A firm has a debt-to-equity ratio of 25%. What is its debt-to-assets ratio?
The debt-to-equity ratio can be divided by one without changing its value : ###\dfrac{D}{E} = 0.25 = \dfrac{0.25}{1}###
So debt ##(D)## could be 0.25 and equity ##(E)## could be 1. Therefore the value of assets ##(V)## could be: ###\begin{aligned} V &= D+E \\ &= 0.25+1 \\ &= 1.25 \\ \end{aligned}###
To find the debt-to-assets ratio: ###\dfrac{D}{V} = \dfrac{0.25}{1.25} = 0.2###
The more mathematically rigorous approach is to use simultaneous equations and algebra:
###\dfrac{D}{E} = 0.25### ##E = \dfrac{D}{0.25}##Substitute this into:
###\begin{aligned} V &= D+E \\ &= D + \dfrac{D}{0.25} \\ &= \dfrac{0.25D}{0.25} + \dfrac{D}{0.25} \\ &= \dfrac{1.25D}{0.25} \\ \end{aligned}### ###D = \dfrac{0.25V}{1.25}### ###\dfrac{D}{V} = \dfrac{0.25}{1.25} = 0.2###A firm has a debt-to-assets ratio of 20%. What is its debt-to-equity ratio?
The debt-to-assets ratio can be divided by one without changing its value : ###\dfrac{D}{V} = 0.2 = \dfrac{0.2}{1}###
So debt ##(D)## could be 0.2 and assets ##(V)## could be 1. Now the value of equity ##(E)## can be found using the market value balance sheet formula: ###V = D+E ### ###\begin{aligned} E &= V-D \\ &= 1 - 0.2 \\ &= 0.8 \\ \end{aligned}###
To find the debt-to-equity ratio: ###\dfrac{D}{E} = \dfrac{0.2}{0.8} = 0.25###
The more mathematically rigorous approach is to use simultaneous equations and algebra:
###\dfrac{D}{V} = 0.2### ##V = \dfrac{D}{0.2}##Substitute this into the market value balance sheet formula and seek to re-arrange the terms to show D/E on the left hand side:
###V = D+E ### ###\dfrac{D}{0.2} = D + E ### ###D = 0.2 D + 0.2 E ### ###D - 0.2 D = 0.2 E ### ###0.8 D = 0.2 E ### ###\begin{aligned} \dfrac{D}{E} &= \dfrac{0.2}{0.8} \\ &= 0.25 \\ \end{aligned}###Your friend just bought a house for $400,000. He financed it using a $320,000 mortgage loan and a deposit of $80,000.
In the context of residential housing and mortgages, the 'equity' tied up in the value of a person's house is the value of the house less the value of the mortgage. So the initial equity your friend has in his house is $80,000. Let this amount be E, let the value of the mortgage be D and the value of the house be V. So ##V=D+E##.
If house prices suddenly fall by 10%, what would be your friend's percentage change in equity (E)? Assume that the value of the mortgage is unchanged and that no income (rent) was received from the house during the short time over which house prices fell.
Remember:
### r_{0\rightarrow1}=\frac{p_1-p_0+c_1}{p_0} ###
where ##r_{0-1}## is the return (percentage change) of an asset with price ##p_0## initially, ##p_1## one period later, and paying a cash flow of ##c_1## at time ##t=1##.
The key thing to realise in this question is that when house prices fall by 10%, there is no fall in the debt owing. The bank will not take pity and reduce the loan!
In the below table, 'k' means thousand. Filling in the values for all except the equity value at t=1, we can calculate that E = V - D = 360k - 320k = 40k, so equity should be 40k.
| Asset, Debt and Equity Values | |||
| Millions of dollars | |||
| Time | V | D | E |
| 0 | 400k | 320k | 80k |
| 1 | 360k | 320k | 40k |
The fall in equity from 80k (=400k-320k) to 40k (=360k-320k) corresponds to a 50% fall in equity:
###\begin{aligned} r_{\text{E, }0\rightarrow1} &= \frac{p_1-p_0+c_1}{p_0} \\ &= \frac{40k-80k+0}{80k} \\ &= \frac{-40k}{80k} \\ &= -0.5 = -50\% \\ \end{aligned} ###
One year ago you bought a $1,000,000 house partly funded using a mortgage loan. The loan size was $800,000 and the other $200,000 was your wealth or 'equity' in the house asset.
