Now or next year?: A hotel that buys rolls from Lena's Bakery offers a deal: it will pay $30,000 for a year's order either today or in twelve months' time. Same amount either way.
Lena takes it today, without a second thought. Why?
Because $30,000 in the bank today can do something for a year. It can pay off part of a loan that costs the bakery 8% a year (3.2.3), or earn interest. By next year it has become more than $30,000.
So a dollar that arrives later is worth less than a dollar now. Business people call this the time value of money.
What next year's $30,000 is worth today, at 8%
Pick the rate
8%: what the bakery pays on a bank loan, so what money tied up for a year costs it. This is the discount rate.
Read the factor
For one year at 8% the table gives 0.9259. A dollar arriving in one year is worth about 93 cents today. This number is the discount factor.
Multiply
$30,000 × 0.9259 = $27,777. That is the present value of the payment.
Check it makes sense
Put $27,777 in the bank today at 8% and a year later it has grown to about $30,000. The two are the same money, seen from different days.
Shrinking future money into today's money like this is called discounting. The further away the money, the more it shrinks, because it has longer to miss out on interest.
| $30,000 received at the end of year | Discount factor at 8% | Present value |
|---|---|---|
| 1 | 0.9259 | $27,777 |
| 2 | 0.8573 | $25,719 |
| 3 | 0.7938 | $23,814 |
| 4 | 0.7350 | $22,050 |
| 5 | 0.6806 | $20,418 |
The factors are always given: You never work out a discount factor yourself. The question prints a table of them, by year and by rate, and you read off the one you need.
A higher rate gives smaller factors, and a later year gives smaller factors. Both shrink the money more.
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Is Station Road worth $120,000?: Fitting out the fourth shop on Station Road costs $120,000 now. Lena and Marco expect it to bring in net cash flows of $30,000, $32,000, $34,000, $36,000 and $38,000 over its first five years.
That is $170,000 back for $120,000 paid. But most of it arrives years from now, so it is worth less than $170,000 today. How much less?
Net present value answers that. Turn each year's net cash flow into today's money, add them up, and take away the cost. What is left is the net present value (NPV).
The formula: NPV = sum of the present values of the returns − the original cost
Present value = net cash flow × discount factor. The formula is on the formula sheet, and the discount factors are printed in the question.
| Year | Net cash flow | Discount factor at 8% | Present value |
|---|---|---|---|
| 0 | ($120,000) | 1.000 | ($120,000) |
| 1 | $30,000 | 0.9259 | $27,777 |
| 2 | $32,000 | 0.8573 | $27,434 |
| 3 | $34,000 | 0.7938 | $26,989 |
| 4 | $36,000 | 0.7350 | $26,460 |
| 5 | $38,000 | 0.6806 | $25,863 |
| Total, years 1 to 5 | $134,523 |
Reading the table
Year 0 is today
The $120,000 is paid now, so it is already in today's money: its factor is 1.000. Brackets mean money going out.
Each later year gets its own factor
Year 2's $32,000 × 0.8573 = $27,434, rounded to the nearest dollar. Never use one factor for every year.
Add the present values
$27,777 + $27,434 + $26,989 + $26,460 + $25,863 = $134,523, the total present value.
Take away the cost
NPV = $134,523 − $120,000 = $14,523.
What does $14,523 mean? After allowing for the 8% the money could have earned elsewhere, the shop still brings in $14,523 more than it costs, in today's money. It clears the bar with room to spare.
NPV above zero
- The project earns more than the discount rate
- Worth doing on the figures
- Station Road at 8%: $14,523
NPV of zero
- The project earns exactly the discount rate
- It pays back its cost and nothing more
NPV below zero
- The project earns less than the discount rate
- The money would do better elsewhere
- Not worth doing on the figures alone
Choosing between projects: When two projects compete for the same money, the one with the higher NPV adds more value in today's money. The figures are not the whole decision, though: 3.8.3 showed what else goes into the choice.
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Lena and Marco worked out the Station Road NPV at 8%, the rate the bank charged them. Then the news reports that interest rates are going up. Does their $14,523 still hold?
A higher rate, a lower NPV: When interest rates rise, money tied up in a project could earn more elsewhere, or costs more to borrow. So the business should use a higher discount rate.
