Investment Appraisal

Investment appraisal uses cash flows, not profits, because cash is what is actually available for distribution or reinvestment. The key principles:

• Incremental: Only cash flows that CHANGE as a result of the project — existing cash flows continuing regardless are irrelevant
• Cash-based: Exclude non-cash items (depreciation, amortisation, profit/loss on disposal). Depreciation is NOT a cash flow — the relevant item is the purchase cost itself.
• After tax: Tax is a genuine cash outflow; project cash flows must be calculated post-tax. Include tax savings from capital allowances.
• Opportunity costs INCLUDED: The value that could be gained from the next-best alternative use of a resource (e.g., if land could be sold for £500k, using it on the project has an opportunity cost of £500k — even though no cash moves)
• Sunk costs EXCLUDED: Costs already incurred that cannot be reversed are not relevant (e.g., market research already paid for)
• Committed costs EXCLUDED: Costs already legally committed regardless of the project decision
• Financing costs EXCLUDED from the cash flows: Interest on borrowings is captured in the discount rate (WACC), NOT in the cash flows — otherwise it would be double-counted

Typical relevant cash flows:

Working capital:

• At the start: the project typically needs extra working capital (inventory, receivables less payables) — cash outflow at t=0
• During the project: if working capital requirement grows (e.g., as sales grow), additional investment each year — cash outflow
• At the end: working capital is "released" — cash inflow. Often assumed to be recovered at the end of the project's life.
• Only the INCREMENTAL change in working capital is a cash flow (not the total working capital balance)

Corporation tax is a cash outflow that must be reflected in project cash flows. But tax is based on taxable profit, not cash flow. The main adjustments:

1. Tax on operating profits:

• Calculate the incremental taxable profit: Revenue − Variable costs − Fixed costs + Tax-deductible items − Depreciation add-back + Capital allowances
• Apply the corporation tax rate
• Deduct the tax in the cash flow statement at the appropriate time

2. Capital allowances (writing-down allowances):

• Although the purchase cost is a lump sum at t=0, the TAX SAVING comes via capital allowances claimed over the asset's life
• UK rule: typically 18% reducing balance (main rate pool) or 6% (special rate pool); Annual Investment Allowance (AIA) at 100% up to a threshold for most new plant and machinery; SBA (Structures and Buildings Allowance) at 3% SL for buildings
• For exam purposes, usually given a rate (e.g., 25% reducing balance)
• The tax SAVING = Capital allowance × Tax rate — this is a positive cash flow
• On disposal: a balancing charge (if disposal proceeds > tax WDV — extra tax) or balancing allowance (if disposal proceeds < tax WDV — extra tax saving) arises

3. Timing of tax cash flows:

• Two common conventions in exam questions:
• Tax paid in the same year as profits arise: tax shown at t=1, 2, 3… in the same year as the profit (simpler assumption)
• Tax paid one year in arrears: tax on year 1 profit paid at t=2 (more realistic for non-large companies; reflects 9 months after year-end payment deadline)
• The question will specify. Always read carefully.

Worked example of tax/capital allowance layout:

Suppose project: Initial investment £100k, life 4 years, residual value £10k, reducing balance capital allowances at 25%, tax 20%, tax paid same year:

In year 4: balancing allowance = opening WDV £42,188 − disposal proceeds £10,000 = £32,188. Tax saving = £32,188 × 20% = £6,438.

When inflation is expected, the analysis must choose between:

• Money (nominal) terms: Cash flows inflated at expected inflation rates; discount using the MONEY (nominal) discount rate
• Real terms: Cash flows in today's prices (no inflation); discount using the REAL discount rate

CONSISTENCY RULE: Money cash flows must be discounted at the money rate; real cash flows must be discounted at the real rate. Mixing them will give the wrong answer.

The Fisher Formula:

Rearranging: Real rate = (1 + money rate) / (1 + inflation rate) − 1

Example: Money rate 10%, inflation 4%. Real rate = 1.10 / 1.04 − 1 = 5.77%.

Or: Real rate 6%, inflation 3%. Money rate = 1.06 × 1.03 − 1 = 9.18%.

