Questions tagged [covariance]
A measure of the degree of linear association between a pair of random variables.
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Prove ρ(X,Z) = ρ
The covariance of two random variables $X$ and $Y$ is defined by:
$$\mathrm{Cov}(X,Y）= \operatorname{E}(X-\operatorname{E}(X))(Y-\operatorname{E}(Y))=\operatorname{E}(XY)-\operatorname{E}(X)\...
2
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1answer
86 views
Co-variance of Portfolio A with Portfolio B
I'm trying to calculate the correlation between two separate portfolios.
I've used A*COV(AB)*B to calculate the co-variance of each portfolio where:
A = Array ...
3
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0answers
46 views
Is Ledoit-Wolf Shrinkage with a Constant Correlation Prior Reasonable for a Stock/Bond Mix?
I've been looking into Ledoit-Wolf shrinkage but I've found the papers concentrate on large numbers of assets that tend to all be highly correlated. Often a universe of large cap stocks.
I'm ...
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1answer
84 views
Covariance and Beta: can anyone explain this calculation?
Let us consider a simple equity portfolio that has exposures to only two factors:
0.5 exposure to value and 0.8 exposure to momentum. Let us assume that the volatilities of the two factors are 3% for ...
2
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1answer
129 views
How to calculate the covariance between two stochastic integrals?
How to calculate the covariance between the integral of a Brownian motion at different times:
$$\text{Cov}\left(\int^{t_1}_0\sigma(t)dW_t,\int^{t_2}_0\sigma(t)dW_t\right)\ ?$$
I know the answer is:
$$\...
2
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0answers
41 views
target correlation for portfolio
Given a long / short equity portfolio, I want to have some net long exposure.
My portfolio volatility is fixed to a target, so in trying to have a certain beta to the market, the only thing I can ...
1
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1answer
75 views
What are the units of the variance of returns?
I am a little confused about the units of the variance of returns. One way to compute that would be to look at the units of returns-
$$r=\frac{1}{\Delta t}\ln\frac{P(t+\Delta t)}{P(t)}=\text{...
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0answers
37 views
VECM Correlation Matrix
I am trying to implement the modified information share price discovery method desribed by Lien.
They start by estimating a VECM of the form
where and
and it's Vector Moving Average ...
3
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2answers
259 views
Filtering smallest eigenvalues
In Risk Budgeting and Diversification Based on Optimized Uncorrelated Factors [1], which introduces minimum torsion bets, Meucci gives an example involving the computation of covariance matrices on ...
2
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3answers
251 views
Negative variance?
Using the formula w*Cov*t(w) I can generate a negative portfolio variance. What are the implications of a negative variance? Should I just assume it's zero? A negative variance is troublesome ...
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1answer
78 views
Variance covariance matrix - number of periods required
Hi I am reviewing the example of Barra risk model in the following document page 23 there is the statement:
"Estimating a covariance matrix for, say, 3,000 stocks requires data
for at least 3,...
8
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3answers
484 views
CAPM model as a regression
The CAPM model states that the returns of a stock are-
$r_s=r_f+\beta (r_m-r_f)+\varepsilon_s$
The $\beta$ defined above is then calculated as $\frac{cov(r_s,r_m)}{var(r_m)}$. My question is ...
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1answer
119 views
Finding a minimum variance portfolio when using a regulariser?
I am aware that the minimum variance portfolio of a market with $n$ securities can be shown to be:
\begin{equation}
w^* = (1^T_n\Sigma^{-1}1_n)^{-1}\Sigma^{-1}1_n, \\ s.t. \ \ 1^T_nw = 1
\end{...
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votes
1answer
26 views
Factor Models: uncorrelated errors don't impact covariances of assets
This question stems from time series factor models (e.g., CAPM, Fama-French, etc.), but is a broader idea.
I am trying to comprehend how adding noise to a time series (e.g., error/residual from a ...
3
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0answers
125 views
Parametric VaR of a portfolio of a stock and an option on that stock
I understand how to calculate the parametric VaR of a stock and an option separately. But I don't understand how one can calculate the VaR of a portfolio of a stock and an option on that stock using ...
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1answer
114 views
Interest rate risk using copulas
In order to simulate an interest rate yield curve, can I just estimate a covariance matrix of historical key rate data, simulate with a normal copula, spline my simulated key rates, then price my ...
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1answer
182 views
Correlation of asset X with a portfolio of asset Y and Z [closed]
I have three assets and a covariance matrix.
How do I calculate the correlation of asset X with a portfolio that includes assets Y and Z?
