Hi guys I have an answer to a question which I don't understand and I need some clarification. Please assist. Thanks.

Question:
Consider a firm that only lasts one period(from t=0 to t=1). The firm has currently 30 units of cash. The firm also has existing assets that will produce 30 units of cash at t=1 if the economy is booming or 0 units of cash if the economy is in crisis. The probability of the economy booming is 0.50 (and the probability of the economy being in crisis is therefore 0.50 too). The firm also has debt with face value 40 that matures at t=1. All agents are risk neutral and the discount rate is zero.

a) Calculate the value at t=0 of the equity and debt of the firm as well as the firm's total value.

Answer given:
Total cash flow for the firm = (0.50)(30+30) + (0.5)(30+0) = 45
value of equity = (0.5)[(30+30) - 40] + (0.50)(0) = 10
value of debt = (0.50)(40) + (0.50)(30) = 35
Firm's total value = value of equity + value of debt = 10 + 35 = 45

My doubts and question:
1. I do not understand why do we have to find the cash flow for the firm in the first place?
2. For the value of equity, I believe the 0.5 is base on the probability of boom and crisis and the first value of 30 is the current cash, the second 30 is the pv of the cash received in yr 1. As for the 40 is due to the debt owned. Am I right for this?
3. For the value of debt, how come for the value of the debt we have to multiple by 0.5 and how do they get the value of 30?

Just think of the payoffs to each of the claimant parties under the boom and bust scenarios.

Boom, the corp ends up with 60 cash units; pays its debt in full (40) and liquidates with the shareholders getting 20.

Bust, the ending cash is 30 which of course goes to the debt in its entirety. Shareholders get zippo.

So the debtholder gets either 40 or 30; shareholder gets either 20 or 0; and each with a probability of 1/2. From that, the expected values of each as given in the answer (35 and 10, resp.) immediately follow.

As to your first question/doubt, I suppose it's to point out that the total value of the firm's assets (1/2 x 30 + 1/2 x 60) is always equal to the total value of all the claimholders' interests (35 + 10). That's an important identity to keep in mind.

By the way, these asymmetric payoff 'profiles' to the debt and equity that you see here will be important later if you study option theory. Best of luck!