Different inflation rates for different items:

• If all cash flows inflate at the same rate, you can use EITHER the real approach (simpler — cash flows don't change) OR the money approach
• If cash flows inflate at DIFFERENT rates (e.g., revenue +4%, variable costs +5%, labour +6%), you MUST use the money approach because real cash flows would change over time
• Tax cash flows use the MONEY amounts (tax authority taxes nominal profits) — so when using the real approach, tax complications arise
• In practice, the money approach is safer when tax or differential inflation is involved

Typical exam scenario:

1. Given: nominal cash flows (or information to inflate to nominal), nominal discount rate, inflation rate(s)
2. Approach: use nominal throughout (discount nominal cash flows at nominal rate)
3. If asked to show the real approach, deflate the cash flows (divide by (1+inflation)ⁿ) and discount at the real rate

Perpetuity (constant cash flow forever):

Example: £10,000 per year forever at 8% = £10,000 / 0.08 = £125,000.

Growing perpetuity (constant growth rate g forever):

Example: Next year £10,000, growing 3% per year, discount 8%: £10,000 / (0.08 − 0.03) = £200,000.

Warning: formula breaks down if g ≥ r. Growth rate must be below discount rate for the formula to give a meaningful (finite) PV.

Annuity (constant cash flow for n years):

Example: £10,000 per year for 5 years at 8%. Annuity factor = (1 − 1.08⁻⁵) / 0.08 = 3.9927. PV = £10,000 × 3.9927 = £39,927.

Delayed annuity: For cash flows starting in year k+1 for n years: PV = (Annual CF × AF for n years) × (1+r)⁻ᵏ. Multiply the annuity by the single-period discount factor for the delay.

Equivalent Annual Cost (EAC) or Equivalent Annual Benefit (EAB):

Used to compare projects with different lives. Converts the PV of the project's total cash flows into an equivalent annual amount:

Example: Project A: 5-year machine, PV of all costs £30,000. Project B: 8-year machine, PV of all costs £40,000. Cost of capital 10%.

• Project A EAC: £30,000 / AF(5, 10%) = £30,000 / 3.7908 = £7,913 per year
• Project B EAC: £40,000 / AF(8, 10%) = £40,000 / 5.3349 = £7,498 per year
• Project B has a lower EAC — chosen (better value per year)

Alternative: compare NPV over the lowest common multiple of lives (5 and 8 → 40 years of chained replacements). EAC is much simpler and gives the same ranking.

The IRR is the discount rate at which NPV = 0. It represents the project's intrinsic rate of return. Accept projects where IRR > cost of capital; reject if IRR < cost of capital.

Calculating IRR by linear interpolation:

Calculate NPV at two discount rates (one giving positive NPV, one negative):

Where a is the lower rate (gives positive NPV) and b is the higher rate (gives negative NPV).

Example: At 10%, NPV = £20,000. At 20%, NPV = −£5,000.
IRR ≈ 10% + [£20,000 / (£20,000 − (−£5,000))] × (20% − 10%) = 10% + (£20,000 / £25,000) × 10% = 10% + 8% = 18%.

Limitations of IRR:

1. Multiple IRRs: Non-conventional cash flows (more than one sign change in cash flows — e.g., a project with a large clean-up cost at the end after positive cash flows) can produce multiple IRRs. NPV remains well-defined, IRR becomes ambiguous.
2. Mutually exclusive projects: When choosing between mutually exclusive projects, IRR can give wrong rankings because it ignores SCALE. A smaller project with higher IRR may have lower NPV than a larger project with lower IRR.
3. Reinvestment assumption: IRR implicitly assumes cash flows are reinvested at the IRR itself — often unrealistic, especially if IRR is very high.
4. Single-number summary can be misleading for complex projects.

Modified IRR (MIRR): Addresses the reinvestment assumption. Assumes cash INFLOWS are reinvested at the cost of capital (rather than at the IRR) and OUTFLOWS are financed at the cost of capital. Gives a more realistic measure of the project's return.

Where: Terminal value = sum of inflows compounded forward to year n at cost of capital; PV of outflows = sum of outflows discounted to year 0 at cost of capital.

