Problem on optimal capital structure

Question

XYZ Company has debt/assets ratio 50%, that is too high and it must be at 45% to be optimal. This debt reduction must also reduce the bankruptcy costs by $30 million. At present, XYZ has 5 million shares of common stock selling at $50 each. The tax rate of XYZ is 30%. How many shares of stock must the company sell, and buy back bonds from the proceeds, to achieve its optimal capital structure?

Answer 1

Given,
Value of equity = 5 million*$50 = $250 million
Debt/assets ratio = 50%
Hence value of debt = $250 million
Value of Altoona Company = $500 million
Value of firm after recapitalization, V = 500 – 0.45*V*.3 + 43.35
1.135 V = 543.35
Value of firm = 478.72 million
Value of equity = 263.3 million
Number of shares = 5.266 million
Hence 265,900 shares must be issued.