MANAGEMENT ACCOUNTING CONCEPTS AND TECHNIQUES
By Dennis Caplan,
CHAPTER 19: Capital Budgeting
Chapter Contents:
 Overview
 Time value of money
 Payback period
 Net present value
 Internal rate of return
 Net present value and internal rate of return, compared
 The discount rate
 Accounting rate of return
 Depreciation expense, income taxes, and capital budgeting
 Present value tables
 Exercises and problems
Overview:
Capital projects involve the acquisition of assets that generate returns over multiple periods. Examples are the construction of a factory or the purchase of a new machine. In this context, a dollar saved is as good as a dollar earned. Hence, capital investments that reduce operating expenses are equivalent to capital investments that generate additional revenues.
This chapter describes four performance measures for capital projects. These performance measures can use budgeted data as a planning tool, to decide whether to invest in a proposed capital project or for choosing among proposed projects. Also, these performance measures can be used retrospectively, to evaluate a capital project against planned performance or against other projects.
A characteristic feature of capital projects is that the bulk of the cash outflows precede the cash inflows. Although a capital project may involve cash outflows that occur over time, and cash inflows that vary from year to year, our discussion will often assume a typical scenario in which there is a single cash outflow for the acquisition of the asset that occurs at the beginning of year one (called “time zero”), followed by a series of equal cash inflows that occur at the end of each year for the life of the project. This series of cash inflows is called an annuity.
Time Value of Money:
A dollar today is worth more than a dollar one year from now. The reason for this appreciation is that cash is an asset, and like any asset, it can be invested to earn a return over time. The discount rate is a measure of the time value of money; it measures how much more a dollar is worth today than a dollar one year from now. For example, if you are indifferent between receiving $1.00 today and $1.20 one year from now, your discount rate is 20%. The time value of money has nothing to do with inflation, which works in the opposite direction. Inflation refers to the declining purchasing power of the dollar that occurs when prices of goods and services rise over time.
Software spreadsheet applications and financial calculators include present value functions that calculate the present value of any amount received (or paid) at any time in the future. These tools also provide the future value, for any point in time in the future, of any amount received (or paid) today. Before these electronic resources were commonplace, tables were widely available that allowed one to easily calculate present values and future values for frequentlyused discount rates and time periods. Although such tables are unnecessary in practice today, we will use them in this chapter, because they visually illustrate the relevant concepts.
Table 1 at the end of this chapter is a present value table. It provides present value factors for selected discount rates that range from 6% to 20%, and time periods that range from one period to twenty periods. If the interest rates are expressed per annum, then the time periods represent years. For example, to determine the present value of any amount X received five years from now, at an interest rate of 8% per annum, one would find the factor at the intersection of Row 5 and the Column for 8% (the factor is 0.6806), and multiply this factor by the amount X.
Many situations involve a stream of equal payments or receipts over a consecutive number of periods. For example, financing the purchase of an automobile might require monthly payments of $1,000 for the next three years, or a proposed capital acquisition might increase revenues by $10,000 every year for the next seven years. Such streams of cash inflows and outflows are called annuities.
Software spreadsheet applications and financial calculators include functions that calculate the present value and future value of annuities. Again, before these electronic resources were widely available, tables were used to calculate the present value or future value of an annuity by multiplying the annual annuity amount by the factor in the table. Table 2 at the end of this chapter is a present value table for annuities. In order to use the table for an annuity of monthly payments or receipts (such as the example of monthly payments for the financing of an automobile), one can treat the rows as months if the interest rates in the column headings are treated as monthly percentages. For example, if the annual interest rate on the car loan is 24%, the monthly interest rate is 2%, and one would need to use the column for 2% (which is not shown in Table 2, but would have been included in tables used by practitioners).
