Probability of Negative Portfolio Return via CLT

A portfolio consists of $n = 50$ stocks with equal weight $1/n$. The annual returns $R_1, \ldots, R_{50}$ are independent, each with mean $\mu = 0.08$ (i.e., $8\%$) and standard deviation $\sigma = 0.20$.

The portfolio return is $\bar{R} = \frac{1}{50}\sum_{i=1}^{50} R_i$.

(a) State what the LLN implies about $\bar{R}$ as $n \to \infty$.

(b) Using the CLT, approximate $P(\bar{R} < 0)$.

You may use $\Phi(2.83) \approx 0.9977$.