Investment Appraisal: Time Value of Money and Project Decisions

Learning objectives

By the end of this chapter you should be able to:

• Explain why timing affects the economic value of cash flows.
• Calculate future values and present values using compounding and discounting.
• Build an incremental project cash-flow schedule, including opportunity costs and excluding sunk costs.
• Apply and interpret payback, discounted payback, accounting rate of return (ARR), net present value (NPV), and internal rate of return (IRR).
• Apply decision logic to recommend whether to proceed and explain the limitations of each method.
• Identify common appraisal errors, including incorrect timing, double counting, and inclusion of irrelevant items.

Overview & key concepts

Investment appraisal is the process of deciding whether a proposed investment should be undertaken, based on the incremental cash flows it generates and the return required for investments of similar risk.

Two principles drive most exam questions:

1. Timing matters.Cash received sooner can be reinvested sooner and is usually less uncertain than cash received later.
2. Only incremental cash flows matter.Include cash flows that change because the decision is taken, and exclude items that do not change with the decision.

Time value of money

Why money has a time value

A cash amount received today is worth more than the same nominal amount received later because today’s cash can be invested to earn a return. Discounting captures this by converting future amounts into today’s equivalents.

Compounding and discounting

Compounding moves a value forward in time; discounting moves a value back to today.

FV = PV × (1 + r)^n

PV = FV ÷ (1 + r)^n

Where:

• PV = present value
• FV = future value
• r = annual interest/discount rate
• n = number of years

Discount rate and discount factor

The discount rate is the required return used to value the project’s cash flows. The discount factor is the multiplier applied to a future cash flow to obtain its present value.

Discount factor in year n = 1 ÷ (1 + r)^n

Nominal vs real cash flows (consistency point)

If inflation is ignored, treat figures as “in today’s money” and use a consistent discount rate. If inflation is included, be consistent:

• use nominal cash flows with a nominal discount rate, or
• use real cash flows with a real discount rate.

Mixing real and nominal inputs will distort NPV and IRR.

Building project cash flows

Relevant (incremental) cash flows

Relevant cash flows are those that occur only if the project proceeds, or those that change because the project proceeds. Typical components:

• Initial investment (purchase, installation, incremental working capital).
• Incremental operating cash flows (extra receipts minus extra payments).
• Terminal cash flows (disposal proceeds, release of working capital, final operating flows).

Items to exclude

• Sunk costs:past cash flows already incurred; they do not change with the decision.
• Non-cash accounting charges(for example, depreciation), unless needed to derive tax cash flows in an after-tax appraisal.
• Costs/revenues that will arise anyway and are unaffected by the decision.

Opportunity costs (often examined)

If the project uses an existing resource that could generate cash elsewhere, the foregone cash is an opportunity cost and should be included as a project outflow.

Examples:

• using premises that could be rented out,
• using staff time that could be charged to other profitable work,
• diverting sales from an existing product (cannibalisation).

Investment appraisal methods

1) Payback period

Payback is the time taken for cumulative undiscounted net cash inflows to recover the initial investment.

Decision logic: where a maximum payback is set, a project is acceptable if it recovers its investment within that limit.

Strengths:

• simple liquidity-focused measure,
• highlights how quickly funds are recovered.

Limitations:

• ignores the time value of money,
• ignores cash flows after payback, so it is not a value measure.

2) Discounted payback period

Discounted payback repeats the payback calculation using discounted cash flows (present values).

Decision logic: acceptable if discounted recovery occurs within the target time.

Strengths:

• recognises the time value of money.

Limitations:

• still ignores cash flows after the recovery point,
• may be “not achieved” within the project life even when the project is attractive by NPV (it is a liquidity proxy, not a value test).

3) Accounting rate of return (ARR)

ARR compares average annual accounting profit with an investment base. The method used must be stated, as conventions differ.

Common conventions include:

ARR = Average annual accounting profit ÷ Initial investment

or

ARR = Average annual accounting profit ÷ Average investment

Average investment is often approximated as:

Average investment = (Initial investment + Residual value) ÷ 2

Profit is accounting profit (not cash flow), typically after depreciation, and may be stated before or after tax depending on how the question defines it. Use the convention given, or clearly state the convention adopted.

Strengths:

• uses profit measures that may align with accounting performance targets.

Limitations:

• ignores the time value of money,
• depends on accounting policies and definitions (profit measure and investment base).

4) Net present value (NPV)

NPV measures value added after allowing for the required return by discounting all incremental cash flows to present values.

