Estimate IRR by interpolation
Estimate the zero-NPV rate between trial rates with opposite NPV signs.
Section 4.4 · Investment appraisal
Exam scope and assessment notes
Cambridge International AS & A Level Accounting 9706, 2026–2028 syllabus, Version 2. This topic covers future net cash flows, payback, ARR, NPV, IRR and supported investment decisions. Discounted payback and questions involving a residual value at the end of a project are excluded by Cambridge. Practice is self-checked, not an official Cambridge grade. Full timed Papers 1–4 and human subject-teacher review remain outstanding.
Before you begin
- Use AS Topics 1.5–1.6 for profit, depreciation and financial analysis, and Topic 2.1 for contribution. Keep accounting profit, project cash flow and the time value of money distinct.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
NPV is +$1,000 at 8% and −$1,000 at 12%. Estimate IRR.
What you will learn
- Identify the lower and higher trial rates
- Apply interpolation
- Explain why the result is approximate
Interpolate between opposite signs
With NPV positive at lower rate L and negative at higher rate H, approximate IRR = L + [NPV at L ÷ (NPV at L − NPV at H)] × (H − L). Keeping the negative NPV signed is equivalent to adding its absolute value in the denominator.
This estimates where NPV crosses zero. It is approximate because the NPV curve is not exactly a straight line across a wide interval.
Check the rate and sign
The fraction should lie between 0 and 1 when trial NPVs have opposite signs; the estimate should lie between the rates.
Use the same currency and period assumptions at both trial rates. Two NPVs with the same sign do not bracket IRR.
Pause and explain
NPV is +$3,000 at 5% and −$1,000 at 9%. What fraction of the rate interval is added above 5%?
Worked example
NPV is +$2,400 at 10% and −$1,600 at 15%. Estimate IRR.
Show the worked solution
- Rate interval = 5 percentage points.
- Positive share = 2,400 ÷ (2,400 + 1,600) = 0.6.
- Estimate = 10% + 0.6 × 5% = 13%.
Answer Estimated IRR is approximately 13%.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
NPV is +$1,800 at 11% and −$1,200 at 16%. Estimate IRR.
Give me a hint
Use positive NPV divided by the absolute sign-change total.
Compare my reasoning
- Fraction = 1,800 ÷ 3,000 = 0.6.
- Interval = 5 percentage points.
- Estimate = 11% + 0.6 × 5% = 14%.
Estimated IRR is approximately 14%.
Look for these in your work
- Fraction correct
- Interval correct
- Rate within bracket
NPV is +$5,000 at 9% and −$3,000 at 17%. Estimate IRR.
Give me a hint
Find the positive-NPV share of the sign change.
Compare my reasoning
- Fraction = 5,000 ÷ 8,000 = 0.625.
- Interval = 8 percentage points.
- Estimate = 9% + 0.625 × 8% = 14.0%.
Estimated IRR is 14.0%.
Look for these in your work
- Signed values handled
- Interval correct
- Result rounded
An IRR estimate is 18.5%, but trial rates were 10% and 15%. What should you check?
Give me a hint
A valid interpolation remains between trial rates.
Compare my reasoning
- Recheck the NPV signs and rate order.
- Check the formula and opposite-sign requirement.
- Recalculate before using the estimate.
The result is outside the bracket, so recheck signs, rate order or arithmetic.
Look for these in your work
- Range check used
- Potential error located
- Decision held
Common mistakes
- Using a wrong sign in the interpolation ratio
- Using only positive NPV as denominator
- Reporting a rate outside the trial interval
What can you explain now?
For interpolation, what should the trial NPVs normally show?
Compare with the explanation
Opposite signs
Opposite signs place zero between the tested rates.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
State interpolation in words and give one reason the result is approximate.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.