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I was recently learning about value at risk and how to calculate it, and one of the steps was to calculate the covariance of the returns of the securities making up the portofolio. This makes sense because if we consider the return of the securities as random variables ( and dont assume independence between them ) and the return of the portofolio is a random variable which is a weighted sum of the security random variables, we would need the covariance matrix. However, if all we need is the portofolio returns variance, why not just estimate directly? i.e get returns of the securities, weight sum them and use the sum along with a simple variance estimator to estimate the variance. I ran a couple of dummy simulation with 2 securities ( returns generated from a normal distribution), the first one where the 2 distributions are indepdent and the second where there was a covariance between them and both method estimated almost the same variance.

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    $\begingroup$ Remember that equity returns are non-gaussian before you attempt to use this for anything important (like managing real money). $\endgroup$
    – amdopt
    Jul 22, 2023 at 11:46
  • $\begingroup$ yea, i am currently looking at other distributions ( like johnson SU) that better fits the returns $\endgroup$ Jul 23, 2023 at 9:42

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They are equivalent but your method gives you no idea how much risk is coming from which asset. Also in the real world portfolio composition keeps changing so past returns are not representative of the current composition.

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  • $\begingroup$ if i understood correctly, you meant the securities i am.holding will change so my portofolio returns will change, but in that case i will get the past returns for say the last year for the securities i have, and make up the hypothetical returns if i held this position for the past year. $\endgroup$ Jul 22, 2023 at 7:16
  • $\begingroup$ In that case it's the same exact thing. You don't need covariance assumptions to calculate portfolio vol, although covariance will affect the portfolio vol anyway. Usually portfolio weights are the result of an optimization and that requires a covariance matrix. $\endgroup$
    – Arshdeep
    Jul 22, 2023 at 8:45
  • $\begingroup$ i understand, i am mainly concerned with computing some performance monitoring metrics so computing speed is an issue i need to address $\endgroup$ Jul 23, 2023 at 0:08
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To find out the factors and marginal change if one of the constraints or assumption breaks.

Understanding the covariance of a portfolio is important because it allows investors to assess the risk of the portfolio and make informed decisions about how to allocate their investments.

When you make decision to put more money into one stock, you have to sacrifice the alternative investment. That may reduce your maximum return of your portfolio

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