NPV vs IRR — which should we use? NPV is theoretically superior — gives absolute value in £, consistent with shareholder wealth maximisation, always unique, handles non-conventional cash flows. IRR has intuitive appeal (a rate of return) and doesn't require knowing the discount rate in advance. Most finance professionals use NPV for final decisions but report IRR for communication.

Payback period: The time required for cumulative cash inflows to equal the initial investment. "How long before we get our money back?"

• Simple payback: ignores time value of money — just cumulative undiscounted cash flows
• Discounted payback: uses discounted cash flows — always longer than simple payback
• Advantages: simple, intuitive, useful for liquidity-constrained businesses, emphasises early cash flows (risk focus)
• Disadvantages: ignores cash flows AFTER the payback period (a project could fail just after payback and the measure wouldn't capture it), subjective cut-off, simple payback ignores time value

Accounting Rate of Return (ARR): Average accounting profit / Average or initial investment. Uses PROFIT (not cash flow), including depreciation. Ignores time value.

• Formula: Average annual accounting profit / Average investment × 100%
• Where: Average investment = (Initial cost + Residual value) / 2
• Accept if ARR ≥ target rate
• Disadvantages: uses profit not cash flow; ignores time value; target is arbitrary; based on accounting conventions that vary

Profitability Index (PI): The ratio of NPV of future cash flows to initial investment.

• PI = PV of future cash inflows / Initial investment. Accept if PI > 1.
• Alternatively: PI = NPV / Initial investment — and then accept if PI > 0
• Use: Primarily for ranking projects under CAPITAL RATIONING (see below)

Comparison table of methods:

Capital rationing arises when the entity cannot invest in all positive-NPV projects — typically because of constraints on available capital (internal budget limits or external market constraints).

Hard rationing: Imposed by external capital markets (e.g., cannot raise more debt or equity — lenders have refused). Often contentious in theory (an efficient market would fund positive-NPV projects).

Soft rationing: Imposed by management — internal budget limits, desire to avoid excessive borrowing, limits on management time/attention, risk appetite. More common in practice.

Single-period capital rationing (only one period is constrained):

• Use the Profitability Index (PI): Rank projects by PI (NPV per £1 of initial investment, or PV of inflows per £1 of investment)
• Accept projects in descending order of PI until capital is exhausted
• If project is divisible (can do part of a project): take fractional shares of the marginal project
• If projects are indivisible (all-or-nothing): test combinations of projects (possibly forgoing the best-PI project if its size doesn't fit with other good projects)
• If projects are mutually exclusive: can only choose one from a set

Worked example: £100,000 available; four projects:

With divisible projects: Take B (£30k, PI 0.60), then A (£40k, PI 0.50), then 75% of D (£30k of £40k, PI 0.40). Total NPV = £18k + £20k + 75% × £16k = £50k.

With indivisible projects: Try combinations — B+A = £70k; B+A+D would be £110k (over budget); B+D = £70k; A+D = £80k; B+A = £70k leaves £30k idle. Best might be B+D (£70k, NPV £34k) if we can't complete D and need NPV maximisation. Or A+D (£80k, NPV £36k). Test all feasible combinations.

Multi-period capital rationing (constraint on more than one period): Requires Linear Programming (LP):

• Objective: maximise total NPV
• Subject to: capital constraints in each period; x values between 0 and 1 (proportions)
• Solved using LP software (Excel Solver, specialised packages)
• At FM level: understand the LP formulation; not typically required to solve a full LP manually

Risk vs uncertainty:

• Risk: Possible outcomes are known and probabilities can be assigned
• Uncertainty: Possible outcomes may not be fully known; probabilities cannot be assigned

In practice, the distinction is blurry; techniques overlap.

1. Sensitivity analysis: How much can each INPUT variable change before NPV becomes zero (the project becomes unacceptable)?

• Calculate the sensitivity percentage = NPV / PV of the variable's cash flows × 100%
• Low sensitivity = small change makes NPV zero → high risk variable; pay close attention
• High sensitivity = large change needed → more robust to this variable

Example: NPV = £50,000. PV of revenue cash flows = £500,000. Sensitivity to revenue = £50,000 / £500,000 = 10%. Revenue can fall by 10% before NPV hits zero. A 10% fall is significant risk if demand is volatile.

