Question

Suppose a large institutional investor Alliance’s portfolio has a beta of 0.5. Another institutional investor Best’s portfolio has an expected return of 10 percent and a standard deviation of 15 percent. Both portfolios have the same Sharpe ratio of 0.6, and the market portfolio can be described as a portfolio comprising these two with equal weights. Suppose there is a firm called Cunning corporation, whose stock’s beta is 2 and it can borrow at the risk free rate. Cunning’s equity value is £1 million and its debt is £1.5 million. The present value of Cunning’s tax shield is £0.3 million. Assuming that both CAPM and the Modigliani-Miller theorem with corporate taxes hold, answer the following questions. a) What is the risk free rate? b) What is the expected return on the market portfolio? c) What is the after-tax WACC of Cunning

Answer 1

(a) Let the risk-free rate be Rf, Alliance's Expected return be Ra, Alliance's Standard Deviation be Sa, best's Expected Return be Rb and Best's standard deviation be Sb. Further, let the market portfolio's return be Rm and its standard deviation be Sm.

As Sharpe's ratio of both the portfolios is 0.6, we have Ra - Rf / Sa = 0.6 = Rb - Rf / Sb

Also, Rb = 10 % and Sb = 15 %

Therefore, (10 - Rf) / 15 = 0.6

Rf = 1 %

(b) Alliance's Beta = Ba = 0.5, As CAPM is assumed to hold, we have Ra = Rf + Ba x (Rm - Rf)

Ra = 1 + 0.5 x (Rm - 1)

Ra = 0.5 + 0.5Rm => 2Ra =1+Rm ------- (A)

and since the market portfolio is a mixture of Alliance and Best in equal weights, we have Ra x 0.5 + Rb x 0.5 = Rm

0.5 x Ra + 10 x 0.5 = Rm => 0.5Ra + 5 = Rm ----------- (B)

Solving equation (A) and (B) we get:

2Ra - 1 = 0.5Ra + 5

Ra = 4 %

Rm = 2 x 4 - 1 = 7 %

(c) Let the tax rate be T, Value of Equity = E = £ 1 million , Value of Debt = D = £ 1.5 million and Firm Value = V = D + E = 1.5 + 1 = £ 2.5 million

Cunning Corps' Beta = B = 2 and Present value of interest tax shields = PV(ITS) = £ 0.3million

Now, PV(ITS) = T x D

0.3 = T x 1.5

T = 0.2 or 20 %

As the firm can borrow at the risk-free rate, the firm's cost of debt is kd = 1 %

As CAPM holds, the firm's cost of equity can be calculated as ke = Rf + B x (Rm - Rf) = 1 + 2 x (7 - 1) = 13 %

Therefore, after-tax WACC = (E/V) x ke + (1-T) x (D/V) x kd = (1/2.5) x 13 + (1-0.2) x (1.5/2.5) x 1 = 5.68 %