26
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 ...
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
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 ...
9
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 ...
8
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 ...
8
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 ...
8
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 $\text{corr}(Z_1(t),Z_2(t)) = E[Z_1(t)Z_2(t)]=\rho t$. ...
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
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
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 ...
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
Here is a working paper that you may be interested in.
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 ...
5
Accurately stated: Diversification helps during turmoil, but helps less as what would be expected by using $w^T \Omega w$ as the portfolio variance where the off-diagonal covariances are estimated during tranquil periods.
This is because correlations and covariances change during turmoil, typically increasing. This reduces the benefit of diversification ...
5
If $\Sigma$ is the covariance matrix of all assets and $w$ is the column vector of weightings of the asset in a certain portfolio. Then
$$
w^T \Sigma w = VAR
$$
is the variance of the portfolio. The contribution to volatility of asset $i$ is given by
$$
w_i (\Sigma w)_i/\sqrt{VAR},
$$
where $(\Sigma w)_i$ is the $i_{th}$ entry in the vector $\Sigma w$.
Note ...
5
Let $(X_t)_{t\geq 0}$ denote a Geometric Brownian Motion
$$ \frac{dX_t}{X_t} = \mu_X dt + \sigma_X dW^X_t,\ \ \ X(0) = X_0$$
such that $X_t$ is lognormally distributed $\forall t > 0$
$$ X_t = X_0 e^{(\mu_X - \frac{1}{2}\sigma_X ^2)t + \sigma_X W_t^X}$$
Let $(Y_t)_{t\geq 0}$ denote an Arithmetic Brownian Motion
$$ dY_t = \mu_Y dt + \sigma_Y dW_t^Y,\ \ \ ...
5
You could, and it doesn't hurt for you to test this yourself. Some of my best work has come from drawing the opposite conclusion to conventional wisdom or stylized "facts" in publications.
That said, it's trivial to construct an example where you won't be able to spread a correlated pair. Suppose the underlying data generation process is $y_t = x_t^2$, you ...
5
here is how to get covariance matrix from correlations:
5
If you assume that a financial asset price has a change that is a wiener process then you can view the future value of that asset as the initial value plus the sum of the independent daily changes (for equity or returns based then you would need log version of this):
$$ S_t = S_0 + \sum \Delta S_i $$
where $\Delta S_i = S_i - S_{i-1} $ is a wiener process.
...
4
If you only need to pick 5 out of 10 and want equal weights then just enumerate all 252 possibilities (as pointed out above) and compute the portfolio volatility
$(\textbf{1}'K^{(i)}\textbf{1})^{1/2} = \left( \sum_{ij}K^{(i)}_{ij} \right)^{1/2}$,
where $K^{(i)}$ is the covariance matrix for the $i$th subset. Then use whatever subset gives the lowest ...
4
Simple rule: correlation coefficient R of N samples is statistically significant if:
$|R| > 2 / \sqrt{N}$
http://capone.mtsu.edu/dwalsh/436/CORRSIG.pdf
But watch out for spurious correlations. It is possible to find statistically significant correlation for non stationary data series even though there is no correlation.
http://www.investopedia.com/terms/...
4
Let's start by replacing $\sigma$ by its estimator formula $\sigma^2=\frac{1}{n}\sum^n_{i=1}(x_i-\mu)^2$. Now, by replacing $\mu$ by its estimator $\mu=\frac{1}{n}\sum^n_{i=1}x_i$ in the formula for the variance we obtain:
$\sigma^2=\frac{1}{2n^2}\sum^n_{j=1}\sum^n_{i=1}(x_i-x_j)^2$.
For the individual asset, the variance will write $\sigma^2_s=\frac{1}{2n^...
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