BFC2140: Corporate Finance 1 AWS
BFC2140: Corporate Finance 1 Table of Contents Topic 1: Introduction to Financial Mathematics ....................................................................................... 2 Topic 2: Financial Mathematics II .............................................................................................................. 5 Topic 3: Valuation of Bonds & Equities ................................................................................................... 9 Topic 4: Project Evaluation I .................................................................................................................... 13 Topic 5: Project Evaluation II ................................................................................................................... 19 Topic 6: Project Evaluation III .................................................................................................................. 29 Topic 7: Risk, Return & Investment I ....................................................................................................... 35 Topic 8: Risk, Return & Investment II ..................................................................................................... 38 Topic 9: Cost of Capital ............................................................................................................................ 43 Topic 10: Capital Structure ...................................................................................................................... 48 Topic 11: Dividend Policy ........................................................................................................................ 55
1
3.6 Dividend Valuation 1. Constant dividend valuation o Use the perpetuity formula o Can be applied for shares that pay a constant dividend (e.g. preference shares) 𝑃0 =
𝐷 𝑅
Example: Worley Ltd is expected to pay a constant annual dividend of $2.50 per share indefinitely. If the discount rate is 8% p.a., what is the value of the share? 𝑃0 =
2.50 = $31.25 0.08
2. Constant dividend growth valuation o If dividends are expected to grow at a constant rate 𝑃0 =
𝐷0 (1 + 𝑔) (𝑅 − 𝑔)
𝑔 = 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑔𝑟𝑜𝑤𝑡ℎ 𝑟𝑎𝑡𝑒 𝑖𝑛 𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 Example: Coco Ltd has just paid an annual dividend of $0.30 per share, which is expected to grow at 5% indefinitely. If the RROR is 8%, how much would you be willing to pay for the share? 𝑃0 =
0.30(1 + 0.05) = $10.50 (0.08 − 0.05)
3. Variable dividend growth valuation o Allow for different growth rates o It is possible for dividends to grow at a high rate for a number of years but not indefinitely o Assume dividend will grow at a constant rate some time in the future Example: Dave Ltd has just paid a $0.50 annual dividend. The required return on Dave’s shares is 10%. Exceptional growth is forecasted for Dave Ltd for the next 3 years at 12% p.a. After year 3, growth rate (g) will settle at 5% indefinitely. What is the value of Dave Ltd shares today? 𝑃0 = =
𝐷0 (1 + 𝑔′ ) 𝐷0 (1 + 𝑔′ )2 𝐷0 (1 + 𝑔′ )3 1 𝐷0 (1 + 𝑔)3 (1 + 𝑔) + + + × (1 + 𝑟) (1 + 𝑟)2 (1 + 𝑟)3 (1 + 𝑟)3 (𝑟 − 𝑔)
0.50(1.12) 0.50(1.12)2 0.50(1.12)3 1 0.50(1.12)3 (1.05) + + + × = $12.64 1.10 (1.10)2 (1.10)3 (1.10)3 (0.10 − 0.05)
3.7 Earnings Based Valuation 𝑉𝑎𝑙𝑢𝑒 =
𝑃 × 𝐸𝑃𝑆 𝐸
𝑃 = 𝑃𝑟𝑖𝑐𝑒 𝑡𝑜 𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝑟𝑎𝑡𝑖𝑜 𝐸𝑃𝑆 = 𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 𝐸
Earnings and dividends are related A company’s after-tax earnings (profit) must be either retained or paid out as dividends 𝐷𝑡 = (1 − 𝑏)𝐸𝑡 𝑏 = 𝑃𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛 𝑜𝑓 𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝑟𝑒𝑡𝑎𝑖𝑛𝑒𝑑
𝐸𝑡 = 𝑃𝑒𝑟𝑖𝑜𝑑 𝑡 𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒
Example: Alex has forecasted that the EPS of Richie Ltd to be at $5. Alex’s market research suggests that an appropriate PE ratio is 8. What is the value of Richie Ltd’s share? 2
𝑉𝑎𝑙𝑢𝑒 =
𝑃 × 𝐸𝑃𝑆 = 8 × 5 = $40 𝐸
Factors influencing PE ratio o Growth opportunities: The greater opportunities for growth, the higher the PE ratio o Risk: The riskier the investment, the lower the PE ratio o Earnings tends to be more volatile than dividends: Requires skill and experience to forecast
Topic 4: Project Evaluation I 4.1 Capital Budgeting Introduction
Capital budgeting decisions are the most important investment decisions made by management The goal of these decisions is to select capital projects that will maximise shareholders’ wealth Capital investments are important because they involve substantial cash outlays and, once made, are not easily reversed Help management to systematically analyse potential business opportunities in order to decide which are worth undertaking Capital budgeting (investment): Cash outlay(s) now in the expectation of benefits (net cash inflows) later Sources of information: o Most of the information needed is generated internally (i) Beginning with the sales force; (ii) Then the production team is involved; (iii) Followed by the accountants o All this information is then reviewed by the financial managers who evaluate the feasibility of the project
4.2 Classification of Investment Projects (i) Independent projects o Projects are independent when their cash flows are unrelated o If two projects are independent, accepting or rejecting one project has no bearing on the decision on the other (ii) Mutually exclusive projects o When two projects are mutually exclusive, accepting one automatically precludes the other (iii) Contingent projects o Contingent projects are those where the acceptance of one project is dependent on another project o These are two types of contingencies situations: (a) Mandatory projects (b) Optional projects
4.3 Overview of Project Evaluation Techniques
Discounted Cash Flow (i) Net Present Value (NPV) (ii) Internal Rate of Return (IRR) (iii) Benefit-Cost Ratio (Profitability Index) Non-Discounting (i) Accounting Rate of Return (ARR) (ii) Payback Period
3
4.4 NPV
Finding the present value entails calculating the equivalent value today of a set of promised (or forecast) future cash flows 𝑁𝐶𝐹1 𝑁𝐶𝐹2 𝑁𝐶𝐹𝑛 𝑁𝑃𝑉 = + + ⋯ + − 𝑁𝐶𝐹0 (1 + 𝑘)1 (1 + 𝑘)2 (1 + 𝑘)𝑛 𝑛
𝑁𝑃𝑉 = ∑ 𝑡=1
𝑁𝐶𝐹𝑡 − 𝑁𝐶𝐹0 (1 + 𝑘)𝑡
𝑁𝐶𝐹0 = 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑐𝑎𝑠ℎ 𝑜𝑢𝑡𝑙𝑎𝑦 𝑁𝐶𝐹𝑡 = 𝑁𝑒𝑡 𝑐𝑎𝑠ℎ 𝑓𝑙𝑜𝑤 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑒𝑑 𝑏𝑦 𝑝𝑟𝑜𝑗𝑒𝑐𝑡 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡 𝑛 = 𝐿𝑖𝑓𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑟𝑜𝑗𝑒𝑐𝑡 𝑘 = 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑟𝑒𝑡𝑢𝑟𝑛 The 5-step approach to calculate NPV 1. Determine the cost of the project Identify and add up all expenses related to the cost of the project While most of the project’s costs occurs at the start of the project, some projects may have costs occurring beyond the first year The cash flow in year 0 (NCF0) is negative, indicating a cost 2. Estimate the project’s future cash flows over its forecasted life Both cash inflows (CIF) and cash outflows are likely in each year of the project Estimate the net cash flow (NCFt) = CIFt – COFt for each year of the project Cash inflows: Receipts from sales of goods and services; receipts from sale of physical assets Cash outflows: Expenditure on materials, labour, and indirect expenses for manufacturing; selling and administrative; inventory and taxes Remember to recognize any salvage value from the project in its terminal year 3. Determine the riskiness of the project and estimate the appropriate cost of capital The cost of capital is the discount rate used in determining the present value of the future expected cash flows The riskier the project, the higher the cost of capital for the project 4. Calculate the project’s NPV Determine the difference between the present value of the expected cash flows from the project and the cost of the project 5. Make a decision Accept the project if it produces a positive NPV or reject the project if the NPV is negative NPV method is consistent with the company’s objective of maximizing shareholders’ wealth. A project with a positive NPV will leave the company better off than before the project. Other things being equal, the market value of the company’s shares should increase Feature of the NPV method: o Consistent with maximising shareholder wealth o Consistent with requiring a proposed investment to achieve at least the rate of return required by investors o Focuses on the incremental effects of an investment. Provides a direct measure of how much a capital project is expected to increase the dollar value of a company Decision rule for the NPV method: o Independent projects NPV > 0 then accept project NPV < 0 then reject project o Mutually exclusive projects Accept project with the highest NPV Advantages o Use DCF valuation technique to adjust for time value of money o Provide direct dollar measure of how much a capital project will increase the value of the firm o Consistent with the goal of maximising shareholder wealth 4
Disadvantages o Can be difficult to understand without an accounting or finance background
Example: An investment promises to pay $670 000 one year from today and a further $1 240 000 four year from today. What is it worth today if the required rate of return is 15 % p.a.? Suppose that we have to outlay $1.2 million to get the right to receive the cash inflows. 𝑃𝑉 =
670,000 1,240,000 + = $1,291,583 1.15 1.154
𝑁𝑃𝑉 = $1,291,583 − $1,200,000 = $91,583
∴ 𝑁𝑃𝑉 > 0 ⇒ 𝐴𝑐𝑐𝑒𝑝𝑡
Example: What project should you accept if the projects were independent? What project should you accept if the projects are mutually exclusive? NPV $1,500 $2,500
Project A Project B
Accept both projects if A and B are independent. If projects are mutually exclusive then accept B.
