--- title: "Investment appraisal: NPV, IRR and payback" description: "The time value of money A dollar, rupee or dirham today is worth more than the same amount in the future, because it can be invested and because the…" url: https://optimizeall.com/learn/project-finance-and-financial-modelling/npv-irr-payback updated: 2026-10-05 --- Project Finance & Financial Modelling · Project finance foundations · lesson 3 of 20 · 16 min # Investment appraisal: NPV, IRR and payback ## The time value of money A dollar, rupee or dirham today is worth more than the same amount in the future, because it can be invested and because the future is uncertain. **Discounting** converts future cash flows into today's value: ``` Present value (PV) = Cash flow in year t / (1 + r)^t NPV = Σ [CF_t / (1 + r)^t] for t = 0 ... n (CF_0 is usually the negative investment) ``` The discount rate *r* reflects the cost of capital and the risk of the cash flows. ## Net present value (NPV) NPV is the sum of all discounted cash flows. **Decision rule:** accept projects with NPV > 0 (they create value above the required return); among mutually exclusive projects, prefer the higher NPV, all else being equal. ## Worked example: Project A *Illustrative.* Investment 1,000 (in thousands of any currency) now; cash inflows of 300, 350, 400 and 300 in years 1–4; discount rate 10%. | Year | Cash flow | Discount factor (10%) | PV | |---|---|---|---| | 0 | −1,000 | 1.0000 | −1,000.0 | | 1 | 300 | 0.9091 | 272.7 | | 2 | 350 | 0.8264 | 289.3 | | 3 | 400 | 0.7513 | 300.5 | | 4 | 300 | 0.6830 | 204.9 | | **NPV** | | | **67.4** | NPV is positive, so the project earns more than 10%. ## Internal rate of return (IRR) The **IRR** is the discount rate at which NPV = 0. For Project A, trial and error (or a spreadsheet IRR function) gives an IRR of about **13.0%**. Since 13.0% exceeds the 10% required return, the project is acceptable, consistent with the positive NPV. Useful cautions: - IRR ignores scale: a small project can have a higher IRR but create less value than a large one. - Non-conventional cash flows (negative, positive, then negative again, e.g., decommissioning) can produce multiple IRRs. - IRR implicitly assumes interim cash flows are reinvested at the IRR, which may be unrealistic; **modified IRR (MIRR)** addresses this. - In project finance, you will see both **project IRR** (on total project cash flows, before financing) and **equity IRR** (on sponsors' cash flows after debt service). Leverage typically makes equity IRR higher than project IRR, at the cost of higher risk. ## Payback period **Payback** is the time until cumulative cash flows turn positive. For Project A: cumulative −1,000, −700, −350, +50. Payback = 2 + 350/400 = **2.9 years**. **Discounted payback** uses discounted cash flows: cumulative −1,000, −727, −438, −138, +67. Discounted payback = 3 + 137.5/204.9 ≈ **3.7 years**. Payback is simple and useful as a liquidity and risk indicator, but it ignores cash flows after the payback point and (in its simple form) the time value of money. Use it alongside NPV, not instead of it. ## Why NPV is the primary measure *Illustrative.* Project B costs 1,000 and returns 0, 0, 200, 600, 700 in years 1–5. Undiscounted, B returns 1,500 versus A's 1,350, which looks better. At 10%, B's NPV is about **−5**, while A's is +67. B's cash arrives too late to cover the cost of capital. NPV captures this; simple totals and payback do not. ## Choosing the discount rate - For a company, a common starting point is the **weighted average cost of capital (WACC)**, adjusted for the specific project's risk. - For project finance equity, sponsors set a target equity return (hurdle rate) reflecting project and country risk. - Higher-risk markets or merchant (uncontracted) revenues justify higher rates. - Be consistent: nominal cash flows (including inflation) use nominal rates; real cash flows use real rates. ## Appraisal template ``` Project: ______ Currency: ______ Nominal/real: ______ Discount rate: ____% (basis: ______) Year: 0 1 2 3 ... n Capex Revenue Opex Tax Net cash flow Discount factor PV NPV: ____ IRR: ____ Payback: ____ Discounted payback: ____ Key sensitivities: ______ ``` ## Common mistakes - Mixing