Fight Finance

Answer: Good choice. You earned $10. Poor choice. You lost $10.

The debt-to-assets ratio should actually be 37.5%, not 40%. To find the debt-to-assets ratio based on the debt-to-equity ratio (D/E), divide the D/E ratio by one which won't change its value :

$$\dfrac{D}{E} = 0.6 = \dfrac{0.6}{1}$$

So debt $(D)$ could be 0.6 and equity $(E)$ could be 1. Therefore the value of assets $(V)$ could be:

$$\begin{aligned} V &= D+E \\ &= 0.6+1 \\ &= 1.6 \\ \end{aligned}$$

Now find the debt-to-assets ratio: $$\dfrac{D}{V} = \dfrac{0.6}{1.6} = 0.375$$

The more mathematically rigorous approach is to use simultaneous equations and algebra:

$$\dfrac{D}{E} = 0.6$$ $E = \dfrac{D}{0.6}$

Substitute this into:

$$\begin{aligned} V &= D+E \\ &= D + \dfrac{D}{0.6} \\ &= \dfrac{0.6D}{0.6} + \dfrac{D}{0.6} \\ &= \dfrac{1.6D}{0.6} \\ \end{aligned}$$ $$D = \dfrac{0.6V}{1.6}$$ $$\dfrac{D}{V} = \dfrac{0.6}{1.6} = 0.375$$

To find the earnings per share (EPS):

$$\begin{aligned} \text{EPS} &= \text{Earnings per share} \\ &= \dfrac{\text{Earnings}}{\text{NumberOfShares}} = \dfrac{100m}{20m} = 5 \\ \end{aligned}$$

Since similar firms have a PE ratio of 12, we can value our firm's current market value of equity $E_0$ based on the earnings:

$$\begin{aligned} E_0 &= \text{Earnings} \times \text{PriceToEarningsRatioOfSimilarFirms} \\ &= 100m \times 12 = 1200 \text{ million} = 1.2 \text{ billion} \\ \end{aligned}$$

Alternatively, the valuation can be done on a per share basis to find the current share price $P_0$:

$$\begin{aligned} P_0 &= \text{EPS} \times \text{PriceToEarningsRatioOfSimilarFirms} \\ &= 5 \times 12 = 60 \\ \end{aligned}$$

The debt's market value $D_0$ can be found based on the debt-to-equity ratio $D/E$ and the market capitalisation of equity $E_0$ found above:

$$\begin{aligned} D_0 &= E_0 \times \dfrac{D_0}{E_0} \\ &= 1.2b \times 0.6 = 0.72b \\ \end{aligned}$$

Answer: Good choice. You earned $10. Poor choice. You lost $10.

The forward looking price-to-earnings ratio is 20, not 10. But it takes many steps to verify this since we can't find the earnings directly from the EBITDA due to not knowing the interest expense, depreciation and amortisation:

$$\text{Earnings} = (\text{EBITDA} - \text{Interest} - \text{DepreciationAndAmortisation})\times (1-\text{TaxRate})$$

Instead we have to follow a number of steps to find the enterprise value, then asset value, equity value, share price, dividend, EPS and finally PE ratio. It's a long and difficult question!

First find the Enterprise Value (EV) from the EV/EBITDA ratio:

$$\text{EVToEBITDARatio} = \dfrac{EV}{EBITDA}$$ $$10 = \dfrac{EV}{200m}$$ $$\begin{aligned} EV &= 10 \times 200m \\ &= 2000m \\ \end{aligned}$$

Add the cash to the EV to get the market value of assets:

$$\text{EnterpriseValue} = \text{Assets} - \text{Cash}$$ $$2000m = \text{Assets} - 100m$$ $$\begin{aligned} \text{Assets} &= 2000m + 100m \\ &= 2100m \\ \end{aligned}$$

Multiply the asset market value by the market debt-to-assets ratio to get the debt market value:

$$\text{DebtToAssetsRatio} = \dfrac{\text{Debt}}{\text{Assets}}$$ $$\dfrac{1}{3} = \dfrac{\text{Debt}}{2100m}$$ $$\begin{aligned} \text{Debt} &= \dfrac{1}{3} \times 2100m \\ &= 700m \\ \end{aligned}$$

