Capital Budgeting and NPV for the CPA BAR Exam
Kyle Ashcraft, CPA · 2019 CPA Exam Scores · 95 FAR · 98 BEC · 91 REG · 90 AUD
Capital budgeting carries real weight on BAR, and it is one of the few areas where a single missed step turns an otherwise correct answer into a wrong one. The arithmetic is not hard. The sequencing is.
This guide covers the full toolkit in order: time value of money, payback period, discounted payback, accounting rate of return, net present value, internal rate of return and the Black-Scholes inputs. It closes with a full simulation worked step by step, including the two places candidates most often go wrong.
In this guide
How does the BAR exam test capital budgeting?
It sits in Area I, Business Analysis, under prospective analysis. The calculations are tested at the application level, and the comparisons and sensitivity questions a step higher at the analysis level.
| Representative task | Skill level |
|---|---|
| Calculate the present value of future cash flows or the NPV of an investment | Application |
| Determine how changes in assumptions affect value, including Black-Scholes inputs | Analysis |
| Compare investment alternatives using payback, NPV, IRR or related metrics | Analysis |
The comparison questions are worth more of your attention than they usually get, because many of them require no arithmetic at all. Know which methods ignore the time value of money, which one runs on accounting income rather than cash, which one answers in dollars and which one answers in a percentage, and a meaningful share of the topic area is already handled.
What does the time value of money actually do here?
A dollar today beats a dollar in five years, because today's dollar can be put to work in the meantime. Every method below either respects that or ignores it, and knowing which is which is half the topic.
| Pattern | What it is | Example at 5% |
|---|---|---|
| PV of a lump sum | One future payment brought back to today | $100,000 in 5 years is about $78,350 today |
| PV of an ordinary annuity | Equal payments at the end of each period | $20,000 a year for 5 years is about $86,600 today |
| PV of an annuity due | Equal payments at the beginning of each period | The same stream paid in advance is worth more |
| Future value | Today's money grown forward | $5,000 today at 5% for 30 years is about $21,600 |
Three rate terms get used loosely and mean different things:
| Term | Meaning |
|---|---|
| Discount rate | Any rate used to bring future cash flows back to present value |
| Hurdle rate | The minimum return management insists on. Usually the discount rate in an NPV problem |
| Internal rate of return | The return the investment actually delivers |
The trap. Reaching for the annuity factor when the cash flow is a one-off. Salvage value, a lump-sum receipt in year four, a one-time tax refund: all of these get the PV of $1 factor for their specific year, not the annuity factor. Match the factor to the shape of the cash flow, and the exam usually hands you both factors precisely because it expects some candidates to grab the wrong one.
How does the payback period work?
It answers one question: how long until we have our money back? Nothing more.
Discounted payback is the same idea with the cash flows discounted first. Because discounted inflows are smaller, the discounted payback period is always longer than the plain one.
| Method | Uses TVM | Strength | Weakness |
|---|---|---|---|
| Payback period | No | Fast, and a rough read on risk | Ignores TVM and everything after payback |
| Discounted payback | Yes | More realistic | Still ignores everything after payback |
Why the blind spot matters. Both versions stop counting the moment the investment is recovered. A project that repays in three years and then produces nothing looks identical to one that repays in three years and then earns for a decade. That is the limitation an analysis-level question is testing when it asks why payback should not be the deciding metric.
What is the accounting rate of return?
ARR is the odd one out, because it runs on accounting income rather than cash flow. That means depreciation is subtracted, even though it is not a cash outflow. It also ignores the time value of money.
With annual cash inflows of $25,000 and depreciation of $10,000 on a $100,000 asset over ten years:
| Annual cash inflows | $25,000 |
| Less depreciation, $100,000 over 10 years | ($10,000) |
| Accounting income | $15,000 |
| ARR, $15,000 / $100,000 | 15% |
Two things to watch. ARR uses a single year of accounting income, not lifetime cash flow. And some textbooks divide by average investment rather than initial investment, which produces a different percentage from the same facts. Read what the question asks for.
How do you calculate net present value?
NPV discounts every future cash flow back to today at the hurdle rate and compares the total against what you have to spend now. Positive means accept, negative means reject. It is the method that uses the most information and the one the exam trusts most.