The interest rate on the home loan was 4% pa.
Over the year, the house produced a net rental yield of 2% pa and a capital gain of 2.5% pa.
Assuming that all cash flows (interest payments and net rental payments) were paid and received at the end of the year, and all rates are given as effective annual rates, what was the total return on your wealth over the past year?
Hint: Remember that wealth in this context is your equity (E) in the house asset (V = D+E) which is funded by the loan (D) and your deposit or equity (E).
There are a few ways to think about this problem. One is to think of the house asset as being financed by a portfolio of debt and equity, where the total historical return on the house asset equals the weighted average total historical return on the debt and equity. Note that the total historical return on the house is 4.5%, the sum of the 2% net rental yield plus the 2.5% capital yield.
###\begin{aligned} r_V &= r_D.\dfrac{D}{V} + r_E.\dfrac{E}{V} \\ 0.045 &= 0.04 \times \dfrac{0.8m}{1m} + r_E.\dfrac{0.2m}{1m} \\ \end{aligned}### ###\begin{aligned}r_E &= \left( 0.045 - 0.04 \times \dfrac{0.8}{1} \right) \times \dfrac{1}{0.2} \\ &= 0.065 \\ \end{aligned}###Alternatively, a table can be used. After filling in all of the known values, the unknown return on equity from time -1 to 0 can be calculated.
| Price and Income Values | ||||||
| Time | V | D | E | |||
| -1 | 1m | 0.8m | 0.2m | |||
| 0 | 1.045m | 0.832m | 0.213m | |||
The capital and income components of the equity rose from 0.2m to 0.213m (=1.045m-0.832m) which corresponds to a total return on equity of:
###\begin{aligned} r_{\text{E, }-1 \rightarrow 0} &= \frac{P_0-P_{-1}+C_0}{P_{-1}} \\ &= \frac{0.213m - 0.2m+0}{0.2m} \\ &= \frac{0.013m}{0.2m} \\ &= 0.065 = 6.5\% \\ \end{aligned} ###
Use the below information to value a levered company with constant annual perpetual cash flows from assets. The next cash flow will be generated in one year from now, so a perpetuity can be used to value this firm. Both the operating and firm free cash flows are constant (but not equal to each other).
| Data on a Levered Firm with Perpetual Cash Flows | ||
| Item abbreviation | Value | Item full name |
| ##\text{OFCF}## | $100m | Operating free cash flow |
| ##\text{FFCF or CFFA}## | $112m | Firm free cash flow or cash flow from assets (includes interest tax shields) |
| ##g## | 0% pa | Growth rate of OFCF and FFCF |
| ##\text{WACC}_\text{BeforeTax}## | 7% pa | Weighted average cost of capital before tax |
| ##\text{WACC}_\text{AfterTax}## | 6.25% pa | Weighted average cost of capital after tax |
| ##r_\text{D}## | 5% pa | Cost of debt |
| ##r_\text{EL}## | 9% pa | Cost of levered equity |
| ##D/V_L## | 50% pa | Debt to assets ratio, where the asset value includes tax shields |
| ##t_c## | 30% | Corporate tax rate |
What is the value of the levered firm including interest tax shields?
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}###Use the below information to value a levered company with annual perpetual cash flows from assets that grow. The next cash flow will be generated in one year from now. Note that ‘k’ means kilo or 1,000. So the $30k is $30,000.
| Data on a Levered Firm with Perpetual Cash Flows | ||
| Item abbreviation | Value | Item full name |
| ##\text{OFCF}## | $30k | Operating free cash flow |
| ##g## | 1.5% pa | Growth rate of OFCF |
| ##r_\text{D}## | 4% pa | Cost of debt |
| ##r_\text{EL}## | 16.3% pa | Cost of levered equity |
| ##D/V_L## | 80% pa | Debt to assets ratio, where the asset value includes tax shields |
| ##t_c## | 30% | Corporate tax rate |
| ##n_\text{shares}## | 100k | Number of shares |
Which of the following statements is NOT correct?
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}###Question 772 interest tax shield, capital structure, leverage
A firm issues debt and uses the funds to buy back equity. Assume that there are no costs of financial distress or transactions costs. Which of the following statements about interest tax shields is NOT correct?