4.5 Internal Rate of Return
The Internal Rate of Return (IRR) is the rate of return that equates the present value of projected cash flows with the initial cash outlay The IRR is the discount rate which makes the project break even (NPV=0) 𝑛
𝑁𝐶𝐹0 = ∑ 𝑡=1
𝑁𝐶𝐹𝑡 (1 + 𝑟)𝑡
Decision rule: Accept if IRR > RROR Advantages: o Intuitively easy to understand o Based on DCF technique Problems with IRR: o May not always lead to the same decision as the NPV rule when projects are mutually exclusive o Reinvestment rate is assumed to be the IRR, which might be unrealistic o Multiple IRRs – A project may have more than one IRR which complicates decision making o Indeterminate IRR – A project may have no IRR o It ignores the size of the project
5
Example: An investment of $1,000 today yields a return of $300 in year 1, $400 in year 2 and $500 in year 3. Calculate the IRR of this investment? If the required rate of return is 13%, should the project proceed? $1,000 =
300 400 500 + + 2 (1 + 𝑟) (1 + 𝑟) (1 + 𝑟)3
Use the trial and error method: 𝐼𝑓 𝑟 = 10%: 𝑁𝑃𝑉 = −21.0368
𝐼𝑓 𝑟 = 8%: 𝑁𝑃𝑉 = 17.6294
Interpolation where: 𝑟 = 𝑟1 + (𝑟2 − 𝑟1 ) [ = 0.10 + (0.08 − 0.10) [
𝑇1 ] 𝑇1 − 𝑇2
−21.0368 ] = 0.0891 = 8.9% −21.0368 + (−17.6294)
4.6 NPV and IRR When IRR and NPV agree The methods will always agree when the projects are: (a) Independent; and (b) The projects’ cash flows are conventional Conventional cash flows: After the initial investment is made (cash outflow), all the cash flows in each future year are positive (inflows) When IRR and NPV disagree The IRR and NPV methods can produce different accept/reject decisions if a project either has; (a) Unconventional cash flows; or Multiple IRRs can result. The maximum number of IRR is equal to the number of sign reversals in cash flows (b) The projects are mutually exclusive IRR cannot be used to rank mutually exclusive projects because the project with the highest IRR may not be the project that would add the greatest value to the company if accepted – that is, the project with the highest NPV Example: Consider the following cash flows for two mutually exclusive projects A and B
Year 0
Year 1
Year 2
NPV (k=15%)
IRR
A
-10,000
2,500
12,500
1,626
25%
B
-10,000
12,000
1,000
1,191
27.8%
Choose project A because it has a higher NPV and thus maximises shareholder wealth. NPV and IRR give different results (ranking) because of the timing of cashflows.
6
Example: Different timing of cash flows
4.7 Modified IRR
IRR assumes that the cash flows from the project are reinvested at the IRR, while the NPV assumes that they are invested at the company’s cost of capital. This optimistic assumption in the IRR method leads to some projects being accepted when they should not Under the MIRR: o Each operating cash flow is reinvested at the company’s cost of capital o The compounded values are summed up to get the project’s terminal value o The MIRR is the interest rate which equates the project’s cost to the terminal value at the end of the project 𝑃𝑉 𝑜𝑓 𝑐𝑜𝑠𝑡 =
𝑇𝑉 (1 + 𝑀𝐼𝑅𝑅)𝑛
4.8 Benefit-Cost Ratio (Profitability Index) 𝐵𝐶𝑅 =
𝑃𝑉 𝑜𝑓 𝑐𝑎𝑠ℎ𝑓𝑙𝑜𝑤𝑠 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑐𝑎𝑠ℎ 𝑜𝑢𝑡𝑙𝑎𝑦
Decision rule: BCR > 1 then accept project For mutually exclusive projects, the benefit-cost ratio may produce a different ranking of projects than that provided by the NPV method
7
Example: A 10,000 8,000 2,000 10,000/8,000=1.25
PV of cashflows Initial cash outlay NPV BCR
B 4,000 2,200 1,800 4,000/2,200=1.81
4.9 Accounting Rate of Return
The ARR is the earnings from a project (after deducting depreciation & income tax) expressed as a percentage of the investment outlay It is based on accounting numbers rather than cash flows. As such, it is not a true rate of return. Instead of discounting a project’s cash flows over time, it simply gives us a number based on average figures from the income and balance sheet Decision rule: ARR is compared to the RROR. If the ARR > RRR then the project is accepted 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑛𝑒𝑡 𝑖𝑛𝑐𝑜𝑚𝑒 𝐴𝑅𝑅 = 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑏𝑜𝑜𝑘 𝑣𝑎𝑙𝑢𝑒 Strengths: o Useful screening measure to ensure that new investment will not adversely affect net incomes o Easy to calculate and understand Weaknesses: o Arbitrary: The choice of depreciation schedule and inventory valuation will significantly affect earnings estimates o Timing of cash flows: The time value of money is not taken into account in the calculation o Project size: ARR does not account for project size when a choice between two projects of different sizes must be made Three variants to ARR: 1. ARR based on initial investment 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝐴𝑅𝑅 = 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 2. ARR based on average book value 𝐴𝑅𝑅 =
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑏𝑜𝑜𝑘 𝑣𝑎𝑙𝑢𝑒
3. ARR based on the initial and final capital value 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝐴𝑅𝑅 = 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑐𝑎𝑝𝑖𝑡𝑎𝑙 Example: Different ways to calculate ARR Item Earnings after income tax and depreciation
Year 1 $10,000
Year 2 $13,500
Year 3 $18,000
Average $13,833
$50,000 $40,000 $45,000
$40,000 $30,000 $35,000
$30,000 $20,000 $25,000
$35,000
Book value (assume 20% straight line dep.) Jan 1st Dec 31st Average
8
ARR based on initial investment:
13,833 50,000
ARR based on average book value:
= 27.7%
13,833 35,000
= 39.52%
ARR based on initial and final capital value:
13,833 50,000+20,000 2
= 39.52%
4.10 Payback Period
The time it takes for the initial cash outlay on a project to be recovered from a project’s net cash flows Decision rule: Projects are accepted if the payback period is less than some given period 𝑃𝐵 = 𝑌𝑒𝑎𝑟𝑠 𝑡𝑜 𝑟𝑒𝑐𝑜𝑣𝑒𝑟 𝑐𝑜𝑠𝑡 +
𝑅𝑒𝑚𝑎𝑖𝑛𝑖𝑛𝑔 𝑐𝑜𝑠𝑡 𝑡𝑜 𝑟𝑒𝑐𝑜𝑣𝑒𝑟 𝐶𝑎𝑠ℎ 𝑓𝑙𝑜𝑤 𝑑𝑢𝑟𝑖𝑛𝑔 𝑡ℎ𝑒 𝑦𝑒𝑎𝑟
Strengths: o Provides an indication of a project’s risk and liquidity o Easy to calculate and understand Weaknesses: o Ignores time value of money o Ignores cash flows occurring after the payback period o Is not a measure of profitability or shareholder wealth o Arbitrary cut off point Year 0 1 2 3 4 Payback
A -10,000 3,000 3,000 4,000 4,000 3 years
B -10,000 5,000 4,000 2,000 2,000 2.5 years
C -10,000 3,000 3,000 4,000 20,000 3 years
4.12 Discounted Payback Period
One of the weaknesses of the ordinary payback period is that it does not take into account the time value of money The discounted payback period calculation calls for the future cash flows to be discounted by the company’s cost of capital Advantage: Tells management how long it takes a project to reach a positive NPV Disadvantage: Still ignores all cash flows after the arbitrary cut-off period, which is a major flaw
Topic 5: Project Evaluation II 5.1 Cash Flow
The focus is on incremental operating cash flows which are cash flows that will occur only if the project is undertaken (i.e. Estimating the cash flow changes that will occur as a result of investing in the project) Typical operating cash flows include wages paid, materials purchased, sales revenue and taxes In project evaluation, we are comparing two alternative scenarios for the future: o One with the investment o One without the investment The accept/ reject decision is made relative to the existing scenario.
