1
$\begingroup$

How do I get t-day covariance matrix from daily return data?

I have an idea of how to calculate t-day variance from daily return data. From 14.6, 14.7 in Options, Futures and Other Derivatives by John Hull, I know that assuming stock price follows Ito's process, t-day log return follows a normal distribution and its expected value and variance are just those of daily log return scaled by t. Based on these and the properties of lognormal distributions, I can do the following step:

  1. compute daily log return from daily return
  2. compute the expected value and variance of daily log return $$\operatorname{E}[\log r_d] = \mu$$ $$\operatorname{var}(\log r_d) = \sigma^2$$
  3. compute the expected value and variance of t-day log return by scaling daily values by t $$\operatorname{E}[r_t] = \mu t$$ $$\operatorname{var}(\log r_t) = \sigma^2t$$
  4. compute the expected value and variance of return by the property of log normal distributions $$\operatorname{E}[r_t] = \exp(\mu t + \frac{\sigma^2}{2})$$ $$\operatorname{var}(r_t) = [\exp(\sigma^{2})-1]\exp(2\mu +\sigma ^{2})$$

If the above are valid, what are the similar reasonings for covariance matrix if I am looking at more than a single stock?

I see in practice some people seem to just scale covariance by 252 when calculating annualized covariance. Also, they don't seem to handle the difference of return and log return. But I don't understand why.

(I can't find appropriate tags for this question)

$\endgroup$

1 Answer 1

1
$\begingroup$

Let $M_d$ be the covariance matrix for the daily log-values.

Then $M_t = t M_d$ is the covariance matrix for the log-values after $t$ days.

From this, standard formulas give the covariances of the values, as in this answer.

One way to write those formulas is that if $X$ and $Y$ are jointly normal, then $$\text{Cov}[e^X,e^Y]\, =E[e^X]\: E[e^Y]\:(e^{\text{Cov}(X,Y)}-1)$$

$\endgroup$
2
  • $\begingroup$ But I already know people scale up the log of daily covariance, what I don't understand is why this is valid. For variance, John Hull gave the reasoning starting from the assumption that stock price follows Ito's process. I want similar reasoning for covariance. $\endgroup$
    – Sara
    Jun 9, 2022 at 8:13
  • 1
    $\begingroup$ If you want to know reasoning that’s parallel to Hull’s argument about variance, then it makes sense to post a new question which includes or summarizes or paraphrases that argument, and ask for the analog with covariance there. $\endgroup$
    – Matt F.
    Jun 9, 2022 at 12:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.