# Covariance between the linear combination of lognormal random variables

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

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.

• Thank you @psboonstra. You made the exact point that I was missing. Truly appreciate your help. Mar 31, 2021 at 15:07