Let a firm start at time \(t=0\) and have projects with payoffs \(\Gamma_{j}(T)=\) [10 86 6 for the three different states of nature at time \(t=T\). The firm is partially financed with debt with face value \(F=5\) and pays taxes at rate \(\tau=0.4\). Assume that the three different states of nature have equal assigned probabilities. There are no arbitrage opportunities and the vector of state prices is \(\psi=\left[\begin{array}{ll}0.3 & 0.25 \\ 0.44\end{array}\right]\).

(a) Check propositions MM1 and MM2. Compute the weighted average of the costs of debt and equity.

(b) Suppose now that \(F=7\), that is, the firm will be bankrupt if state of nature \(j=3\) occurs. How do the values of debt, equity, and the firm change? Compute the weighted average of the costs of debt and equity and compare them with the previous case.