Charta is on Google Play: practise on the go. Get the app
Charta

Work out net present value and the internal rate of return

CFA® Level I · Quantitative Methods · The Time Value of Money in Finance · QM.3.a

The idea

Net present value discounts every cash flow of a project at the required return and subtracts the outlay, so a positive NPV means the project adds value at that required return. The internal rate of return is the discount rate that sets NPV to zero, which answers a different question: not how much value, but at what rate. The two agree on accept-or-reject for a single conventional project and can disagree on ranking two, because IRR implicitly assumes cash flows are reinvested at the IRR itself. When they conflict, follow NPV — it is denominated in money, which is what shareholders actually receive.

Formulas

Net present value

NPV = Σ CFₜ ÷ (1 + r)ᵗ

CFₜ
the cash flow at time t, with the initial outlay as a negative CF₀
r
the required rate of return
t
the period, counted from 0

Accept if NPV is positive. Where NPV and IRR disagree on mutually exclusive projects, NPV wins.

A practice question

From the question bank, with the reason each wrong option is wrong.

Net present value is the present value of a project's future cash flows minus:

  1. the initial investment — correct
  2. the discount rate appliedWhy not: A rate cannot be subtracted from an amount of money. The rate does its work inside the discounting, before anything is netted off.
  3. its internal rate of returnWhy not: Also a rate, and also already used elsewhere: IRR is what you compare against the required return, not something you subtract.

NPV measures value added in money, so the outlay has to be netted off the discounted inflows.

A harder one

A project costs $1,000 today and pays $1,210 in two years. At a 10% required return, its NPV is closest to:

  1. $210Why not: The undiscounted surplus, 1,210 − 1,000. Waiting two years for that 1,210 costs exactly the 210.
  2. $100Why not: One year of discounting at 10% on the outlay, not a present value of the inflow. Discount 1,210 by 1.10² and it comes back to 1,000.
  3. $0 — correct

Discounting 1,210 at 10% for two years gives exactly 1,000, so it breaks even — which also makes 10% its IRR.

The hardest tier

A project costs $500 now and pays $200 at the end of each of the next three years. At an 8% required return, its NPV is closest to:

  1. $100Why not: 600 − 500, the undiscounted surplus. Discounting the three inflows takes the total from 600 to 515.42.
  2. -$15Why not: The right magnitude with the sign reversed. The discounted inflows come to 515.42, which is above the 500 outlay, not below it.
  3. $15 — correct

The three inflows discount to 185.19, 171.47 and 158.77, a total of 515.42 against a 500 outlay.

In the app: 18 practice questions on this outcome, a narrated walkthrough with diagrams, spaced review that brings it back before you forget it.

Take the free diagnostic →

More in The Time Value of Money in Finance