A company has:   • $7 million market value of equity   • $5 million market value of debt    •...

Given

Market value of equity E=$7mE = \$7mE=$7m

Market value of debt D=$5mD = \$5mD=$5m

Total value V=D+E=12mV = D+E = 12mV=D+E=12m

WACC (geared) =9.375%= 9.375\%=9.375%

Tax rate T=15%T = 15\%T=15%

Let gearing in terms of D/ED/ED/E:

g=DE=57=0.7143g = \frac{D}{E} = \frac{5}{7} = 0.7143g=ED​=75​=0.7143

Under Modigliani & Miller with tax, the relationship between WACC of a geared firm (WACC_g) and the ungeared cost of equity kuk_uku​ is:

WACCg=ku [1+g(1−T)]1+g\text{WACC}_g = \frac{k_u \,[1 + g(1-T)]}{1+g}WACCg​=1+gku​[1+g(1−T)]​

Rearrange for kuk_uku​:

ku=WACCg⋅1+g1+g(1−T)k_u = \text{WACC}_g \cdot \frac{1+g}{1+g(1-T)}ku​=WACCg​⋅1+g(1−T)1+g​

Substitute:

g=0.7143g = 0.7143g=0.7143

1+g=1.71431+g = 1.71431+g=1.7143

g(1−T)=0.7143×0.85=0.6071g(1-T)=0.7143 \times 0.85 = 0.6071g(1−T)=0.7143×0.85=0.6071

1+g(1−T)=1.60711+g(1-T)=1.60711+g(1−T)=1.6071

ku=0.09375×1.71431.6071≈0.10=10%k_u = 0.09375 \times \frac{1.7143}{1.6071} \approx 0.10 = 10\%ku​=0.09375×1.60711.7143​≈0.10=10%

So the ungeared cost of equity is 10%.