Rate of Return Analysis: IRR, MIRR, Payback Period, Comparison

TAKEAWAYS:

• IRR is the discount rate that makes the Net Present Value (NPV) of cash flows zero—it measures a project’s profitability without external benchmarks.
• MIRR adjusts IRR by assuming reinvestment at the minimum attractive rate of return (MARR), making it more realistic for projects with uneven cash flows.
• Payback Period tells how long it takes to recover the initial investment—useful for liquidity but ignores time value of money beyond the cutoff.
• IRR vs. MIRR vs. Payback differ in reinvestment assumptions, cash flow handling, and suitability for multiple projects.
• Real-world use: Banks (loan interest rates), NEPSE (stock returns), and Pathao (ride-hailing profitability) rely on these metrics.
• Exam focus: Worked examples (IRR calculation by trial-and-error or formula), comparisons, and interpreting results for accept/reject decisions.

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1. Internal Rate of Return (IRR): The Discount Rate That Makes NPV Zero

IRR is the interest rate at which the present value of all cash inflows equals the present value of cash outflows, making NPV = 0. It answers: "What return does this project generate if we invest today?"

How IRR Works

• Solve for $r$ in: $$\text{NPV}=\sum _{t=0}^n\frac{C{F}_t}{(1+r{)}^t}=0$$ where $C{F}_t$ = cash flow at time $t$, $n=$ project life.
• Graphical method: Plot NPV vs. discount rate; IRR is where the curve crosses zero.
• Trial-and-error: Test discount rates until NPV ≈ 0 (common in exams).

Worked Example: NEPSE Stock Investment

Suppose you buy a stock for Rs. 50,000 and expect cash flows of Rs. 15,000/year for 5 years. What is the IRR?

Cash Flow Diagram:

Year 0: -50,000
Year 1:  15,000
Year 2:  15,000
Year 3:  15,000
Year 4:  15,000
Year 5:  15,000 + salvage value (if any)

Trial-and-Error Calculation:

Interpretation:

• If your MARR (minimum attractive rate) is 10%, IRR (12%) > MARR → Accept the project.
• If MARR were 15%, IRR (12%) < MARR → Reject.

Limitations of IRR

• Multiple IRRs: Possible if cash flows change signs more than once (e.g., initial outflow, inflow, then outflow again).
• Reinvestment assumption: IRR assumes intermediate cash flows are reinvested at the IRR itself (often unrealistic).
• Cannot rank mutually exclusive projects: IRR may conflict with NPV for projects of different scales.

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2. Modified Internal Rate of Return (MIRR): Fixing IRR’s Reinvestment Flaw

MIRR adjusts IRR by:

1. Reinvesting intermediate cash inflows at the MARR (not IRR).
2. Financing cash outflows at the MARR (borrowing cost).

Formula: $$\text{MIRR}={\left(\frac{\text{Future Value of Inflows}}{\text{Present Value of Outflows}}\right)}^{\frac{1}{n}}-1$$ where:

• Future Value of Inflows = ${\sum }_{t=1}^{n}C{F}_t\times (1+\text{MARR}{)}^{n-t}$
• Present Value of Outflows = Initial investment.

Worked Example: Daraz E-Commerce Expansion

Daraz invests Rs. 1,000,000 in a new warehouse. Cash inflows:

• Year 1: Rs. 300,000
• Year 2: Rs. 400,000
• Year 3: Rs. 500,000 Assume MARR = 10%.

Step 1: Calculate Future Value of Inflows (reinvest at 10%)

• Year 1 → Year 3: $300,000\times (1.10{)}^2=363,000$
• Year 2 → Year 3: $400,000\times (1.10{)}^1=440,000$
• Year 3: $500,000$ Total FV Inflows = 363,000 + 440,000 + 500,000 = Rs. 1,303,000

Step 2: Present Value of Outflows Only initial investment: Rs. 1,000,000

Step 3: Compute MIRR $$\text{MIRR}={\left(\frac{1,303,000}{1,000,000}\right)}^{\frac{1}{3}}-1=8.8\%\approx 9\%$$

Comparison with IRR:

• IRR (trial-and-error) ≈ 15% (overestimates due to reinvestment assumption).
• MIRR (9%) is more conservative and realistic.

Advantages of MIRR

• Single answer: No multiple IRR problem.
• Realistic reinvestment: Uses MARR, not IRR.
• Better for ranking: Works well with NPV for mutually exclusive projects.

When to Use MIRR Over IRR

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3. Payback Period: How Long to Recover Investment?

Payback Period (PP) is the time required to recover the initial investment from cash inflows. It ignores:

• Time value of money (after payback year).
• Cash flows after payback.

