9.8: Capital investment analysis
- Page ID
- 44265
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)One of a company’s most significant financial decisions involves the purchase of property, plant, and equipment that will be used in business operations. The costs of these assets are often very high, and they will be in place for many years to come.
Before acquiring a capital asset such as equipment, machinery, or a building, which involves a large expenditure and long-term commitment, a company should evaluate how effectively it is expected to generate a return on investment for the business. Capital investment analysis is a form of differential analysis used to determine (1) whether a fixed asset should be purchased at all, or (2) which fixed asset among a number of choices is the best investment. Three commonly used methods for evaluating capital investments will be discussed.
The first two, the average rate of return method and the cash payback method, are relatively straightforward calculations that are often used to determine whether a proposed investment meets a minimum standard for it even to be considered further.
The average rate of return method is the percentage return of net income from the proposed investment. It is calculated as follows:
\(\ \frac{\text{Average annual income}}{\text{Average investment}}\)
Each of the two amounts must first be calculated independently.
\(\ \begin{array}{c}\text { Average annual income } \\ \text { (numerator) }\end{array}=\frac{\text { Total estimated income over the asset's useful life }}{\text { Number of years in the asset's useful life }}\)
\(\ \begin{array}{l}\text { Average investment } \\ \text { (denominator) }\end{array}=\frac{\text { Book value at the beginning of the first year }^{1}+\text { Book value at the end of the last year }^{2}}{2}\)
1 The book value at the beginning of the first year is the asset’s cost.
2 The book value at the end of the last year is the asset’s residual value.
As an example, a new piece of equipment that is being considered for purchase costs $90,000 and has a residual value of $10,000. It is expected to generate revenue of $75,000 over its estimated useful life of 5 years.
\(\ \text{Average annual income} =\frac{\$ 75,000}{5}=\$ 15,000\)
\(\ \text{Average investment} =\frac{\$110,000 + $10,000}{2}=\$ 60,000\)
\(\ \bf{\text{Average investment}} =\frac{\$15,000}{$60,000}=\ 25\)%
The average rate of return of 25% should then be compared to the minimum rate of return that management requires. If the average rate of return is greater than the minimum acceptable rate, the equipment should be evaluated further since it seems promising. If it does not even meet this standard, however, it should not be purchased.
The cash payback method looks at the annual net cash inflow from the use of an asset to determine how many years it will take to recover the cost of the asset. Net cash flow includes all cash revenue generated minus all cash expenditures paid from using the asset. Depreciation is not a cash expenditure, so it would not be considered in determining net cash flow.
As an example, a new piece of equipment that is being considered for purchase costs $80,000. It is expected to generate $25,000 cash revenue each year and require cash expenditures of $5,000 to maintain.
Cash payback period $=\frac{\text { cost }}{\text { Annual nest cash flow }}=\frac{\$ 80,000}{\$ 25,000-\$ 5,000}=4$ years
The cash payback period of four years should be compared to the maximum period that management desires. If the cash payback period of four years is more than an acceptable payback period of three years, for example, the purchase should no longer be considered. If a payback period of five years is acceptable, the purchase should be looked into further.
If annual net cash flows are not expected to be equal each year, the cash payback period is determined by adding the annual expected cash flows year by year until the sum equals the initial cost of the asset.
For example, a new piece of equipment that is being considered for purchase costs $80,000. Its expected annual cash flows are as follows:
|
Year |
Net Cash Flow |
Cash Flow to Date |
|
1 |
$12,000 |
$12,000 |
| 2. |
18,000 |
30,000 |
|
3 |
24,000 |
54,000 |
|
4 |
26,000 |
80,000 |
|
5 |
30,000 |
110,000 |
|
6 |
34,000 |
144,000 |
In this case, net cash flows recover the initial cost of $80,000 after four full years. The average rate of return and the cash payback methods are relatively simple to calculate, yet they yield rather general results. Since neither considers the time value of money, they are more effective for shorter-term investments. They are often used as an initial screening to see if an investment should be immediately disqualified. If not, the investment may be analyzed further using more robust
analyses.