9
Incremental cash flows 1. Include cash flows and only cash flows in your calculations 2. Include the impact of the project on cash flows from other product lines o Consider if the product associated with a project is expected to cannibalize or boost sales of another product in the analysis 3. Include all opportunity costs (i.e. the cost of giving up the next best alternative) 4. Forget sunk costs (past investments are irrelevant) 5. Include only after-tax cash flows in the cash flow calculations Cash flows not profits
Focus is on cash not ‘profit’ which is an accounting concept Only cash, not ‘profit’ can be spent, reinvested, paid out in dividends etc Profit includes: o Some non-cash items (e.g. depreciation) o Some cash items which are non-operating (e.g. interest payments on borrowed funds)
Initial and subsequent outlays
Should be included at appropriate dates
Example: A machine costs $25 000 to buy and a further $2 000 to install. There is a 3-month waiting list for buyers. The installation cost is payable in cash on installation. A deposit of $5 000 is required on placement of an order and the balance within two months of delivery/ installation. What is the total initial outlay? If the required rate of return is 1% per month, what is the present value of the initial outlay(s)? Month 0 3 5
Action Order placed Delivery/installation Balance due
𝑃𝑉 = −5,000 −
Cash flow -$5,000 -$2,000 -$20,000
2,000 2,000 − = −$25,970.49 3 (1.01) (1.01)5
5.2 Free Cash Flow Calculation What to include in the calculation? 1. Operating cash flows 2. Opportunity costs o Cash inflows without the investment that are foregone (and hence become cash outflows) o Example: A firm owns a warehouse which it is not currently using. A proposal is put forward to convert the warehouse into a number of residential townhouses. A DCF analysis of this proposal should include the opportunity cost of selling the warehouse. The estimated sale price should be included as a cash outflow 3. Salvage value o This is the amount received at the end of a project. May be scrap value only (e.g. From the sale of equipment) and may be negative (e.g. An environmental clean-up) 4. Additional working capital in the final year o The FCF in the last, or terminal, year of a project also includes another cash flow item that is not typically included in the calculations for other years such as working capital (Additional WC) o Additional WC = Cash and cash equivalents + accounts receivable + inventories – accounts payable o In the final year of the project, the working capital that has been invested in year 0 may be recovered 10
o o
The principle behind including only these cash flows in y0 is that investments must be made before any cash flow from operations are realised When a project ends, the cash and cash equivalents are no longer needed, the accounts receivable are collected, the inventories are sold and the accounts payable are paid. I.e. the company recovers the net working capital that has been invested in the project. To reflect this in the FCF calculation, the cash flow in the last year of the project includes a negative investment in working capital that equals the cumulative investment in the working capital over the project life
What to exclude from the calculation? 1. Allocated costs o Costs already allocated by the firm and will be paid irrespective of whether or not the project proceeds o Examples: Rates, head office costs, rent, corporate image advertising. o But sometimes allocated costs can be incremental and should be included 2. Financing charges o Example: Interest expense o The required rate of return covers interest and return to equity 3. Sunk costs o These are costs already borne by the firm. They are a cost to the firm irrespective of whether or not the project proceeds. o Example: The cost of research & development on a previous product Incremental After-Tax FCF Calculation: Method 1 Calculation Revenue Revenue − Op Ex Less: Cash operating expenses = Earnings before interest, tax, depreciation & amortisation EBITDA Less: Depreciation & amortisation −D&A = Earning before interest and tax EBIT EBIT*(1- Company Tax Rate) × (1−TC) = Net operating profit after tax NOPAT Add back: Depreciation & amortisation +D&A = Cash flow from operations CF Opns − Cap Exp Less: Capital expenditures Less: Additions to working capital
Notation
Addition to working capital = Cash & cash equiv. + Accounts −Add WC receivables + Inventories – Accounts payable Note: This additional WC is assumed to be recovered in the terminal year = Free cash flow FCF Incremental After-Tax FCF Calculation: Method 2 Calculation Revenue Revenue − Op Ex Less: Cash operating expenses = Earnings before interest, tax, depreciation & amortisation EBITDA EBITDA*(1 – Company Tax Rate) × (1−TC) = Net cash flow after tax NCFAT Plus: Depreciation tax shield + (D × TC) = Cash flow from operations CF Opns − Cap Exp Less: Capital expenditures Less: Additions to working capital −Add WC = Free cash flow FCF 11
Notation
5.3 Depreciation and Tax Impact of company tax on depreciation
When a firm purchases equipment it will lose value (depreciation) over time. Although depreciation is not a cash outflow, companies can claim depreciation on an asset as a deduction in determining company income Methods to account for the tax effect of depreciation in the cash flow table: 1. Deduct depreciation, account for the company tax and then add back depreciation 2. Add depreciation tax shield to the after-tax cash flow Depreciation tax shield = Depreciation x TC Depreciation produces tax savings known as depreciation tax shield
Depreciation calculation 1. Straight line depreciation o In this unit, we assume that the remaining value is zero in year n 1 𝑑= 𝑛 2. Reducing balance depreciation o Unless specified by the question, the depreciation rate for reducing balance is 150% of straight line 1.5 𝑑= 𝑛 Example: An asset is acquired for $100 000 and has an expected economic life of 4 years 𝑆𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝑙𝑖𝑛𝑒 𝑟𝑎𝑡𝑒 = 𝑅𝑒𝑑𝑢𝑐𝑖𝑛𝑔 𝑏𝑎𝑙𝑎𝑛𝑐𝑒 𝑟𝑎𝑡𝑒 =
Year # 1 2 3 4
1 = 25% 4 1.5 = 37.5% 4
Reducing balance method Written down value Depreciation during the (start of the year) year $100,000 0.375*100,000 = $37,500 $62,500 0.375*62,500 = $23,438 $39,062 0.375*39,062 = $14,638 $24,414 0.375*24,414 = $9,155
Straight line method Depreciation this year $25,000 $25,000 $25,000 $25,000
𝑊𝑟𝑖𝑡𝑡𝑒𝑛 𝑑𝑜𝑤𝑛 𝑣𝑎𝑙𝑢𝑒 (𝑒𝑛𝑑 𝑜𝑓 𝑦𝑒𝑎𝑟 4): 𝑆𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝑙𝑖𝑛𝑒 𝑚𝑒𝑡ℎ𝑜𝑑: 𝑉𝑎𝑙𝑢𝑒 = 0 𝑅𝑒𝑑𝑢𝑐𝑖𝑛𝑔 𝑏𝑎𝑙𝑎𝑛𝑐𝑒 𝑚𝑒𝑡ℎ𝑜𝑑: 𝑉𝑎𝑙𝑢𝑒 = 24,414 − 9,155 = $15,259 Impact of taxation on disposal value
