Question

Suppose the portfolio of a large institutional investor ‘Animal’ has a beta of 1.25, and the standard deviation of the rate of return on its portfolio is 15 percent and its expected rate of return is 15 percent. The portfolio of another institutional investor ‘Beast’ has a beta of 0.75. The market portfolio may be expressed as a portfolio comprising the portfolios of Animal and Beast. Suppose there is a firm called ‘Cunning corporation’, whose stock’s beta is 2 and it can borrow at the risk free rate, which is 2.5 percent. Cunning’s equity value is £1.5 million and its debt is £1 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 expected return on the market portfolio? b) What is the standard deviation of the rate of return on Beast’s portfolio? c) What is the weight attached to Beast if we express the market portfolio as a portfolio comprising the portfolios of Animal and Beast? d) What is the after-tax WACC of Cunning?

Answer 1

Part (a)

Risk free rate, Rf = 2.5%

Beta of Animal = B_A = 1.25

If Rm is the expected return from the market then,

E(R_A) = 15% = Rf + B_A x (Rm - Rf) = 2.5% + 1.25 x (Rm - 2.5%)

Hence, Rm = (15% - 2.5%) / 1.25 + 2.5% = 12.50%

Part (b)

Beta should be proportional to standard deviation.

Hence, $\sigma$_B / $\sigma$_A = B_B / B_A; Hence, $\sigma$_B = $\sigma$_A x B_B / B_A = 15% x 0.75 / 1.25 = 9.00%

Part (c)

Let's say, the market portfolio can be expressed as a portfolio comprising of proportion p of Animal and proportion (1-p) of Beast.

Hence, resultant beta of the portfolio = p x B_A + (1-p) x B_B = p x 1.25 + (1 - p) x 0.75 = 0.75 + 0.50 x p = beta of the market portfolio = 1

hence, p = (1 - 0.75) / 0.5 = 0.5

So the weight attached to Beast = 1 - p = 1 - 0.5 = 0.5 = 50%.

Part (d)

Cost of equity for Cunning Corporation, Ke = Rf + B_C x (Rm - Rf) = 2.5% + 2 x (12.50% - 2.50%) = 22.50%

Present value of tax shield = 0.3 = Tax rate, T x Debt, D = T x 1

hence, tax rate, T = 0.3 / 1= 0.3 = 30%

Pre tax cost of debt, Kd = Rf= 2.5%

Debt, D = 1

Equity, E = 1.5

Hence, post tax WACC = D / (D + E) x Kd x (1 - T) + E / (D + E) x Ke = 1 / (1 + 1.5) x 2.5% x (1-30%) + 1.5 / (1 + 1.5) x 22.50% = 14.20%