A company can invest in one of two mutually exclusive projects. The first one is safe and its assets will have a value of EUR $1,000,000$ in one year. The other project is risky and its assets, in one year, will have a value of EUR $1,800,000$ or EUR $100,000$ with equal probability. The initial investment for either projects is $500,000.$ The company plans to raise the EUR $500,000$ by issuing a zero$-$coupon bond $($the project is financed $100\%$ with debt$)$ that will be paid back in one year. The firm cannot commit ex$-$ante to choose a project. All agents are risk neutral and the interest rate is $0.$

a$)$ What is the face value of debt?

b$)$ What is the increase in the equity value after the issuance of the bond is announced?

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