So the (gross) debt is 700m but the net debt (net of cash) is gross debt less the 100m cash:

$$\begin{aligned} \text{NetDebt} &= \text{GrossDebt} - \text{Cash} \\ &= 700m - 100m \\ &= 600m \\ \end{aligned}$$

The equity market capitalisation is the asset market value less the (gross) debt market value. Using the (market value, not book value) balance sheet formula:

$$\text{Assets} = \text{Debt} + \text{Equity}$$ $$2100m = 700m + \text{Equity}$$ $$\begin{aligned} \text{Equity} &= 2100 - 700 \\ &= 1400m \\ \end{aligned}$$

The equity market capitalisation equals the share price multiplied by the number of shares:

$$\text{Equity} = \text{SharePrice} \times \text{NumberOfShares}$$ $$1400m = \text{SharePrice} \times 2m$$ $$\text{SharePrice} = \dfrac{1400m}{2m} = 700$$

Now that we know the current share price, we can find next year's dividend based on the dividend yield:

$$r_\text{dividend} = \dfrac{\text{Dividend}_1}{\text{SharePrice}_0}$$ $$0.04 = \dfrac{\text{Dividend}_1}{700}$$ $$\text{Dividend}_1 = 700 \times 0.04 = 28$$

To find the forward price-to-earnings ratio, we first need the earnings per share (EPS), based on the payout ratio:

$$\text{PayoutRatio}_1 = \dfrac{\text{Dividend}_1}{\text{EPS}_1}$$ $$0.4 = \dfrac{28}{\text{EPS}_1}$$ $$\text{EPS}_1 = \dfrac{28}{0.4} = 70$$

For the forward price-to-earnings ratio:

$$\text{ForwardPriceToEarningsRatio} = \dfrac{\text{SharePrice}_0}{\text{EPS}_1} = \dfrac{700}{70} = 10$$

Answer: Good choice. You earned $10. Poor choice. You lost $10.

The cash flows continue forever so we'll use the perpetuity formula to price the company's assets $(V)$.

$$V=\dfrac{\text{FreeCashFlow}}{r_\text{WACC}-g}$$

'Textbook method' of firm valuation with interest tax shields

The textbook method includes the interest tax shields in the discount rate by discounting the operating free cash flow (OFCF) by the weighted average cost of capital after tax:

$$\begin{aligned} V_L &= \dfrac{\text{OFCF}}{\text{WACC}_\text{AfterTax} - g} \\ &= \dfrac{100m}{0.0625 - 0} \\ &= 1600m \\ \end{aligned}$$

'Harder method' of firm valuation with interest tax shields

The harder method includes the interest tax shields in the cash flow by discounting the firm free cash flow (FFCF) by the weighted average cost of capital before tax:

$$\begin{aligned} V_L &= \dfrac{\text{FFCF}}{\text{WACC}_\text{BeforeTax} - g} \\ &= \dfrac{112m}{0.07 - 0} \\ &= 1600m \\ \end{aligned}$$

Answer: Good choice. You earned $10. Poor choice. You lost $10.

The cash flows continue forever so we'll use the perpetuity formula to price the company's assets $(V)$.

$$V=\dfrac{\text{FreeCashFlow}}{r_\text{WACC}-g}$$

'Textbook method' of firm valuation with interest tax shields

The textbook method includes the interest tax shields in the discount rate by discounting the operating free cash flow (OFCF) by the weighted average cost of capital after tax:

$$\begin{aligned} V_L &= \dfrac{\text{OFCF}}{\text{WACC}_\text{AfterTax} - g} \\ &= \dfrac{48.5m}{0.097 - 0} \\ &= 500m \\ \end{aligned}$$

'Harder method' of firm valuation with interest tax shields

The harder method includes the interest tax shields in the cash flow by discounting the firm free cash flow (FFCF) by the weighted average cost of capital before tax:

$$\begin{aligned} V_L &= \dfrac{\text{FFCF}}{\text{WACC}_\text{BeforeTax} - g} \\ &= \dfrac{50m}{0.1 - 0} \\ &= 500m \\ \end{aligned}$$

Answer: Good choice. You earned $10. Poor choice. You lost $10.

