Study Case $$Asymmetric information in the financial system.$$

Context

Suppose we have an imaginary economy with a lot of entrepreneurs. Each entrepreneur needs external funding of IDR $100$ million for a project. There are two risk categories for the project: high$-$risk and low$-$risk. The cash flow returns of a high$-$risk project equals IDR $500$ million $($probability $=0\mathrm{.}25)$ and IDR $0$ probability$=1-0\mathrm{.}25=0\mathrm{.}75).$ On the other hand, the low$-$risk project

generates cash flows IDR $125$ million, always $($probability$=1).$ By assuming risk$$neutrality $($universal$)$ and the risk$$free rate is zero $($no discounting$),$ each project creates IDR $125$ million on average. The entrepreneurs have positive NPV$,$ since $125-100>0.$

However, the bank does not know the probability to generate the cash flow for each project, but the entrepreneur knows. Suppose the bank can$$t find a way to know the riskiness of a project by screening the project. The bank only knows the estimated probability for low$-$risk projects, which equals to $20\%$ of all cases. The entrepreneur takes his project only if he has a strictly positive

expected payoff, by taking into account what he has to pay back to the bank. We may assume that the economy has perfect competition in the credit market.

Questions

$1.$ Suppose that the bank has following statement: $$With probability $0\mathrm{.}8$ the project is a high$-$risk, the project fails to generate cash flow with probability $0\mathrm{.}75,$ and with probability $0\mathrm{.}2$ the project is a low$-$risk project. So$,$ to create a break even condition, the bank needs to charge an interest rate r by using: $0\mathrm{.}8\backslash$times $0\mathrm{.}25\backslash$times $100(1+$r$)+0\mathrm{.}2\backslash$times $100(1+$ r$)=100.$ Is there any creditworthy entrepreneur who

would accept the interest rate? What is wrong with this statement?

$2.$ By assuming debt financing, what interest rate the bank will charge in the equilibrium condition? and what kind of projects are financed by the bank? Who is credit rationed?

$3.$ By using equity financing, the bank holds some share alpha $($a$)($alpha $($a$)>0)$ of the firm, which implies the bank will receive share of alpha $($a$)$ from the firm$$s cash flows. Proof that the bank would finance both high and low risk projects. In addition, both high and low risk project holders pay the $$fair$$ capital. explain!