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

The lady's hard work studying and attending school during her childhood is a sunk cost since there's nothing she can do to get that time back. It's in the past, it's spent, she should forget about it. Her decision to attend university should ignore this sunk cost because regardless of her decision, this cost can't be recouped. Only the incremental benefits and costs should be included in her decision. Economists were the first to discover the importance of marginal changes, and the period is known as the Marginalist Revolution.

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

Since most of the cash flows are weekly, it's easier to work with effective weekly rates than annual ones so convert the effective annual rate to an effective weekly rate:

###\begin{aligned} r_\text{eff wkly} &= (1+r_\text{eff yrly})^{1/52}-1 \\ &= (1+0.098)^{1/52}-1 \\ &= 0.001799508 \\ &\approx 0.0018 \\ \end{aligned} ###

Since all discount rates and cash flows are real, there is no need to do any conversions using inflation.

University fees will be ##4 \times $1,277 = $5,108## per semester, paid at t=0 for the first semester and again at t=19 weeks for the second semester.
The present value of university fees for one year is:

###\begin{aligned} V_\text{0, annual fee} &= (\text{First semester cost now}) + \frac{(\text{Second semester cost in 19 weeks})}{(1+0.001799508)^{19}} \\ &= 4 \times 1,277 + \frac{4 \times 1,277}{(1+0.001799508)^{19}} \\ &= 10,044.45768 \\ \end{aligned} ###

But as well as this explicit annual cost there is also the implicit opportunity cost which is that students can't work full-time while they are studying full-time. Students can still work during the summer holidays, but they can't work from t=0 to 38 weeks.
At $20/hr and 35hrs/wk they miss out on $700/wk paid in arrears. The present value of this annual opportunity cost is:

###\begin{aligned} V_\text{0, annual wages foregone} &= \frac{C_\text{wage 1,2,..38}}{r} \left(1-\frac{1}{(1+r)^{38}} \right) \\ &= \frac{700}{0.001799508} \left(1-\frac{1}{(1+0.001799508)^{38}} \right) \\ &= 25,688.58368 \\ \end{aligned} ###

Adding up the present value of the annual explicit and implicit costs:

###\begin{aligned} V_\text{0, annual cost} &= V_\text{0, annual fee} + V_\text{0, annual wages foregone} \\ &= 10,044.45768 + 25,688.58368 \\ &= 35,733.04136 \\ \end{aligned} ###

The annual costs are expected to be constant every year, so the present value of the costs over the whole 3 year degree is:

###\begin{aligned} V_\text{0, 3yr cost} &= \frac{V_\text{0,1,2, annual cost}}{r_\text{eff yrly}} \left(1-\frac{1}{(1+r_\text{eff yrly})^{3}} \right)(1+r_\text{eff yrly})^1 \\ &= \frac{35,733.04136}{0.098} \left(1-\frac{1}{(1+0.098)^{3}} \right)(1+0.098)^1 \\ &= $97,915.91465 \\ \end{aligned} ###

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

Using the Cash Flow From Assets Equation,

### CFFA = NI + Depr - CapEx - \Delta NWC + IntExp ###

Capital expenditure (CapEx) can be calculated as the change in Net Fixed Assets (NFA) plus depreciation. Note that NFA is the same thing as the carrying amount of property, plant and equipment (PPE).

###\begin{aligned} CapEx &= PPE_\text{now} - PPE_\text{before} + Depr \\ &= 700-680+34 \\ &= 54 \\ \end{aligned}###

CapEx is positive, so the firm must have bought more capital assets than it sold.

To find the change in net working capital (##\Delta NWC##), take the difference between the NWC now and before. Note that current assets includes inventory, trade debtors and rent paid in advance. Current liabilities only includes trade creditors in this instance.

###\begin{aligned} \Delta NWC &= CA_\text{now} - CL_\text{now} - (CA_\text{before} - CL_\text{before}) \\ &= (70+11+4-11) - (50+16+3-19) \\ &= 74 - 50 \\ &= 24 \\ \end{aligned}###

Now just substitute the values:

###\begin{aligned} CFFA &= NI + Depr - CapEx - \Delta NWC + IntExp \\ &= 147 + 34 - 54 - 24 + 39 \\ &= 142 \\ \end{aligned}###

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

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.

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

Buying less capital assets (non-current assets) such as land, buildings and trucks will decrease CapEx and increase CFFA.