Example: Suppose the asset is sold for $S at the end of year 3. $S is the cash inflow at the end of year 3. What are the tax implications? o The answer depends on the sale price compared to the written down value at the time of sale o Written down value = Total cost of investment – Accumulated depreciation o Assume w = $640 o Scenario 1: s = $540 Loss on sale = 640 – 540 = 100 This loss is deductible for tax purposes (at company tax rate of 30%) Tax saving = 100 x 0.3 = $30 After-tax net cash flow from disposal = Sale price + Tax saved = 540 + 30 = 570 12
o
Scenario 2: s = $1000 Gain on sale = 1000 – 640 = 360 Tax liability on gain on sale = 360 x 0.3 = 108 After-tax net cash flow from disposal = Sale price + Tax liability = $1000 – 108 = 892
5.4 Past Exam Example
Bellco Ltd has identified a new market for its products. To increase the output level, the company is considering the purchase of some new machinery at a cost of $400,000. The machinery is imported from the United States and the delivery and installation costs are $20,000. The current estimated before-tax net operating cash revenue for the coming 3 years are: $260,000 in the first year, $240,000 in the second year, and $200,000 in the third year. The purchase of the new machinery is expected to increase the expected before-tax net operating cash revenue for the next 3 years by 80% of the current estimated value. Bellco will need to obtain financing from the bank to fund this investment. The interest payment on the loan is $85,000 per annum. The machinery will be sold at the end of the third year and its market value at that time is estimated to be $55,000. The company tax rate is 30% and reducing balance depreciation at 50% per year is allowed. The operating cash flows should be considered to occur at year-end. The required rate of return is 15% per year Y0
Y1 260,000*0.8 = 208,000 208,000 420,000*0.5 = 210,000 (2,000) (600) (1400) 210,000 210,000 + (1400) = 208,600
Net operating cash revenue EBITDA Less: Depreciation EBIT Less: Company tax NOPAT Add: Depreciation CF Opns Less: Cap Exp FCF
(420,000) (420,000)
208,600
Y2 240,000*0.8 = 192,000 192,000 210,000*0.5 = 105,000 87,000 26,100 60,900 105,000
Y3 200,000*0.8 = 160,000 160,000
165,900
127,750
165,900
54,250 182,000
52,500 107,500 32,250 75,250 52,500
𝐴𝑡 𝑌0: 𝐶𝑎𝑝 𝐸𝑥 = 400,000 + 20,000 = 420,000 𝐴𝑡 𝑌3 (𝑇𝑒𝑟𝑚𝑖𝑛𝑎𝑙 𝑦𝑒𝑎𝑟): 𝐶𝑎𝑝 𝐸𝑥𝑝 = 55,000 − (0.3 ∗ 𝐺𝑎𝑖𝑛 𝑜𝑛 𝑠𝑎𝑙𝑒) = 55,000 − (0.3 ∗ 2,500) = 54,250 𝑁𝑃𝑉 = −420,000 +
208,600 165,900 182,000 + + = $6,503.49 1.15 1.152 1.153
Example 2: Outdoor performing arts centre
Evaluating a project to increase the number of seats by building four new box seating areas and adding 5000 seats for the general public. Each box seating area is expected to generate $400,000 in incremental annual revenue, while each of the new seats for the general public will generate $2,500 in incremental annual revenue The incremental expenses for the new boxes and seating will be 60% of the revenues. These expenses include hiring additional personnel to handle concessions, ushering and security The new construction will cost $10m and will be fully depreciated on a straight-line basis over 10 years. The centre will have to invest $1m in additional working capital immediately. This working capital is recovered in the last year of the project. The tax rate is 30% 13
5.5 Mutually Exclusive Projects with Different Lives
A firm may be considering undertaking two or more projects at the same time but may be limited to choosing only one of these projects because of: Limited levels of debt finance, office or factory capacity, or skilled personnel Mutually exclusive projects cannot occur at the same time Mutually exclusive projects with different lives can be evaluated using the following methods: a. Constant Chain of Replacement i. Lowest Common Multiple ii. NPV Perpetuity method b. Equivalent Annual Value/ Cost Method
Example: The table shows the cash outflows for 2 machines. Assume both have the same annual cash inflows. The aim is to buy the machine that results in the lower present value of outflows. Assume a RROR of 13% p.a. Year 0 1 2 3 4
Machine A $100,000 $170,000 $180,000 $200,000 $220,000
Machine B $150,000 $200,000 $220,000
𝑃𝑉 (𝐶𝑜𝑠𝑡𝑠 𝑜𝑓 𝐴) = $664,949 𝑃𝑉(𝐶𝑜𝑠𝑡𝑠 𝑜𝑓 𝐵) = $499,283 Machine B appears to be cheaper but the comparison is flawed. Consider replicating machine B in years 3 and 4 Year 0 1 2 3 4
Machine A $100,000 $170,000 $180,000 $200,000 $220,000
Machine B $150,000 $200,000 $220,000 + $150,000 = $370,000 $200,000 $220,000
𝑃𝑉(𝐶𝑜𝑠𝑡𝑠 𝑜𝑓 𝐵 𝑡𝑤𝑖𝑐𝑒) = $890,296 𝑃𝑉(𝐶𝑜𝑠𝑡𝑠 𝑜𝑓 𝐴) = $664,949 Therefore, the correct decision is to buy machine A due to lower cash outflow 14
6.7 Decision Tree Analysis
Used to evaluate investment options involving a series of decisions over a period of time Decision Tree Analysis involves taking account of the probability of various events occurring and the effect of those decisions on the NPV of the project Decision tree analysis then involves calculating the NPV of each decision working backwards through the tree. (Assume a discount rate of 10% pa)
Example: The management of ELEC P/L are considering investing in a 1-year research project that will attempt to develop an electric mop. The research program will initially cost $500,000 and if successful will yield a cash flow of $150,000 in perpetuity. Management also believes there is only a 33% chance of successfully developing an electric mop. Assuming a discount rate of 12% pa, should ELEC P/L proceed with the project?
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑝𝑎𝑦𝑜𝑓𝑓 (𝑎𝑡 𝑡 = 1) = 1,250,000 × 0.33 + 0 × 0.67 = $412,500 𝑁𝑃𝑉 (𝑎𝑡 𝑡 = 0) = −500,000 +
412,500 = −131,696.43 1.12
∴ 𝐷𝑜𝑛′ 𝑡 𝑝𝑟𝑜𝑐𝑒𝑒𝑑
Example: More complex problem
Management of a silicon chip manufacturing firm are faced with the decision of investing in two alternative chip producing machines, both of which have an operational life of 8 years. Machine A costs $4m and machine B costs $3m. If machine B is chosen, management may after 3 years choose to upgrade it for a cost of $3m so that it has the same capacity as machine A. However, this will only occur if demand is high. Management also believes that the probability of high demand for chips in the first 3 years will be 0.7. However, if demand is high in the first 3 years management believes there is a 0.8 probability that demand will remain high in the following 5 years. However, if demand is low, management believes that there is a 0.6 probability that it will remain low. Discount rate is 10%
Let H3 = event of high demand in first 3 years
L3 = event of low demand in first 3 years,
H5 = event of high demand in next 5 years
L5 = event of low demand in next 5 years
P(H3)=0.7
P(L3)=0.3
P(H5|H3)=0.8
P(L5|H3)=0.2
P(L5|L3)=0.6
P(H5|L3)=0.4
Management has also estimated the following annual cash flows from the decisions Machine A A B B Upgrade B Upgrade B