The net present value (NPV) method for evaluating a potential investment
also looks at estimated future net cash flows generated by the asset. It compares the purchase price (investment amount) to the present value of all the future net cash flows from using the asset. The investment is considered viable if the present value of the future net cash flows is greater than the purchase price. Otherwise, the investment should be avoided.
Present value factors the timing of future net cash inflows and the effect of a prevailing interest rate. An amount of cash received in the future is worth less than the same amount of cash received today. This is because cash received now may be invested at a given interest rate that causes its value to grow over time compounding, where interest is earned both on principal and on interest that has already been earned. The opportunity to invest dollars received in the future rather than today is postponed, missing out on time available to earn interest.
Determining the future value of a current amount is calculated by multiplying the amount by itself plus the interest rate. For example, the future value of $1.00 in 3 years at an interest rate of 6% would be calculated as follows:
$1.00 x 1.06 = $1.06 x 1.06 = $1.12 x 1.06 = $1.19
Note that interest is calculated on interest previously earned. This process is called compounding.
Present value works in the opposite direction. An amount in the future is known or estimated (such as a net cash inflow), and the calculation backs that amount up to its current value. The process is called discounting. For example, the present value of $1.00 to be received in 3 years at an interest rate of 6% would be calculated as follows:
\(\ \frac{\$ 1.00}{1.06}=\frac{\$ 0.94}{1.06}=\frac{\$ 0.89}{1.06}=\$ 0.84 (\text{rounded to the nearest cent})\)
The following table summarizes the present value of $1 for 10 periods for three interest rates: 6%, 8%, and 10%. Amounts are rounded to five decimal places rather than to the nearest cent.
|
Period |
6% |
8% |
10% |
|
1 |
0.94340 |
0.92593 |
0.90909 |
|
2 |
0.89000 |
0.85734 |
0.82645 |
|
3 |
0.83962 |
0.79383 |
0.75132 |
|
4 |
0.79209 |
0.73503 |
0.68302 |
|
5 |
0.74725 |
0.68058 |
0.62093 |
|
6 |
0.70495 |
0.63017 |
0.56448 |
|
7 |
0.66505 |
0.58349 |
0.51316 |
|
8 |
0.62741 |
0.54027 |
0.46651 |
|
9 |
0.59190 |
0.50025 |
0.42410 |
|
10 |
0.55840 |
0.46319 |
0.38555 |
Note that all amounts in the present value table are less than $1.00 since all represent a future cash receipt rather than the $1.00 today. The further into the future the $1.00 will be received for a given interest rate, the lower its present value.
Clearly not all future cash receipts are for $1.00. To get the present value of a different value, multiply the actual number of dollars by the present value of $1 amount given in the table at the intersection of a specified interest rate and number of years.
Examples 1 and 2 illustrate the process of discounting the future net cash flows to determine their total and comparing it to the cost of the asset.
A company is considering purchasing equipment #1 for $100,000. It is expected to provide net cash flows of $24,000 per year for the next six years for a total of $144,000. The minimum desired rate of return on the investment is 6%.
|
Year |
Undiscounted Net Cash Flow |
Present Value of $1 at 6% |
Discounted Net Cash Flow |
Since the undiscounted net cash flow amount is the same each year, the total discounted net cash flow could also be calculated by using the present value of an annuity of $1, as follows: $24,000 x 4.91731 = $118,016 Rather than multiplying $24,000 six times by six different factors, $24,000 can be multiplied once by the sum of all the factors (4.91731). The result is the same. |
|
1 |
$24,000 |
0.94340 |
$22,642 |
|
|
2 |
24,000 |
0.89000 |
21,360 |
|
|
3 |
24,000 |
0.83962 |
20,151 |
|
|
4 |
24,000 |
0.79209 |
19,010 |
|
|
5 |
24,000 |
074725 |
17,934 |
|
|
6 |
24,000 |
0.70495 |
16,919 |
|
|
Total |
$144,000 |
4.91731 |
$118,016 |
|
|
Cost |
(100,000) |
|||
|
NPV |
$18,016 |
In this case, the net present value of the future cash flows of $18,016 is greater than the cost of the asset, $100,000. The investment may be accepted since it more than pays for itself over time.