###CFFA=NI+Depr-CapEx - \varDelta NWC+IntExp###

There will be less depreciation and therefore a lower depreciation tax shield, causing a decrease in CFFA, but this is likely to be a small effect compared to the fall in CapEx.

###\begin{aligned} CFFA &= NI+Depr-CapEx - \varDelta NWC+IntExp \\ &= (Rev-COGS-FC-Depr-IntExp).(1-t_c)+Depr-CapEx - \varDelta NWC+IntExp \\ &= (Rev-COGS-FC).(1-t_c)+\mathbf{Depr.t_c} -CapEx - \varDelta NWC+IntExp.t_c \\ \end{aligned}###

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

Firm free cash flow (FFCF) should equal net payments to debt and equity holders.

Since the firm is unlevered, there are no cash flows to or from debt holders which makes it easier.

Note that the capital expenditure of $0.8m is a red herring since CapEx is already subtracted in FFCF, so the CapEx can be ignored.

###\begin{aligned} FFCF &= (\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}) \\ 1m &= 0 - 0 + (1.8m + 1.3m) - (\text{new equity raisings}) \\ \end{aligned} ### ###\begin{aligned} (\text{new equity raisings}) &= 0 - 0 + (1.8m + 1.3m) - 1m \\ &= 2.1m \\ \end{aligned} ###

Commentary regarding the retained profits

Since "the firm has sufficient retained profits to pay the dividend and complete the buy back", why does the firm need to raise money through equity financing?

Remember that all things on the liabilities (L) and owners' equity (OE) side of the balance sheet (L+OE), such as retained earnings, are just records of how the assets (A=L+OE) were funded.

These liabilities and owners' equity are not physical things such as cash on the assets side of the balance sheet.

So a firm can have lots of retained profits but no cash to pay things such as dividends. For example, the assets might be all property, plant and equipment (PPE) rather than cash.

Also, profit (on the income statement or P&L) does not necessarily correspond with cash flows. For example, depreciation (and often interest expense) are accruals that are non-cash items. So high profits doesn't necessarily mean high cash flow.

The note only mentions the sufficient retained earnings to afford the dividend and buyback due to the constraint on companies that most legal systems disallow firms from paying out equity through dividends and buybacks unless they have current or retained profits. This is designed to stop Ponzi schemes.

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

Equity free cash flow (EFCF) should equal net payments to equity holders (which is also equal to firm free cash flow (FFCF) less net payments to debt holders).

The sales and net income are irrelevant in this question. They are already included in the EFCF figure.

###\begin{aligned} EFCF &= FFCF - (\text{net payments to debt holders}) \\ &= (\text{net payments to equity holders}) \\ &= (\text{payments to equity holders}) - (\text{receipts from equity holders}) \\ &= (\text{dividend and buyback payments}) - (\text{new equity raisings}) \\ 2m &= (1m + 1.3m) - (\text{new equity raisings}) \\ \end{aligned} ### ###\begin{aligned} (\text{new equity raisings}) &= (1m + 1.3m) - 2m \\ &= 0.3m \\ \end{aligned} ###

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

Initially at time zero, no time has elapsed so there is zero revenue, expenses, income, depreciation and so on. But there is an initial change in net working capital due to the inventory purchase.

###\begin{aligned} \Delta NWC_0 &= \Delta NWC_\text{now} - \Delta NWC_\text{before} \\ &= CA_\text{now} - CL_\text{now} - (CA_\text{before} - CL_\text{before}) \\ &= 0.8m - 0 - (0 - 0) \\ &= 0.8m \\ \end{aligned}###

There is CapEx, which is the $6m cost of the equipment. Using the Cash Flow From Assets equation:

###\begin{aligned} CFFA_0 &= NI_0 + Depr_0 - CapEx_0 - \Delta NWC_0 + IntExp_0 \\ &= 0 + 0 - 6m - 0.8m + 0 \\ &= -6.8m \\ \end{aligned}###

At time one the Net Income, CapEx and change in NWC must be calculated and then substituted into the CFFA equation.

###\begin{aligned} \Delta NWC_1 &= CA_\text{now} - CL_\text{now} - (CA_\text{before} - CL_\text{before}) \\ &= 0.8m - 0 - (0.8m - 0) \\ &= 0 \\ \end{aligned}### ###\begin{aligned} NI_1 &= (Rev_1 - COGS_1 - FC_1 - Depr_1 - IntExp_1)(1-t_c) \\ &= (4m \times 8 - 4m \times 3 - 1.5m - 1m)(1-0.3) \\ &= 12.25m \\ \end{aligned}###

Note that CapEx is zero since no equipment was bought or sold.