There is an important relationship between Table 1 and Table 2. The present value of any annuity can be calculated by using Table 1 separately for each period over which the annuity occurs, and then summing these individual amounts. Table 2 (or the annuity present value function on a calculator) simplifies the task, by calculating the present value of the entire stream of payments or receipts at once. This relationship implies that one can always “build” Table 2, row by row, by summing the entries for the corresponding column in Table 1, down to that row. For example:
Table 1: Present value of $1 received (or paid) n years from now 

N 
6% 
7% 
8% 
9% 
1 2 3 4 
0.9434 0.8900 0.8396 0.7921 
0.9346 0.8734 0.8163 0.7629 
0.9259 0.8573 0.7938 0.7350 
0.9174 0.8417 0.7722 0.7084 
Table 2: Present value of an annuity of $1 for the next n years 

N 
6% 
7% 
8% 
9% 
1 2 3 4 
0.9434 1.8334 2.6730 3.4651 
0.9346 1.8080 2.6243 3.3872 
0.9259 1.7833 2.5771 3.3121 
0.9174 1.7591 2.5313 3.2397 
0.9346 + 0.8734 + 0.8163 = 2.6243
Hence, an annuity of $1 for three years at 7% equals $2.6243, which can be derived either by adding the three annual amounts provided in Table 1, or more simply by using the factor in row 3 of Table 2.
Next we examine four methods for evaluating capital projects.
Payback Period:
The payback period measures the time required to recoup the initial investment in the capital asset. Consider the following two examples.
Project 
Initial Cost 
Cash Inflows in Year 

1 
2 
3 
4 
5 
6 
7 

A 
$10,000 
$2,000 
$2,000 
$1,000 
$3,000 
$2,000 
$1,500 
$0 
B 
$10,000 
$2,000 
$2,000 
$2,000 
$3,000 
$2,000 
$2,000 
$2,000 
The payback period for Project A is five years, because the sum of cash inflows for years one through five is $10,000 and $10,000 is also the initial cost of the project. The payback period for Project B is greater than four years but less than five years, because the sum of cash inflows through year four is $9,000, and the sum of cash inflows through year five is $11,000, while the initial cost is $10,000. In this situation, the payback period could be expressed as 4˝ years.
If cash inflows are constant from year to year during the life of the project, the payback period can be calculated as follows:

= 
Initial Investment 
Payback Period 


Annual Cash Inflow 
The payback period has two drawbacks. First, it ignores the time value of money. However, this drawback is somewhat mitigated by the fact that, in any case, the payback period tends to favor projects that recover the initial investment quickly. The second drawback is that the payback period ignores cash inflows that occur after the end of the payback period. The following example illustrates these issues:
Project 
Initial Cost 
Cash Inflows in Year 

1 
2 
3 
4 
5 
6 
7 

C 
$8,000 
$2,000 
$2,000 
$1,000 
$3,000 
$0 
$0 
$0 
D 
$8,000 
$2,000 
$2,000 
$2,000 
$2,000 
$2,000 
$2,000 
$2,000 
Both projects have a payback period of four years. However, Project D is clearly preferred to Project C, both because Project D generates more cash inflows earlier during the payback period ($2,000 in year three versus $1,000 for Project C, which is offset in year four), and because Project D continues to generate returns after the payback period is over.
The payback period is a heuristic. A heuristic is a decisionaid that is easily understood and easily communicated, but that might not always result in the best decision.
Net Present Value:
The net present value (NPV) of a capital project answers the following question:
What is the project worth in today’s dollars?
The NPV is the sum of the present value of all current and future cash inflows and outflows. Since the present value of a cashflow that occurs today is its face value, the NPV of a project is the sum of any cashflows that occur at time zero plus the present value of all future cashflows.