For example, assume I want to calculate the correlation of ...
7
votes
1answer
300 views
How can I use a more efficient volatility estimator to improve the co-variance matrix?
Using mean-variance, I need to estimate a co-variance matrix $\Sigma$ to obtain the best weights in my portfolio.
However, there are other ways to compute the volatility $\sigma$ than historical ...
2
votes
2answers
77 views
Explanations regarding Minimum Variance Portfolio
I am sorry in advance if this question seems a bit stupid but during my class my lecturer said that:
"The traditional estimator of the variance-covariance matrix is the sample covariance. However ...
1
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0answers
69 views
Deriving Cox, Ingersoll and Ross expression for the relationship between forwards and futures, how do they conclude a specific step?
I'm trying to derive a specific relationship about the relationship between forwards and futures from "The relationship between forward and futures prices", written 1981 by Cox, Ingersoll and Ross (...
2
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1answer
3k views
Using CAPM to find correlation of two assets with each other
I stumpled upon an exercise in an investments book:
The data below describe a three-stock financial market that satisfies
the single-index model.
...
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1answer
577 views
How to efficiently get covariance matrices from a rolling window in Matlab?
I'am trying to produce a rolling window to estimate a covariance matrix using a for-loop. I have my returns under the variable returns_sec and I have 260 ...
2
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1answer
228 views
Variance of a straddle (Black Scholes)
I am trying to determine the variance of the payout of a straddle. For puts and calls individually:
Var[P] = E[P^2] - E[P]^2
Var[C] = E[C^2] - E[C]^2
where:
$$
E[...
4
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2answers
266 views
Are two stochastic processes independent if the Wiener processes inside are uncorrelated
Assume there are two stochastic processes:
$dx_t = \alpha_1(x_t,t)dt + \beta_1(x_t,t)dW^1_t$
and $dy_t = \alpha_2(y_t,t)dt + \beta_2(y_t,t)dW^2_t$.
Does $dW^1_t\times{dW^2_t} = 0$ imply that $\...
2
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0answers
87 views
Ledoit-Wolf, expected order of optimal shrinkage intensity
I have a question regarding the optimal shrinkage intensity derived in the Ledoit-Wolf method. Specifically, I'm referring to their version concerned with the target defined as the single index factor ...
3
votes
1answer
961 views
What is the covariance of two correlated Ornstein-Uhlenbeck processes?
What is the covariance of two correlated Ornstein-Uhlenbeck processes? I was trying correlation(1,2)*Var1^(1/2)*Var2^(1/2), but I am not sure! I took Var1=(sigma1^2/(2*speedofmeanreversion1))*(1-exp(-...
4
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0answers
191 views
Using Kendall rank correlation to construct a covariance matrix?
I am wondering if it's mathematically 'correct' to employ a correlation matrix based on Kendall-correlation when constructing a covariance matrix?
I.e., is it wrong to multiply standard deviations of ...
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0answers
190 views
Marchenko–Pastur, Student distribution and returns
I have a question regarding random matrix theory. I've been studying various papers and I found some confusing definitions of Marchenko-Pastur law. The most clear was the one on wiki:
wiki-Pastur-...
3
votes
1answer
134 views
Correlation -1 and standard deviation [closed]
My book says that for a portfolio of two stocks:
$\sigma_p = \sqrt{w_A^2 \sigma_A^2 + (1-w_A)^2 \sigma_B^2 + 2 w_A (1 - w_A) \rho_{AB} \sigma_A \sigma_B}$
Elsewhere it says that if the correlation ...
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1answer
65 views
how to find the weights in a portfolio? [closed]
Compute the weights in a portfolio consisting of two kinds of stocks if the expected return on the portfolio is to be $E(K_v)=10\%$, given the following information on the returns on stock 1 and 2:
$$
...
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1answer
773 views
Variance covariance matrix for a portfolio containing bonds also with other asset classes
What should we take for a bond or a zero coupon bond in order to make a variance covariance matrix?
For example:-
Equities - we take the market price
Cash - we take the spot rates
Bonds - Do we take ...
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vote
1answer
183 views
Calculate control variate for monte carlo simulation
For an exercise I need to calculate $\mathbb{E}[X]$ with a Monte Carlo simulation. I need to use control variate $Y$ with $\text{Var}(Y)=2$ and $\text{Cov}(X,Y)=1$.
I am asked to give the optimale ...
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0answers
144 views
Fourier transform covariance estimator
I am estimating realized variance and covariance by the estimator described in this paper, and relying on Fourier Transform.
