Internal Rate of Return (IRR) & Modified IRR

Internal Rate of Return (IRR) and Modified Internal Rate of Return (MIRR)

Internal Rate of Return (IRR) Definition: The Internal Rate of Return (IRR) is the discount rate that makes the Net Present Value (NPV) of a project equal to zero. It represents the expected annualized return of an investment.

Formula: The IRR is found by solving the following equation:

Sum of (Cash Flow at time t / (1 + IRR)^t) = 0

where:

Cash Flow at time t = Cash flow at time t

IRR = Internal Rate of Return

t = Time period

Explanation:

If IRR is greater than the Cost of Capital, the project is accepted as it generates a return higher than the required rate.

If IRR is less than the Cost of Capital, the project is rejected as it fails to meet the required return.

Example: Consider an investment requiring an initial cost of $10,000, with cash inflows of $4,000, $4,000, and $4,000 over three years. To find IRR, we solve:

0 = -10,000 + (4,000 / (1+IRR)^1) + (4,000 / (1+IRR)^2) + (4,000 / (1+IRR)^3)

Using numerical methods, we find IRR is approximately 18.82%.

Modified Internal Rate of Return (MIRR) Definition: The Modified Internal Rate of Return (MIRR) is an improvement over IRR that assumes positive cash flows are reinvested at the firm's cost of capital (rather than IRR itself), thereby providing a more realistic measure of investment profitability.

Formula:

MIRR = [(Future Value of positive cash flows / Present Value of negative cash flows)^(1/n)] - 1

where:

Future Value of positive cash flows = Future value of positive cash flows compounded at the reinvestment rate.

Present Value of negative cash flows = Present value of negative cash flows discounted at the financing cost.

n = Number of years.

Explanation:

Unlike IRR, MIRR eliminates multiple IRR issues and provides a single, realistic rate of return.

MIRR considers both financing cost (for cash outflows) and reinvestment return (for cash inflows), making it a better decision-making tool.

Example: Using the same example as above:

Assume the cost of capital is 10%.

The future value of cash inflows at 10% reinvestment rate:
- Year 1: 4,000 × (1.1)^2 = 4,840
- Year 2: 4,000 × (1.1)^1 = 4,400
- Year 3: 4,000 (no compounding needed)
- Total Future Value of positive cash flows = 4,840 + 4,400 + 4,000 = 13,240

The present value of the initial investment remains $10,000.

Now applying the MIRR formula:

MIRR = [(13,240 / 10,000)^(1/3)] - 1

MIRR = (1.324)^(1/3) - 1

MIRR is approximately 10.49%.

Key Differences Between IRR and MIRR:

Feature	IRR	MIRR
Assumption on Reinvestment	Reinvests at IRR	Reinvests at cost of capital
Number of Possible Rates	Multiple IRRs possible	Only one MIRR
More Realistic?	Less realistic	More realistic
Decision Criterion	Compare with cost of capital	More accurate profitability measure

Conclusion:

IRR is useful but can sometimes give misleading results due to multiple IRRs.

MIRR provides a more accurate reflection of a project’s profitability by assuming reinvestment at a more realistic rate.

MIRR is preferred for making capital budgeting decisions when comparing multiple projects.