Log-normal and normal distribution conversion

Question

$c_R,R_f$ are known (constant part of return and risk free rate). Let $R=R_f(e^r-1)=c_R+\epsilon_R$, $r\sim N(\mu_r,\sigma_r)$, how to specify the distribution of $\epsilon_R\sim Lognormal(?,?)$ by mean and variance of a normal distribution?

Answer 1

Using your notation, we have:

$$R=R_{f}(e^{r}-1)=R_{f}\,e^{r}-R_{f}$$

where

$$r\sim N(\mu_{r},\sigma_{r})$$

Now, we also know that if $r$ has a normal distribution, then $e^{r}$ is log-normally distributed:

$$e^{r}\sim LN(\mu_{r},\sigma_{r})$$

Furthermore, by scaling $e^{r}$ by $R_{f}$ we arrive at:

$$R_{f}\,e^{r}\sim LN(\mu_{r}+\text{ln}(R_{f}),\sigma_{r})$$

Finally, we subtract the constant $R_{f}$ from the above term:

$$R_{f}\,e^{r}-R_{f}$$

Now, because the standard log-normal distribution is a 2-parameter distribution with support on $x\in(0,\infty)$, it needs to be adapted to account for the negative shift of $R_{f}$.

So, we now consider the 3-parameter log-normal distribution, which includes a location parameter, $\theta$. Essentially, this adjusts the support of the distribution to be $x\in(\theta,\infty)$. It has probability density function:

$$f_{X}(x;\mu,\sigma,\theta)=\frac{1}{(x-\theta)\sigma\sqrt{2\pi}}e^{\frac{(\text{ln}(x-\theta)-\mu)^{2}}{2\sigma^{2}}}$$

where $x>\theta,\,\sigma>0$. The mean and variance of the shifted log-normal distribution are easy enough to calculate. The mean is equal to the mean of the non-shifted log-normal plus the shift:

$$E[X+c]=E[X]+c$$

Similarly, the variance is equal to the variance of the non-shifted log-normal:

$$\text{Var}(X+c)=\text{Var}(X)$$

So we arrive at:

$$R=R_{f}(e^{r}-1)\sim LN_{3}(\mu_{r}+\text{ln}(R_{f}),\sigma_{r},-R_{f})$$

with probability density function:

$$f_{R}(r;\mu,\sigma,\theta)=\frac{1}{(r+R_{f})\sigma_{r}\sqrt{2\pi}}e^{\frac{\big(\text{ln}(r+R_{f})-\mu_{r}-\text{ln}(R_{f})\big)^{2}}{2\sigma_{r}^{2}}}$$

So to answer your question, for the equation:

$$R=c_{R}+\epsilon_{R}$$

the value of $c_{R}$ is completely dependent on how you specify the parameters of $\epsilon_{R}$. You can have:

$$c_{R}=0,\,\,R\sim LN_{3}(\mu_{r}+\text{ln}(R_{f}),\sigma_{r},-R_{f})$$

or

$$c_{R}=-R_{f},\,\,R\sim LN_{3}(\mu_{r}+\text{ln}(R_{f}),\sigma_{r},0)$$

etc.

Answer 2

You specify the distribution of a Lognormal variable with the mean and the variance (or standard deviation) with the parameters of the Normal distribution used to derive the Lognormal distribution.

By definition of the Lognormal variable (https://en.wikipedia.org/wiki/Log-normal_distribution) you would have: $$r\sim N(\mu_r, \sigma_r) \Rightarrow e^r =\epsilon_r \sim LogNormal (\mu_r, \sigma_r )$$

I never came across an author that specifies $Y\sim Lognormal(E[Y],\sqrt{Var[Y]})$