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.

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:

• www.accountingcoach.com
• www.principlesofaccounting.com

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

Accountants use the 'effective interest method' to calculate interest expense which is the yield on the debt multiplied by its book value at the beginning of the period, with accrual adjustments if the debt matures during the year. Mathematically this is:

###IntExp_1 = r_D.D_0###

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.

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

###NI=(Rev-COGS-FC-Depr-IntExp).(1-t_c)###

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

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

###\begin{aligned} CFFA &= (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+\mathbf{IntExp.t_c} \\ \end{aligned}###

The last bold term is called the interest tax shield (##IntExp.t_c##) which is the tax saving per year. Clearly, an increase in interest expense (IntExp) will lead to an increase in CFFA. This is because higher IntExp results in lower tax payments to the government since before-tax NI is lower. But IntExp does not affect CFFA since IntExp is a funding cost and has nothing to do with the firm's assets themselves, similarly for dividends.

It's quite surprising that higher interest 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 depreciation tax shield (##Depr.t_c##) is also shown in bold in the above equation. It's the tax saving per year from paying less tax to the government due to depreciation.

With regards to answers a and e, cash flows to creditors and equity holders equal CFFA, so if dividends or interest payments fall then CFFA must have fallen. Note that interest payments are not necessarily equal to interest expense, for example a zero coupon bond has no interest payments until maturity yet it has interest expense every year.

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.

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.

Tax accountants apply the 'effective interest method' as below.

First find the effective annual yield of the zero coupon bond (##r_\text{eff, yearly}##). Note that the price is simply the present value of the principal since there are no coupons:

###\begin{aligned} P_0 &= \frac{F_T}{(1 + r_{\text{eff yrly, 0}\rightarrow T})^{T_\text{years}}} \\ &= \frac{F_\text{0.5yr}}{(1 + r_{\text{eff yrly, 0}\rightarrow 0.5yr})^{0.5}} \\ 950 &= \frac{1,000}{(1 + r_{\text{eff yrly, 0}\rightarrow 0.5yr})^{0.5}} \\ \end{aligned} ###

###\begin{aligned} r_{\text{eff yrly, 0}\rightarrow 0.5yr} &= \left( \frac{1,000}{950} \right)^{1 / 0.5} -1 \\ &=0.108033241 \\ \end{aligned} ###

Multiply the book value of the bond at the start of the period by the yield just calculated to find the annual interest expense. But since this bond matures in 6 months, only half of the interest expense will be 'accrued'. Note that tax accountants use 'accrual methods' to allocate the interest expense. They don't find present or future values, they would just multiply the annual amount by half to get 6 months' worth of interest expense:

###\begin{aligned} \text{IntExp}_\text{6mths} &= \text{IntExp}_\text{annual}.\frac{1}{2} \\ &= D_\text{book value}.r_\text{eff, yearly}.\frac{1}{2} \\ &= 950 \times 0.108033241 \times \frac{1}{2} \\ &=102.6315789 \times \frac{1}{2} \\ &= 51.31578947 \\ \end{aligned} ###

Note that this version of interest expense is only used for accounting purposes. It is not the true 'cost of debt' that a finance person would calculate. But it is important to know how accountants find this amount since it is relevant for the calculation of tax which is a cash flow, not an imaginary accrual item like interest expense or depreciation.

Commentary

The 'finance version' of the true cost of debt in dollar terms in 6 months when the bond matures is simply the increase in the value of the debt liability. Since it's a zero-coupon bond there's no coupons and only the principal to pay, and in 6 months the bond matures and has a value of the principal, so the dollar cost of debt in 6 months is $50 (=1,000 - 950).

Mathematically, this is the future value of the bond price and coupons less then current bond price. This is also equal to the total return of the bond multiplied by the current price. Remembering that the coupon (##c_\text{0.5yrs}##) is zero:

###\begin{aligned} V_\text{0.5yrs, cost of debt} &= p_0.r_{0 \rightarrow \text{0.5yrs}} \\ &= p_0.\frac{(p_\text{0.5yrs} - p_0 + c_\text{0.5yrs})}{p_0} \\ &= p_\text{0.5yrs} - p_0 + c_\text{0.5yrs} \\ &= 1,000 - 950 + 0 \\ &= 50 \\ \end{aligned} ###

To make it a value in one year which is when the business prepares its accounts, find the future value in another 6 months. Assume that the yield curve is flat, so the forward rate applying from 6 months to one year is the same as the spot 6 month rate which is the total return of the bond, so :

### r_{\text{eff yrly, 0}\rightarrow 0.5yr} = r_{\text{eff yrly, 0.5yr}\rightarrow 1yr} = 0.108033241 ###

###\begin{aligned} V_\text{1yr, cost of debt} &= V_\text{0.5yrs, cost of debt}(1+r_{\text{eff yrly, 0.5yr}\rightarrow 1yr})^{0.5} \\ &= 50(1+0.108033241)^{0.5} \\ &= 52.63157895 \\ \end{aligned} ###

The difference between the accounting and finance versions of the cost of debt in this example is quite small and is all due to accrual differences stemming from how the bond matures half way through the year. If the bond matured in more than one year then both the accounting and finance versions would be the same.

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 \\ &= 325 - 365 + 40 \\ &= 0 \\ \end{aligned}###

Since CapEx is zero, the firm either didn't spend or sell any of its capital assets, or it sold the exact same amount of assets that it bought or upgraded.

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

###\begin{aligned} CapEx &= GFA_\text{now} - GFA_\text{before} \\ &= 400 - 400 \\ &= 0 \\ \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}) \\ &= 200-150 - (230-205) \\ &= 50 - (25) \\ &= 25 \\ \end{aligned}###

Now just substitute the values:

###\begin{aligned} CFFA &= NI + Depr - CapEx - \Delta NWC + IntExp \\ &= 35 + 40- 0 - 25 + 10 \\ &= 60 \\ \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 &= NFA_\text{now} - NFA_\text{before} + Depr \\ &= 90 - 100+ 20 \\ &= 10 \\ \end{aligned}###

Since CapEx is positive, the firm must have spent more on capital assets than it sold.

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

###\begin{aligned} CapEx &= GFA_\text{now} - GFA_\text{before} \\ &= 150- 140 \\ &= 10 \\ \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}) \\ &= 120-75 - (80-65) \\ &= 45 - (15) \\ &= 30 \\ \end{aligned}###

Now just substitute the values:

###\begin{aligned} CFFA &= NI + Depr - CapEx - \Delta NWC + IntExp \\ &= 21+ 20 -10 - 30 + 20 \\ &= 21 \\ \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 &= NFA_\text{now} - NFA_\text{before} + Depr \\ &= 320- 290+ 10 \\ &= 40 \\ \end{aligned}###

Since CapEx is positive, the firm must have spent more on capital assets than it sold.

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

###\begin{aligned} CapEx &= GFA_\text{now} - GFA_\text{before} \\ &= 360- 320\\ &= 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}) \\ &= 120-110 - (90-60) \\ &= 10 - (30) \\ &= -20 \\ \end{aligned}###

Now just substitute the values:

###\begin{aligned} CFFA &= NI + Depr - CapEx - \Delta NWC + IntExp \\ &= 7+ 10 -40 - (-20) + 5 \\ &= 2 \\ \end{aligned}###