Suppose there are two kinds of companies, good and bad ones. Both types of companies are faced with an investment opportunity that requires an investment of $$6$M today $($year $0)$ and can give a high or a low cash flow in year $2,$ but they differ in the probabilities of the high and the low cash flows as follows:

$\backslash$table$[[$Cash flow,Probability of cash flow$],[$Good Company,Bad Company$],[$$$15$M$,0\mathrm{.}8,0\mathrm{.}2],[$$$5$M$,0\mathrm{.}2,0\mathrm{.}8]]$

There are as many good as bad companies around and the discount rate is zero.

Assume that managers $($who are also the initial owners of the companies$)$ know whether their company is good or bad in period $0,$ but investors do not know $$ntil year $1($after the investment is made but before the cash flow uncertainty is resolved$).$ The companies need to raise capital in year $0.$

Managers suffer a huge personal loss if the company goes into financial distress $($they die from a stroke$).$ They want to avoid that at any cost. Therefore, they cannot finance the investment with normal debt.

$($a$)$ Suppose all companies can only raise capital by issuing equity. What proportion of year $2$ cash flows will investors demand to finance the $$6M$ investment in year $0.$

$(6$ marks$)$

$($b$)$ Suppose now all companies can choose to issue convertible debt to raise the $$6$ million. The debt has a face value $F=$$$6M$ but can be converted into a proportion $x$ of the company's equity in year $1($after investors have learnt whether companies are good or bad but before year $2$ cash flow uncertainty is realized$).$ If investors believe in period $0$ that whichever company issuing convertible debt is good, and whichever company issuing equity is bad.

$($i$)$ What proportion of year $2$ cash flows will investors demand from companies that issue equity to be willing to finance the $$6M$ investment year $0?$

$($ii$)$ What proportion of year $2$ cash flows $($this is the $x$ from above$)$ will investors demand from companies that issue convertible debt, if we assume that convertible debt holders will convert into equity in year $1?$

$($c$)$ Check whether it is actually true that in the situation above, managers of companies prefer to issue convertible debt, and managers of bad compar prefer to issue equity. Explain the intuition.