In the typical scenario in which there is an initial cash outlay for the acquisition of an asset, followed by cash inflows throughout the useful life of the asset, the NPV can be calculated as follows:
NPV 
= 
∑ 
cash
inflow (1+k)^{n} 
− initial outlay 
Where k is the discount rate, n is the number of periods from time zero in which the cash inflow occurs, and the summation is over the n periods of the life of the project. If the cash inflows are an annuity over the life of the project, the numerator in the above equation can be moved outside of the summation to obtain the following:
NPV 
= 
annual
cash inflow x ∑ 
1 
−
initial outlay 
(1+k)^{n} 
The summation now depends only on k and n:
∑ 
__1___ (1+k)^{n} 
It is exactly this term that is provided in a present value table for annuities (see Table 2 at the end of this chapter), where k represents the discount rate in the column heading, and n represents the number of years (the row).
________________________________________________________________________
Example: The Sunrise Bakery is considering purchasing a new oven. The oven will cost $1,500, and the owner anticipates that the oven will increase the bakery’s future net cash inflows by $800 per year for the next five years. What is the anticipated NPV of this capital acquisition, if the bakery’s discount rate is 10%?
NPV = ($800 x 3.7908) – $1,500 = $3,033 – $1,500 = $1,533.
The factor 3.7908 comes from Table 2: the intersection of the column for 10% and row 5.
________________________________________________________________________
Because NPV provides an absolute measure of the return from the project, not a ratio, it tends to favor large projects. Also, the NPV calculation implicitly assumes that free cash flows can be reinvested at the discount rate. Despite these potential drawbacks, net present value is usually the most reliable criterion by which to judge capital projects on an individual basis.
Internal Rate of Return:
The internal rate of return (IRR) is the discount rate computed such that the net present value of the project equals zero. Software spreadsheet applications and financial calculators usually include a function that calculates the IRR. The following example illustrates how the IRR was approximated prior to the widespread availability of these electronic tools.
________________________________________________________________________
Example: The Sunrise Bakery is considering an expansion to its outdoor dining space that would require an initial cash outlay of $26,000 and increase net cash inflows by $8,000 per year for four years. The owner of the bakery does not anticipate any benefit from this expansion after year four, because at that time she hopes to finance a major renovation of the building that would expand the indoor dining area into the location of the patio. What is the IRR of the proposed expansion to the current outdoor dining space?
Setting the NPV equal to zero in the NPV equation, and solving for the present value factor:
0 = ($8,000 x the present value factor) – $26,000
→ present value factor = 3.25
Looking in Row 4 of Table 2 (since the life of the annuity is four years), the closest factor to 3.25 is 3.2397 in the column for 9%. Therefore, the IRR is approximately 9%.
________________________________________________________________________
Relative to NPV, the advantage of IRR is that it provides a performance measure that is independent of the size of the project. Hence, IRR can be used to compare projects that require significantly different initial investments.
An important drawback of IRR is that it can induce managers to reject proposed projects that shareholders would like the company to accept. For example, if the manager is evaluated based on the average IRR of all capital projects undertaken, and if a proposed capital project offers an IRR that is above the company’s cost of capital, but below the average of all capital projects undertaken thus far, the proposed project would adversely affect the manager’s performance measure, although it would increase economic returns to shareholders.
IRR implicitly assumes that free cashflows can be reinvested at the computed internal rate of return. This assumption is analogous to the assumption imbedded in the NPV calculation that free cashflows can be reinvested at the discount rate. However, in the context of IRR, the assumption is more problematic than in the context of NPV if the IRR is unusually high or low.
Net Present Value and Internal Rate of
Return, Compared:
There is an important and close relationship between NPV and IRR. The NPV is greater than zero if and only if the IRR is greater than the discount rate. This relationship implies that if a single proposed capital investment is considered in isolation, both NPV and IRR will provide the same answer to the question of whether or not the investment should be undertaken.
However, NPV and IRR need not provide the same answer if projects that require different investments are compared. Consider the following example, comparing two projects each with a oneyear life. Assume a 10% discount rate in the NPV calculation. In this simple setting with a oneyear life, the IRR is easily calculated as the profit divided by the initial investment.