If the cost of the asset had been $130,000 rather than $100,000, the net present value would have been ($11,984), which is $118,016 - $130,000. In this case the NPV is negative and the proposed purchase should be rejected.
A company is considering purchasing equipment #2 for $100,000. It is expected to provide net cash flows of different amounts each year for the next six years for a total of $144,000. The minimum desired rate of return on the investment is 6%.
|
Year |
Undiscounted Net Cash Flow |
Present Value of $1 at 6% |
Discounted Net Cash Flow |
Since the undiscounted net cash flow amounts are different each year, the total discounted net cash flow must be calculated using six individual calculations. Each year the undiscounted net cash flow amount is multiplied by the present value of $1 factor at 6%. |
|
1 |
$34,000 |
0.94340 |
$32,076 |
|
|
2 |
30,000 |
0.89000 |
26,700 |
|
|
3 |
26,000 |
0.83962 |
21,830 |
|
|
4 |
24,000 |
0.79209 |
19,010 |
|
|
5 |
18,000 |
074725 |
13,451 |
|
|
6 |
12,000 |
0.70495 |
8,459 |
|
|
Total |
$144,000 |
$121,526 |
||
|
Cost |
(100,000) |
|||
|
NPV |
$24,526 |
In this case, the net present value of the future cash flows of $21,256 is greater than the cost of the asset, $100,000. The investment may be accepted since it more than pays for itself over time.
Net present value can be used to perform differential analysis to compare results of two or more proposed investments to determine which is more financially beneficial. Examples 3 and 4a show these comparisons.
A company is considering two different proposals for purchasing equipment. Both assets will be useful for six years. The first piece of equipment costs $100,000, and the second costs $140,000. The undiscounted cash flows appear in the two tables that follow.
|
#1 Year |
Undiscounted Net Cash Flow |
Present Value of $1 at 6% |
Discounted Net Cash Flow |
#2 Year |
Undiscounted Net Cash Flow |
Present Value of $1 at 6% |
Discounted Net Cash Flow |
|
|
1 |
$34,000 |
0.94340 |
$32,076 |
1 |
$44,000 |
0.94340 |
$41,510 |
|
|
2 |
30,000 |
0.89000 |
26,700 |
2 |
39,000 |
0.89000 |
34,710 |
|
|
3 |
26,000 |
0.83962 |
21,830 |
3 |
34,000 |
0.83962 |
28,547 |
|
|
4 |
24,000 |
0.79209 |
19,010 |
4 |
31,000 |
0.79209 |
24,555 |
|
|
5 |
18,000 |
074725 |
13,451 |
5 |
23,000 |
074725 |
17,187 |
|
|
6 |
12,000 |
0.70495 |
8,459 |
6 |
16,000 |
0.70495 |
11,279 |
|
|
Total |
$144,000 |
$121,526 |
Total |
$187,000 |
$157,788 |
|||
|
Cost |
(100,000) |
Cost |
(140,000) |
|||||
|
NPV |
$24,526 |
NPV |
$17,788 |
The second piece of equipment has a higher estimated net cash flow each year, but it also costs more to purchase. Both assets yield a positive net present value, but the first piece of equipment has a higher NPV, $21,526, vs. the NPV of the second piece, $17,788. The first piece of equipment should be purchased based on this result.
It is possible that two different investments will span two different periods; that is, one may generate cash flows for more years than the other. In order to perform a differential analysis, the number of years must be the same for both. To make them comparable, the asset with the higher number of years of cash flows is adjusted to assume that it is sold for its residual value amount in the last year that the other asset provides cash flows.
