How do I calculate NPV when the cash flows are unequal each year? I keep getting the wrong answer.
I'm working through time value of money problems for CFA Level I and I can handle the basic annuity and perpetuity formulas just fine. But when I get a problem with different cash flows in each year, I freeze up. For example, a project that pays $5,000 in Year 1, $8,000 in Year 2, and $12,000 in Year 3. Do I need to discount each one separately? Is there a shortcut on the BA II Plus? Any help would be appreciated.
Great question — unequal (or "mixed") cash flow problems are among the most commonly tested TVM concepts on the CFA Level I exam. The key insight is that each cash flow must be discounted individually back to the present, and then you sum them all up.
The Formula:
NPV = -CF_0 + CF_1 / (1 + r)^1 + CF_2 / (1 + r)^2 + ... + CF_n / (1 + r)^n
Worked Example:
Westfield Corp is evaluating a warehouse expansion. The initial outlay is $20,000, and the project generates the following cash flows:
| Year | Cash Flow |
|---|---|
| 0 | -$20,000 |
| 1 | $5,000 |
| 2 | $8,000 |
| 3 | $12,000 |
The required rate of return is 9%.
Step-by-step:
- PV of CF_1 = $5,000 / (1.09)^1 = $5,000 / 1.0900 = $4,587.16
- PV of CF_2 = $8,000 / (1.09)^2 = $8,000 / 1.1881 = $6,733.44
- PV of CF_3 = $12,000 / (1.09)^3 = $12,000 / 1.2950 = $9,266.41
Sum of PVs = $4,587.16 + $6,733.44 + $9,266.41 = $20,587.01
NPV = $20,587.01 - $20,000 = $587.01
Since NPV > 0, Westfield should accept the project.
BA II Plus Shortcut:
Use the CF worksheet:
- Press CF, then 2nd CLR WORK
- CF0 = -20000, ENTER, down arrow
- C01 = 5000, ENTER, down arrow, F01 = 1, ENTER, down arrow
- C02 = 8000, ENTER, down arrow, F02 = 1, ENTER, down arrow
- C03 = 12000, ENTER, down arrow, F03 = 1
- Press NPV, enter I = 9, down arrow, CPT
- You should get 587.01
Common mistakes to avoid:
- Forgetting to set F (frequency) back to 1 if you previously used a different frequency
- Discounting by multiplying instead of dividing
- Using the annuity formula when cash flows are NOT equal
For more practice with TVM problems, check out our CFA Level I question bank — we have dozens of unequal cash flow scenarios with full solutions.
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