Project 
Initial Investment 
Payout at end of year 
Net Present Value 
Internal Rate of Return 
A 
$1,000 
$1,200 
$91 [(1,200 ÷ 1.1) – 1,000] 
20% 
B 
$100 
$200 
$82 [(200 ÷ 1.1) – 100] 
100% 
Hence, NPV favors Project A, while IRR favors Project B. What is the “correct” answer? The answer depends on the opportunity cost associated with the additional $900 required to finance Project A compared with financing Project B. For example, if the company has $1,000 to invest and can replicate Project B ten times, doing so would clearly be preferable to Project A. On the other hand, if the company can earn only 1% on the $900 additional funds available if Project B is chosen over Project A, then the company prefers Project A, calculated as follows:
Project 
NPV 
IRR 
A 
$91, as determined above 
20%, as determined above 
B plus $900 invested at 1% 
$8 [($1,109 ÷ 1.1) – $1,000] 
($1,109  $1,000) ÷ $1,000 = 1.1% 
The $1,109 in the bottom row is the total payout at the end of the year from this option, calculated as $200 from Project B plus $909 from the $900 investment that earns 1%. The NPV of $8 is actually less than the NPV from Project B alone, because the NPV of the $900 invested at 1% is negative.
In conclusion, NPV and IRR need not rank projects equivalently, if the projects differ in size.
The Discount Rate:
The discount rate is critical in determining whether the NPV of a project is positive or negative (and equivalently, whether the project IRR is greater or less than the discount rate). However, the choice of discount rate is seldom obvious.
In most situations, the appropriate discount rate is the company’s cost of capital. The cost of capital is a weighted average of the company’s cost of debt and its cost of equity. Interest rates on borrowings provide information about the cost of debt. Determining the cost of equity is more difficult, and constitutes an important topic in the area of finance. The Weighted Average Cost of Capital (WACC) is a concept from corporate finance that frequently serves as an appropriate discount rate for capital budgeting decisions. In some cases, however, the company would benefit from distinguishing between the existing average cost of capital, and the marginal cost of capital, because the cost of debt generally increases as companies become more highly leveraged.
Many companies establish a companywide hurdle rate, to communicate to managers the appropriate discount rate for investment decisions. Often, the hurdle rate seems to exceed the company’s cost of capital, which encourages managers to act conservatively in their capital budgeting decisions: an outcome that is difficult to justify with finance theory.
Another option for the discount rate is the opportunity cost associated with the funds required for the capital project. In most cases, the cost of capital and the opportunity cost should be approximately equal. However, most of us pay a higher rate to borrow funds than we earn on our financial investments. Hence, if a decisionmaker has cash to either invest in a capital project or invest in the financial markets, an appropriate discount rate for the capital project is the opportunity cost of the earnings the decisionmaker would have earned in the financial markets. This rate is probably lower than the cost of raising additional financing for the project.
Accounting Rate of Return:
The accounting rate of return (ARR) is sometimes called the book rate of return. Of the four capital project performance measures discussed in this chapter, the accounting rate of return is the only performance measure that depends on the company’s accounting choices. It is calculated as follows:
Accounting Rate of Return 
= 
Average Incremental Annual Income from the Project 
Average Net Book Investment in the
Project 
In the simple setting in which the capital project consists of the purchase of a single depreciable asset, the numerator is the average incremental annual cash inflow (additional revenues or the reduction in operating expenses) attributable to the asset, minus the annual depreciation expense. The denominator is the net book investment in the asset, averaged over the life of the asset.
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Example: A machine costs $12,000 and increases cash inflows by $4,000 annually for four years. The machine has zero salvage value.
Depreciation expense = $12,000 ÷ 4 = $3,000 per year.
Incremental income from the machine = $4,000 – $3,000 = $1,000 per year.
Because income from the machine is identical in each year of its fouryear life, the average income over the life of the asset is also $1,000 annually.