nominal cash flows with real discount rates. - Using accounting profit instead of cash flow. - Ignoring terminal or residual values, or decommissioning costs. - Ranking projects by IRR alone. - Including sunk costs (money already spent that cannot be recovered). ## Hands-on: appraisal in Excel, avoiding the NPV trap ```text Row 1: Year 0..4 in B1:F1 Row 2: Cash flow -1000, 300, 350, 400, 300 in B2:F2 Correct NPV =B2+NPV(10%, C2:F2) → 67.4 Common error =NPV(10%, B2:F2) → 61.3 (discounts year 0 by one period) IRR =IRR(B2:F2) → ≈ 13.0% MIRR =MIRR(B2:F2, 10%, 10%) finance and reinvestment rates stated explicitly Cumulative B3 =B2 ; C3 =B3+C2 (fill right) Payback =MATCH(TRUE, INDEX(C3:F3>=0,0), 0) - 1 + ( -INDEX(B3:F3, MATCH(TRUE, INDEX(C3:F3>=0,0),0)) / INDEX(C2:F2, MATCH(TRUE, INDEX(C3:F3>=0,0),0)) ) Dated cash flows =XNPV(Rate, Values, Dates) and =XIRR(Values, Dates) ``` The payback formula finds the first year with non-negative cumulative cash and interpolates within it (2 + 350/400 = 2.9 years). If it feels opaque, use a helper row instead: clarity beats cleverness in a model others must audit. ## Hands-on: the same in Python ```python import numpy as np import numpy_financial as npf cf = np.array([-1000, 300, 350, 400, 300], dtype=float) rate = 0.10 print(f"NPV {npf.npv(rate, cf):.1f}") # numpy-financial treats cf[0] as t = 0 → 67.4 print(f"IRR {npf.irr(cf):.2%}") # ≈ 13.0% for r in (0.08, 0.12): print(f"NPV at {r:.0%}: {npf.npv(r, cf):.1f}") cum = cf.cumsum() k = int(np.argmax(cum >= 0)) # first year cumulative cash is non-negative payback = (k - 1) + (-cum[k - 1] / cf[k]) disc = cf / (1 + rate) ** np.arange(len(cf)) dcum = disc.cumsum() j = int(np.argmax(dcum >= 0)) print(f"Payback {payback:.1f} y; discounted payback {(j - 1) + (-dcum[j - 1] / disc[j]):.1f} y") ``` ## How to measure success - NPV reconciles between hand calculation, spreadsheet and code. - The discount rate, its basis and nominal/real convention are stated on every appraisal. - Decisions cite NPV first, with IRR and payback as supporting indicators. ## Video lecture: Investment appraisal: NPV, IRR and payback Lecture coming soon · 9 chapters · about 9 minutes. Read the full transcript below. 1. Is this project worth doing? 2. Why it matters 3. The concept: discounting 4. Worked example one: Project A 5. Worked example two: why NPV leads 6. Watch me do it: NPV in Excel without the trap 7. Project IRR versus equity IRR 8. Choosing the discount rate 9. Recap and try this now ## Lecture transcript ### Is this project worth doing? Would you rather have a thousand pounds today, or a thousand pounds in five years? Almost everyone chooses today, and they're right, for two reasons. Money today can be invested to earn a return. And the future is uncertain. That simple intuition is the foundation of every investment decision a board, lender or investor makes. In this lecture you'll learn how discounting turns future cash flows into today's value, how to calculate net present value and internal rate of return by hand and in a spreadsheet, how payback and discounted payback work, and why NPV should lead the decision. We'll also cover a spreadsheet trap that catches experienced analysts. By the end, you'll be able to appraise a project and explain the result in two sentences. ### Why it matters Why does this matter? Because nearly every capital decision compares money that moves at different times. Spend now, earn later. And simple approaches mislead. Add up undiscounted cash flows and a project with late returns can look better than one with early returns, when it's actually worse. Use payback alone and you ignore everything after the payback date. And even when people use the right method, small technical errors, such as the wrong discount rate or the wrong spreadsheet function, can flip the answer. Investment appraisal is where finance meets strategy, so getting it right matters far beyond the finance team. ### The concept: discounting Here's the concept. Discounting converts a future cash flow into today's value. The present value of a cash flow in year t is that cash flow divided by one plus the discount rate, raised to the power t. Net present value, NPV, is the sum of all those discounted