Formula: $$\text{PP}=\text{Year before full recovery}+\frac{\text{Remaining cost}}{\text{Cash flow in payback year}}$$

Worked Example: Ncell Tower Installation

Ncell spends Rs. 20,000,000 on a new 5G tower. Annual cash inflows:

• Year 1: Rs. 6,000,000
• Year 2: Rs. 7,000,000
• Year 3: Rs. 8,000,000

Calculation:

• Year 1: Rs. 6M → Remaining = 20M - 6M = 14M
• Year 2: Rs. 7M → Remaining = 14M - 7M = 7M
• Year 3: Rs. 8M → Full recovery in Year 3.

But since Year 2 only covers 7M, the exact PP is: $$\text{PP}=2+\frac{7,000,000}{8,000,000}=2.875\text{ years}$$

Interpretation:

• If Ncell’s payback cutoff is 3 years, this project is acceptable.
• Limitation: Ignores cash flows after Year 3 (e.g., Rs. 8M in Year 3 is more valuable than Rs. 6M in Year 1).

Advantages and Disadvantages of Payback Period

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4. Comparing IRR, MIRR, and Payback Period

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5. Real-World Applications in Nepal

1. Banking (Loan Interest Rates)

• Concept Used: IRR/MIRR to determine loan profitability.
• Example: Nabil Bank evaluates a Rs. 50M commercial loan with repayments of Rs. 12M/year for 5 years.
  • IRR ≈ 14% → If bank’s cost of funds is 10%, the loan is profitable.
  • MIRR (using bank’s deposit rate as MARR) gives a more accurate return.

2. NEPSE (Stock Market Investments)

• Concept Used: IRR for expected returns.
• Example: An investor buys a stock for Rs. 10,000 and expects dividends of Rs. 1,500/year for 4 years, then sells for Rs. 12,000.
  • IRR ≈ 11% → If the investor’s expected return is 10%, this stock is a good buy.

3. Pathao (Ride-Hailing Profitability)

• Concept Used: Payback Period for driver investments.
• Example: A driver spends Rs. 200,000 on a bike and earns Rs. 8,000/month.
  • PP = 200,000 / (8,000 × 12) ≈ 2.1 years.
  • If Pathao’s policy requires PP < 3 years, the driver’s investment is acceptable.

4. Infrastructure Projects (Roads, Hydroelectricity)

• Concept Used: MIRR for government projects.
• Example: The government invests Rs. 10B in a hydropower plant with Rs. 3B/year for 5 years.
  • MIRR (using 8% MARR) ≈ 12% → Justifies the project if social benefits exceed costs.

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6. Visualizing Cash Flow Patterns

IRR vs. MIRR Reinvestment Assumptions

Payback Period vs. NPV

IRR Curve (NPV vs. Discount Rate)

• X-axis: Discount rate (%)
• Y-axis: NPV (Rs.)
• Key Point: IRR is where NPV = 0 (here, 12%).

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7. Exam Tip: How to Score Full Marks

1. Always draw a cash flow diagram before calculations (worth 2–3 marks).
2. For IRR:
   • Show trial-and-error steps (even if using a calculator).
   • State whether to accept/reject based on MARR.
3. For MIRR:
   • Clearly label MARR and show reinvestment at MARR.
   • Compare with IRR in your answer.
4. For Payback Period:
   • Calculate exact years (not just whole years).
   • Discuss limitations (ignores TVM, post-PP cash flows).
5. Comparison Questions:
   • Use a table (like above) to contrast IRR, MIRR, and PP.
   • Mention when each is preferred (e.g., PP for liquidity, MIRR for ranking).

Common Mistakes to Avoid:

• Forgetting to compare with MARR in IRR/MIRR questions.
• Misapplying the payback formula (e.g., not dividing by the cash flow in the payback year).
• Assuming IRR > MARR always means accept (check NPV if projects are mutually exclusive).

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8. Practice Problem (Exam-Style)

Question: A company is evaluating two projects:

• Project X: Initial cost = Rs. 500,000; Cash flows = Rs. 150,000/year for 5 years.
• Project Y: Initial cost = Rs. 800,000; Cash flows = Rs. 200,000/year for 4 years. MARR = 10%.

1. Calculate IRR and MIRR for both projects.
2. Which project should be chosen if they are mutually exclusive? Justify using NPV and MIRR.
3. What is the payback period for each? Is it a reliable metric here?

Solution Outline:

1. IRR:
   • Project X: ≈ 12% (trial-and-error).
   • Project Y: ≈ 10%.
2. MIRR (using 10% MARR):
   • Project X: ≈ 11%.
   • Project Y: ≈ 10%.
3. NPV at 10%:
   • Project X: NPV ≈ +Rs. 25,000.
   • Project Y: NPV ≈ +Rs. 30,000.
4. Payback Period:
   • Project X: 3.33 years.
   • Project Y: 4 years.
5. Decision:
   • Choose Project Y (higher NPV and MIRR).
   • Payback is less reliable because it ignores cash flows after Year 4 (Project Y) vs. Year 5 (Project X).