The higher interest expense does lower profit which appears bad, but this actually leads to lower tax payments and since assets are unchanged there will be higher cash flows from assets which shareholders will benefit from, leading to a higher share price.
A retail furniture company buys furniture wholesale and distributes it through its retail stores. The owner believes that she has some good ideas for making stylish new furniture. She is considering a project to buy a factory and employ workers to manufacture the new furniture she's designed. Furniture manufacturing has more systematic risk than furniture retailing.
Her furniture retailing firm's after-tax WACC is 20%. Furniture manufacturing firms have an after-tax WACC of 30%. Both firms are optimally geared. Assume a classical tax system.
Which method(s) will give the correct valuation of the new furniture-making project? Select the most correct answer.
The unlevered manufacturing business's cash flows should be discounted by other manufacturing firms' WACC after tax.
There are two parts to this question. The first is about how to take the interest tax shield into account. The second is about how to adjust for the higher systematic risk of the manufacturing project compared to the existing retail business.
Treatment of the interest tax shieldTo find the value of the levered project (##V_L##), the unlevered CFFA should be discounted by the WACC after tax. This is called the 'text book' method of valuation. If the unlevered CFFA will occur in perpetuity with no growth, then:
###V_L = \frac{CFFA_U}{r_\text{WACC after tax}} = \frac{NI+Depr-CapEx - \varDelta NWC}{r_D.(1-t_c).\frac{D}{V_L} + r_{EL}.\frac{E_L}{V_L}}###The tax shield is taken into account in the discount rate (##r_\text{WACC after tax}##), not the cash flow (##CFFA_U##). The after tax WACC takes the interest tax shield into account by reducing the cost of debt by the corporate tax rate: ###r_\text{WACC after tax} = r_D.\mathbf{(1-t_c)}.\frac{D}{V_L} + r_{EL}.\frac{E_L}{V_L}###
The unlevered cash flow (##CFFA_U##) excludes interest expense (##IntExp=0##) and therefore doesn't take the interest tax shield into account:
###CFFA_U=NI+Depr-CapEx - \varDelta NWC### Treatment of the higher systematic risk of the projectThe WACC after (and before) tax is supposed to reflect the systematic risk of the cash flows. Since the project is in the more systematically risky manufacturing industry, the high WACC of a similar manufacturing firm should be used, not the low WACC of the retail firm which has less systematic risk.
Additionally, discounting by the WACC after tax only works if the firm always has the same proportion of debt. If the debt-to-assets ratio changes then the amount of tax shields will change and the after-tax WACC must be recalculated every year.
Why is Capital Expenditure (CapEx) subtracted in the Cash Flow From Assets (CFFA) formula?
###CFFA=NI+Depr-CapEx - \Delta NWC+IntExp###
Capital expenditure (CapEx) is equal to 'net capital expenditure' which is the cash spent on (non-current) assets less the cash received from selling them. It is subtracted in the cash flow from assets (CFFA) equation to make up for how depreciation is added back. Since depreciation (Depr) is added back, no cost has been allocated to the assets bought such as land, buildings, factories and trucks, so it is subtracted in CFFA as CapEx.
The sum of the un-discounted Depr and CapEx amounts will cancel each other out, but there is a timing difference which is important. Depreciation allocates the asset cost over its life and this has nothing to do with cash flows, ignoring the time value of money. CapEx reflects when the money is actually spent, usually at the start when the asset is bought, taking the time value of money into account.
Interest expense (IntExp) is an important part of a company's income statement (or 'profit and loss' or 'statement of financial performance').
How does an accountant calculate the annual interest expense of a fixed-coupon bond that has a liquid secondary market? Select the most correct answer:
Annual interest expense is equal to:
According to the 'effective interest method' which is the standard used by accountants when preparing financial reports, the interest expense for each debt contract equals the yield multiplied by its book value at the beginning of the period, with accrual adjustments if the debt matures during the year. Since the bond trades in a liquid market, the firm's accountants will 'mark to market' the bond price, so the bond's book value will be equal to its market value. Mathematically, the interest expense will be:
###IntExp_1 = r_{\text{D, 0}\rightarrow 1}.D_0###Some good articles on the effective interest method:
Cash Flow From Assets (CFFA) can be defined as:
Cash flow from assets (CFFA), also known as free cash flow to the firm (FCFF), is the cash generated from the firm's assets. It is the income component of the total dollar return from the assets, in the same way as shares, bonds and land have income returns called dividends, coupons and rent respectively. The idea is that the value of a project or business is the present value of its cash flow from assets in the same way that the value of a stock is the present value of its dividends.