A higher rate means smaller discount factors, smaller present values and so a lower NPV. The cash flows have not changed at all; what they are worth today has.
| Discount rate | Total present value, years 1 to 5 | NPV of Station Road |
|---|---|---|
| 6% | $142,241 | $22,241 |
| 8% | $134,523 | $14,523 |
| 10% | $127,444 | $7,444 |
| 20% | $99,531 | ($20,469) |
At 10% the shop is still worth doing, but it is worth half as much. At 20% it would lose money in today's terms. The same five cash flows give four different answers, so an NPV is only as good as the rate chosen for it.
Three reasons an NPV may be wrong
The discount rate is too low
If rates rise from 8% to 10%, the NPV falls from $14,523 to $7,444. A small NPV can turn negative with a small change in the rate.
The cost is higher than planned
If the fit-out costs $135,000 instead of $120,000, the NPV at 8% is $134,523 − $135,000 = ($477). Any cost above $134,523 makes it negative.
The cash flows are forecasts
If every year brings 10% less, the NPV at 8% falls to $1,070. Years four and five are the hardest to forecast, and they are years away.
Pick the reason that belongs to NPV: Forecasts that may be wrong are a weakness of payback and ARR too. The discount rate is the one that belongs to NPV alone.
Asked why an NPV may be inaccurate, say what would change, show that it lowers the NPV, and use the figures to show how far: here, whether it could turn negative.
Three ways to look at one shop: Lena and Marco now have three figures for Station Road. Payback: 3 years 8 months (3.8.1). ARR: 8.33% a year (3.8.2). NPV at 8%: $14,523.
Each one answers a different question about the same $120,000.
| Method | The question it answers | Station Road |
|---|---|---|
| Payback period | How quickly does the money come back? | 3 years 8 months |
| ARR | What is the average yearly profit, as a % of the cost? | 8.33% |
| NPV | What is the project worth in today's money, after the cost? | $14,523 at 8% |
What NPV does well
- It allows for the time value of money. Year five's $38,000 counts as $25,863, not $38,000. Payback and ARR count a dollar in year five the same as a dollar in year one.
- It uses every year. Payback stops counting once the $120,000 is back; NPV counts all five years.
- It gives a clear rule. Above zero, the project beats the discount rate; below zero, it does not.
- It can include the cost of borrowing. Discounting at the bank's 8% tests the shop against what the money costs the bakery.
Where NPV falls short
- The rate is a choice. At 10% the NPV halves; at 20% it turns negative. The answer depends on a guess.
- It is harder to work out and explain. A shop manager understands 3 years 8 months more easily than $14,523 in today's money.
- It says nothing about speed. A project with a higher NPV may take longer to pay back, which matters when cash is short.
- It rests on forecasts, like the other methods, and leaves out what numbers cannot show.
The figure informs the decision: An NPV is a well-made figure, not a decision. Lena and Marco would still ask whether the shop fits their plans, whether they can manage four shops, and how much cash they can spare while it gets going.
Used together, the three methods show value, profit and speed. Each covers a gap in the others.
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How this comes up: A table of net cash flows with the discount factors printed beside them. You complete the present value column, total it, and calculate the NPV, for two marks, showing your working. Sometimes you first work out each year's net cash flow from income and expenses, for three.
Then a short explain: how a rise in interest rates would affect your NPV, or one reason why it may be inaccurate, for two marks, with the figures.
The two-mark calculation
- Multiply each year by its own factor. Year 1 with the year-1 factor, year 2 with year 2.
- Write every present value in the table. A negative year stays negative.
- Add them up. That is the total present value; it is worth a mark on its own.
- Take away the cost. NPV = total present value − cost. The cost is never discounted.
- Keep the units. If the table is in $000s or millions, say so, or write the answer out in full.
The traps: Adding the cash flows without discounting. 4 × $18,000 − $45,000 = $27,000 is not an NPV: it ignores the time value of money.
Discounting the cost, or taking it off twice. It is paid today. If year 0 is in the table at 1.000, leave it out of the total and subtract it once, at the end.
Complete Table 1 by calculating the present value of the second deck oven for its first four years, using a discount rate of 8%. Using your completed table, calculate the net present value of the oven (show all your working).
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