Now, as my data is one day of data in ultra high frequency, so that the ...
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2answers
414 views
What is the preferred GARCH method in practice?
My advance apologies, if this question is too naive or basic. Please be patient with my first experiences with SE; ask for clarification, if needed.
I recognize there are many (often-criticized) ...
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3answers
1k views
How to estimate variance-covariance matrix of assets with different length of historical data? [duplicate]
Consider you have 4 assets A, B, C and D, where
Asset <...
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0answers
216 views
Finding mean vector and covariance matrix for annual returns given quarterly returns
I am currently trying to calculate a vector for the mean annual returns of 4 different asset classes along with their 4x4 covariance matrix in excel. However, I am having problems since the data I ...
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0answers
41 views
Portfolio Hedging under Uncertain Correlations
I have a portfolio ($w_0=1$) and two hedging assets ($w_1,w_2$) and a co-variance matrix for the three $\Sigma$. However the co-variance $\Sigma$ is only an estimate.
For fairly well behaved assets (...
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0answers
47 views
How to calculate the estimation error of portfolio variance using propagation results?
I am trying to find a conservative approximation for the propagated estimation error of a investment portfolio's variance (comprising two assets), given we know the estimation error for the variance ...
1
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2answers
837 views
Ledoit-Wolf Shrinkage estimator not giving positive definite covariance matrix
I used ten year daily data for 407 stocks and computed the daily and monthly covariance matrices. Since I have more variables than observations for the monthly matrix, I wasn't surprised to find the ...
4
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1answer
718 views
Beta between stock and option
In Black Scholes model I would like to compute
$$
\beta_K = \frac{\mathrm{cov}(C_{K,T},S_T)}{\mathrm{cov}(S_T,S_T)} = \frac{\mathrm{cov}((S_T - K)^+,S_T)}{\mathrm{cov}(S_T,S_T)}
$$
with respect to say ...
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0answers
763 views
How to calculate beta against a multi-asset benchmark
Lets say that I have a benchmark, $BM$ that consists of 3 assets- 30% asset $A$, 30% asset $B$ and 40% asset $C$. Now, lets further assume I am trying to construct a portfolio that uses $BM$ as its ...
2
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3answers
1k views
Semi-variance/Downside Risk, what about the rest of the covariance matrix?
I just bumped into a rather interesting article from wikipedia :
http://en.wikipedia.org/wiki/Downside_risk
where they define the semi-variance also called Downside risk, which bascially only ...
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2answers
982 views
Garch for covariance matrix?
I have seen plenty of literature about GARCH on estimation volatility. how about covariance? There are plenty of risk models depending on the covariance matrix.
I guess we can assume the correlation ...
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0answers
222 views
modeling regime switching for Correlation matrix
I am trying to estimate covariance in multiple time series. However, I want to do this using a regime-switching framework. So, I start with fitting a GARCH(1,1) model and then de-volatalize the series....
3
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1answer
589 views
Markowitz Mean-Variance Implied Returns
What is the closed form solution for the following inverse Markowitz problem?
Given a mean-variance optimized fully invested portfolio $X$, a risk aversion parameter $\lambda$ and a var-covar ...
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1answer
3k views
Can the Minimum Variance Hedge ratio be greater than 1?
The Minimum Variance Hedge ratio is defined as: $h = \rho * \frac{\sigma_S}{\sigma_F}$
For correlation $\rho$ and $\sigma_S , \sigma_F$ for S.D. of changes in asset and future prices accordingly.
...
4
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1answer
1k views
Regime Switching for Dynamic Correlations
I would like to implement a Regime Switching for Dynamic Correlations in an out-of-sample analysis using MATLAB.
After looking at the literature on the subject, they all refer to an article by Denis ...
1
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1answer
229 views
How can I forecast future correlation?
There are some standard models for forecasting volatility (e.g., GARCH) and for forecasting returns (e.g., factor models). What kind of standard models exist for forecasting future correlation between ...
3
votes
1answer
2k views
Covariance matrix and Cholesky decomposition
I am simulating a spread option with stochastic volatility using Monte Carlo simulation. I have the positive-definite covariance matrix
$$
\rho = \left( \begin{array}{cccc}
1 & \rho_{1,2} & \...
3
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2answers
520 views
How to get Multivariate Betas from an Estimated EWMA co variance Matrix?
I have a portfolio of 4 assets. I also have returns for 3 indices. I want to get the multivariate betas for these 4 assets-based on these assets. I only have the 7 x 7 covariance matrix estimated by a ...