Demand H L H L H L 15
Cashflow p.a. 1m 0.5m 0.6m 0.2m 1m 0.5m
Decision 2
Upgrade B 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑝𝑎𝑦𝑜𝑓𝑓 𝑝. 𝑎. = 1𝑚 × 0.8 + 0.5 × 0.2 = 0.9𝑚 0.9𝑚 1 𝑁𝑃𝑉(𝑎𝑡 𝑡 = 3) = −3 + (1 − ) = 0.4117𝑚 0.1 1.15 Do not upgrade B 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑝𝑎𝑦𝑜𝑓𝑓 𝑝. 𝑎. = 0.6𝑚 × 0.8 + 0.2𝑚 × 0.2 = 0.52𝑚 0.52𝑚 1 𝑁𝑃𝑉(𝑎𝑡 𝑡 = 3) = (1 − ) = 1.971𝑚 0.1 1.15 ∴ 𝐷𝑜 𝑛𝑜𝑡 𝑢𝑝𝑔𝑟𝑎𝑑𝑒 𝑖𝑓 𝑑𝑒𝑚𝑎𝑛𝑑 𝑖𝑠 ℎ𝑖𝑔ℎ 𝑖𝑛 𝑓𝑖𝑟𝑠𝑡 3 𝑦𝑒𝑎𝑟𝑠
Decision 1
Buy machine A Assume high demand in first 3 years 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑝𝑎𝑦𝑜𝑓𝑓 𝑝. 𝑎. = 1𝑚 × 0.8 + 0.5 × 0.2 = 0.9𝑚 0.9𝑚 1 𝑁𝑃𝑉(𝑎𝑡 𝑡 = 3) = (1 − ) = 3.412𝑚 0.1 1.15 Assume low demand in first 3 years 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑝𝑎𝑦𝑜𝑓𝑓 𝑝. 𝑎. = 1𝑚 × 0.4 + 0.5 × 0.6 = 0.7𝑚 𝑁𝑃𝑉(𝑎𝑡 𝑡 = 3) = 𝑁𝑃𝑉(𝑎𝑡 𝑡 = 0) = −4𝑚 + 0.7 (
3.142𝑚 1.13
1𝑚
0.7𝑚 1 (1 − ) = 2.654𝑚 0.1 1.15 1
2.654𝑚 1.13
+ 0.1 (1 − 1.13 )) + 0.3 (
+
0.5𝑚 1 (1 − 1.13 )) 0.1
= $0.5065𝑚
Buy machine B Assume high demand in first 3 years 0.52𝑚 1 𝑁𝑃𝑉(𝑎𝑡 𝑡 = 3) = (1 − ) = 1.971𝑚 [𝑓𝑟𝑜𝑚 𝑑𝑒𝑐𝑖𝑠𝑖𝑜𝑛 2] 0.1 1.15
16
Assume low demand in first 3 years 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑝𝑎𝑦𝑜𝑓𝑓 𝑝. 𝑎. = 0.6𝑚 × 0.4 + 0.2 × 0.6 = 0.36𝑚 0.36𝑚 1 𝑁𝑃𝑉(𝑎𝑡 𝑡 = 3) = (1 − ) = 1.365𝑚 0.1 1.15 N𝑃𝑉(𝑎𝑡 𝑡 = 0) = −3𝑚 + 0.7 (
1.971𝑚 1.13
+
0.6𝑚 1 (1 − 1.13 )) + 0.1
1.365𝑚 1.13
0.3 (
+
0.2𝑚 1 (1 − 1.13 )) 0.1
= −$0.4062𝑚
∴ 𝐵𝑢𝑦 𝑚𝑎𝑐ℎ𝑖𝑛𝑒 𝐴
Topic 7: Risk, Return & Investment I 7.1 Realised Returns 𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑟𝑒𝑡𝑢𝑟𝑛 =
𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 + 𝐸𝑛𝑑𝑖𝑛𝑔 𝑚𝑎𝑟𝑘𝑒𝑡 𝑣𝑎𝑙𝑢𝑒 − 𝐵𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝑚𝑎𝑟𝑘𝑒𝑡 𝑣𝑎𝑙𝑢𝑒 𝐵𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝑚𝑎𝑟𝑘𝑒𝑡 𝑣𝑎𝑙𝑢𝑒 = 𝑑𝑖𝑑𝑖𝑑𝑒𝑛𝑑 𝑦𝑖𝑒𝑙𝑑 + 𝑐𝑎𝑝𝑖𝑡𝑎𝑙 𝑔𝑎𝑖𝑛𝑠 𝑦𝑖𝑒𝑙𝑑 𝑅𝑻 =
𝐶𝐹1 + 𝑃1 − 𝑃0 𝑃0
Example: Suppose you bought 100 shares of ANZ one year ago at $34.99. At the end of the year, you sell the stock for $30.02. How did you do assuming you had 100 shares? 𝑅𝑒𝑡𝑢𝑟𝑛 𝑖𝑛 % =
(30.02 × 100) − (34.99 × 100) = −0.14204 = −14.204% 34.99 × 100
7.2 Average Returns (𝑅1 + ⋯ + 𝑅𝑇 ) 𝑅̅ = 𝑇
∑𝑛𝑖=1 𝑅𝑖 𝑜𝑟 𝐸(𝑅𝑎𝑠𝑠𝑒𝑡 ) = 𝐸(𝑅) = 𝑛
𝑛
𝑜𝑟 𝐸(𝑅) = 𝑅̅ = ∑ 𝑝𝑖 𝑅𝑖 𝑖=1
𝑤ℎ𝑒𝑟𝑒 𝑝𝑖 =
1 𝑛
Note: pi can also be a probability How to use the calculator: 1. Clear the statistics memory using “-m/C STAT” key 2. Enter the first value and press “∑+” 3. Continue entering the values and pressing “∑+” 4. Find mean by pressing “𝑥̅ , 𝑦̅” 5. Find standard deviation by press “𝑆𝑥 , 𝑆𝑦 ”
7.3 Holding Period returns
Also known as the compounded realised returns 𝐻𝑃𝑅 = (1 + 𝑅1 ) × (1 + 𝑅2 ) × … × (1 + 𝑅𝑇 ) − 1
7.4 What is risk?
Risk can be defined as the uncertainty of the future outcomes or the probability of an adverse outcome Alternatively, it is a chance of a financial loss or the variability of returns associated with an asset A common measure of risk is the variance or standard deviation of expected returns
7.5 Variance and Standard Deviation 𝑉𝑎𝑟(𝑅) =
1 [(𝑅1 − 𝑅̅ )2 + (𝑅2 − 𝑅̅ )2 + ⋯ + (𝑅𝑇 − 𝑅̅ )2 ] 𝑇−1 17
1
3.6 Dividend Valuation 1. Constant dividend valuation o Use the perpetuity formula o Can be applied for shares that pay a constant dividend (e.g. preference shares) 𝑃0 =
𝐷 𝑅
Example: Worley Ltd is expected to pay a constant annual dividend of $2.50 per share indefinitely. If the discount rate is 8% p.a., what is the value of the share? 𝑃0 =
2.50 = $31.25 0.08
2. Constant dividend growth valuation o If dividends are expected to grow at a constant rate 𝑃0 =
𝐷0 (1 + 𝑔) (𝑅 − 𝑔)
𝑔 = 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑔𝑟𝑜𝑤𝑡ℎ 𝑟𝑎𝑡𝑒 𝑖𝑛 𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 Example: Coco Ltd has just paid an annual dividend of $0.30 per share, which is expected to grow at 5% indefinitely. If the RROR is 8%, how much would you be willing to pay for the share? 𝑃0 =
0.30(1 + 0.05) = $10.50 (0.08 − 0.05)
3. Variable dividend growth valuation o Allow for different growth rates o It is possible for dividends to grow at a high rate for a number of years but not indefinitely o Assume dividend will grow at a constant rate some time in the future Example: Dave Ltd has just paid a $0.50 annual dividend. The required return on Dave’s shares is 10%. Exceptional growth is forecasted for Dave Ltd for the next 3 years at 12% p.a. After year 3, growth rate (g) will settle at 5% indefinitely. What is the value of Dave Ltd shares today? 𝑃0 = =
𝐷0 (1 + 𝑔′ ) 𝐷0 (1 + 𝑔′ )2 𝐷0 (1 + 𝑔′ )3 1 𝐷0 (1 + 𝑔)3 (1 + 𝑔) + + + × (1 + 𝑟) (1 + 𝑟)2 (1 + 𝑟)3 (1 + 𝑟)3 (𝑟 − 𝑔)
0.50(1.12) 0.50(1.12)2 0.50(1.12)3 1 0.50(1.12)3 (1.05) + + + × = $12.64 1.10 (1.10)2 (1.10)3 (1.10)3 (0.10 − 0.05)
3.7 Earnings Based Valuation 𝑉𝑎𝑙𝑢𝑒 =
𝑃 × 𝐸𝑃𝑆 𝐸
𝑃 = 𝑃𝑟𝑖𝑐𝑒 𝑡𝑜 𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝑟𝑎𝑡𝑖𝑜 𝐸𝑃𝑆 = 𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 𝐸
Earnings and dividends are related A company’s after-tax earnings (profit) must be either retained or paid out as dividends 𝐷𝑡 = (1 − 𝑏)𝐸𝑡 𝑏 = 𝑃𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛 𝑜𝑓 𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝑟𝑒𝑡𝑎𝑖𝑛𝑒𝑑
𝐸𝑡 = 𝑃𝑒𝑟𝑖𝑜𝑑 𝑡 𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒
Example: Alex has forecasted that the EPS of Richie Ltd to be at $5. Alex’s market research suggests that an appropriate PE ratio is 8. What is the value of Richie Ltd’s share? 2
𝑉𝑎𝑙𝑢𝑒 =
𝑃 × 𝐸𝑃𝑆 = 8 × 5 = $40 𝐸
Factors influencing PE ratio o Growth opportunities: The greater opportunities for growth, the higher the PE ratio o Risk: The riskier the investment, the lower the PE ratio o Earnings tends to be more volatile than dividends: Requires skill and experience to forecast
Topic 4: Project Evaluation I 4.1 Capital Budgeting Introduction
Capital budgeting decisions are the most important investment decisions made by management The goal of these decisions is to select capital projects that will maximise shareholders’ wealth Capital investments are important because they involve substantial cash outlays and, once made, are not easily reversed Help management to systematically analyse potential business opportunities in order to decide which are worth undertaking Capital budgeting (investment): Cash outlay(s) now in the expectation of benefits (net cash inflows) later Sources of information: o Most of the information needed is generated internally (i) Beginning with the sales force; (ii) Then the production team is involved; (iii) Followed by the accountants o All this information is then reviewed by the financial managers who evaluate the feasibility of the project
