A firm is looking to finance a new project but it has no funds available. The required

investment for the new project consists of $5$ million. The investment promises a return

of $50\%$ if it is successful and it pays zero if it is unsuccessful. There are two types of firms

in the economy: good types and bad types. The success rate of good firms is $80\%$ and the

success rate of bad firms is $20\%.$ The market has a prior probability of a firm being of the

good type equal to $0\mathrm{.}3.$

$1.$ Under the scenario of perfect information, what contractual terms would the good

borrower offer to a risk$-$neutral investor?

$2.$ Under asymmetric information, will a risk$-$neutral investor be willing to lend the

required funds? If so$,$ what terms will be offered by the firm?