I have two lognormally distributed random variables $Y_i=e^{X_i}$ where $X_i \sim \mathcal{N}\big(\mu_i, \: \sigma_i^2 \big)$ for $i=1,2$, and $X_1$ and $X_2$ are correlated by $\rho_{12}$. Now, Let $Z=\alpha_2 Y_2 - \alpha_1 Y_1$.

Question : What is the covariance between $Y_1$ and $Z$?

What I understand so far are the general results for $Y_1$ and $Y_2$:

$\mathbb{E}[\:Y_i \:]=\text{exp}(\mu_i+\frac{\sigma_i^2}{2})$

$\mathbb{Var}[\:Y_i \:]=\text{exp}(2\mu_i) \big[\text{exp}(\sigma_i^2) -1 \big] \: \text{exp}(\sigma_i^2)$

$\mathbb{Cov}[Y_1 \: Y_2]=\mathbb{E}[\:Y_1 \: Y_2]- \mathbb{E}[\:Y_1 \:]\mathbb{E}[\:Y_2 \:]\\ \; \;\;\; \: \quad \quad \quad \:=\text{exp}(\mu_1 +\mu_2)\text{exp}(\frac{1}{2}(\sigma_1^2+2\rho_{12}\sigma_1\sigma_2+\sigma_2^2))\\ \; \;\;\; \: \quad \quad \quad \quad \qquad-\text{exp}(\mu_1+\frac{\sigma_1^2}{2}) \:\text{exp}(\mu_2+\frac{\sigma_2^2}{2})\\ \; \;\;\; \: \quad \quad \quad \:=\text{exp}(\mu_1+\mu_2+\frac{1}{2}(\sigma_1^2+\sigma_2^2))\big[\text{exp}(\rho_{12}\sigma_1\sigma_2)-1\big]$

How can I utilise these results to compute $\mathbb{Cov}[Y_1, Z]$?


1 Answer 1


You can use what you've already derived to answer your question with just a few more steps: \begin{align} \mathbb{Cov}[Y_1, Z] &= \mathbb{Cov}[Y_1, \alpha_2 Y_2 - \alpha_1 Y_1]\\ &=\mathbb{Cov}[Y_1, \alpha_2 Y_2] - \mathbb{Cov}[Y_1, \alpha_1 Y_1]\\ &=\alpha_2\mathbb{Cov}[Y_1, Y_2] - \alpha_1\mathbb{Var}[Y_1] \end{align} Now just plug in your expressions.

  • 1
    $\begingroup$ Thank you @psboonstra. You made the exact point that I was missing. Truly appreciate your help. $\endgroup$
    – yufiP
    Mar 31, 2021 at 15:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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