4.2 Classification of Investment Projects (i) Independent projects o Projects are independent when their cash flows are unrelated o If two projects are independent, accepting or rejecting one project has no bearing on the decision on the other (ii) Mutually exclusive projects o When two projects are mutually exclusive, accepting one automatically precludes the other (iii) Contingent projects o Contingent projects are those where the acceptance of one project is dependent on another project o These are two types of contingencies situations: (a) Mandatory projects (b) Optional projects
4.3 Overview of Project Evaluation Techniques
Discounted Cash Flow (i) Net Present Value (NPV) (ii) Internal Rate of Return (IRR) (iii) Benefit-Cost Ratio (Profitability Index) Non-Discounting (i) Accounting Rate of Return (ARR) (ii) Payback Period
3
4.4 NPV
Finding the present value entails calculating the equivalent value today of a set of promised (or forecast) future cash flows 𝑁𝐶𝐹1 𝑁𝐶𝐹2 𝑁𝐶𝐹𝑛 𝑁𝑃𝑉 = + + ⋯ + − 𝑁𝐶𝐹0 (1 + 𝑘)1 (1 + 𝑘)2 (1 + 𝑘)𝑛 𝑛
𝑁𝑃𝑉 = ∑ 𝑡=1
𝑁𝐶𝐹𝑡 − 𝑁𝐶𝐹0 (1 + 𝑘)𝑡
𝑁𝐶𝐹0 = 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑐𝑎𝑠ℎ 𝑜𝑢𝑡𝑙𝑎𝑦 𝑁𝐶𝐹𝑡 = 𝑁𝑒𝑡 𝑐𝑎𝑠ℎ 𝑓𝑙𝑜𝑤 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑒𝑑 𝑏𝑦 𝑝𝑟𝑜𝑗𝑒𝑐𝑡 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡 𝑛 = 𝐿𝑖𝑓𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑟𝑜𝑗𝑒𝑐𝑡 𝑘 = 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑟𝑒𝑡𝑢𝑟𝑛 The 5-step approach to calculate NPV 1. Determine the cost of the project Identify and add up all expenses related to the cost of the project While most of the project’s costs occurs at the start of the project, some projects may have costs occurring beyond the first year The cash flow in year 0 (NCF0) is negative, indicating a cost 2. Estimate the project’s future cash flows over its forecasted life Both cash inflows (CIF) and cash outflows are likely in each year of the project Estimate the net cash flow (NCFt) = CIFt – COFt for each year of the project Cash inflows: Receipts from sales of goods and services; receipts from sale of physical assets Cash outflows: Expenditure on materials, labour, and indirect expenses for manufacturing; selling and administrative; inventory and taxes Remember to recognize any salvage value from the project in its terminal year 3. Determine the riskiness of the project and estimate the appropriate cost of capital The cost of capital is the discount rate used in determining the present value of the future expected cash flows The riskier the project, the higher the cost of capital for the project 4. Calculate the project’s NPV Determine the difference between the present value of the expected cash flows from the project and the cost of the project 5. Make a decision Accept the project if it produces a positive NPV or reject the project if the NPV is negative NPV method is consistent with the company’s objective of maximizing shareholders’ wealth. A project with a positive NPV will leave the company better off than before the project. Other things being equal, the market value of the company’s shares should increase Feature of the NPV method: o Consistent with maximising shareholder wealth o Consistent with requiring a proposed investment to achieve at least the rate of return required by investors o Focuses on the incremental effects of an investment. Provides a direct measure of how much a capital project is expected to increase the dollar value of a company Decision rule for the NPV method: o Independent projects NPV > 0 then accept project NPV < 0 then reject project o Mutually exclusive projects Accept project with the highest NPV Advantages o Use DCF valuation technique to adjust for time value of money o Provide direct dollar measure of how much a capital project will increase the value of the firm o Consistent with the goal of maximising shareholder wealth 4
Disadvantages o Can be difficult to understand without an accounting or finance background
Example: An investment promises to pay $670 000 one year from today and a further $1 240 000 four year from today. What is it worth today if the required rate of return is 15 % p.a.? Suppose that we have to outlay $1.2 million to get the right to receive the cash inflows. 𝑃𝑉 =
670,000 1,240,000 + = $1,291,583 1.15 1.154
𝑁𝑃𝑉 = $1,291,583 − $1,200,000 = $91,583
∴ 𝑁𝑃𝑉 > 0 ⇒ 𝐴𝑐𝑐𝑒𝑝𝑡
Example: What project should you accept if the projects were independent? What project should you accept if the projects are mutually exclusive? NPV $1,500 $2,500
Project A Project B
Accept both projects if A and B are independent. If projects are mutually exclusive then accept B.
4.5 Internal Rate of Return
The Internal Rate of Return (IRR) is the rate of return that equates the present value of projected cash flows with the initial cash outlay The IRR is the discount rate which makes the project break even (NPV=0) 𝑛
𝑁𝐶𝐹0 = ∑ 𝑡=1
𝑁𝐶𝐹𝑡 (1 + 𝑟)𝑡
Decision rule: Accept if IRR > RROR Advantages: o Intuitively easy to understand o Based on DCF technique Problems with IRR: o May not always lead to the same decision as the NPV rule when projects are mutually exclusive o Reinvestment rate is assumed to be the IRR, which might be unrealistic o Multiple IRRs – A project may have more than one IRR which complicates decision making o Indeterminate IRR – A project may have no IRR o It ignores the size of the project
5
Example: An investment of $1,000 today yields a return of $300 in year 1, $400 in year 2 and $500 in year 3. Calculate the IRR of this investment? If the required rate of return is 13%, should the project proceed? $1,000 =
300 400 500 + + 2 (1 + 𝑟) (1 + 𝑟) (1 + 𝑟)3
Use the trial and error method: 𝐼𝑓 𝑟 = 10%: 𝑁𝑃𝑉 = −21.0368
𝐼𝑓 𝑟 = 8%: 𝑁𝑃𝑉 = 17.6294
Interpolation where: 𝑟 = 𝑟1 + (𝑟2 − 𝑟1 ) [ = 0.10 + (0.08 − 0.10) [
𝑇1 ] 𝑇1 − 𝑇2
−21.0368 ] = 0.0891 = 8.9% −21.0368 + (−17.6294)
4.6 NPV and IRR When IRR and NPV agree The methods will always agree when the projects are: (a) Independent; and (b) The projects’ cash flows are conventional Conventional cash flows: After the initial investment is made (cash outflow), all the cash flows in each future year are positive (inflows) When IRR and NPV disagree The IRR and NPV methods can produce different accept/reject decisions if a project either has; (a) Unconventional cash flows; or Multiple IRRs can result. The maximum number of IRR is equal to the number of sign reversals in cash flows (b) The projects are mutually exclusive IRR cannot be used to rank mutually exclusive projects because the project with the highest IRR may not be the project that would add the greatest value to the company if accepted – that is, the project with the highest NPV Example: Consider the following cash flows for two mutually exclusive projects A and B
Year 0
Year 1
Year 2
NPV (k=15%)
IRR
A
-10,000
2,500
12,500
1,626
25%
B
-10,000
12,000
1,000
1,191
27.8%
Choose project A because it has a higher NPV and thus maximises shareholder wealth. NPV and IRR give different results (ranking) because of the timing of cashflows.
