###### Option Greeks

The Greeks are a group of mathematical derivatives applied to help manage or... **Read More**

An expansion project is a capital project that involves a company increasing its business size. Expansion projects are independent projects because they do not affect the cash flows of the rest of the company. Cash flows of expansion projects can be categorized as follows:

- Initial investment outlay
- After-tax operating cash flows over the project’s life
- Terminal year after-tax non-operating cash flows

This is the cost required to start a given project.

$$\text{Outlay}=\text{FCInv}+\text{NWCInc}$$

Where:

\(\text{FCInv} =\) investment in new fixed capital

\(\text{NWCInv} =\) investment in net working capital

\(\text{NWCInv} =\Delta\text{non-cash current assets}-\Delta\text{non-debt current liabilities}\)

A positive \(\text{NWCInv}\) shows cash outflow while a negative represents cash inflow.

\(\text{CF} = (\text{S} – \text{C} – \text{D}) (1 – \text{T}) + \text{D}\) or

\(\text{CF} = (\text{S} – \text{C}) (1 – \text{T}) + \text{TD}\)

Where:

\(S =\) sales

\(C =\) cash operating expenses

\(D =\) depreciation charge

\(T =\) Marginal tax rate

$$\text{TNOCF}=\text{Sal}_{T}+\text{NWCInv}-\text{T}(\text{Sal}_{T}-\text{B}_{T})$$

Where:

\(\text{Sal}_{T} =\) cash proceeds from sale of fixed capital on *termination date*.

\(\text{B}_{T} =\) book value of fixed capital on termination date.

Consider the following information relating to an expansion project:

- Initial investment outlay = $300,000 of which:
- Non-depreciable land = $37,500
- Equipment = $262,500

- Net working capital = $40,000
- Annual sales = $320,000
- Annual cash operating expenses = $100,000
- Income tax = 40%
- Project life = 5 years
- Required rate of return = 10%
- Equipment will be depreciated using the straight line to zero method over the project life.
- Assume that the fixed capital assets will be sold off at the end of year 5 at $75,000.

The cashflows for expansion projects can be organized in various ways:

- Table format with cash flows collected by year.
- Table format with cash flows collected by type.
- Using equations given by the formulas above.

$$ \textbf{Table 1: Table Format with Cash Flows Collected by Year} $$

$$\small{\begin{array}{l|c|c|c|c|c} \textbf{Year} & \textbf{0} & \textbf{1} & \textbf{2} & \textbf{3} & \textbf{4} & \textbf{5} \\ \hline\textbf{Investment outlays:} & & & & & & \\ \hline\text{Fixed capital} & -300,000 & & & & & \\ \hline\text{Net Working Capital} & -40,000 & & & & & \\ \hline\text{Total} & -340,000 & & & & & \\ \hline\textbf{After-tax operating cash flows:} & & & & & & \\ \hline\text{Sales} & & 320,000 & 320,000 & 320,000 & 320,000 & 320,000 \\ \hline\text{Cash operating expenses} & & 100,000 & 100,000 & 100,000 & 100,000 & 100,000 \\ \hline\text{Depreciation}^{1} & & 52,500 & 52,500 & 52,500 & 52,500 & 52,500 \\ \hline\textbf{Operating income before tax} & & \textbf{167,500} & \textbf{167,500} & \textbf{167,500} & \textbf{167,500} & \textbf{167,500} \\ \hline\text{Taxes on Operating income (40%)} & & 67,000 & 67,000 & 67,000 & 67,000 & 67,000 \\ \hline\textbf{Operating income after tax} & & \textbf{100,500} & \textbf{100,500} & \textbf{100,500} & \textbf{100,500} & \textbf{100,500}\\ \hline\text{Add back depreciation} & & 52,500 & 52,500 & 52,500 & 52,500 & 52,500 \\ \hline\textbf{After tax operating cash flow} & & \textbf{153,000} & \textbf{153,000} & \textbf{153,000} & \textbf{153,000} & \textbf{153,000}\\ \hline\textbf{Terminal year after tax operating cash flows:} & & & & & & \\ \hline\text{After tax salvage value}^{2} & & & & & & 60,000 \\ \hline\text{Return of net working capital} & & & & & & 40,000 \\ \hline\textbf{Total} & & & & & & \textbf{100,000} \\ \hline\text{Total after tax cash flows} & -340,000 & 153,000 & 153,000 & 153,000 & 153,000 & 253,000 \\ \hline\text{NPV}^{3} & 302,083 & & & & & \\ \hline\text{IRR}^{4} & 38.35\% & & & & &\\ \end{array}}$$

$$\text{Depreciation}=\bigg(\frac{1}{5}\times$262,500\bigg)=$52,500$$

$$\text{Gain after selling off fixed capital assets} = $75,000 – $37,500 = $37,500$$

$$\text{Tax on gains} (40\%\times$37,500)=$15,000$$

$$\text{After-tax value}= $75,000- $15,000 = $60,000$$

NPV is the present value of the future after-tax cash flows less the initial outlay.

