, each component has standard deviation $\sigma_i$, $i \in \{1,\dots,n\}$, and pairwise correlations are $\rho_{ij}$, $i \not= j$. Thus:
$$\sigma_B^2=\sum_{i=1}^nw_i^2\sigma_i^2+2\sum_{i=1}^n\sum_{1=j}^iw_iw_j\sigma_i\sigma_j\rho_{ij}$$
The implied ...
5
The following paper (Identifying Small Mean Reverting) is not directly related to portfolio risk minimization but it provides a method to build tradable mean reverting portfolios based on a multivariate co-integration approach. It has the advantages of providing a theoretical framework along with two algorithms. It also takes into account financial strict ...
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