6
Example: Different timing of cash flows
4.7 Modified IRR
IRR assumes that the cash flows from the project are reinvested at the IRR, while the NPV assumes that they are invested at the company’s cost of capital. This optimistic assumption in the IRR method leads to some projects being accepted when they should not Under the MIRR: o Each operating cash flow is reinvested at the company’s cost of capital o The compounded values are summed up to get the project’s terminal value o The MIRR is the interest rate which equates the project’s cost to the terminal value at the end of the project 𝑃𝑉 𝑜𝑓 𝑐𝑜𝑠𝑡 =
𝑇𝑉 (1 + 𝑀𝐼𝑅𝑅)𝑛
4.8 Benefit-Cost Ratio (Profitability Index) 𝐵𝐶𝑅 =
𝑃𝑉 𝑜𝑓 𝑐𝑎𝑠ℎ𝑓𝑙𝑜𝑤𝑠 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑐𝑎𝑠ℎ 𝑜𝑢𝑡𝑙𝑎𝑦
Decision rule: BCR > 1 then accept project For mutually exclusive projects, the benefit-cost ratio may produce a different ranking of projects than that provided by the NPV method
7
Example: A 10,000 8,000 2,000 10,000/8,000=1.25
PV of cashflows Initial cash outlay NPV BCR
B 4,000 2,200 1,800 4,000/2,200=1.81
4.9 Accounting Rate of Return
The ARR is the earnings from a project (after deducting depreciation & income tax) expressed as a percentage of the investment outlay It is based on accounting numbers rather than cash flows. As such, it is not a true rate of return. Instead of discounting a project’s cash flows over time, it simply gives us a number based on average figures from the income and balance sheet Decision rule: ARR is compared to the RROR. If the ARR > RRR then the project is accepted 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑛𝑒𝑡 𝑖𝑛𝑐𝑜𝑚𝑒 𝐴𝑅𝑅 = 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑏𝑜𝑜𝑘 𝑣𝑎𝑙𝑢𝑒 Strengths: o Useful screening measure to ensure that new investment will not adversely affect net incomes o Easy to calculate and understand Weaknesses: o Arbitrary: The choice of depreciation schedule and inventory valuation will significantly affect earnings estimates o Timing of cash flows: The time value of money is not taken into account in the calculation o Project size: ARR does not account for project size when a choice between two projects of different sizes must be made Three variants to ARR: 1. ARR based on initial investment 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝐴𝑅𝑅 = 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 2. ARR based on average book value 𝐴𝑅𝑅 =
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑏𝑜𝑜𝑘 𝑣𝑎𝑙𝑢𝑒
3. ARR based on the initial and final capital value 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝐴𝑅𝑅 = 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑐𝑎𝑝𝑖𝑡𝑎𝑙 Example: Different ways to calculate ARR Item Earnings after income tax and depreciation
Year 1 $10,000
Year 2 $13,500
Year 3 $18,000
Average $13,833
$50,000 $40,000 $45,000
$40,000 $30,000 $35,000
$30,000 $20,000 $25,000
$35,000
Book value (assume 20% straight line dep.) Jan 1st Dec 31st Average
8
ARR based on initial investment:
13,833 50,000
ARR based on average book value:
= 27.7%
13,833 35,000
= 39.52%
ARR based on initial and final capital value:
13,833 50,000+20,000 2
= 39.52%
4.10 Payback Period
The time it takes for the initial cash outlay on a project to be recovered from a project’s net cash flows Decision rule: Projects are accepted if the payback period is less than some given period 𝑃𝐵 = 𝑌𝑒𝑎𝑟𝑠 𝑡𝑜 𝑟𝑒𝑐𝑜𝑣𝑒𝑟 𝑐𝑜𝑠𝑡 +
𝑅𝑒𝑚𝑎𝑖𝑛𝑖𝑛𝑔 𝑐𝑜𝑠𝑡 𝑡𝑜 𝑟𝑒𝑐𝑜𝑣𝑒𝑟 𝐶𝑎𝑠ℎ 𝑓𝑙𝑜𝑤 𝑑𝑢𝑟𝑖𝑛𝑔 𝑡ℎ𝑒 𝑦𝑒𝑎𝑟
Strengths: o Provides an indication of a project’s risk and liquidity o Easy to calculate and understand Weaknesses: o Ignores time value of money o Ignores cash flows occurring after the payback period o Is not a measure of profitability or shareholder wealth o Arbitrary cut off point Year 0 1 2 3 4 Payback
A -10,000 3,000 3,000 4,000 4,000 3 years
B -10,000 5,000 4,000 2,000 2,000 2.5 years
C -10,000 3,000 3,000 4,000 20,000 3 years
4.12 Discounted Payback Period
One of the weaknesses of the ordinary payback period is that it does not take into account the time value of money The discounted payback period calculation calls for the future cash flows to be discounted by the company’s cost of capital Advantage: Tells management how long it takes a project to reach a positive NPV Disadvantage: Still ignores all cash flows after the arbitrary cut-off period, which is a major flaw
Topic 5: Project Evaluation II 5.1 Cash Flow
The focus is on incremental operating cash flows which are cash flows that will occur only if the project is undertaken (i.e. Estimating the cash flow changes that will occur as a result of investing in the project) Typical operating cash flows include wages paid, materials purchased, sales revenue and taxes In project evaluation, we are comparing two alternative scenarios for the future: o One with the investment o One without the investment The accept/ reject decision is made relative to the existing scenario.
9
Incremental cash flows 1. Include cash flows and only cash flows in your calculations 2. Include the impact of the project on cash flows from other product lines o Consider if the product associated with a project is expected to cannibalize or boost sales of another product in the analysis 3. Include all opportunity costs (i.e. the cost of giving up the next best alternative) 4. Forget sunk costs (past investments are irrelevant) 5. Include only after-tax cash flows in the cash flow calculations Cash flows not profits
Focus is on cash not ‘profit’ which is an accounting concept Only cash, not ‘profit’ can be spent, reinvested, paid out in dividends etc Profit includes: o Some non-cash items (e.g. depreciation) o Some cash items which are non-operating (e.g. interest payments on borrowed funds)
Initial and subsequent outlays
Should be included at appropriate dates
Example: A machine costs $25 000 to buy and a further $2 000 to install. There is a 3-month waiting list for buyers. The installation cost is payable in cash on installation. A deposit of $5 000 is required on placement of an order and the balance within two months of delivery/ installation. What is the total initial outlay? If the required rate of return is 1% per month, what is the present value of the initial outlay(s)? Month 0 3 5
Action Order placed Delivery/installation Balance due
𝑃𝑉 = −5,000 −
Cash flow -$5,000 -$2,000 -$20,000
2,000 2,000 − = −$25,970.49 3 (1.01) (1.01)5
5.2 Free Cash Flow Calculation What to include in the calculation? 1. Operating cash flows 2. Opportunity costs o Cash inflows without the investment that are foregone (and hence become cash outflows) o Example: A firm owns a warehouse which it is not currently using. A proposal is put forward to convert the warehouse into a number of residential townhouses. A DCF analysis of this proposal should include the opportunity cost of selling the warehouse. The estimated sale price should be included as a cash outflow 3. Salvage value o This is the amount received at the end of a project. May be scrap value only (e.g. From the sale of equipment) and may be negative (e.g. An environmental clean-up) 4. Additional working capital in the final year o The FCF in the last, or terminal, year of a project also includes another cash flow item that is not typically included in the calculations for other years such as working capital (Additional WC) o Additional WC = Cash and cash equivalents + accounts receivable + inventories – accounts payable o In the final year of the project, the working capital that has been invested in year 0 may be recovered 10
o o