Because a firm's assets (V) are owned by the debt (D) and equity (E) holders who fund it (V=D+E), the CFFA is indirectly owed to debt and equity holders. CFFA is the cash available to distribute to the debt and equity holders. CFFA is equal to the debt's coupon and principal payments less new debt raisings plus the equity's dividend and buy back payments less new equity raisings. The CFFA must equal the net payments to the debt and equity holders who fund the assets because if the CFFA is not paid out, the cash must have been kept by the business and thus increases net working capital, making CFFA zero. Here are some other examples which also hinge on the way the increase in net working capital (NWC) is subtracted from CFFA.
- If a firm pays a dividend to shareholders then the firm's cash will fall, causing a decrease in NWC which will increase CFFA. This increase is exactly equal to the dividend payment to shareholders.
- If a firm raises equity in an initial public offering (IPO), then the cash raised will increase current assets (cash) and will be subtracted in the CFFA equation as an increase in net working capital. The cash payment by the shareholders to the company for the shares in the IPO is a negative cash flow from the company to the shareholders, and this is equal to the negative CFFA due to the large increase in NWC.
- A firm earns positive net income and generates excess cash but no debt or equity is raised or paid out. It seems like there is a positive CFFA and it is not equal to the zero payments to debt or equity holders. But in fact, if no cash flows are paid to or received from debt or equity holders, then the excess cash will just sit in the bank account, increasing net working capital, and causing CFFA to be zero, which is equal to the zero payments to debt and equity holders. So actually CFFA was not positive, it was zero. We just overlooked that the excess cash increases NWC.
Summarizing with an equation,
###\begin{aligned} CFFA &= (\text{net payments to debt holders}) + (\text{net payments to equity holders}) \\ &= (\text{payments to debt holders}) - (\text{receipts from debt holders}) + (\text{payments to equity holders}) - (\text{receipts from equity holders}) \\ &= (\text{coupon and principal payments}) - (\text{new debt raisings}) + (\text{dividend and buyback payments}) - (\text{new equity raisings}) \\ \end{aligned} ###
A company increases the proportion of debt funding it uses to finance its assets by issuing bonds and using the cash to repurchase stock, leaving assets unchanged.
Ignoring the costs of financial distress, which of the following statements is NOT correct:
In a capital restructure where debt is issued to replace equity:
- The gearing ratios ##(D/V_L)## and ##(D/E_L)## increase , so statement (a) is true.
- Less taxes are paid due to higher interest expense and therefore lower before-tax profit, so statement (b) is true. ###\text{Corporate Tax}=(Rev-COGS-FC-Depr-\mathbf{IntExp}).t_c###
- Net income (NI) falls for the same reason as above, so statement (c) is true. ###NI=(Rev-COGS-FC-Depr-IntExp).(1-t_c)###
- Levered firm free cash flow (FFCF), same as levered cash flow from assets (CFFA) increases due to higher interest tax shields, so statement (d) is true. ###\begin{aligned} FFCF &= NI+Depr-CapEx - \varDelta NWC+IntExp \\ &= (Rev-COGS-FC).(1-t_c)+Depr.t_c - CapEx - \varDelta NWC + \mathbf{IntExp.t_c} \\ \end{aligned}###
- Equity free cash flow (EFCF, also called cash flow to equity) is not likely to stay the same, so statement (e) is false.
EFCF is the FFCF less net payments to debt holders (coupons and principal payments less debt raisings). This should also equal the dollar value of dividends and buybacks less equity raisings.
###\begin{aligned} EFCF &= \text{dividend payments} + \text{buybacks} - \text{equity raisings}\\ &= FFCF - \text{debt cash flows} \\ &= FFCF - (\text{coupon payments} + \text{principal repayments} - \text{debt raisings}) \\ \end{aligned}###Equity free cash flow (EFCF) may increase for the same reason as firm free cash flow (FFCF) increases: the higher amount of interest tax shields, which means more money is available to pay dividends and undertake buy backs.