NPV = Σ (Cash flow in year t × Discount factor in year t) − Initial outflow

Decision logic (value first):

• if the discounted value of incremental inflows exceeds the discounted value of incremental outflows, the project is worthwhile at the required return (NPV > 0),
• if it falls short, it is not worthwhile at that required return (NPV < 0).

Strengths:

• uses time value of money,
• includes all relevant cash flows,
• measures value created in money terms.

Limitations:

• depends on forecasts and the discount rate,
• less intuitive than a percentage return.

5) Internal rate of return (IRR)

IRR is the discount rate at which NPV equals zero for the project’s incremental cash flows.

Decision logic:

• IRR above the required return is consistent with a positive NPV,
• IRR below the required return is consistent with a negative NPV.

Strengths:

• produces a percentage return, which is easy to communicate.

Limitations:

• can mis-rank mutually exclusive projects (NPV is the better value guide),
• may give multiple IRRs with non-standard cash-flow patterns,
• implicitly assumes reinvestment at the IRR.

Worked example

Walkthrough example: cash-flow mapping and decision

TechGear Ltd is assessing a new production line.

• Immediate cash payment at time 0: £76,000
• Net incremental operating cash inflow: £20,000 each year for years 1–5
• Disposal proceeds at the end of year 5: £10,000

The project would use a storage facility already owned. If the project does not proceed, the facility could be rented out for £3,000 per year. Treat the foregone rental income as an opportunity cost of the project. A feasibility study costing £5,000 has already been paid for.

The required return for projects of this risk is 8.9%.

Timing assumptions:

• unless stated otherwise, cash flows occur at each year-end,
• the initial payment is at time 0,
• ignore inflation,
• this illustration ispre-tax: no separate tax computations are required.

Sign convention reminder:

• treat inflows as positive numbers and outflows as negative numbers.

Tasks

1. Build the incremental cash-flow schedule (include opportunity costs; exclude sunk costs).
2. Calculate payback and discounted payback.
3. Calculate NPV at 8.9% and estimate IRR.
4. Recommend whether to proceed, explaining what each measure does and does not tell you.

Solution

Step 1: Build the incremental cash flows

Relevant cash flows:

• Time 0: initial investment outflow = (76,000)
• Years 1–5: operating inflow = +20,000 each year
• Years 1–5: opportunity cost (foregone rent) = (3,000) each year
• Year 5: disposal proceeds inflow = +10,000
• Feasibility study = sunk cost (exclude)

Net annual incremental inflow:

Net annual inflow (Years 1–5) = 20,000 − 3,000 = 17,000

So cash flows are:

• Year 0: (76,000)
• Years 1–4: +17,000 each year
• Year 5: +27,000 (17,000 + 10,000)

Step 2: Payback period (undiscounted)

Cumulative undiscounted cash flow:

• Year 0: (76,000)
• Year 1: (59,000)
• Year 2: (42,000)
• Year 3: (25,000)
• Year 4: (8,000)
• Year 5: 19,000

Payback occurs between Years 4 and 5.

Unrecovered after Year 4 = 8,000
Year 5 inflow = 27,000

Payback period = 4 + (8,000 ÷ 27,000) = 4.30 years (approx.)

Step 3: Discounted payback (8.9%)

Discount factors at 8.9% (rounded to three decimals):

• Year 1: 0.918
• Year 2: 0.843
• Year 3: 0.774
• Year 4: 0.711
• Year 5: 0.653

Discounted cash flows:

• Year 0: (76,000)
• Year 1: 17,000 × 0.918 = 15,606
• Year 2: 17,000 × 0.843 = 14,331
• Year 3: 17,000 × 0.774 = 13,158
• Year 4: 17,000 × 0.711 = 12,087
• Year 5: 27,000 × 0.653 = 17,631

Cumulative discounted cash flow:

• Year 0: (76,000)
• Year 1: (60,394)
• Year 2: (46,063)
• Year 3: (32,905)
• Year 4: (20,818)
• Year 5: (3,187)

Discounted payback is not achieved within five years.

Exam comment: discounted payback can be “not achieved” even when a project is only slightly value-creating in longer-life scenarios; it is designed to comment on discounted liquidity, not total value created.

Step 4: Net present value (NPV)

Total PV of inflows (rounded factors/present values):

15,606 + 14,331 + 13,158 + 12,087 + 17,631 = 72,813

NPV = 72,813 − 76,000 = (3,187) (approx.)

Note: small differences can arise due to rounding of discount factors.