The weighted average cost of capital (WACC) before tax is:

$$\begin{aligned} r_\text{WACC before tax} &= r_D.\frac{D}{V_L} + r_{EL}.\frac{E_L}{V_L} \\ &= 0.04 \times 0.8 + 0.163 \times (1-0.8) \\ &= 0.0646 \\ \end{aligned}$$ $$\begin{aligned} r_\text{WACC after tax} &= r_D.\mathbf{(1-t_c)}.\frac{D}{V_L} + r_{EL}.\frac{E_L}{V_L} \\ &= 0.04 \times (1 - 0.3) \times 0.8 + 0.163 \times (1-0.8) \\ &= 0.055 \\ \end{aligned}$$

The cash flows continue forever so we'll use the perpetuity formula to price the company's assets $(V)$.

$$V=\dfrac{\text{FreeCashFlow}}{r_\text{WACC}-g}$$

'Textbook method' of firm valuation with interest tax shields

The textbook method includes the interest tax shields in the discount rate by discounting the operating free cash flow (OFCF) by the weighted average cost of capital after tax:

$$\begin{aligned} V_L &= \dfrac{\text{OFCF}}{\text{WACC}_\text{AfterTax} - g} \\ &= \dfrac{30k}{0.055 - 0.015} \\ &= 750k \\ \end{aligned}$$

The current value of debt equals the current value of assets multiplied by the debt-to-assets ratio:

$$\begin{aligned} D &= V_L \times \dfrac{D}{V_L} \\ &= 750k \times 0.8 \\ &= 600k \\ \end{aligned}$$

The benefit from interest tax shields in the first year is equal to the interest expense that year multiplied by the corporate tax rate:

$$\begin{aligned} \text{BenefitFromInterestTaxShields}_1 &= \text{InterestExpense}_1 \times t_c \\ &= D_0 \times r_D \times t_c \\ &= 600k \times 0.04 \times 0.3\\ &= 24k \times 0.3 \\ &= 7.2k \\ \end{aligned}$$

To find the market capitalisation of equity, use the market value balance sheet formula:

$$V_L = D + E$$ $$750k = 600k + E$$ $$\begin{aligned} E &= 750k - 600k \\ &= 150k \end{aligned}$$

The share price $P$ can be found based on the market capitalisation of equity formula:

$$E = P \times n_\text{shares}$$ $$\begin{aligned} P &= \dfrac{E}{n_\text{shares}} \\ &= \dfrac{150k}{100k} \\ &= 1.5 \\ \end{aligned}$$

Answer: Good choice. You earned $10. Poor choice. You lost $10.

The cash flows continue forever so we'll use the perpetuity formula to price the company's assets $(V)$.

$$V_0=\dfrac{\text{FreeCashFlow}_1}{r_\text{WACC}-g}$$

'Textbook method' of firm valuation with interest tax shields

The textbook method includes the interest tax shields in the discount rate by discounting the operating free cash flow (OFCF) which excludes the interest tax shields by the weighted average cost of capital after tax which includes interest tax shields:

$$\begin{aligned} V_{0L} &= \dfrac{\text{OFCF}_1}{\text{WACC}_\text{AfterTax} - g} \\ &= \dfrac{12.5m}{0.0825 - 0.02} \\ &= 200m \\ \end{aligned}$$

'Harder method' of firm valuation with interest tax shields

The harder method includes the interest tax shields in the cash flow by discounting the firm free cash flow (FFCF) which includes interest tax shields by the weighted average cost of capital before tax which excludes interest ta shields:

$$\begin{aligned} V_{0L} &= \dfrac{\text{FFCF}_1}{\text{WACC}_\text{BeforeTax} - g} \\ &= \dfrac{14m}{0.09 - 0.02} \\ &= 200m \\ \end{aligned}$$

