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