Question

a) Two firms, ABC Ltd and XYZ Ltd, are identical in every respect apart from their capital structure. Both will earn $284 million if the market swings upwards and $100 million in a downward swing. There is an even chance of the market swinging upwards or downwards. ABC Ltd has no debt. XYZ Ltd has issued $800 million of its debt at an interest rate of 10% and hence, $80 million of its income is paid out as interest. Assume that investors can borrow and lend at the same rate as the corporation. The WACC of both firms is 16%.

I) Suppose that you have $40 million to invest in XYZ Ltd shares. Is there an alternative investment in ABC Ltd that would generate the same payoff?

Answer 1

All financials below are in $ million.

Value of the firm ABC or XYZ, V_F = Operating income / WACC

Hence, in case of upswing, probability of upswing = P_Upswing = 50%; V_{F, Upswing} = Operating income_Upswing / WACC

= 284 / 16% = 1,775

in case of downswing, probability of downswing = P_Downswing = 50%; V_{F, Downswing} = Operating income_Downswing / WACC

= 100 / 16% = 625

Hence, expected value of firm, V_F = P_Upswing x V_{F, Upswing} + P_Downswing x V_{F, Downswing} = 50% x 1,775 + 50% x 625 = 1,200

In case of ABC, Value of equity, V_{E, ABC} = V_F = 1,200 as it's unlevered

In case of XYZ, Value of equity, V_{E, XYZ} = V_F - Debt = 1,200 - 800 = 400

If you invest $40 million in XYZ Ltd shares, your ownership = Investment amount / V_{E, XYZ} = 40 / 400 = 10%

Net income of XYZ = Operating income - interest

Net income in case of upswing = 284 - 10% x 800 = 204

Net income in case of downswing = 100 - 10% x 800 = 20

Expected net income = 50% x 204 + 50% x 20 = 112

Hence, expected payoff to you from XYZ = Ownership x expected net income = 10% x 112 = 11.2

Let's say the equivalent position in ABC be:

• Borrow an amount B
• Invest B + $ 40 mn in ABC

Expected Net income of ABC = 50% x 284 + 50% x 100 = 192

Proportion ownership in ABC = Investment in ABC / V_{E, ABC} = (B + 40) / 1,200

Equity income = Proportion owned in ABC x Expected net income of ABC = (B + 40) / 1,200 x 192

Interest to be paid on borrowing = B x 10%

Hence, total payoff on this position = Equity income - interest to be paid = (B + 40) / 12,00 x 192 - B x 10%

If we force the payoff from position in ABC to be same as that from position in XYZ then,

(B + 40) / 1,200 x 192 - B x 10% = 11.2

Hence, 0.16B + 6.40 - 0.1B = 11.2

Hence, B = (11.2 - 6.40) / (0.16 - 0.1) = 80

Hence, the alternative investment in ABC Ltd that would generate the same payoff will be:

• Borrow an amount of $ 80 million and
• Invest B + $ 40 mn = $ 120 mn in ABC