Even cash flows
Invest $100,000, receive $25,000 a year for ten years, hurdle rate 10%, annuity factor 6.144.
| PV of inflows, $25,000 × 6.144 | $153,600 |
| Less initial investment | ($100,000) |
| Net present value | $53,600 |
Uneven cash flows
The annuity factor only works when every payment is identical. If the amounts differ year to year, discount each one separately with the PV of $1 factor for its own year and add the results.
The depreciation tax shield
Depreciation never leaves the bank account, but it reduces taxable income, and the tax you avoid is real money you keep.
$45,000 of depreciation at a 30% rate shields $13,500 of tax each year.
The single most important rule in this article. If the question gives you pre-tax cash flows, you have two jobs, not one. Tax the operating inflows, then add the shield back. If it gives you after-tax cash flows, both adjustments are already inside the number and you add nothing.
Getting half of this right is worse than getting none of it right, because adding the shield to an untaxed inflow inflates NPV twice over. The worked simulation at the end of this article shows exactly what that looks like.
Salvage value
Salvage is a single receipt in the final year, so it gets the PV of $1 factor for that year. Never the annuity factor.
One subtlety worth knowing. When the depreciable base is cost less salvage, the asset's book value at the end of its life equals the salvage value. Selling it for exactly that produces no gain and therefore no tax, so the full salvage amount is discounted. If a question depreciates the full cost instead, book value ends at zero, the sale creates a taxable gain, and only the after-tax proceeds get discounted.
What is the internal rate of return?
IRR is the discount rate at which NPV comes out to exactly zero. It answers the same question NPV answers, in a different unit: NPV gives you dollars, IRR gives you a percentage.
| Comparison | Decision | What NPV would say |
|---|---|---|
| IRR above the hurdle rate | Accept | NPV is positive |
| IRR below the hurdle rate | Reject | NPV is negative |
| IRR equals the hurdle rate | Indifferent | NPV is zero |
Take the earlier project: $100,000 out, $25,000 a year for ten years. Its IRR is about 21.4%, comfortably above a 10% hurdle rate, which is another way of saying the same thing the $53,600 positive NPV already said.
You will rarely be asked to compute IRR by hand, because doing it properly means solving iteratively. Expect to be given an IRR and asked to act on it, or to be asked what IRR means when it sits above or below the hurdle rate.
How do the methods compare?
| Method | Answer is in | Uses TVM | Weakness |
|---|---|---|---|
| Payback period | Years | No | Ignores TVM and later cash flows |
| Discounted payback | Years | Yes | Ignores later cash flows |
| Accounting rate of return | Percentage | No | Ignores TVM, uses income not cash |
| Net present value | Dollars | Yes | More steps to get wrong |
| Internal rate of return | Percentage | Yes | Harder to compute, can mislead on scale |
Why NPV wins when the two disagree. IRR is a rate, so it is blind to size. A tiny project returning 40% has a higher IRR than a large one returning 15%, but the large one may add far more value in dollars. When a question sets up a conflict between NPV and IRR on mutually exclusive projects, NPV is the answer.
What do you need to know about Black-Scholes?
Enough to reason about it, not enough to compute it. The model prices European-style options, which can only be exercised at expiration, and BAR tests the direction each input pushes the value.
| Input | What it is | If it rises, call value |
|---|---|---|
| Current stock price | What the underlying trades at today | Increases |
| Strike price | The price you would pay to exercise | Decreases |
| Risk-free rate | Return on a risk-free investment | Increases |
| Time to maturity | How long until expiration | Increases |
| Volatility | How much the stock price moves | Increases |
Note that the option premium is not an input. It is what the model produces. A question listing it among the inputs is testing whether you know the difference between what goes in and what comes out.
For a call, the strike price is the only one of the five that moves value in the opposite direction, which makes it easy to remember: everything up except the price you have to pay.
A full worked simulation
This has the shape of a real BAR task-based simulation. It folds together a depreciation tax shield, an annuity and a salvage lump sum, and it contains the trap discussed above.
The question. ClearWater is considering a water treatment system costing $250,000 that produces uniform pre-tax cash inflows of $90,000 a year for five years. Residual value is $25,000. Useful life is five years, straight-line depreciation, tax rate 30%, cost of capital 12%.