For the calculation of the Net Book Investment in the denominator, even though the asset life is four years, five points in time must be considered: time zero (the beginning of year one), and the end of years one through four. At the time the machine is purchased (time zero), the net book investment equals the purchase price of $12,000. As the machine is depreciated, the accumulated depreciation account balance increases, and the net book investment decreases.
Year 
Historical Cost 
Accumulated Depreciation 
Net Book Investment 
0 1 2 3 4 
$12,000 12,000 12,000 12,000 12,000 
$ 0 3,000 6,000 9,000 12,000 
$12,000 9,000 6,000 3,000 0 
The denominator in the accounting rate of return is calculated as
$12,000 + $9,000 + $6,000 + $3,000 + $0 
= 
$6,000 
5 
The accounting rate of return is
$1,000 
= 
16.7% 
$6,000 
This calculation depends on the company’s depreciation method. For example, if the company used doubledeclining depreciation, the accounting rate of return would exceed 16.7% (the numerator does not change, but the average net book investment decreases).
________________________________________________________________________
When straightline depreciation is used, the calculation of the denominator simplifies, because the average of any straight line is the midpoint of that line. The midpoint is calculated as
Initial
book value + ending book value
2
For the numerical example above, the calculation is
$12,000 + $0 
= 
$6,000 
2 
Graphically, this is illustrated as follows:
If the machine has a salvage value, and if the company accounts for that salvage value by decreasing the depreciable basis of the asset, the salvage value has a counterintuitive effect on the denominator of the ARR calculation: it actually increases the company’s net book investment.
For example, assume that the machine in the example above has a salvage value of $4,000. In this case, the annual depreciation expense is ($12,000 – $4,000) ÷ 4 = $2,000. The schedule of net book investment is as follows:
Year 
Historical Cost 
Accumulated Depreciation 
Net Book Investment 
0 1 2 3 4 
$12,000 12,000 12,000 12,000 12,000 
$ 0 2,000 4,000 6,000 8,000 
$12,000 10,000 8,000 6,000 4,000 
The denominator in the accounting rate of return is then calculated as
$12,000 + $10,000 + $8,000 + $6,000 + $4,000 
= 
$8,000 
5 
The accounting rate of return is then
$4,000  $2,000 
= 
25% 
$8,000 
Again, because straightline depreciation is used, the denominator can be calculated more simply as
Initial
book value + ending book value
2
which is now
$12,000 + $4,000 
= 
$8,000 
2 
and graphically
To illustrate how the accounting rate of return depends on the company’s choice of accounting policies, assume that instead of treating the salvage value as a reduction in the depreciable basis of the asset, the company treats the salvage value as income in the year of disposal. In this case, the average annual income from the asset is calculated as follows:
$1,000 + $1,000 + $1,000 + $5,000 
= 
$2,000 
4 years 
The average net book investment is $6,000, as in the original example. The accounting rate of return is now
$2,000 
= 
33.3% 
$6,000 
Hence, depending on how the company chooses to treat the salvage value of the machine, the accounting rate of return is either 25% or 33.3%.
The accounting rate of return can also be calculated year by year, instead of averaging over the life of the project. In this case, the ARR provides information about the impact of the project on the company’s (or division’s) return on investment, which is an important performance measure discussed in Chapter 22.
Depreciation Expense, Income Taxes, and
Capital Budgeting:
Because net present value and internal rate of return focus on cashflows, and depreciation expense is not a cashflow, depreciation does not enter NPV and IRR calculations directly. However, if income taxes are incorporated into the capital budgeting decision (as should normally be the case), then depreciation expense becomes relevant, because depreciation expense reduces taxable income, and hence, reduces tax expense. Obviously, the capital budgeting analysis should incorporate depreciation expense as determined for tax reporting purposes, not for financial reporting purposes, if there is a booktax difference.
The reduction in taxes generated by depreciation expense is sometimes called the depreciation tax shield.
