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 ...
  • 6,394
8 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: $$ Z_1 = w_{11} X_1 + w_{12} X_2 $$ $$ Z_2 = w_{21} X_1 + w_{22} X_2 $$ ...
  • 8,099
7 votes

What is the preferred GARCH method in practice?

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 ...
  • 4,247
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 (...
  • 8,099
6 votes
Accepted

Covariance matrix and Cholesky decomposition

I am not sure if I understood your question correctly but I will try to answer it anyway. If you have a standard normal random vector $z \sim N(\mathbb{0},I_n)$ (where $z,0 \in \mathbb{R}^{n\times1}$ ...
  • 2,894
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}...
  • 5,311
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, ...
  • 2,856
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. ...
  • 4,123
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 ...
  • 6,394
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

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 ...
  • 2,825
4 votes
Accepted

Garch for covariance matrix?

I think you're looking for multivariate GARCH models of which this is an overview paper. Multivariate GARCH models have one big drawback: they are pretty hard to estimate due to the number of ...
  • 7,731
4 votes
Accepted

Markowitz Mean-Variance Implied Returns

The formula is $$ \mu = \lambda CX $$ in your notation. You find it in many places, e.g. here. The assumption is that you know $\lambda$ which is a strong assumption. Furthermore it only holds if ...
  • 13.3k
4 votes
Accepted

Regime Switching for Dynamic Correlations

The clearest and most intuitive article I have seen so far is Kritzman et al., Regime Shifts: Implications for Dynamic Strategies in FAJ (May / June 2012) It not only shows how you can use HMM for ...
  • 27k
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

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
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 ...
  • 6,923
4 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}...
  • 436
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

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 ...
  • 5,938
3 votes

What is the preferred GARCH method in practice?

Interesting question, as All the answers (including mine) could not be generalized unfortunately. As far as I am concerned, I use a univariate EGARCH for risk modelling purposes (Filtered Historical ...
  • 315
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 ...
  • 141
3 votes

Ledoit-Wolf Shrinkage estimator not giving positive definite covariance matrix

The problem with Ledoit-Wolf is that it's very sensitive to outliers. You should try these: DCC GARCH unfortunately, not available in Python Exponentially weighed moving average (EWMA) gives ...
  • 319
3 votes

Semi-variance/Downside Risk, what about the rest of the covariance matrix?

There are 2 issues that come to mind What is the correct definition of semi-covariance $$ \frac{1}{n}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {\min \left( {{r_i},0} \right)} } \min \left( {{r_j}...
  • 603
3 votes

Garch for covariance matrix?

Not sure your question is about having a process for covariance or to have multivariate GARCH. The standard viewpoint on a stochastic volatility for covariance is to use a Whishart process. See for ...
  • 10.7k
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

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 ...
  • 4,123
3 votes

Handling Missing values in stocks returns when estimating the co variance matrix

One really nice book that comes to my mind is Little, Rubin, Statistical Analysis with Missing Data I read part of it but probably it is too much information in your case. For your application, i ...
  • 2,894
3 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^...
  • 2,372
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. ...
  • 14.1k

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