Or cash flow to equity could fall because there is less equity to actually pay after the debt raising and equity buy-back. On the other hand, the systematic risk of the remaining equity will be higher since there is more leverage, so the total required return of equity will be higher. But this higher total return could be achieved by capital gains in the share price (not included in EFCF) or dividends and buy-backs (part of EFCF), so there won't necessarily be an increase in EFCF if higher capital gains are realised.
The equity free cash flow is also affected by the coupon and principal payments to debt holders. Note that coupon payments are not necessarily equal to interest expense (IntExp), since accountants define interest expense according to the 'effective interest method' which is yield times book value of debt at the start of the year, so even zero-coupon bonds have interest expense.
Note that the cash flow to equity holders will be very high during the time when the equity is bought back, because clearly this will be a large equity buy-back. But the question is asking how the cash flow to equity will change after the capital restructure, not during the restructure.
There are many different ways to value a firm's assets. Which of the following will NOT give the correct market value of a levered firm's assets ##(V_L)##? Assume that:
- The firm is financed by listed common stock and vanilla annual fixed coupon bonds, which are both traded in a liquid market.
- The bonds' yield is equal to the coupon rate, so the bonds are issued at par. The yield curve is flat and yields are not expected to change. When bonds mature they will be rolled over by issuing the same number of new bonds with the same expected yield and coupon rate, and so on forever.
- Tax rates on the dividends and capital gains received by investors are equal, and capital gains tax is paid every year, even on unrealised gains regardless of when the asset is sold.
- There is no re-investment of the firm's cash back into the business. All of the firm's excess cash flow is paid out as dividends so real growth is zero.
- The firm operates in a mature industry with zero real growth.
- All cash flows and rates in the below equations are real (not nominal) and are expected to be stable forever. Therefore the perpetuity equation with no growth is suitable for valuation.
Where:
###r_\text{WACC before tax} = r_D.\frac{D}{V_L} + r_{EL}.\frac{E_L}{V_L} = \text{Weighted average cost of capital before tax}### ###r_\text{WACC after tax} = r_D.(1-t_c).\frac{D}{V_L} + r_{EL}.\frac{E_L}{V_L} = \text{Weighted average cost of capital after tax}### ###NI_L=(Rev-COGS-FC-Depr-\mathbf{IntExp}).(1-t_c) = \text{Net Income Levered}### ###CFFA_L=NI_L+Depr-CapEx - \varDelta NWC+\mathbf{IntExp} = \text{Cash Flow From Assets Levered}### ###NI_U=(Rev-COGS-FC-Depr).(1-t_c) = \text{Net Income Unlevered}### ###CFFA_U=NI_U+Depr-CapEx - \varDelta NWC= \text{Cash Flow From Assets Unlevered}###Answer (e) double counts the interest tax shields and thus over-estimates the value of the levered firm's assets.
The cash flow from assets is big since it's levered and therefore adds the benefit of the interest tax shields from the debt ##(IntExp.t_c)## since:
###CFFA_L = CFFA_U + IntExp.t_c###The discount rate is small since it subtracts the benefit of the proportional interest tax shield since
###r_\text{WACC after tax} = r_\text{WACC before tax} - D.r_D.t_c/V_L### Because the perpetuity equation's cash flows are bigger and the discount rate is smaller, the benefit of the interest tax shields are double counted and we've over-valued the levered business's assets ##(V_L)##.All other answers give the correct valuation of the levered firm's assets ##(V_L)##. They are all equivalent. They count the benefit of interest tax shields only once.
Thanks to Shahzada for correcting an error in the solutions.
One formula for calculating a levered firm's free cash flow (FFCF, or CFFA) is to use earnings before interest and tax (EBIT).
###\begin{aligned} FFCF &= (EBIT)(1-t_c) + Depr - CapEx -\Delta NWC + IntExp.t_c \\ &= (Rev - COGS - Depr - FC)(1-t_c) + Depr - CapEx -\Delta NWC + IntExp.t_c \\ \end{aligned} \\###
The above FFCF includes the benefit of interest tax shields.
Because there is a plus ##IntExp.t_c## term, there is an interest tax shield being included in this FFCF. The firm's free cash flow with interest tax shields can be represented by ##FFCF_\text{wITS}##.