The effect of income taxes can also be incorporated into the payback period and the accounting rate of return in a straightforward manner. In other words, any of these capital budgeting techniques can be applied on a pretax or a posttax basis.
Exercises and Problems:
191:
A) A project requires an initial cash outlay of $800,
and returns $1,000 at the end of year 3 (nothing at the end of years 1 or 2).
What is the approximately Net Present Value of this project, using a cost of
capital of 10%?
B) A project requires an initial cash outlay of $5,000,
and returns $1,000 at the end of each year from Year 1 through Year 10. What is
the project’s approximate Internal Rate of Return.
C) Refer
to the project in part (B). What is the Payback Period of the Project? Also,
what is the Accounting Rate of Return on the average net investment, assuming
that the $5,000 purchase price is for a machine that is depreciated using
straightline depreciation over 10 years, with zero salvage value?
192: A machine costs $4,000 (paid out at the beginning of year 1), and generates yearend net cash inflows of $2,000 per year for five years (this is the useful life of the machine). The machine has zero salvage value. The company uses straightline depreciation and a 10% cost of capital.
Required:
A) What is the payback period for this project?
B) What is the net present value of this project?
C) What is the accounting rate of return for this project?
193: A company purchases an asset for $40,000. The asset has a useful life of seven years. The salvage value is expected to be $10,000. For purposes of computing the accounting rate of return (i.e., the book rate of return), what is the average net investment in the asset?
194: What
is the net present value of a project that requires an initial cash outlay of
$1,000, and returns $7,000 at the end of seven years, using a discount rate of
12%?
195:
An investment of $400 will yield a
single, lumpsum payoff of $1,000 after 10 years. At a discount rate of 7%,
what is the Net Present Value of this project?
196:
A machine that costs $1,000 will save
$200 in operating costs every year for the next seven years. What is the
approximate Internal Rate of Return of this capital project?
197:
A project requires an initial
investment of $500, and will return $100 per year for 11 years. Using a
discount rate of 9%, what is this project’s Net Present Value?
198: A machine costs $50,000 (paid out at the beginning of year 1), and generates yearend net cash inflows of $8,834 per year for 12 years (this is the useful life of the machine). The machine has zero salvage value. The company uses straightline depreciation.
Required:
A) What is the internal rate of return?
B) What is the net present value, using a discount rate of 10%?
C) What is the accounting rate of return?
199:
Using an 11% discount rate, what is
the net present value of a project that requires cash outlays of $10,000 at the
beginning of years one, three and five, and provides cash inflows of $20,000 at
the end of years one, three and five? You may assume that the present value of
a cashflow at the end of year X is equivalent to the cashflow of an equivalent
amount at the beginning of year X + 1 (e.g., December 31, 2005 is the same as
January 1, 2006).
1910: The Seven Flags over the
Required: Calculate the net present value of the new roller coaster, as of January 1, 2007.
1911: Consider a capital project with a oneyear life. The cash outlay for the equipment occurs at the beginning of year one, and a single cash inflow occurs at the end of year one. The equipment has zero salvage value. Straightline depreciation is used. Indicate which of the following statements are true.
A) If the
payback period is less than one, the net present value will be greater than
zero.
B) The internal rate of return is half of the accounting (book)
rate of return.
C) The inverse of the payback period equals the internal rate of
return plus one.
D) The payback period is the inverse of the internal rate of
return.
E) If the
internal rate of return is greater than the discount rate, the net present
value will be greater than zero.
1912:
Consider the following two possible
capital projects:
Project 
Initial Cost (incurred at
beginning of year 1) 
Project Life 
Salvage Value 
Positive annual cash flow
(received at the end of each year) 
A 
$200,000 
16 years 
$0 
$28,000 
B 
$56,000 
14 years 
$0 
$9,783 
Using the above information, it can be
shown that the Internal Rate of Return of Project B is higher than the Internal
Rate of Return of project A.
