Hello, I would like to check the answer for the following exercise. In particular, I would need an explanation step by step on how the new firm value 4,708.24 is obtained according to the solution below. Thanks a lot.

Limitas Inc. is a publicly traded chemical company with 200 million shares trading at $20 a share and $ 1 billion in outstanding debt; the market interest rate on the debt is 7%.The firm has a cost of capital of 10.44% and the marginal tax rate is 40%. The firm is considering issuing new equity and retiring all of its debt. Estimate the new value for the firm if it goes through with this transaction? (The riskfree rate is 5% and the market risk premium is 4%. You can also assume no growth in perpetuity)

Solutiongiven

Current market value of equity = 4000

Market value of debt = 1000

Value of firm = 5000

Cost of capital = 10.44% = X (.80) + 4.2% (.20)

Cost of equity = 0.12

Current beta = 1.75

After debt is repaid

New beta = 1.52173913

New cost of equity = 11.09% ! Equal to cost of capital

New firm value = $4,708.24 ! Change in firm value = (.1044-.1109) 5000/.1109