###\begin{aligned} FFCF_\text{wITS} &= (EBIT)(1-t_c) + Depr - CapEx -\Delta NWC + IntExp.t_c \\ &= (Rev - COGS - Depr - FC)(1-t_c) + Depr - CapEx -\Delta NWC + IntExp.t_c \\ \end{aligned} \\###On the one hand, since earnings before interest and tax (EBIT) is higher than pre-tax earnings (or earnings before tax EBT), the tax on EBIT will be higher than the tax on pre-tax earnings. This means that the interest tax shield is not being taken into account when tax is calculated on EBIT. But because the interest tax shield is added on separately at the end of the above FFCF equation, which increases FFCF by the amount of the tax saving from having debt and interest expense, the interest tax shield is taken into account.
Note that EBIT is also called operating profit if there are no non-operating items such as shut-down business segments. Net operating profit equals ##(EBIT)(1-t_c)##.
Note that the firm free cash flow with interest tax shields ##(FFCF_\text{wITS})## should be discounted by the WACC before tax ##(r_\text{WACC before tax})##. This will give the value of the levered firm or project with interest tax shields ##(V_\text{wITS})##. The weighted average cost of capital before tax is:
###r_\text{WACC before tax} = \dfrac{D}{V}.r_D + \dfrac{E}{V}.r_E###One formula for calculating a levered firm's free cash flow (FFCF, or CFFA) is to use net operating profit after tax (NOPAT).
###\begin{aligned} FFCF &= NOPAT + Depr - CapEx -\Delta NWC \\ &= (Rev - COGS - Depr - FC)(1-t_c) + Depr - CapEx -\Delta NWC \\ \end{aligned} \\###
The above FFCF excludes the benefit of interest tax shields.
This is because interest expense is completely ignored in this FFCF, it appears no where. The firm's free cash flow excluding interest tax shields can be represented by ##FFCF_\text{xITS}##.
Note that there is no interest expense subtracted from pre-tax operating profit, so the pre-tax operating profit will be higher than the actual pre-tax profit which subtracts interest expense, and this means that the NOPAT subtracts a higher amount of tax than will actually paid in reality. This is why using NOPAT in the above FFCF formula excludes the benefit of interest tax shields; there's too much tax.
###\begin{aligned} FFCF_\text{xITS} &= NOPAT + Depr - CapEx -\Delta NWC \\ &= (Rev - COGS - Depr - FC)(1-t_c) + Depr - CapEx -\Delta NWC \\ \end{aligned} \\###To calculate FFCF with interest tax shields, the above formula can be adjusted by simply adding ##IntExp.t_c##:
###\begin{aligned} FFCF_\text{wITS} &= NOPAT + Depr - CapEx -\Delta NWC + IntExp.t_c \\ &= (Rev - COGS - Depr - FC)(1-t_c) + Depr - CapEx -\Delta NWC + IntExp.t_c \\ \end{aligned} \\###Note that net operating profit after tax equals EBIT after tax, ##NOPAT = (EBIT)(1-t_c)##, if there are no non-operating items such as shut-down business segments.
Note that the firm free cash flow excluding interest tax shields ##(FFCF_\text{xITS})## should be discounted by the WACC after tax ##(r_\text{WACC after tax})##. This will give the value of the levered firm or project with interest tax shields ##(V_\text{wITS})##. The weighted average cost of capital after tax is:
###r_\text{WACC after tax} = \dfrac{D}{V}.r_D.(1-t_c) + \dfrac{E}{V}.r_E###There are many ways to calculate a firm's free cash flow (FFCF), also called cash flow from assets (CFFA). Some include the annual interest tax shield in the cash flow and some do not.
Which of the below FFCF formulas include the interest tax shield in the cash flow?
###(1) \quad FFCF=NI + Depr - CapEx -ΔNWC + IntExp### ###(2) \quad FFCF=NI + Depr - CapEx -ΔNWC + IntExp.(1-t_c)### ###(3) \quad FFCF=EBIT.(1-t_c )+ Depr- CapEx -ΔNWC+IntExp.t_c### ###(4) \quad FFCF=EBIT.(1-t_c) + Depr- CapEx -ΔNWC### ###(5) \quad FFCF=EBITDA.(1-t_c )+Depr.t_c- CapEx -ΔNWC+IntExp.t_c### ###(6) \quad FFCF=EBITDA.(1-t_c )+Depr.t_c- CapEx -ΔNWC### ###(7) \quad FFCF=EBIT-Tax + Depr - CapEx -ΔNWC### ###(8) \quad FFCF=EBIT-Tax + Depr - CapEx -ΔNWC-IntExp.t_c### ###(9) \quad FFCF=EBITDA-Tax - CapEx -ΔNWC### ###(10) \quad FFCF=EBITDA-Tax - CapEx -ΔNWC-IntExp.t_c###The formulas for net income (NI also called earnings), EBIT and EBITDA are given below. Assume that depreciation and amortisation are both represented by 'Depr' and that 'FC' represents fixed costs such as rent.