A) Is
the NPV for Project A higher than, equal to, or lower than, the NPV for Project
B, assuming a 10% discount rate?
B) Is
the Payback Period for Project A better than, equal to, or worse than, the
Payback Period for Project B?
C) Is
the Accounting Rate of Return for Project A higher than, equal to, or lower
than, the Accounting Rate of Return for Project B, assuming straightline
depreciation?
D) If
the discount rate is 11% instead of 10%, which project has the higher NPV?
E) If the
discount rate is 10%, and both projects have a salvage value of $66,000 (i.e.,
the equipment for Project B actually appreciates),
which project would have the higher NPV?
1913: A machine with a useful life of five years and a
salvage value of $4,000 is purchased for $20,000. The benefit of the machine is
that it reduces normal cash operating expenses by $5,000 per year during the
first two years of the machine’s life, and by $4,000 for each of the following
three years.
Required:
A) Calculate the accounting rate of return for the
project, assuming that the full $20,000 purchase price is depreciated using the
straightline method, so that at the end of year five, the machine has a book
value of zero, and the salvage value is treated as income in year five.
B) Calculate the accounting rate of return for the project, assuming that the net cost of the machine (purchase price less salvage value) is depreciated using the straightline method.
1914: Plain Vanilla Industries purchases a machine for
$120,000. The machine has a six year life and a salvage value of zero. The
company depreciates the machine using the sumoftheyearsdigits method, which
results in depreciation expense in each year as follows:
Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 
$34,286 28,571 22,857 17,143 11,429 5,714 $120,000 
The machine increases cash inflows to
the firm by $30,000 in year one, $35,000 in year two, $40,000 in year three,
$35,000 in year four, and $25,000 in year five.
Required: Calculate the accounting rate of return from the
investment in the machine.
PRESENT VALUE TABLES:
Table 1: Present value of $1 received (or paid) n years from now 

n 
6% 
7% 
8% 
9% 
10% 
11% 
12% 
13% 
14% 
15% 
20% 
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 
0.9434 0.8900 0.8396 0.7921 0.7473 0.7050 0.6651 0.6274 0.5919 0.5584 0.5268 0.4970 0.4688 0.4423 0.4173 0.3936 0.3714 0.3503 0.3305 0.3118 
0.9346 0.8734 0.8163 0.7629 0.7130 0.6663 0.6227 0.5820 0.5439 0.5083 0.4751 0.4440 0.4150 0.3878 0.3624 0.3387 0.3166 0.2959 0.2765 0.2584 
0.9259 0.8573 0.7938 0.7350 0.6806 0.6302 0.5835 0.5403 0.5002 0.4632 0.4289 0.3971 0.3677 0.3405 0.3152 0.2919 0.2703 0.2502 0.2317 0.2145 
0.9174 0.8417 0.7722 0.7084 0.6499 0.5963 0.5470 0.5019 0.4604 0.4224 0.3875 0.3555 0.3262 0.2992 0.2745 0.2519 0.2311 0.2120 0.1945 0.1784 