$$\text{NPV}=\frac{153,000}{1.10}+\frac{153,000}{1.10^{2}}+\frac{153,000}{1.10^{3}}+\frac{153,000}{1.10^{4}}+\frac{153,000}{1.10^{5}}-340,000=$302,083$$

NPV > 0 implies that the expansion project adds value, thus, we should invest in this project.

IRR is the rate of return that results in \(\text{NPV} = 0\).

In this case, IRR is determined using a financial calculator as **38.35%**.

\(\text{IRR}(38.35\%)>\text{RRR}(10\%)\), thus, we should invest in this project.

We can also find the NPV of cashflows of Table 1 using rows as shown in the table below.

$$ \textbf{Table 2: Table Format with Cash Flows collected by Type} $$

$$\small{\begin{array}{l|c|c|c|c} \textbf{Time} & \textbf{Type of Cash Flow} & \textbf{Before-tax Cash Flow} & \textbf{After-tax Cash Flow} & \textbf{PV at 10%} \\ \hline

0 & \text{Fixed capital} & -300,000 & -300,000 & -300,000 \\ \hline0 & \text{Net working capital} & -40,000 & -40,000 & -40,000 \\ \hline1-5 & \text{Sales minus cash expense} & 320,000-100,000=220,000 & 220,000(1-0.40) =132,000 & 500,384 \\ \hline1-5 & \text{Depreciation tax savings} & 0 & 52,500(0.4) =21,000 & 79,607 \\ \hline5 & \text{After-tax salvage value} & 75,000 & 75,000-0.40(75,000-37,500) =60,000 & 37,255 \\ \hline5 & \text{Return of net working capital} & 40,000 & 40,000 & 24,837 \\& & & \textbf{NPV} & \textbf{302,083}\\ \end{array}}$$

As illustrated in Table 2, the NPV of both methods is the same $302,083.

Instead of using tables to organize cash flows you can use equations as follows:

$$\begin{align*}\text{Outlay}&=\text{FCInv}+\text{NWCInc}\\&=300,00-+40,000\\&=$340,000\end{align*}$$

$$\begin{align*}\text{F}&=(\text{S}-\text{C}-\text{D})(1-\text{T})+\text{D}\\&=(320,000-100,000-52,500) (1-0.40) +52,500\\&= $153,000\end{align*}$$

$$\begin{align*}\text{TNOCF}&=\text{Sal}_{T}+NWCInv-\text{T}(\text{Sal}_{\text{T}}-\text{B}_{\text{T}})\\&= 75,000 + 40,000 – 0.4(75,000-37,500) \\&= $100,000\end{align*}$$

The NPV of the project is the present value of the cash flows- an outlay of $340,000 at time zero, an annuity of $153,000 for five years and a single payment of $100,000 in five years.

$$\text{NPV}=-340,000+\sum_{t=1}^{5}\frac{153,000}{(1.10)^{t}}+\frac{100,000}{(1.10)^{5}}=$302,083$$

We can clearly see that whether we use the tabular format for collecting cash flows by year, by type, or an equation format we will arrive the same NPV.

Depreciation is a source of tax savings. The appropriate depreciation expense to use is the expense allowed for tax purposes. Higher depreciation leads to lower income and higher cash flows.

The accelerated depreciation method results in higher after-tax cash flows in the early life of the project and lower after-tax cash flows in the later life as relative to the straight-line depreciation method. Therefore, the accelerated depreciation method increases the NPV of the project relative to the straight-line depreciation method.

The modified Accelerated Cost Recovery System (MACRS) method is the relevant depreciation in the United States. Under MACRS, capital assets are grouped into 3-, 5-, 7-, or 10-year classes that are depreciated using a double declining balance method, with an optimal switch to straight-line and half a year of depreciation the first year.