###NI=(Rev - COGS - Depr - FC - IntExp).(1-t_c )### ###EBIT=Rev - COGS - FC-Depr### ###EBITDA=Rev - COGS - FC### ###Tax =(Rev - COGS - Depr - FC - IntExp).t_c= \dfrac{NI.t_c}{1-t_c}###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}###To value a business's assets, the free cash flow of the firm (FCFF, also called CFFA) needs to be calculated. This requires figures from the firm's income statement and balance sheet. For what figures is the balance sheet needed? Note that the balance sheet is sometimes also called the statement of financial position.
Current assets and current liabilities are needed to find the change in net working capital, an input into the firm's free cash flow equation. These two quantities are found in the balance sheet.
Capital expenditure is needed in the firm's free cash flow equation, and the CapEx can be found using the balance sheet's change in net fixed assets (usually property, plant and equipment (PPE)) plus depreciation. Net fixed assets is found in the balance sheet.
Thanks to Shahzada for assisting with this question.
A firm has a debt-to-assets ratio of 50%. The firm then issues a large amount of equity to raise money for new projects of similar systematic risk to the company's existing projects. Assume a classical tax system. Which statement is correct?
Because the company's new projects are of similar systematic risk to the company's existing projects, the systematic risk of the assets will be constant. Therefore the WACC before tax, also called the required return on assets or the cost of capital, which takes only the time value of money and systematic risk into account, should remain the same, so answer (e) is correct.
Since the company is issuing equity, the leverage ratios will fall, not rise, so (a) and (b) are incorrect.
Because no more debt is being issued, the value of the firm's interest tax shields will remain constant. Therefore the value of the firm will not increase due to interest tax shields, so answer (c) is incorrect. However, the value of the firm will increase if the new projects are positive NPV.
The firm's after-tax WACC is likely to increase since there will be proportionally less interest tax shields because equity is increasing but debt is constant. So there will be a lower weight on the after-tax cost of debt ##\left(r_D.(1-t_c)\right)##. Therefore answer (d) is incorrect.
Note that the WACC's before and after tax take the time value of money and the systematic risk of the project into account. But the WACC after tax also takes interest tax shields into account as well. This is why the after-tax WACC rises when the proportion of equity funding rises because the proportion (or weight) of debt funding and thus interest tax shields falls.
Therefore the higher the weight of debt ##(D/V_L)##,
- The lower the after-tax WACC.
- No effect on the before-tax WACC, it is independent of capital structure.
A firm is considering a new project of similar risk to the current risk of the firm. This project will expand its existing business. The cash flows of the project have been calculated assuming that there is no interest expense. In other words, the cash flows assume that the project is all-equity financed.
In fact the firm has a target debt-to-equity ratio of 1, so the project will be financed with 50% debt and 50% equity. To find the levered value of the firm's assets, what discount rate should be applied to the project's unlevered cash flows? Assume a classical tax system.
In this case the after tax WACC is the correct answer. Let's look at the pre tax WACC first to examine why.
The pre tax WACC is very useful since it accounts for the time value of money and systematic risk. To find the levered value of a firm's project, levered cash flows are discounted by the pre tax WACC, so the value of the interest tax shield is included in the cash flow. In this question, we are given the unlevered cash flows, the cash flows with no interest expense and therefore no debt. If these unlevered cash flows are discounted by the pre tax WACC, the value of the firm's project will be undervalued since it will exclude the value of interest tax shields.
The after tax WACC is exactly the same as the pre tax WACC, except it also reduces the cost of debt by the amount of the interest tax shield. It uses the after-tax cost of debt (##r_D(1-t_c)##) rather than the pre-tax cost of debt (##r_D##). To find the levered value of the firm's project, unlevered cash flows are discounted by the after tax WACC. In this case the value of the interest tax shield will be included in the discount rate, the after tax WACC. This will value the levered firm's project including the value of the interest tax shields, giving an accurate valuation.