M&M Prop 2 - cost of equity

ke = kei + (1 – T)(kei – kd )(Vd / Ve) Moondog Co is a company with a 20:80 debt: equity ratio. Using CAPM, its cost of equity has been calculated as 12%. It is considering raising some debt finance to change its gearing ratio to 25:75 debt to equity. The expected return to debt holders is 4% per year, and the rate of corporate tax is 30%. Required: Calculate the theoretical cost of equity in Moondog Co after the refinancing. ANSWER: Using M+M's Proposition 2 equation, we can degear the existing ke and then regear it to the new gearing level: Degearing: ke = kei + (1 – T)(kei – kd)(Vd/Ve) 12% = kei + (1 – 0.30)(kei – 4% )(20/80) Now we need to rearrange this formula: 0.12 = kei + (1-0.30)( kei – 0.04)(20/80) 0.12 = kei + (0.7)( kei – 0.04)(0.25) 0.12 = kei + (0.175)( kei – 0.04) 0.12 = kei + 0.175 kei – 0.007 0.12 = 1.175 kei – 0.007 0.127 = 1.175 kei (0.127/1.175) = kei So rearranging carefully gives kei = 0.108 (10.8%) Now regearing: ke = 10.8% + (1 – 0.30)(10.8% – 4%)(25/75) ke = 12.4% QUERY: Please can you take me through this formula step by step. I cannot get my head around this. How is the '0.007' calculated? Where does the 1.175 come from? I can work out how to get the 0.175 (0.7*0.25). Please clarify the above. Many thanks in advance Rory