Assume that the equipment from the previous expansion project falls into a MACRS 3-year class. The depreciation rates under US MACRS 3-year class are as follows:

$$\small{\begin{array}{c|c} \textbf{Year} & \textbf{Depreciation Rate}\\ \hline1 & 33.33\% \\ \hline2 & 44.45\% \\ \hline3 & 14.81\% \\ \hline4 & 7.41\% \\ \end{array}}$$

We can recalculate the NPV of the expansion project as follows:

$$\small{\begin{array}{l|c|c|c|c|c|c} \textbf{Year} & \textbf{0} & \textbf{1} & \textbf{2} & \textbf{3} & \textbf{4} & \textbf{5}\\ \hline \bf\textit{Investment outlays:} & & & & & & \\ \hline\text{Fixed capital} & -300,000 & & & & & \\ \hline\text{Working Capital} & -40,000 & & & & & \\ \hline\textbf{Total} & \textbf{-340,000} & & & & & \\ \hline \bf\textit{After-tax operating cash flows:}& & & & & & \\ \hline\text{Sales} & & 320,000 & 320,000 & 320,000 & 320,000 & 320,000 \\ \hline \text{Cash operating expense} & & 100,000 & 100,000 & 100,000 & 100,000 & 100,000 \\ \hline\text{Depreciation}^{1} & & 87,491 & 116,681 & 38,876 & 19,451 & 0 \\ \hline\textbf{Operating income before tax} & & \textbf{132,509} & \textbf{103,319} & \textbf{181,124} & \textbf{200,549} & \textbf{220,000}\\ \hline\text{Taxes on Operating income} & & 53,004 & 41,328 & 72,450 & 80,220 & 88,000 \\ \hline\textbf{Operating income after tax} & & \textbf{79,505} & \textbf{61,991} & \textbf{108,674} & \textbf{120,329} & \textbf{132,000} \\ \hline\text{Add back depreciation} & & 87,491 & 116,681 & 38,876 & 19,451 & 0 \\ \hline\textbf{After tax operating cash flow} & & \textbf{166,996} & \textbf{178,672} & \textbf{147,550} & \textbf{139,780} & \textbf{132,000} \\ \hline \bf\textit{Terminal year after tax operating cash flows:}& & & & & & \\ \hline\text{After tax salvage value} & & & & & & 60,000 \\ \hline\text{Return of net working capital} & & & & & & 40,000 \\ \hline\textbf{Total} & & & & & & \textbf{100,000} \\\hline\text{Total after tax cash flows} & -340,000 & 166,996 & 178,672 & 147,550 & 139,780 & 232,000 \\ \hline\text{NPV at 10% RRR} & 309,860 & & & & & \\ \hline\text{IRR} & 40.64\% & & & & &\\ \end{array}}$$

Depreciation for Year 1 = 33.33% × $262,500 = $87,491

Depreciation for Year 2 = 44.45% × $262,500 = $116,681

Depreciation for Year 3 = 14.81% × $262,500 = $38,876

Depreciation for Year 4 = 7.41 % × $262,500 = $19,451

Notice that using the MACRS depreciation method increases operating income after taxes in years 3, 4 and 5 and reduces it in years 1 and 2. The NPV increases by $7,777 and the IRR also increases from 38.35% to 40.64%.

Tax savings are calculated as follows:

$$\small{\begin{array}{c|c|c|c} \textbf{Year} & \textbf{Depreciation} & \textbf{Tax savings} & \textbf{PV @ 10} \\ \hline\textbf{1} & 87,491 & 0.4 × 87,491= 34,996 & 31,815 \\ \hline\textbf{2} & 116,681 & 0.4 × 116,681=46,672 & 38,572 \\ \hline\textbf{3} & 38,876 & 0.4 × 38,876 = 15,550 & 11,683 \\ \hline\textbf{4} & 19,451 & 0.4 ×19,451= 7,780 & 5,314 \\ \hline\textbf{5} & 0 & 0.4 × 0 & 0 \\ \hline& & & \textbf{\$87,384}\\ \end{array}}$$

Replacement projects are capital projects that deal with the replacement of old equipment with new equipment. Replacement decisions are often complex and require detailed analysis of cash flows.

