# Internal Rate of Return (IRR)

## Definition

IRR is the discount rate at which the Net Present Value (NPV) of a project is exactly zero. Equivalently, it is the rate of return at which:

> Present value of cash inflows = Present value of cash outflows

IRR is the project's own intrinsic rate of return, found by trial and error.

## Why trial and error?

There is no direct algebraic formula for IRR with uneven cash flows, so we guess two discount rates, compute NPV at each, and then interpolate to pin down the exact rate.

## Steps to calculate IRR

Step 1 — Find two bracketing rates (trial & error).

Discount the cash flows at a trial rate and observe the NPV, then adjust:

If NPV > 0 → the rate is too low → increase the discount rate.
If NPV < 0 → the rate is too high → decrease the discount rate.

Keep going until you have one rate giving a positive NPV (Lower rate, L) and one giving a negative NPV (Higher rate, H). The true IRR lies between them.

Step 2 — Interpolate between the two rates.

```

IRR = L + [ NPV_L / (NPV_L − NPV_H) ] × (H − L)

```

Where:

`L₹ = Lower rate
`H₹ = Higher rate
`NPV_L₹ = NPV at the lower rate (positive)
`NPV_H₹ = NPV at the higher rate (negative)

> Note: `NPV_H₹ is negative, so `NPV_L − NPV_H₹ becomes `NPV_L + |NPV_H|₹ — the full spread of NPVs across the two rates.

## Decision use

IRR is later used as a benchmark: a project is acceptable when its IRR exceeds the required rate of return (cost of capital).

Worked example

### Example 1

IRR by interpolation: A project's NPV is +₹500 at 10% and −₹300 at 15%.

`IRR = L + [NPV_L / (NPV_L − NPV_H)] × (H − L)`

`IRR = 10 + [500 / (500 − (−300))] × (15 − 10)`

`IRR = 10 + [500 / 800] × 5`

`IRR = 10 + 3.125 = 13.125%`

So the project's internal rate of return is approximately 13.13%.

### Example 2

Choosing trial rates: At a first trial of 12% you get NPV = +₹120 (positive → increase rate). At 16% you get NPV = −₹40 (negative → you now bracket the IRR between 12% and 16%, ready to interpolate).

⚠️ Common exam mistakes

Sign errors in interpolation — NPV_H is negative, so the denominator (NPV_L − NPV_H) must be larger than NPV_L, not smaller.
Picking two trial rates that give NPVs of the same sign; the IRR must be bracketed by one positive and one negative NPV.
Treating the interpolated IRR as exact — interpolation gives an approximation; wider gaps between L and H reduce accuracy.

Reference:

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Internal Rate of Return (IRR) and the interpolation method

Worked example

⚠️ Common exam mistakes