The market capitalisation of equity can be found using two different methods. We can discount the equity free cash flow by the required return on equity:

$$\begin{aligned} E_{0} &= \dfrac{\text{EFCF}_1}{r_E - g} \\ &= \dfrac{11m}{0.13 - 0.02} \\ &= 100m \\ \end{aligned}$$

Alternatively, the market capitalisation of equity is equal to the asset value multiplied by the equity-to-assets ratio, which is one less the debt-to-assets ratio:

$$\begin{aligned} E_{0} &= V_0 \times \dfrac{E_0}{V_0} \\ &= V_0 \times \left( 1-\dfrac{D_0}{V_0} \right) \\ &= 200m \times (1-0.5) \\ &= 100m \\ \end{aligned}$$

The share price is then:

$$E_{0} = n_\text{shares} \times P_{0\text{,shares}}$$ $$100m = 1m \times P_{0\text{,shares}}$$ $$\begin{aligned} P_{0\text{,shares}} &= \dfrac{100m}{1m} \\ &= 100 \\ \end{aligned}$$

The debt cash flow (DebtCF) at the end of the first year can be found in a few different ways. It should be the difference between the firm free cash flow (FFCF) including interest tax shields and the equity free cash flow (EFCF):

$$\text{FFCF}_1 = \text{EFCF}_1 + \text{DebtCF}_1$$ $$14m = 11m + \text{DebtCF}_1$$ $$\begin{aligned} \text{DebtCF}_1 &= 14m-11m \\ &= 3m \\ \end{aligned}$$

Since the 50% debt-to-assets ratio must be constant over time, the assets, equity and debt should all grow by the same 2% pa growth rate. However, the debt will actually grow by its 5% pa yield to maturity ignoring the coupons which are a part of DebtCF. Therefore to make the debt grow by only 2% pa rather than 5% pa, the debt cash flow must be 3% pa (=5% - 2%) of the 100m current debt value which is $3m (=0.03*100m).

To find the net amount of debt that needs to be repaid or bought back:

$$\begin{aligned} \text{DebtCF} &= \text{DebtCoupons} + \text{DebtFaceValueRepayments} - \text{DebtRaisings} \\ &= \text{DebtCoupons} + \text{NetDebtRepaymentsExcludingCoupons} \\ \end{aligned}$$ $$3m = 1.2m + \text{NetDebtRepaymentsExcludingCoupons}_1$$ $$\begin{aligned} \text{NetDebtRepaymentsExcludingCoupons}_1 &= 3m-1.2m \\ &= 1.8m \\ \end{aligned}$$

Answer: Good choice. You earned $10. Poor choice. You lost $10.

The interest tax shield per year is $IntExp.t_c$, and the odd numbered equations include it. Let the firm free cash flow with the interest tax shield be $FFCF_\text{wITS}$ and the cash flow excluding the interest tax shield be $FFCF_\text{xITS}$. Then:

$$\begin{aligned} FFCF_\text{wITS}&=NI + Depr - CapEx -ΔNWC + IntExp \\ &=EBIT.(1-t_c )+ Depr- CapEx -ΔNWC+IntExp.t_c \\ &=EBITDA.(1-t_c )+Depr.t_c- CapEx -ΔNWC+IntExp.t_c \\ &=EBIT-Tax + Depr - CapEx -ΔNWC \\ &=EBITDA-Tax - CapEx -ΔNWC \\ \end{aligned}$$ $$\begin{aligned} FFCF_\text{xITS}&=NI + Depr - CapEx -ΔNWC + IntExp.(1-t_c) \\ &=EBIT.(1-t_c) + Depr- CapEx -ΔNWC \\ &=EBITDA.(1-t_c )+Depr.t_c- CapEx -ΔNWC \\ &=EBIT-Tax + Depr - CapEx -ΔNWC-IntExp.t_c \\ &=EBITDA-Tax - CapEx -ΔNWC-IntExp.t_c \\ \end{aligned}$$