Alternative Investment Appraisal Rules

Payback Period

• Definition: The number of years required to recover the initial investment through accumulated future cash flows.

• Decision Rule: Accept projects that recover their initial outlay within a specified number of years (X-year payback).

• Example:

  • New investment costs £10,000.

  • Future cash flows: £3,000 (years 1-3), £4,000 (years 4-5), £6,000 (year 6).

  • Investor's required return: 14%.

  • Firm's payback period: 3 years.

  • Payback period = 3.25 years (calculated by 3 + (1000/(4000)).

  • Decision: Reject the project based on the payback rule.

  • Note: NPV is £4,144 > 0.

Advantages of Payback Period

• Easy to compute and understand, making it useful as a coarse screening tool.

• Encourages cash generation.

• Values early cash flows over late cash flows.

Disadvantages of Payback Period

• Ignores the time value of money.

• The choice of cutoff period is arbitrary.

• Ignores cash flows after the cutoff period.

• Biased towards rejecting long-lived projects, even if they have positive NPVs.

• Biased towards accepting short-lived projects, even if they have negative NPVs.

Discounted Payback Period

• Definition: The number of years required to recover the initial investment through DISCOUNTED accumulated future cash flows.

• Addresses: Time value of money issue.

• Example: Using the same cash flows as the payback period example.

  • Cash Flows: -£10,000 (t=0), £3,000 (t=1), £3,000 (t=2), £3,000 (t=3), £4,000 (t=4), £4,000 (t=5), £6,000 (t=6).

  • $−10,000 + \frac{3,000}{1.14^1} + \frac{3,000}{1.14^2} + \frac{3,000}{1.14^3} = −3,035.10$

Internal Rate of Return (IRR)

• Definition: The discount rate that results in a zero Net Present Value (NPV).

• $−10,000 + \frac{3,000}{(1 + 𝐼𝑅𝑅)^1} + \frac{3,000}{(1 + 𝐼𝑅𝑅)^2} + \frac{3,000}{(1 + 𝐼𝑅𝑅)^3} + \frac{4,000}{(1 + 𝐼𝑅𝑅)^4} + \frac{4,000}{(1 + 𝐼𝑅𝑅)^5} + \frac{6,000}{(1 + 𝐼𝑅𝑅)^6} = 0$

• Example:

  • Using the same cash flow data.

  • IRR = 26.46%.

• Decision Rule: Accept the project if IRR > Investor's required rate of return.

• In this case, accept the project, since 26.46% > 14%.

NPV and IRR Relationship

• For conventional investment projects (initial negative cash outflow followed by positive cash inflows):

  • Accept the project if IRR > r* (investor's required rate of return).

  • Both IRR and NPV provide the same accept/reject decision.

• IRR does not indicate the increase to the value of the firm; it is a percentage.

Problems with IRR

• Non-conventional Cash Flows: IRR does not work well with non-conventional cash flow projects (cash flows that change signs more than once).

• Example:

  • Year 0: -£4,500; Year 1: £6,000; Year 2: £6,000; Year 3: -£8,000.

  • NPV is zero at both 15.47% and 33.3%, resulting in multiple IRRs.

  • Using a cutoff rate of 14% would lead to accepting the project, even though the NPV is negative on either side of the two IRRs.

• Mutually Exclusive Projects: IRR may lead to incorrect decisions when comparing mutually exclusive projects (projects where only one can be chosen).

  • Example:

    • Project A: £1m investment, £1.2m cash flow in year 1. IRR = 20%.

    • Project B: £3m investment, £3.45m cash flow in year 1. IRR = 15%.

    • NPV Project A = $−1m + \frac{1.2m}{1.12}$ = £71,429.

    • NPV Project B = $−3m + \frac{3.45m}{1.12}$ = £80,357.

    • NPV suggests Project B is better, while IRR suggests Project A is better.

• The crossover rate is 12.5%.

Accounting Rate of Return (ARR)

• Definition: The ratio of the project’s average annual profits to the average annual book value of assets.

• Also referred to as Average Accounting Return (AAR) or Book Rate of Return (BRR).

• $𝐴𝑅𝑅 = \frac{𝐴𝑣𝑒𝑟𝑎𝑔𝑒 \space 𝑜𝑓 \space 𝑎𝑐𝑐𝑜𝑢𝑡𝑖𝑛𝑔 \space 𝑝𝑟𝑜𝑓𝑖𝑡𝑠 \space 𝑒𝑎𝑐ℎ \space 𝑦𝑒𝑎𝑟}{𝐴𝑣𝑒𝑟𝑎𝑔𝑒 \space 𝑜𝑓 \space 𝑏𝑜𝑜𝑘 \space 𝑣𝑎𝑙𝑢𝑒 \space 𝑜𝑓 \space 𝑎𝑠𝑠𝑒𝑡𝑠 \space 𝑖𝑛𝑣𝑒𝑠𝑡𝑒𝑑 \space 𝑒𝑎𝑐ℎ \space 𝑦𝑒𝑎𝑟}$

• Example:

  • Initial investment in assets: £50,000

    Year	Depreciation	Closing Book Value	Revenues less Operating Expenses	Depreciation	Earnings Before Tax	Taxes (33%)	Earnings for Ordinary
    t=0		50,000
    t=1	5,000	45,000	20,000	5,000	15,000	4,950	10,050
    t=2	10,000	35,000	19,000	10,000	9,000	1,970	6,030
    t=3	15,000	20,000	18,500	15,000	3,500	1,155	2,345
    t=4	20,000	0	21,000	20,000	1,000	330	670

  • $Average\space Profit = \frac{10050 + 6030 + 2345 + 670}{4} = £4773.75$

  • $Average \space BV = \frac{50000 + 45000 + 35000 + 20000}{4} = £37500$

  • $ARR = \frac{4773.75}{37500} = 12.73\%$

  • If target ARR = 12%, accept. If target ARR = 13%, reject.

Advantages of ARR

• Easy to compute because the firm already collects accounting information for budgeting and planning.

Disadvantages of ARR

• Ignores the time value of money.

• The choice of target ARR is arbitrary.

• Based on earnings, not cash flows.

  • Uses accounting depreciation.

  • Tax charge based on accounting earnings.

• No standard calculation method.

• Not a true return on investment.

Profitability Index (PI)

• Definition: The ratio of the present value of future cash flows of a project to its initial investment.

• $𝑃𝐼 = \frac{𝑃𝑉 \space 𝑜𝑓 \space 𝑓𝑢𝑡𝑢𝑟𝑒 \space 𝑐𝑎𝑠ℎ \space 𝑓𝑙𝑜𝑤𝑠}{𝐼𝑛𝑖𝑡𝑖𝑎𝑙 \space 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡}$

• Decision Rule: Accept a project if PI > 1.

• Like IRR, PI does not work well when dealing with mutually exclusive projects.