0.9091 0.8264 0.7513 0.6830 0.6209 0.5645 0.5132 0.4665 0.4241 0.3855 0.3505 0.3186 0.2897 0.2633 0.2394 0.2176 0.1978 0.1799 0.1635 0.1486 
0.9009 0.8116 0.7312 0.6587 0.5935 0.5346 0.4817 0.4339 0.3909 0.3522 0.3173 0.2858 0.2575 0.2320 0.2090 0.1883 0.1696 0.1528 0.1377 0.1240 
0.8929 0.7972 0.7118 0.6355 0.5674 0.5066 0.4523 0.4039 0.3606 0.3220 0.2875 0.2567 0.2292 0.2046 0.1827 0.1631 0.1456 0.1300 0.1161 0.1037 
0.8850 0.7831 0.6931 0.6133 0.5428 0.4803 0.4251 0.3762 0.3329 0.2946 0.2607 0.2307 0.2042 0.1807 0.1599 0.1415 0.1252 0.1108 0.0981 0.0868 
0.8772 0.7695 0.6750 0.5921 0.5194 0.4556 0.3996 0.3506 0.3075 0.2697 0.2366 0.2076 0.1821 0.1597 0.1401 0.1229 0.1078 0.0946 0.0829 0.0728 
0.8696 0.7561 0.6575 0.5718 0.4972 0.4323 0.3759 0.3269 0.2843 0.2472 0.2149 0.1869 0.1625 0.1413 0.1229 0.1069 0.0929 0.0808 0.0703 0.0611 
0.8333 0.6944 0.5787 0.4823 0.4019 0.3349 0.2791 0.2326 0.1938 0.1615 0.1346 0.1122 0.0935 0.0779 0.0649 0.0541 0.0451 0.0376 0.0313 0.0261 
Table 2: Present value of an annuity of $1 received (or paid) each year for the next n years 

n 
6% 
7% 
8% 
9% 
10% 
11% 
12% 
13% 
14% 
15% 
20% 
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 
0.9434 1.8334 2.6730 3.4651 4.2124 4.9173 5.5824 6.2098 6.8017 7.3601 7.8869 8.3838 8.8527 9.2950 9.7122 10.1059 10.4773 10.8276 11.1581 11.4699 
0.9346 1.8080 2.6243 3.3872 4.1002 4.7665 5.3893 5.9713 6.5152 7.0236 7.4987 7.9427 8.3577 8.7455 9.1079 9.4466 9.7632 10.0591 10.3356 10.5940 
0.9259 1.7833 2.5771 3.3121 3.9927 4.6229 5.2064 5.7466 6.2469 6.7101 7.1390 7.5361 7.9038 8.2442 8.5595 8.8514 9.1216 9.3719 9.6036 9.8181 
0.9174 1.7591 2.5313 3.2397 3.8897 4.4859 5.0330 5.5348 5.9952 6.4177 6.8052 7.1607 7.4869 7.7862 8.0607 8.3126 8.5436 8.7556 8.9501 9.1285 
0.9091 1.7355 2.4869 3.1699 3.7908 4.3553 4.8684 5.3349 5.7590 6.1446 6.4951 6.8137 7.1034 7.3667 7.6061 7.8237 8.0216 8.2014 8.3649 8.5136 
0.9009 1.7125 2.4437 3.1024 3.6959 4.2305 4.7122 5.1461 5.5370 5.8892 6.2065 6.4924 6.7499 6.9819 7.1909 7.3792 7.5488 7.7016 7.8393 7.9633 
0.8929 1.6901 2.4018 3.0373 3.6048 4.1114 4.5638 4.9676 5.3282 5.6502 5.9377 6.1944 6.4235 6.6282 6.8109 6.9740 7.1196 7.2497 7.3658 7.4694 
0.8850 1.6681 2.3612 2.9745 3.5172 3.9975 4.4226 4.7988 5.1317 5.4262 5.6869 5.9176 6.1218 6.3025 6.4624 6.6039 6.7291 6.8399 6.9380 7.0248 
0.8772 1.6467 2.3216 2.9137 3.4331 3.8887 4.2883 4.6389 4.9464 5.2161 5.4527 5.6603 5.8424 6.0021 6.1422 6.2651 6.3729 6.4674 6.5504 6.6231 
0.8696 1.6257 2.2832 2.8550 3.3522 3.7845 4.1604 4.4873 4.7716 5.0188 5.2337 5.4206 5.5831 5.7245 5.8474 5.9542 6.0472 6.1280 6.1982 6.2593 
0.8333 1.5278 2.1065 2.5887 2.9906 3.3255 3.6046 3.8372 4.0310 4.1925 4.3271 4.4392 4.5327 4.6106 4.6755 4.7296 4.7746 4.8122 4.8435 4.8696 