###\begin{aligned} CFFA_1 &= NI_1 + Depr_1 - CapEx_1 - \Delta NWC_1 + IntExp_1 \\ &= 12.25m + 1m - 0 - 0 + 0 \\ &= 13.25m \\ \end{aligned}### For the final cash flow at time 2, the Net Income will be the same but the CapEx and change in NWC must be calculated. ###\begin{aligned} \Delta NWC_2 &= CA_\text{now} - CL_\text{now} - (CA_\text{before} - CL_\text{before}) \\ &= 0 - 0 - (0.8m - 0) \\ &= -0.8m \\ \end{aligned}### ###\begin{aligned} CapEx_2 &= -(P_\text{mkt} - \text{CapitalGainsTax}) \\ &= -(P_\text{mkt} - (P_\text{mkt} - P_\text{book}).t_c) \\ &= -(0.9m - (0.9m - 4m) \times 0.3) \\ &= -1.83m \\ \end{aligned}###

Note that change in NWC and CapEx are both negative, so net cash was gained rather than spent from selling the inventory (NWC) and selling the equipment (CapEx).

###\begin{aligned} CFFA_2 &= NI_2 + Depr_2 - CapEx_2 - \Delta NWC_2 + IntExp_2 \\ &= 12.25m + 1m - (-1.83m) - (-0.8m) + 0 \\ &= 15.88m \\ \end{aligned}###

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

Since all cash flows are real and the and the discount rate is real, there is no need to convert rates or cash flows which is a relief.

About note 1. The firm's current assets and liabilities are irrelevant for evaluating our project since they will not change if we go ahead with the project or not. They are not incremental cash flows. But the increase in current assets and liabilities are incremental cash flows and will lead to a change in net working capital. Here is the general equation for calculating the change (or increase) in net working capital:

###\begin{aligned} \varDelta NWC &= NWC_\text{now} - NWC_\text{before} \\ &= (CA_\text{now} - CL_\text{now}) - (CA_\text{before} - CL_\text{before}) \\ &= \varDelta CA - \varDelta CL \\ \end{aligned}###

The increases in NWC at each time will be:

###\begin{aligned} \varDelta NWC_0 &= \varDelta CA - \varDelta CL \\ &= 2m - 0 \\ &= 2m \\ \varDelta NWC_\text{1} &= \varDelta CA - \varDelta CL \\ &= 0.2m - 0.1m \\ &= 0.1m \\ \varDelta NWC_\text{2} &= \varDelta CA - \varDelta CL \\ &= (-2m-0.2m) - (-0.1m) \\ &= -2.1m \\ \end{aligned}###

Note that ##\varDelta NWC_\text{2}## is negative because at the end of the project all of the working capital (which is probably mostly inventory) is sold. This is stated at the end of note 1. The lower inventory means current assets falls, which is a negative change.

About note 2. The research and development cost is a sunk cost so it should be ignored. It can not be recovered and it has to be paid whether we go ahead with the project or not, so it's irrelevant to our decision and evaluation.

When the equipment is sold at t=2 for $0.6m there will be negative capital expenditure (CapEx). Complicating matters is the capital gains tax (CGT) effect since the book carrying value (original cost less accumulated depreciation) of the equipment will be 0, so there will be a capital gain of $0.6m on which we have to pay CGT. Since the 50% CGT discount applies we can reduce the corporate tax rate so the CGT will be less. The capital expenditure will be:

###\begin{aligned} CapEx &= -(P_\text{mkt} - CGT) \\ &= -(P_\text{mkt} - (P_\text{mkt}-P_\text{book}).(1-\text{CGTDiscount}).t_c) \\ &=-(0.6m - (0.6m-0) \times(1-0.5) \times 0.3) \\ &=-0.51m \\ \end{aligned}###

Note that selling a capital asset is a negative capital expenditure (CapEx), that's why the figure is negative (-0.51m).

To find the Net Income (NI) which will be paid at the end of each year (t=1 and 2),

###\begin{aligned} NI &= (Rev-COGS-FC-Depr-IntExp).(1-t_c) \\ &= (Q(P-VC)-FC-Depr-IntExp).(1-t_c) \\ &= (4m(8-5)-1m-3m-0).(1-0.3) \\ &= 5.6m \end{aligned}###

To find the CFFA at each time period.