The analysis of a replacement project differs from that of an expansion project in that for a replacement decision, we must:

a) Consider the sale of the old asset in the calculation of the initial outlay.

$$\text{Outlay}=\text{FCInv}+\text{NWCInv}-\text{Sal}_{0}+\text{T}(\text{Sal}_{0}-\text{B}_{0})$$

Where:

\(\text{Sal}_{0}=\) Salvage value (cash proceeds) from the sale of old fixed capital

\(\text{B}_{0}=\) Book value of old fixed capital

b) Determine the incremental operating cashflows as the difference between the cash flows from the new asset and cash flows from the old asset.

$$\text{CF}=(\Delta\text{S}-\Delta\text{C})(1-\text{T})+\Delta\text{DT}$$

Or

$$\text{CF}=(\Delta\text{S}-\Delta\text{C}-\Delta\text{D})(1-\text{T})+\Delta{D}$$

Where:

\(\Delta\text{S}=\) change in sales or incremental sales

\(\Delta\text{C}=\) change in cash operating expenses or incremental cash operating expenses

\(\Delta\text{D}=\) change in depreciation expense or incremental depreciation expense

\(\text{T}=\) marginal tax rate

c) Calculate the terminal year non-operating cash flow as:

$$\text{TNOCF}=\text{Sal}_{T}+NWCInv-\text{T}(\text{Sal}_{\text{T}}-\text{B}_{\text{T}})$$

\(\text{Sal}_{T}=\) cash proceeds from sale of fixed capital on Termination date.

\(\text{B}=\) book value of fixed capital on termination date.

Establish whether to accept a replacement project given the following information.

$$\small{\begin{array}{lr|lr} \textbf{Old Equipment} & & \textbf{New Equipment} & \\ \hline \text{Current book value} & \$300,000 & & \\ \hline\text{Current market value} & \$500,000 & \text{Acquisition cost} & \$900,000 \\ \hline\text{Remaining life} & \text{10 years} & \text{Life} & \text{10 Years}\\ \hline\text{Annual sales} & \$ 350,000 & \text{Annual sales} & \$ 500,000 \\ \hline\text{Cash operating expense} & \$150,000 & \text{Cash operating expense} & \$170,000 \\ \hline\text{Annual depreciation} & \$ 35,000 & \text{Annual depreciation} & \$95,000 \\ \hline\text{Accounting salvage value} & \$0 & \text{Accounting salvage value} & \$0 \\ \hline\text{Expected salvage value} & \$ 110,000 & \text{Expected salvage value} & \$220,000\\ \end{array}}$$

- Additional net working capital required: $72,000.
- Tax rate: 30%.
- Required rate of return: 8%.

The initial outlay is computed as follows:

$$\begin{align*}\text{Outlay}&=\text{FCInv}+\text{NWCInv}-\text{Sal}_{0}+\text{T}(\text{Sal}_{0}-\text{B}_{0})\\&=900,000+ 72,000 – 500,000 + 0.3(500,000 – 300,000) \\&= $532,000\end{align*}$$

$$\begin{align*}\text{CF}&=(\Delta\text{S}-\Delta\text{C}-\Delta\text{D})(1-\text{T})+\Delta{D}\\&= [(500,000 – 350,000) – (170,000 – 150,000) – (95,000 – 35,000)] (1 – 0.3) + (95,000 – 35,000)\\&= $ 109,000\end{align*}$$

And the terminal year incremental after-tax non-operating cash flow is:

$$\begin{align*}\text{TNOCF}&=\text{Sal}_{T}+NWCInv-\text{T}(\text{Sal}_{\text{T}}-\text{B}_{\text{T}})\\&=(220,000-110,000)+72,000-0.3(220,000-110,000)-0\\&=$149,000\end{align*}$$

$$\begin{align*}\text{NPV}&=-532,000+\sum_{t=1}^{10}\frac{109,000}{(1.08)^{t}}+\frac{149,000}{(1.08)^{10}}\\&=$268,415\end{align*}$$

The NPV is positive and the IRR = 17.32% > 8%; therefore, replacement project is attractive.

## Question

Which of the following is the main advantage of using an accelerated depreciation method relative to the straight-line depreciation method when calculating cash flows for a replacement project?

- There is no advantage since the results will be the same.
- The NPV for project is likely to increase.
- The taxes payable will increase in the early years and decrease in the later years.
## Solution

The correct answer is B.The accelerated depreciation method reduces tax outflows in the early years and increases them in the later years leading to the after-tax cash flows increasing in early years and decreasing in later years. This results to a higher NPV.

A is incorrect. The accelerated depreciation method offers a higher depreciation tax saving than the straight-line method.

C is incorrect. Tax payable will decrease in the early years and increase the later years as depreciation charge decreases as the years go by.

*Reading 19: Capital Budgeting*

*LOS 19 (a) Calculate the yearly cash flows of expansion and replacement capital projects and evaluate how the choice of depreciation method affects those cash flows.*