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9 votes
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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: $$ Z_1 = w_{11} X_1 + w_{12} X_2 $$ $$ Z_2 = w_{21} X_1 + w_{22} X_2 $$ ...
Attack68's user avatar
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8 votes
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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 ...
Matthew Gunn's user avatar
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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 (...
Attack68's user avatar
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6 votes
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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}...
John's user avatar
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6 votes
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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, ...
StackG's user avatar
  • 3,016
6 votes
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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}...
R. Rayl's user avatar
  • 466
6 votes
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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. ...
Dave Harris's user avatar
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5 votes
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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) ...
Daneel Olivaw's user avatar
5 votes
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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 ...
Matthew Gunn's user avatar
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5 votes
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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 ...
develarist's user avatar
  • 3,000
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 ...
Kermittfrog's user avatar
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4 votes
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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)+\...
Comp_Warrior's user avatar
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 \...
Chris Degnen's user avatar
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^...
M. Jeunesse's user avatar
  • 2,422
4 votes
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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 ...
phdstudent's user avatar
  • 8,306
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 ...
nbbo2's user avatar
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4 votes
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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 ...
demully's user avatar
  • 5,061
4 votes
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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 $n^2-n$ ...
Pleb's user avatar
  • 4,276
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 ...
Enrico Schumann's user avatar
4 votes
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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,...
André Bittencourt's user avatar
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 ...
Alex C's user avatar
  • 9,372
3 votes
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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 ...
skoestlmeier's user avatar
  • 2,916
3 votes
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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 ...
Hans-Peter Schrei's user avatar
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 ...
Richi Wa's user avatar
  • 13.7k
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 ...
Dave Harris's user avatar
  • 4,299
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 ...
Matifou's user avatar
  • 141
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
Steve Satchell's user avatar
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 ...
simzoor's user avatar
  • 383
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. ...
Quantuple's user avatar
  • 14.6k
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 ...
Antoine's user avatar
  • 51

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