###\begin{aligned} CFFA_0 &= NI+Depr-CapEx - \varDelta NWC+IntExp \\ &= 0 +0 -6m - 2m +0 \\ &= -8m \\ CFFA_1 &= NI+Depr-CapEx - \varDelta NWC+IntExp \\ &= 5.6m +3m -0 - 0.1m +0 \\ &= 8.5m \\ CFFA_2 &= NI+Depr-CapEx - \varDelta NWC+IntExp \\ &= 5.6m +3m -(-0.51m) - (-2.1m) +0 \\ &= 11.21m \\ \end{aligned}###

The project value is the present value of the CFFA.

###\begin{aligned} V_\text{0, project} &= CFFA_0 + \frac{CFFA_1}{(1+r)^1} + \frac{CFFA_2}{(1+r)^2} \\ &= -8m + \frac{8.5m}{(1+0.1)^1} + \frac{11.21m}{(1+0.1)^2} \\ &= -8m + 7.72727273m + 9.26446281m \\ &= 8.99173553m \\ \end{aligned}###

Since the net present value is positive, the project should be accepted. The project will increase the value of the firm's assets by $8.99m.

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

The higher chance of meeting a clever spouse is a positive side effect that should be included in the decision to go to university. It's not a sunk cost since the chance of meeting a clever young man depends on her decision to go to university or not. Also, it's not an opportunity cost since it's actually a benefit from going to university.

It's funny how the word 'opportunity' in the last sentence of the question makes people sub-consciously think opportunity cost, even though it is clearly not a cost.

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

The hours that he could work in the future are an opportunity cost of his decision to take a day off and relax, assuming that working is the 'next best alternative' to relaxing. The hours that he could spend working are not a sunk cost yet because they are in the future, they are an incremental cost of staying home to relax.

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

The hours that he took off from work are now a sunk cost because they are in the past and there is nothing that he can do to get them back, he should forget about them. As the saying goes, "don't cry over spilt milk".

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

Using the Cash Flow From Assets equation,

### CFFA = NI + Depr - CapEx - \Delta NWC + IntExp ###

Capital expenditure (CapEx) can be calculated as the change in Net Fixed Assets (NFA) plus depreciation. Note that NFA is the same thing as the carrying amount of property, plant and equipment (PPE).

###\begin{aligned} CapEx &= NFA_\text{now} - NFA_\text{before} + Depr \\ &= 240 - 300 + 20 \\ &= -40 \\ \end{aligned}###

Note that CapEx is negative. This means that the firm must have sold more capital assets than it bought.

Another way to calculate CapEx is to look at the difference in the undepreciated cost:

###\begin{aligned} CapEx &= FA_\text{now} - FA_\text{before} \\ &= 300 - 340 \\ &= -40 \\ \end{aligned}###

To find the change in net working capital (##\Delta NWC##), take the difference between the NWC now and before:

###\begin{aligned} \Delta NWC &= CA_\text{now} - CL_\text{now} - (CA_\text{before} - CL_\text{before}) \\ &= 220-175 - (180-190) \\ &= 45 - (-10) \\ &= 55 \\ \end{aligned}###

Now just substitute the values:

###\begin{aligned} CFFA &= NI + Depr - CapEx - \Delta NWC + IntExp \\ &= 77 + 20 - (-40) - (55) + 10 \\ &= 77 + 20 + 40 - 55 + 10 \\ &= 92 \\ \end{aligned}###

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

Using the Cash Flow From Assets Equation,

### CFFA = NI + Depr - CapEx - \Delta NWC + IntExp ###

Capital expenditure (CapEx) can be calculated as the change in Net Fixed Assets (NFA) plus depreciation. Note that NFA is the same thing as the carrying amount of property, plant and equipment (PPE).

###\begin{aligned} CapEx &= PPE_\text{now} - PPE_\text{before} + Depr \\ &= 400 - 400 + 20 \\ &= 20 \\ \end{aligned}###

CapEx is positive, so the firm must have bought more capital assets than it sold.

To find the change in net working capital (##\Delta NWC##), take the difference between the NWC now and before. Note that current assets includes inventory, trade debtors and rent paid in advance. Current liabilities only includes trade creditors in this instance.

