Question

Consider a firm as follows: Assume that the firm has no debt. The cash flows are received at the end of each year and are perpetual. Cost of equity capital for an unlevered firm, r0, is 20%. The first cash-flow will be received one year from today. All calculations for valuation are done today. Firm value is defined as collective value of debt and equity.

Sales = $ 500,000

Cash Costs = 360,000

Operating Income = 140,000

Tax @ 34% -47,600

Unlevered cash flow (UCF) $ 92,400

To illustrate that financing has no impact on firm value if there are no taxes (i.e. financing affects firm value purely because the interest payments generate tax-savings), and to further illustrate the equivalence of various valuation methods.Assume the the firm has$ 100,000 of debt @ 10% and the tax rate is zero.Find the firm value using
i) WACC method$__________.

Answer 1

The given firm is an all equity firm with unlevered cash flow, UCF = $92,400

The firm is expected to earn above free cash flows perpetually.

For perpetual cash stream, the present value is given by following formula:

PV (CF)= CF / (1+r)^1 + CF/(1+r)^2 + CF / (1+r)^3 + ................infinity

PV (CF) = CF / r

Where r is the discount rate.

The cost of equity capital for the firm is given as 20% which should be used as the discount rate.

Value of the firm = $92,400 / 0.20 = $462,000

1) Illustration for impact of financing when no tax

We shall consider the following three cases: 1) 100% equity 2) 100% debt 3) $100,000 debt (given)

Case (1) - 100% equity

The value of the firm will be greater as compared to above when taxes are zero.

The Profit AFter tax will be equal to Profit before tax which is same as Operating Income for this company, as there are no taxes and interest expenses

Hence, value of the firm = Profit before tax / r% = 140,000 / 20% = $700,000

Case (2) - 100% debt

To illustrate this case, consider that the above company has 100% debt instead of 100% equity.

Since this company has perpetual cash flows and zero growth rate of cash flows, it implies that the firm makes no reinvestment. In other words, the pay-out ratio of the company is 100%.

This means the company pays 100% of its free cash flows as perpetual interest to the debt holders (in the previous case, the company paid 100% of operating income to equity shareholders)

Now, value of the firm = Value of Equity + value of perpetual debt = (0) + Interest Expense / d%

Since value of equity is zero and perpetual debt is valued as interest divided by discount rate.

Value of the firm = 140,000 / d%

The discount rate for valuing the firm should reflect the opportunity cost of capital for the firm. The opportunity cost of capital reflects the return on assets. Since both the firms are same with respect asset side of the balance sheet and earn same return on assets, the opportunity cost of capital should also be same. Hence, the opportunity cost of capital for the 100% debt firm should be same as 100% equity firm.

Hence, discount rate, d% = 20%

Hence, value of the firm = 140,000 / 20% = $700,000

Case (3): With a debt of $100,000 at 10%

The free cash flows can be calculated now as follows

	100% Equity	100% Debt	D=100,000
Sales	500,000	500,000	500,000
Expenses	360,000	360,000	360,000
Operating Income	140,000	140,000	140,000
Interest	0	140,000	10,000
Profit Before Tax	140,000	0	130,000
Tax	0	0	0
Profit After Tax	140,000	0	130,000

We now have to calculate the discount rate applicable for the firm.

As explained in the case (2), the risk involved in the cash flows depends on the asset side of the balance sheet of the firm and depends only on the asset efficiency, operating and marketing performances of the firm. Since these are all same for all three cases, the opportunity cost of capital for the firm will remain as 20% irrespective of the liability side or financing mix of the firm.

Opportunity cost of capital = 20% = Company cost of capital

Company cost of capital, WACC = D/V * cost of debt + E/V * cost of equity

WACC = 20% = (100,000 / (100,000+E)) * 10% + E / (100,000+E) * Ke

Value of E = PAT / ke (since perpetuity)

Value of E = 130,000 / Ke

We have to do a reiterative calculation to find out the cost of equity.

Alternatively, we can use the Goal Seek feature of excel, to calculate the variable Ke which will satisfy the above relationship of opportunity cost of capital.

We can find that for ke = 21.66%, the opportunity cost of capital will be 20% (approx.)

E = 130,000 / 21.65% = $600,215

D = 10,000 /10% = 100,000

V = $700,215 (approx) = $700,000

Thus, the value of the firm is $700,000 which was same as the value of the firm in previous two cases.

	100% Equity	100% Debt	D=100,000
Sales	500,000	500,000	500,000
Expenses	360,000	360,000	360,000
Operating Income	140,000	140,000	140,000
Interest	0	140,000	10,000
Profit Before Tax	140,000	0	130,000
Tax	0	0	0
Profit After Tax	140,000	0	130,000
Valuation
Value of Equity	700000	0	600215.15
Value of Debt	0	700000	100000
Value of the firm	700000	700000	700215.15
Cost of Equity, Ke	22%
Opportunity CC	20%
Value of Equity	600215.1518
Value of Debt	100000
Value of Firm	700215.1518
Interest Rate	10%
WACC	19.994%

(Note: if you cannot use excel function, you have to do reiteration of several cycles with different cost of equity. For each cost of equity (say, 21% as this has to be greater than opportunity cost of capital due to presence of equity), we have to verify whether the above equation of opportunity cost of capital vs WACC is satisfied)

Thus, we have found that the value of the firm is equal to $700,000 in all the following three cases:

a) 100% equity b) 100% debt and c ) with a debt of 100,000

Hence, the financing mix does not have any impact on valuation of the firm when there are no corporate taxes

B) Illustrating equivalence of various methods

We can calculate value of firm in case (3) using several methods:

Method (1) - Using Opportunity cost of capital

In this method, we have to add all cashflows received by each stakeholder of the company and discount by the opportunity cost of capital.

Cash flows received by equity shareholders = 130,000 ( since perpetuity with 100% payout)

Cash flows received by debt holders = 10,000 (perpetuity at 10%)

Total cash flows = 130,000 +10,000 = 140,000

Value of firm = Total CF / Opportunity cost = 140,000 / 20% = $700,000

Method 2: Using values of each security

In this method, we can calculate individual security prices and add up to arrive at value of the firm.

Value of firm = Value of equity + value of debt

Value of equity = Dividends /Cost of equity = 130,000 / 21.65% = $600,215

Value of Debt = Interest expense / interest rate = 10,000 / 10% = $100,000

Value of firm = 600215 + 100000 = $700,215

Method 3 - Using WACC

We can calculate the WACC using cost of equity and debt. However, in this method, we should know the market weights of equity and debt.

Let us assume that we know WACC to be 20% as calculated before.

We now have to calculate the profit after tax of the company ignoring the interest

Profit after tax ignoring interest expense, PAT = 140,000 (refer table given below)

	WACC Method
Sales	500000
Expenses	360000
Operating Income	140000
Interest	0
Profit Before Tax	140000
Tax	0
Profit After Tax	140000
WACC	19.994%
Valuation	700215.15

Value of firm = PAT (without deducting interest) / WACC

V = 140,000 / 19.994% = $700,215

Thus, all three methods give same valuation results.