# NPV and IRR

Business I · Economics of the Firm · https://tryals.app/learn/business-i/npv-and-irr

## NPV and Decision Rules

**Discounting** converts future cash into today's units: $PV = \frac{CF_t}{(1+r)^t}$, where $r$ is the **discount rate** (its opportunity cost). 

**Net Present Value** sums a project's entire cash life in today's euros:

$$NPV = -C_0 + \sum_{t=1}^{T} \frac{CF_t}{(1+r)^t}$$

**Accept if NPV > 0**: the project creates value beyond alternatives. 

**Worked example.** Invest 100k€ now, receive 60k€ in year 1 and 60k€ in year 2, with $r = 10\%$: 

$$NPV = -100 + \frac{60}{1.1} + \frac{60}{1.21} = 4.1\text{k EUR} > 0$$

**Internal Rate of Return** asks: what discount rate makes $NPV = 0$? Accept when $IRR > r$.

| Rule | Question | Accept when |
| --- | --- | --- |
| NPV | Value in today's euros? | $NPV > 0$ |
| IRR | Return the project earns? | $IRR > r$ |

> **Pitfall:** A positive NPV means the project beats the alternative investment, not just "makes money." A project earning 3% when $r = 10\%$ makes money but still destroys value.

## The NPV Profile

Plot a project's NPV against the discount rate and you get its **NPV profile**: a downward-sloping curve holding both decision rules at once. 

**Why it slopes down**: a higher $r$ shrinks every future cash flow, hitting distant cash hardest via $(1+r)^t$.

**Reading the curve**

- The **height at your discount rate** gives the NPV value verdict.
- The **horizontal crossing point** is the IRR, where NPV hits zero.

**Where rankings flip.** Compare two mutually exclusive projects: one pays early, the other pays late and big. The late-cash project has a **steeper** profile, so the two profiles cross. At low discount rates, the big-late project wins on NPV; at high rates, the early-cash project wins.

> **Pitfall:** Ranking mutually exclusive projects by IRR. When profiles cross, the project with the higher IRR can have the **lower** NPV at your actual discount rate. Always let NPV measure value created.

## Practice questions

5 of this lesson's 12 practice questions, with answers. The full set is in the app.

### 1. A project’s IRR is 18% and the firm’s cost of capital is 11%. By the IRR rule, what should the firm do?

A. Accept: the project earns more than the capital costs
B. Accept only if the payback period is under two years
C. Nothing can be concluded without the NPV profile
D. Reject: the IRR should be below the cost of capital

**Answer:** A. Accept: the project earns more than the capital costs

**Why:** IRR 18% > cost of capital 11% → accept; equivalently, NPV at 11% is positive. For a single conventional project the two rules agree.

Page: https://tryals.app/practice/business-i/npv-and-irr/a-projects-irr-is-18-and-the-firms-cost-of-capital-is-11-by-the

### 2. Order the steps of a discounted-cash-flow appraisal.

**Answer:**

1. Forecast the project’s cash flows year by year
2. Choose the discount rate from the opportunity cost of capital
3. Discount each cash flow to present value
4. Sum the PVs and subtract the initial outlay
5. Accept if NPV is positive

**Why:** Forecast → rate → discount → sum → decide: the DCF pipeline behind every serious investment memo.

Page: https://tryals.app/practice/business-i/npv-and-irr/order-the-steps-of-a-discounted-cash-flow-appraisal

### 3. A project costs 200k€ and pays 110k€ at the end of each of years 1 and 2. At $r = 10\%$, set the slider to its NPV in k€.

**Answer:** -9.1 (within ±3)

**Why:** $NPV = -200 + 100 + 90.9 = -9.1$k€. The undiscounted total (+20k€) looked fine, discounting exposes the loss. That gap is the whole reason NPV exists.

Page: https://tryals.app/practice/business-i/npv-and-irr/a-project-costs-200k-and-pays-110k-at-the-end-of-each-of-years-1

### 4. Two projects both have an IRR of 20%. One is a 5,000€ kiosk, the other a 5 million€ factory. Why does the firm still need NPV to choose?

A. Because IRR ignores the time value of money, which NPV incorporates
B. Because the two projects must have different discount rates
C. Because IRR is a rate and says nothing about size
D. Because IRR is unreliable and NPV is always the more accurate calculation

**Answer:** C. Because IRR is a rate and says nothing about size

**Why:** A percentage is scale-blind by construction, and shareholders bank euros rather than rates. Option 3 gets IRR backwards, it is defined by discounting, being the rate that sets NPV to zero. This is why capital budgeting ranks on NPV and reports IRR alongside as a sanity check.

Page: https://tryals.app/practice/business-i/npv-and-irr/two-projects-both-have-an-irr-of-20-one-is-a-5-000-kiosk-the

### 5. Match each appraisal concept to its meaning.

**Answer:**

- Discount rate → Opportunity cost of the invested money
- NPV → Project value in today’s euros
- IRR → Rate at which NPV equals zero
- Payback period → Time until cumulative cash recovers the outlay

**Why:** Four lenses on one project, with NPV as the master metric because it measures value, not speed or percentage.

Page: https://tryals.app/practice/business-i/npv-and-irr/match-each-appraisal-concept-to-its-meaning