Lack of understanding about simple question involving Lognormal distribution

Question

Given that $S_2$ is the accumulated value of a £1 investment over 2 years , with independent rates $i_1, i_2$ in the first and second year respectively (so $S_2 = (1 + i_1)(1 + i_2)$), I was told that $E[S_2] = 1.076$ and $Var[S_2] = 0.0027$.

Now, suppose that $(1+i_1)$ and $(1+i_2)$ are independent, identically distributed, lognormal random variables with parameters $µ$ and $σ^2$.

Find the values of the parameters $µ$ and $σ^2$ for $(1+i_1)$ and $(1+i_2)$ knowing that E[$S^2$] and Var($S^2$) are equal to the values given.

I'm always quite confused by questions involving the lognormal distribution, so any guidance through this question would be appreciated. ( In the check sheet the answer is given as $µ = 0.036$ and $σ^2 = 0.0012$ )

Answer 1

You should probably start by saying that

• the mean of a lognormal distribution is $\exp\left(\mu+\frac{\sigma^2}{2}\right)$ and the variance is $\left(\exp\left(\sigma^2\right)-1\right)\exp\left(2\mu+\sigma^2\right)$
• If for $S_2$ these are $1.076$ and $0.0027$, then $\exp\left(\sigma^2\right) -1 = \frac{0.0027}{1.076^2}$ so $\sigma^2\approx 0.00233$ and $\mu \approx 0.072$
• But those are for two years combined. For one year, using the i.i.d. property, the values are half these, i.e. $\mu \approx 0.036$ and $\sigma^2 \approx 0.0012$