cash flows, including the initial investment in year zero, which is usually negative. The decision rule is simple: accept projects with a positive NPV, because they create value above the required return, and among mutually exclusive options, prefer the higher NPV. The internal rate of return, IRR, is the discount rate at which NPV equals exactly zero. Think of discounting like a currency exchange between years. A pound in year four is converted into today's pounds at an exchange rate set by your required return. NPV simply adds everything up in one currency. ### Worked example one: Project A Let's work Project A from the lesson, in thousands of any currency. Invest one thousand now, then receive three hundred, three fifty, four hundred and three hundred in years one to four. The discount rate is ten per cent. Year one: three hundred times nought point nine zero nine one, about two seventy-three. Year two: three fifty times nought point eight two six four, about two eighty-nine. Year three: four hundred times nought point seven five one three, about three hundred. Year four: three hundred times nought point six eight three, about two oh five. Add them up and subtract the thousand: NPV is about sixty-seven. Positive, so it earns more than ten per cent. The IRR, the rate where NPV hits zero, is about thirteen per cent. Simple payback: cumulative cash turns positive during year three, at about two point nine years. Discounted payback is later, about three point seven years. ### Worked example two: why NPV leads Now the example that shows why NPV should lead. Project B also costs one thousand, but returns nothing in years one and two, then two hundred, six hundred and seven hundred in years three to five. Undiscounted, B returns fifteen hundred against A's thirteen fifty, so it looks better. At ten per cent, B's NPV is about minus five. Its cash arrives too late to cover the cost of capital. Now some IRR cautions. IRR ignores scale: a small project can have a higher IRR but create less value. Cash flows that change sign more than once, for example with decommissioning costs at the end, can produce multiple IRRs. And IRR implicitly assumes interim cash is reinvested at the IRR itself; modified IRR addresses that. In project finance you'll see both project IRR, before financing, and equity IRR, after debt service. Leverage usually lifts equity IRR, at the cost of higher risk. ### Watch me do it: NPV in Excel without the trap Let me show you the trap. Project A's cash flows sit in one row, year zero to year four. The tempting formula is equals NPV of ten per cent and the whole row. It returns about sixty-one. That's wrong, and here's why. Excel's NPV function assumes the first value arrives at the end of period one, so it discounts your year-zero investment by a year. The correct formula is the year-zero cash flow, plus NPV of the rate and years one to four only. That gives sixty-seven point four, matching our hand calculation. IRR takes the whole row, including year zero, and gives about thirteen per cent. And when cash flows fall on real, uneven dates, which they almost always do in project finance, use XNPV and XIRR with a date row. The Python library numpy-financial behaves differently again: its NPV treats the first value as time zero. Always check which convention your tool uses. ### Project IRR versus equity IRR In project finance you'll meet two IRRs, and mixing them up causes real confusion. Project IRR is calculated on the total project cash flows before financing: all the capital expenditure going out, and all the cash flow available for debt service coming back. It tells you how good the asset is, regardless of how it's funded. Equity IRR is calculated on the sponsors' own cash flows: the equity they inject, and the distributions they receive after debt service, reserves and lock-up tests. Because debt usually costs less than the project earns, leverage typically lifts equity IRR above project IRR. The illustrative badges show nine and fourteen per cent. But remember the last lecture's double edge: that higher equity return carries higher risk. So compare project IRR with a project-level hurdle, equity IRR with the sponsor's equity hurdle, and