9
votes
Accepted
Is there a way using matrix algebra to add portfolios to a covariance matrix of assets?
If your two assets are denoted by random variables $X_1$, $X_2$, with 2x2 covariance matrix $\mathbf{Q}$ and the portfolios:
$$ Z_1 = w_{11} X_1 + w_{12} X_2 $$
$$ Z_2 = w_{21} X_1 + w_{22} X_2 $$
...
8
votes
Accepted
CAPM model as a regression
If you really believed the CAPM's prediction that $\alpha=0$, then imposing $\alpha=0$ in your estimation would indeed lead to your 2nd formula.
The problems?
The CAPM doesn't work so imposing a ...
7
votes
Estimate covariance matrix using prices
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 (...
6
votes
Accepted
How can I use a more efficient volatility estimator to improve the co-variance matrix?
Let $s$ be a $N\times1$ vector of standard deviations and $C$ be an $N\times N$ correlation matrix. The covariance matrix is equal to
$$\Sigma=\text{diag}(s) \ C \ \text{diag}(s)$$
where $\text{diag}...
6
votes
Accepted
Simulating covariance matrices with nonzero correlation
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, ...
6
votes
Accepted
Covariance of two Brownian Motions
Since $\text{Cov}(X, Y) = E(XY) - EX EY$, we have
\begin{align}
\text{Cov}(tB_{3t} - B_{2t} + 5, B_s - 1) &= E[tB_{3t}B_s - tB_{3t} - B_{2t}B_s + B_{2t} + 5B_s - 5] - (5)(-1) \\
&= tE[B_{3t}...
6
votes
Accepted
Shrinkage of the Sample Covariance matrix, theory
Yes. It comes from a core theorem of statics, Stein's Lemma. It shook the foundations of the field of statistics when it came out. It blew up an entire way of viewing mathematical statistics. ...
5
votes
Accepted
How to calculate the covariance between two stochastic integrals?
By:
bilinearity of covariance,
independence of Brownian increments, and
Itô's isometry,
we obtain:
$$\begin{align}
& \text{Cov}\left(\int^{t_1}_0\sigma(t)dW_t,\int^{t_2}_0\sigma(t)dW_t\right)
...
5
votes
Accepted
Finding a minimum variance portfolio when using a regulariser?
You're not going to get an analytic formula except in special cases of function $\rho(x)$. And you're probably going to want $\rho$ convex.
If $\rho$ is convex, the problem is a convex optimization ...
5
votes
Accepted
Widely accepted methods for coming up with the co-variance matrix of assets?
Multivariate volatility models for replacing the sample covariance matrix with in the mean-variance portfolio selection model:
RiskMetrics 1996 EWMA (Exponentially weighted moving average) covariance ...
5
votes
Covariance Between Two Frontier Portfolios
Let $\Sigma$ denote the covariance matrix of our asset universe, $\mu$ is the vector of expected returns. Further, $\mathbb{1}$ is a vector of ones. Let's identify the vector of the minimum variance ...
4
votes
Accepted
Using CAPM to find correlation of two assets with each other
The solution provided can be derived using the CAPM. For asset $A$ you have:
$$R_A-R_f = \alpha_A +\beta_A(R_M-R_f)+\epsilon_A$$
Similarly for asset B:
$$R_B-R_f = \alpha_B +\beta_B(R_M-R_f)+\...
4
votes
Correlation -1 and standard deviation
$\sigma_p=\sqrt{\omega_a^2 \sigma_a^2+(1-\omega_a)^2 \sigma_b^2+2 \omega_a (1-\omega_a) \rho_{ab} \sigma_a \sigma_b}$
with
$\rho_{ab}=-1$
the term under the square root simplifies to
$(\omega_a \...
4
votes
Accepted
What is the covariance of two correlated Ornstein-Uhlenbeck processes?
Using https://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process#Solution
$$X^i_t = (X^i_0 + \int_0^t\sigma_i e^{a_i u} dB^i_u)e^{-a_it} $$
and
$$ X^i_t-\mathbb{E}[X^i_t] = e^{-a_it} \int_0^...
4
votes
Accepted
Variance-Covariance Matrix under $\mathbb{P}$ and $\mathbb{Q}$
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 ...
4
votes
Meaning of an identity matrix for the covariance in portfolio optimization
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 ...
4
votes
Accepted
Meaning of an identity matrix for the covariance in portfolio optimization
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 ...
4
votes
Accepted
Show that the following result holds true for the variance of the return of a portfolio of shares
The variance part is correct.
For the covariance part we can observe the following: There are $n$ variance terms in the $n \times n$ covariance matrix. This implies that there must be $n^2-n$ ...
4
votes
Find k of n assets that "minimize" the correlation matrix
(I take it that 5 out of 10 assets is just an example, because in this case all combinations could easily be checked.)
Here would be an example how to do it in R ...
4
votes
Accepted
Sample Variance of Portfolio
Yes, indeed. It's a simple Linear Algebra and Expectation result:
Given:
$Var(w'r) = \mathbb{E}[(w'r)^2] = \mathbb{E}[(w'rr'w)]$
With $w$ and $r$ the vectors of weights and returns. As $w$ is constant,...
3
votes
Co-variance of Portfolio A with Portfolio B
If you take $A^T∗COV∗B$ then the result will be 1 x1 ( a scalar). (1xN * NxN * Nx1 = 1x1).
I believe you forgot to take the transpose of A. The vector which pre-multiplies COV needs to be a row ...
3
votes
Accepted
What are the units of the variance of returns?
There is nothing incorrect with your formulas, so let's look at the units when you annualize the volatility.
As an example, assume you have 252 daily return data. Their dimension is $(time)^{-1}$ and ...
3
votes
Accepted
Filtering smallest eigenvalues
After writing an email to Meucci directly, I posted the question in his LinkedIn Group ARPM - Advanced Risk and Portfolio Management. Below are his answer and the answer of other group members, which ...
3
votes
Negative variance?
As pointed out by other users here your designed covariance matrix appearantly is not positive-definite and therefore you get this strange behaviour.
Please note that this is not just a mathematical ...
3
votes
Covariance estimation: shrinkage, random matrix theory, what else?
I thought I would answer the question of "what am I using." All shrinkage estimators map to a Bayesian estimator that differs only in the prior distributions. In other words, you get a point ...
3
votes
Ledoit-Wolf Shrinkage estimator not giving positive definite covariance matrix
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 ...
3
votes
Semi-variance/Downside Risk, what about the rest of the covariance matrix?
one solution that works is set up the usual correlation matrix and pre- and post multiply by a diagonal matrix with semi standard deviations down the diagonal taking care that they are not zero
3
votes
Using Kendall rank correlation to construct a covariance matrix?
A first hint:
To convert Kendall's $\tau$ to the Pearson correlation coefficient $\rho$, one could use the relationship: $$\rho = \sin\Bigl(\frac{\pi}{2}\tau\Bigr)$$
But keep in mind that this only ...
3
votes
Are two stochastic processes independent if the Wiener processes inside are uncorrelated
[Edit]
My "answer" below is not a really an answer for I have completely misinterpreted your original question. I thought you asked about the covariance of 2 processes over a given time horizon (i.e. ...
3
votes
Why the weight vector of 'global minimum variance' the 'eigenvector' with the minimum eigenvalue?
You will have to add some constraints to get the weight vector of the eigen vector of the smallest eigen values, otherwise 0 is a trivial solution.
Without going in the details of handling those ...
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