There are many banks in a country. Each bank holds a portfolio of corporate loans that will pay off $100M next year as well a portfolio of mortgage loans. Half of the banks have made good mortgage loans; these "good" banks will receive a payoff from their mortgage loans of $80M next year. The other half have made bad mortgage loans; these "bad" banks will receive a payoff from their mortgage loans of $20M next year. Neither the corporate loan nor mortgage loan portfolios generate any payoff after next year. Banks have no cash and no other assets. Each bank owes $90M of debt that it must pay off today by raising $90M of cash. It can do so in one of two ways. It can issue $90M of equity to new investors or it can sell off its corporate loan portfolio for $90M (a $10M discount to its true value of $100M). A bank cannot sell its mortgage portfolio. The bank owes no other debt. Existing shareholders receive a payoff of $0 if the bank defaults on its debt, so the bank will not default. A bank's manager knows whether her bank is a good bank or a bad bank, but outside investors only know that there is an equal probability that any bank is a good bank or a bad bank. Outside investors are also aware of each bank's need to pay off its loan. In making decisions, bank managers maximize value for their existing shareholders. Assume no discounting and no taxes. Assume that investors in the market believe that a good bank will liquidate its corporate loan portfolio while a bad bank will issue equity. This assumption is reasonable since a good bank will naturally worry about the dilutive effects of issuing equity, and the loss from liquidating the corporate loan portfolio is not too big. Demonstrate that, given this assumption by the market, a good bank will, in fact, liquidate its corporate loan portfolio and a bad bank will, in fact, issue equity. Support your answers with calculations.