never quote one as though it were the other. ### Choosing the discount rate Which discount rate should you use? For a company, a common starting point is the weighted average cost of capital, adjusted for the specific project's risk. For project finance equity, sponsors set a target equity return, a hurdle rate, reflecting project and country risk. Higher-risk markets, or merchant revenue without a long-term contract, justify higher rates. And be consistent. Nominal cash flows, which include inflation, go with nominal discount rates. Real cash flows go with real rates. Mixing them is one of the most common appraisal errors I see. The others: using accounting profit instead of cash flow, ignoring residual values or decommissioning costs, ranking projects by IRR alone, and including sunk costs, money already spent that can't be recovered. ### Recap and try this now Let's recap. Money today is worth more than money tomorrow, so discount future cash flows to compare them. NPV adds up all discounted cash flows, and a positive NPV means the project creates value above the required return. IRR is the rate at which NPV equals zero: useful, but beware scale, multiple IRRs and reinvestment assumptions. Payback, simple or discounted, is a useful liquidity and risk indicator, but it ignores what happens after payback, so use it alongside NPV. And watch the spreadsheet trap: Excel's NPV discounts the first value. Your try-this-now: build Project A in a spreadsheet, confirm the NPV and IRR, then test how NPV changes at eight per cent and twelve per cent. At what rate does NPV reach zero, and does that match your IRR? ## Video transcript Welcome. In this lesson we answer a question every investor, lender and board member asks: is this project worth doing? The starting point is the time value of money. Money today is worth more than money tomorrow, because it can earn a return and because the future is uncertain. So we discount future cash flows back to today. Net present value, or NPV, adds up all those discounted cash flows, including the initial investment as a negative number. If NPV is positive, the project earns more than your required return. It creates value. Take our example. We invest one thousand today and receive three hundred, three hundred and fifty, four hundred and three hundred over four years. Discounted at ten percent, those inflows are worth about one thousand and sixty-seven today. So the NPV is about sixty-seven. Positive. Good. Internal rate of return, or IRR, is the discount rate at which NPV becomes zero. Here it is about thirteen percent, comfortably above ten. IRR is intuitive, but be careful. It ignores the size of a project, and unusual cash flow patterns can produce more than one IRR. Payback tells you how quickly you get your money back. Here, a little under three years. It is a useful signal of liquidity risk, but it ignores everything after the payback point. Now the key lesson. Project B returns more cash in total, one thousand five hundred versus one thousand three hundred and fifty. But its cash arrives late. At ten percent, its NPV is slightly negative. Timing matters. So use NPV as your primary measure, IRR and payback as supporting views, keep inflation treatment consistent, and always use cash flows rather than accounting profit. ## Key takeaways - NPV discounts all cash flows at the required return; accept if NPV > 0. - IRR is the rate where NPV = 0; beware scale, multiple IRRs and reinvestment assumptions. - Payback shows liquidity risk but ignores later cash flows; discounted payback adds time value. - Use consistent nominal/real treatment and cash flows, not accounting profit. ## Try it Build the Project A table in a spreadsheet, confirm NPV and IRR, then test how NPV changes at 8% and 12% discount rates. - [Previous: Capex, opex, budgeting and cash flow](https://optimizeall.com/learn/project-finance-and-financial-modelling/capex-opex-and-budgets) - [Next: Capital structure, gearing and the cost of capital](https://optimizeall.com/learn/project-finance-and-financial-modelling/capital-structure) - [All lessons of Project Finance & Financial Modelling](https://optimizeall.com/learn/project-finance-and-financial-modelling)