###\begin{aligned} \Delta NWC &= CA_\text{now} - CL_\text{now} - (CA_\text{before} - CL_\text{before}) \\ &= (49+14+5-4) - (38+2+5-10) \\ &= 64 - 35 \\ &= 29 \\ \end{aligned}###

Now just substitute the values:

###\begin{aligned} CFFA &= NI + Depr - CapEx - \Delta NWC + IntExp \\ &= 21 + 20 - 20 - 29 + 19 \\ &= 11\\ \end{aligned}###

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

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.

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

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} ###

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

The net income (NI) equation and cash flow from assets (CFFA) equations are:

###NI=(Rev-COGS-FC-\mathbf{Depr}-IntExp).(1-t_c)###

###CFFA=NI+\mathbf{Depr}-CapEx - \varDelta NWC+IntExp###

Substituting NI into CFFA and then expanding and collecting like terms,

###\begin{aligned} CFFA &= (Rev-COGS-FC-\mathbf{Depr}-IntExp).(1-t_c)+\mathbf{Depr}-CapEx - \varDelta NWC+IntExp \\ &= (Rev-COGS-FC).(1-t_c)+\mathbf{Depr.t_c} -CapEx - \varDelta NWC+IntExp.t_c \\ \end{aligned}###

The bold term is called the depreciation tax shield (##Depr.t_c##) which is the tax saving per year. This is because higher depreciation results in lower tax payments to the government since before-tax NI is lower. Depreciation is added to CFFA since depreciation is not a cash flow, it's an accrual item made up by accountants who attempt to allocate the lumpy capital expenditure (CapEx) smoothly through time. But CFFA is only supposed to include cash items, so depreciation is added back. The only effect of depreciation on CFFA is the depreciation tax-shield effect.

It's quite surprising that higherdepreciation expense actually leads to a CFFA increase and at the same time a NI decrease. It shows how CFFA (or better, the net present value of CFFA) can give quite a different picture of firm value compared with NI.

Note that the interest tax shield (##IntExp.t_c##) is also shown in the above equation. It's the tax saving per year from paying less tax to the government due to interest expense.

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

An increase in inventories will have no effect on net income because accountants only expense inventory when it is sold. In other words, buying a billion dollars of inventory and leaving it in a warehouse all year will have no impact on net income. The billion dollar cost of the inventory will only affect net income through the account 'cost of goods sold' (COGS) when the inventory is sold, or when the inventory is written down if it becomes old, out-dated or out of fashion and the accountant decides to re-value it.

Obviously any purchase of inventory will have a negative effect on cash flow, and the opportunity cost of having money tied up in inventory rather than in the bank is a cost that must be taken into account. This is why CFFA subtracts the increase in net working capital ##ΔNWC##, which is the increase in current assets less the increase in current liabilities.

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

Initially at time zero, no time has elapsed so there is zero revenue, expenses, income, depreciation and so on. But there is CapEx, which is the $9m cost of building the mine as well as the $3m opportunity cost of foregoing the eco-resort. Using the Cash Flow From Assets equation:

###\begin{aligned} CFFA_0 &= NI_0 + Depr_0 - CapEx_0 - \Delta NWC_0 + IntExp_0 \\ &= 0 + 0 - (9m+3m) - 0 + 0 \\ &= -12m \\ \end{aligned}###

At time one the Net Income and CapEx must be calculated and then substituted into the CFFA equation. Note that revenue at time one includes the 1,000 kilograms of gold as well as the $1m of diamonds mentioned in note 1 which is a side benefit.

###\begin{aligned} NI_1 &= (Rev_1 - COGS_1 - FC_1 - Depr_1 - IntExp_1)(1-t_c) \\ &= ((1,000 \times 0.05m + 1m) - (1,000 \times 0.03m) - 4m - 8m - 0)(1-0.3) \\ &= (51m - 30m - 4m - 8m - 0)(1-0.3) \\ &= 9m \times 0.7 \\ &= 6.3m \\ \end{aligned}### ###\begin{aligned} CapEx_1 &= -(P_\text{mkt} - \text{CapitalGainsTax}) \\ &= -(P_\text{mkt} - (P_\text{mkt} - P_\text{book}).t_c) \\ &= -(2.5m - (2.5m - 1m) \times 0.3) \\ &= -2.05m \\ \end{aligned}###

Note that CapEx is actually negative, so net cash was gained not spent.

###\begin{aligned} CFFA_1 &= NI_1 + Depr_1 - CapEx_1 - \Delta NWC_1 + IntExp_1 \\ &= 6.3m + 8m - (-2.05m) - 0 + 0 \\ &= 16.35m \\ \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{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}###