Advantages: identifies critical variables; intuitive. Disadvantages: only one variable at a time (ignores correlations); doesn't give probability of outcomes; ignores possible positive surprises.

2. Scenario analysis: Evaluate NPV under different sets of assumptions — typically best case, base case, worst case.

• All key variables change together in a consistent way
• Unlike sensitivity analysis, captures CORRELATIONS between variables
• Provides a range of possible NPVs — useful for decision-making

3. Simulation (Monte Carlo):

• Specify probability distributions for each uncertain variable
• Run thousands of simulations (computer-based) — each draws random values from the distributions and calculates NPV
• Build up a probability distribution of NPV outcomes
• Can see: expected NPV, standard deviation, probability of negative NPV, percentiles (5% worst case, etc.)
• Advantages: captures correlations, gives full distribution. Disadvantages: requires good input distribution data; computer-intensive; false precision risk.

4. Expected values (EV): Probability-weighted outcome.

• EV of NPV = Σ (probability × NPV in that scenario)
• Also can calculate standard deviation and coefficient of variation for risk measurement
• Limitation: based on a single estimate that may not match any actual outcome (average may be impossible, like rolling a 3.5 on a dice)
• Assumes decision-maker is risk-neutral

5. Certainty equivalents (CE): Convert risky cash flows to "certain" equivalents, then discount at the risk-free rate.

• CE = Risky cash flow × α (where 0 < α < 1 reflects risk aversion)
• The more risky the cash flow, the lower α (more discount for risk)
• Alternative to using a risk-adjusted discount rate

Risk-adjusted discount rate (RADR): Add a risk premium to the cost of capital for riskier projects. Simple and common in practice. But can distort the analysis because it assumes risk grows exponentially with time.

Real options theory applies option-pricing concepts to managerial decisions. Recognises that investment projects often contain FLEXIBILITY — options to change course as information arrives. Traditional NPV under-estimates value by ignoring this flexibility.

Four main types of real option:

Real option value and uncertainty: Counter-intuitively, HIGHER uncertainty often means HIGHER option value — because there is more upside to capture when things go well, combined with limited downside (you can abandon). This contrasts with traditional NPV where higher uncertainty typically reduces value (via higher discount rate).

Valuation methods:

• Black-Scholes model: The European-option pricing formula applied to real options. Inputs: current value of underlying asset, exercise price (investment cost), risk-free rate, time to expiry, volatility of underlying asset.
• Binomial trees: Simpler; models possible paths of the underlying variable over time. Particularly useful for American options (can be exercised any time) and real options with multiple decision points.
• Monte Carlo simulation: For complex options with multiple sources of uncertainty.

Practical implications for investment appraisal:

• Projects with flexibility may be VALUABLE even when traditional NPV is zero or negative
• Don't kill projects too quickly — consider option to delay or abandon
• "Stage-gate" investment processes formalise real option thinking: stages of investment with decision points
• R&D and exploration often have substantial real option value
• FM exam may ask for CONCEPTUAL understanding rather than numerical Black-Scholes calculation

Post-audit (post-completion audit) is the review of an investment decision AFTER the project has been in operation for a period. Compares actual outcomes with those projected at the time of approval.

Benefits of post-audit:

• Accountability: Managers proposing projects know their forecasts will be scrutinised — reducing optimism bias
• Learning: Identifies systematic errors in forecasting (e.g., consistent over-estimation of demand) — improves future appraisals
• Action: Identifies whether the project should be continued, modified, or abandoned
• Management development: Post-audit discipline encourages rigorous analysis and disciplined implementation

Limitations:

• Costly and time-consuming
• Hindsight bias — outcomes may depend on factors outside the manager's control; unfair to penalise
• May discourage managers from proposing innovative/risky projects for fear of criticism
• Difficult to isolate the project's contribution from other business changes
• Outcome may not be the manager's fault — external factors (economy, competitor actions, regulation)

Best practices for effective post-audit:

• Focus on LEARNING, not blame
• Audit only SIGNIFICANT projects (materiality threshold)
• Timing: typically 6-12 months after completion (enough data; not too late)
• Separate what was controllable from uncontrollable factors
• Report findings formally; incorporate lessons into future appraisal guidance