Step 5: Internal rate of return (IRR)

Cash flows:

• Year 0: (76,000)
• Years 1–4: +17,000
• Year 5: +27,000

Using a calculator/spreadsheet IRR function:

IRR ≈ 7.38%

If a spreadsheet is not assumed, estimate IRR by interpolation:

1. choose two discount rates, one giving a small positive NPV and one giving a negative NPV,
2. then approximate using:

IRR ≈ r1 + [NPV1 ÷ (NPV1 − NPV2)] × (r2 − r1)

Where r1 is the lower rate with NPV1 (typically positive) and r2 is the higher rate with NPV2 (typically negative).

Interpretation and recommendation

• Payback is about 4.30 years, indicating slow recovery over a five-year life.
• Discounted payback is not achieved within five years, reinforcing that discounted recovery is weak.
• NPV at 8.9% is negative (approximately £3,187), meaning the project is not expected to meet the required return.
• IRR is about 7.38%, which is below 8.9%, consistent with the negative NPV.

Recommendation: do not proceed on the current estimates, unless the cash-flow forecasts can be improved or the required return is reconsidered.

Common pitfalls and misunderstandings

• Including sunk costs (past cash flows) as if they were avoidable.
• Omitting opportunity costs (foregone rental income, displaced contribution, or alternative use of resources).
• Mixing profit measures with cash-flow measures (especially when moving between ARR and NPV/IRR).
• Incorrect timing assumptions (time 0 vs year-end; mid-year conventions; working capital timing).
• Ignoring terminal effects (disposal proceeds; release of working capital).
• Inconsistent treatment of inflation (nominal vs real mismatch).
• Treating IRR as superior to NPV for mutually exclusive projects (NPV is the value measure).
• Using payback as a profitability test (it is primarily a liquidity/risk indicator).

Summary

Investment appraisal evaluates whether a project is worthwhile using incremental cash flows and the required return. Discounting converts future cash flows into present values so that timing and risk are recognised.

• Build cash flows on an incremental basis: include opportunity costs and exclude sunk costs.
• Use NPV as the primary value measure.
• Use IRR as a percentage cross-check, recognising its limitations.
• Use payback/discounted payback to comment on liquidity and recovery speed.
• Use ARR only when an accounting-profit perspective is specifically required, stating the convention used.

FAQ

What is the time value of money, and why does it matter in appraisal?

Earlier cash flows can be reinvested sooner and are generally less uncertain. Discounting converts future cash flows into present values so different-year amounts can be compared consistently.

How do you calculate NPV?

Discount each incremental cash flow and subtract the initial outflow.

NPV = Σ (Cash flow in year t × Discount factor in year t) − Initial outflow

A positive result indicates value creation at the required return.

Why can IRR be misleading?

IRR can mis-rank mutually exclusive projects, can produce multiple answers with unusual cash-flow patterns, and embeds an assumption about reinvestment at the IRR. NPV remains the stronger value-based test.

Why are sunk costs excluded?

Sunk costs have already occurred and cannot be changed by the decision. Including them mixes past spending with future incremental outcomes and can bias decisions.

What does discounted payback add compared with payback?

It recognises the time value of money by using present values. However, it still ignores cash flows after recovery and can be “not achieved” even for projects that create value over longer horizons.

Glossary

Time value of money
The idea that money received earlier is worth more than the same nominal amount received later because it can earn a return and is usually less uncertain.

Future value (FV)
The amount a present sum will accumulate to after compounding at a given rate over a given period.

Present value (PV)
The current equivalent of a future cash flow after discounting at a given rate.

Discount rate
The required return used to convert future cash flows into present values.

Discount factor
The multiplier applied to a cash flow in a specific year to obtain its present value: 1 ÷ (1 + r)^n.

Relevant (incremental) cash flows
Cash flows that arise because of the decision, including opportunity costs, measured on a cash basis for appraisal.

Sunk cost
A past cash flow already incurred that does not change with the decision.

Opportunity cost
Cash benefits forgone by using resources in one way rather than their best alternative use.

Net present value (NPV)
The present value of incremental inflows minus the present value of incremental outflows, including the initial investment.

Internal rate of return (IRR)
The discount rate that makes NPV equal to zero for the project’s incremental cash flows.

Payback period
The time taken for cumulative undiscounted net cash inflows to recover the initial investment.

Discounted payback period
The time taken for cumulative discounted net cash inflows (present values) to recover the initial investment.

Accounting rate of return (ARR)
A profit-based performance measure comparing average annual accounting profit with an investment base (initial investment or average investment, depending on convention).