Internal Rate of Return

Internal Rate of Return

The Internal Rate of Return (IRR) is the discount rate that makes an investment's NPV equal to zero:

$\sum_{t=0}^{n} \frac{CF_t}{(1+IRR)^t} = 0$

Finding IRR with Bisection

There is no closed-form solution for IRR in general. We use bisection search:

1. Set bounds: lo = -0.9999, hi = 10.0
2. Compute midpoint: mid = (lo + hi) / 2
3. If npv(mid, cashflows) > 0: IRR is higher → lo = mid
4. If npv(mid, cashflows) < 0: IRR is lower → hi = mid
5. Repeat ~1000 times for convergence

Decision Rule

• IRR > hurdle rate: Accept the investment
• IRR < hurdle rate: Reject the investment

Example

For cash flows [-1000, 300, 400, 500]: IRR ≈ 8.9%

This means the investment earns about 8.9% annually. If your required return is 10%, reject it (matches our NPV result).