A company is considering two different proposals for purchasing equipment. The first asset provides cash flows for four years, and the second one provides cash flows for six years. Both assets cost $100,000 and have a residual value of $10,000. Both have undiscounted net cash flows of $124,000 as shown in the tables that follow.
|
#1 Year |
Undiscounted Net Cash Flow |
Present Value of $1 at 6% |
Discounted Net Cash Flow |
#2 Year |
Undiscounted Net Cash Flow |
Present Value of $1 at 6% |
Discounted Net Cash Flow |
|
|
1 |
$38,000 |
0.94340 |
$35,849 |
1 |
$34,000 |
0.94340 |
$32,076 |
|
|
2 |
34,000 |
0.89000 |
30,260 |
2 |
32,000 |
0.89000 |
28,480 |
|
|
3 |
30,000 |
0.83962 |
25,189 |
3 |
26,000 |
0.83962 |
21,830 |
|
|
4 |
22,000 |
0.79209 |
17,426 |
4 |
22,000 |
0.79209 |
17,426 |
|
|
4 (residual) |
10,000 |
0.79209 |
7,921 |
|||||
|
|
|
|
|
|||||
|
|
|
|
|
|||||
|
Total |
$124,000 |
$108,724 |
Total |
$124,000 |
$107,733 |
|||
|
Cost |
(100,000) |
Cost |
(100,000) |
|||||
|
NPV |
$8,724 |
NPV |
$7,733 |
Note that the cash flow period for the second piece of equipment is adjusted to four years to match that of the first piece of equipment. There are two net cash flows for the second piece of equipment in year four: (1) the inflow from using the equipment, and (2) the proceeds from selling it at its residual value. The cash flows for the fifth and sixth years for the second asset are not considered and are therefore greyed out in the table.
Both assets yield a positive net present value, but the first piece of equipment has a higher NPV, $8,724, vs. the NPV of the second piece, $7,733. The first piece of equipment should be purchased based on this result.
As a final illustration of two companies with different cash flow periods, note that the net present value would be identical if the annual net cash flows were the same. In this case, all are equal in years 1, 2, and 3. In year 4, they also both equal $32000: for the first asset the $32,000 is all operational cash flow, and for the second piece of equipment, the $32,000 includes $22,000 of operational cash flow and $10,000 selling price.
A company is considering two different proposals for purchasing equipment. The first asset provides cash flows for four years, and the second one provides cash flows for six years. Both assets cost $100,000 and have a residual value of $10,000. Both have undiscounted net cash flows of $124,000 as shown in the tables that follow.
|
#1 Year |
Undiscounted Net Cash Flow |
Present Value of $1 at 6% |
Discounted Net Cash Flow |
#2 Year |
Undiscounted Net Cash Flow |
Present Value of $1 at 6% |
Discounted Net Cash Flow |
|
|
1 |
$34,000 |
0.94340 |
$35,076 |
1 |
$34,000 |
0.94340 |
$32,076 |
|
|
2 |
32,000 |
0.89000 |
28,480 |
2 |
32,000 |
0.89000 |
28,480 |
|
|
3 |
26,000 |
0.83962 |
21,830 |
3 |
26,000 |
0.83962 |
21,830 |
|
|
4 |
32,000 |
0.79209 |
25,347 |
4 |
22,000 |
0.79209 |
17,426 |
|
|
4 (residual) |
10,000 |
0.79209 |
7,921 |
|||||
|
|
|
|
|
|||||
|
|
|
|
|
|||||
|
Total |
$124,000 |
$107,733 |
Total |
$124,000 |
$107,733 |
|||
|
Cost |
(100,000) |
Cost |
(100,000) |
|||||
|
NPV |
$7,733 |
NPV |
$7,733 |
Differential analysis is a useful planning tool for projecting relative results among alternatives. It encourages managers to think ahead and analyze the components of alternative outcomes with the goal of more insightful decision making.