Factors given: PV of $1 for 5 years at 12% is 0.567. PV of an ordinary annuity for 5 years at 12% is 3.605.
What is the NPV? Try it before reading on.
- Annual depreciation. The depreciable base is cost less salvage, so ($250,000 − $25,000) / 5 = $45,000 a year. Subtracting salvage first is the step most often skipped.
- Depreciation tax shield. $45,000 × 30% = $13,500 a year.
- Tax the operating inflows. The question said pre-tax, so the $90,000 is not what ClearWater keeps. $90,000 × (1 − 0.30) = $63,000.
- Total after-tax annual cash flow. $63,000 + $13,500 = $76,500.
- Discount the annual stream. It is uniform over five years, so use the annuity factor. $76,500 × 3.605 = $275,782.50.
- Discount the residual value. One receipt in year five, so use the lump-sum factor. $25,000 × 0.567 = $14,175. Book value at the end equals the $25,000 salvage, so there is no gain and no tax on the sale.
- Net present value. $275,782.50 + $14,175 − $250,000 = $39,957.50.
| PV of after-tax annual cash flows | $275,782.50 |
| PV of residual value | $14,175.00 |
| Less initial investment | ($250,000.00) |
| Net present value | $39,957.50 |
NPV is positive, so ClearWater accepts.
What step three catches. Skip it and you add $13,500 of tax shield to an inflow you never taxed, producing an annual figure of $103,500 and an NPV of $137,292.50. That is more than three times the right answer, and it comes from one omitted line.
The reason it is so easy to miss is that the shield feels like the hard part, so finding it feels like finishing. Whenever a question says pre-tax, write both adjustments down before you calculate either one.
Glossary
| Term | Definition |
|---|---|
| Time value of money | A dollar today is worth more than a dollar later |
| Discount rate | Any rate used to bring future cash flows to present value |
| Hurdle rate | The minimum return management requires |
| Lump sum | A single future payment |
| Ordinary annuity | Equal payments at the end of each period |
| Annuity due | Equal payments at the beginning of each period |
| Payback period | Years to recover the investment, undiscounted |
| Discounted payback | The same, using discounted cash flows. Always longer |
| Accounting rate of return | Accounting income divided by investment. Ignores TVM |
| Net present value | PV of inflows less the initial investment |
| Internal rate of return | The discount rate at which NPV equals zero |
| Depreciation tax shield | Depreciation expense times the tax rate |
| Salvage value | Expected end-of-life proceeds. Discounted as a lump sum |
| Black-Scholes model | Option pricing model for European-style options |
| European-style option | Exercisable only at expiration |
| American-style option | Exercisable at any point up to expiration |
Frequently asked questions
Which capital budgeting methods use the time value of money?
NPV, IRR and discounted payback do. Basic payback period and the accounting rate of return do not.
What is the difference between NPV and IRR?
They answer the same question in different units. NPV gives a dollar amount, IRR gives a percentage. When the two conflict on mutually exclusive projects, follow NPV, because a percentage is blind to the size of the investment.
When do I add the depreciation tax shield?
When the question gives pre-tax cash flows. But adding the shield is only half the job: you must also tax the operating inflows. If the question gives after-tax cash flows, both adjustments are already in the number and you add nothing.
How is salvage value treated in an NPV problem?
As a single receipt in the final year, discounted with the PV of $1 factor rather than the annuity factor. If the depreciable base was cost less salvage, book value equals salvage at the end, so there is no taxable gain on the sale.
Why is the depreciable base cost less salvage?
Because depreciation allocates the amount the business expects to consume, and the salvage value is the part it expects to get back. Forgetting to subtract it overstates depreciation and every figure that follows from it.
Will I have to calculate Black-Scholes on BAR?
No. You need the five inputs and the direction each one pushes a call option's value. The option premium is the model's output, not an input.
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Kyle Ashcraft, CPA scored 90 or above on every section of the CPA exam in 2019, including a 95 on FAR. He is the founder and sole instructor of Maxwell CPA Review, a complete CPA review course covering all six sections, where he creates every lecture, textbook and study outline himself.
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