Exercise 6. Bank earnings interest rate - 2

With the interest rate calculated in the last example, we still lose money 50% of the time. What should the interest rate be so that the chance of losing money is 1 in 20?

In math notation, what should the interest rate be so that $\mbox{Pr}(S<0) = 0.05$?

Remember that we can add a constant to both sides of the equation to get: $$ \mbox{Pr}\left(\frac{S - \mbox{E}[S]}{\mbox{SE}[S]} < \frac{ - \mbox{E}[S]}{\mbox{SE}[S]}\right) $$

which is

$$ \mbox{Pr}\left(Z < \frac{- { [lp + x(1-p)]}n}{(x-l) \sqrt{np(1-p)}}\right) = 0.05 $$

Let z = qnorm(0.05) give us the value of z for which: $$ \mbox{Pr}(Z \leq z) = 0.05 $$

Use the qnorm function to compute a continuous variable at given quantile of the distribution to solve for z.
In this equation, $l$, $p$, and $n$ are known values. Once you've solved for z, solve for x.
Divide x by the loan amount to calculate the rate.