The principle behind including only these cash flows in y0 is that investments must be made before any cash flow from operations are realised When a project ends, the cash and cash equivalents are no longer needed, the accounts receivable are collected, the inventories are sold and the accounts payable are paid. I.e. the company recovers the net working capital that has been invested in the project. To reflect this in the FCF calculation, the cash flow in the last year of the project includes a negative investment in working capital that equals the cumulative investment in the working capital over the project life
What to exclude from the calculation? 1. Allocated costs o Costs already allocated by the firm and will be paid irrespective of whether or not the project proceeds o Examples: Rates, head office costs, rent, corporate image advertising. o But sometimes allocated costs can be incremental and should be included 2. Financing charges o Example: Interest expense o The required rate of return covers interest and return to equity 3. Sunk costs o These are costs already borne by the firm. They are a cost to the firm irrespective of whether or not the project proceeds. o Example: The cost of research & development on a previous product Incremental After-Tax FCF Calculation: Method 1 Calculation Revenue Revenue − Op Ex Less: Cash operating expenses = Earnings before interest, tax, depreciation & amortisation EBITDA Less: Depreciation & amortisation −D&A = Earning before interest and tax EBIT EBIT*(1- Company Tax Rate) × (1−TC) = Net operating profit after tax NOPAT Add back: Depreciation & amortisation +D&A = Cash flow from operations CF Opns − Cap Exp Less: Capital expenditures Less: Additions to working capital
Notation
Addition to working capital = Cash & cash equiv. + Accounts −Add WC receivables + Inventories – Accounts payable Note: This additional WC is assumed to be recovered in the terminal year = Free cash flow FCF Incremental After-Tax FCF Calculation: Method 2 Calculation Revenue Revenue − Op Ex Less: Cash operating expenses = Earnings before interest, tax, depreciation & amortisation EBITDA EBITDA*(1 – Company Tax Rate) × (1−TC) = Net cash flow after tax NCFAT Plus: Depreciation tax shield + (D × TC) = Cash flow from operations CF Opns − Cap Exp Less: Capital expenditures Less: Additions to working capital −Add WC = Free cash flow FCF 11
Notation
5.3 Depreciation and Tax Impact of company tax on depreciation
When a firm purchases equipment it will lose value (depreciation) over time. Although depreciation is not a cash outflow, companies can claim depreciation on an asset as a deduction in determining company income Methods to account for the tax effect of depreciation in the cash flow table: 1. Deduct depreciation, account for the company tax and then add back depreciation 2. Add depreciation tax shield to the after-tax cash flow Depreciation tax shield = Depreciation x TC Depreciation produces tax savings known as depreciation tax shield
Depreciation calculation 1. Straight line depreciation o In this unit, we assume that the remaining value is zero in year n 1 𝑑= 𝑛 2. Reducing balance depreciation o Unless specified by the question, the depreciation rate for reducing balance is 150% of straight line 1.5 𝑑= 𝑛 Example: An asset is acquired for $100 000 and has an expected economic life of 4 years 𝑆𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝑙𝑖𝑛𝑒 𝑟𝑎𝑡𝑒 = 𝑅𝑒𝑑𝑢𝑐𝑖𝑛𝑔 𝑏𝑎𝑙𝑎𝑛𝑐𝑒 𝑟𝑎𝑡𝑒 =
Year # 1 2 3 4
1 = 25% 4 1.5 = 37.5% 4
Reducing balance method Written down value Depreciation during the (start of the year) year $100,000 0.375*100,000 = $37,500 $62,500 0.375*62,500 = $23,438 $39,062 0.375*39,062 = $14,638 $24,414 0.375*24,414 = $9,155
Straight line method Depreciation this year $25,000 $25,000 $25,000 $25,000
𝑊𝑟𝑖𝑡𝑡𝑒𝑛 𝑑𝑜𝑤𝑛 𝑣𝑎𝑙𝑢𝑒 (𝑒𝑛𝑑 𝑜𝑓 𝑦𝑒𝑎𝑟 4): 𝑆𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝑙𝑖𝑛𝑒 𝑚𝑒𝑡ℎ𝑜𝑑: 𝑉𝑎𝑙𝑢𝑒 = 0 𝑅𝑒𝑑𝑢𝑐𝑖𝑛𝑔 𝑏𝑎𝑙𝑎𝑛𝑐𝑒 𝑚𝑒𝑡ℎ𝑜𝑑: 𝑉𝑎𝑙𝑢𝑒 = 24,414 − 9,155 = $15,259 Impact of taxation on disposal value
Example: Suppose the asset is sold for $S at the end of year 3. $S is the cash inflow at the end of year 3. What are the tax implications? o The answer depends on the sale price compared to the written down value at the time of sale o Written down value = Total cost of investment – Accumulated depreciation o Assume w = $640 o Scenario 1: s = $540 Loss on sale = 640 – 540 = 100 This loss is deductible for tax purposes (at company tax rate of 30%) Tax saving = 100 x 0.3 = $30 After-tax net cash flow from disposal = Sale price + Tax saved = 540 + 30 = 570 12
o
Scenario 2: s = $1000 Gain on sale = 1000 – 640 = 360 Tax liability on gain on sale = 360 x 0.3 = 108 After-tax net cash flow from disposal = Sale price + Tax liability = $1000 – 108 = 892
5.4 Past Exam Example
Bellco Ltd has identified a new market for its products. To increase the output level, the company is considering the purchase of some new machinery at a cost of $400,000. The machinery is imported from the United States and the delivery and installation costs are $20,000. The current estimated before-tax net operating cash revenue for the coming 3 years are: $260,000 in the first year, $240,000 in the second year, and $200,000 in the third year. The purchase of the new machinery is expected to increase the expected before-tax net operating cash revenue for the next 3 years by 80% of the current estimated value. Bellco will need to obtain financing from the bank to fund this investment. The interest payment on the loan is $85,000 per annum. The machinery will be sold at the end of the third year and its market value at that time is estimated to be $55,000. The company tax rate is 30% and reducing balance depreciation at 50% per year is allowed. The operating cash flows should be considered to occur at year-end. The required rate of return is 15% per year Y0
Y1 260,000*0.8 = 208,000 208,000 420,000*0.5 = 210,000 (2,000) (600) (1400) 210,000 210,000 + (1400) = 208,600
Net operating cash revenue EBITDA Less: Depreciation EBIT Less: Company tax NOPAT Add: Depreciation CF Opns Less: Cap Exp FCF
(420,000) (420,000)
208,600
Y2 240,000*0.8 = 192,000 192,000 210,000*0.5 = 105,000 87,000 26,100 60,900 105,000
Y3 200,000*0.8 = 160,000 160,000
165,900
127,750
165,900
54,250 182,000
52,500 107,500 32,250 75,250 52,500
𝐴𝑡 𝑌0: 𝐶𝑎𝑝 𝐸𝑥 = 400,000 + 20,000 = 420,000 𝐴𝑡 𝑌3 (𝑇𝑒𝑟𝑚𝑖𝑛𝑎𝑙 𝑦𝑒𝑎𝑟): 𝐶𝑎𝑝 𝐸𝑥𝑝 = 55,000 − (0.3 ∗ 𝐺𝑎𝑖𝑛 𝑜𝑛 𝑠𝑎𝑙𝑒) = 55,000 − (0.3 ∗ 2,500) = 54,250 𝑁𝑃𝑉 = −420,000 +
208,600 165,900 182,000 + + = $6,503.49 1.15 1.152 1.153
Example 2: Outdoor performing arts centre
Evaluating a project to increase the number of seats by building four new box seating areas and adding 5000 seats for the general public. Each box seating area is expected to generate $400,000 in incremental annual revenue, while each of the new seats for the general public will generate $2,500 in incremental annual revenue The incremental expenses for the new boxes and seating will be 60% of the revenues. These expenses include hiring additional personnel to handle concessions, ushering and security The new construction will cost $10m and will be fully depreciated on a straight-line basis over 10 years. The centre will have to invest $1m in additional working capital immediately. This working capital is recovered in the last year of the project. The tax rate is 30% 13
5.5 Mutually Exclusive Projects with Different Lives
A firm may be considering undertaking two or more projects at the same time but may be limited to choosing only one of these projects because of: Limited levels of debt finance, office or factory capacity, or skilled personnel Mutually exclusive projects cannot occur at the same time Mutually exclusive projects with different lives can be evaluated using the following methods: a. Constant Chain of Replacement i. Lowest Common Multiple ii. NPV Perpetuity method b. Equivalent Annual Value/ Cost Method
Example: The table shows the cash outflows for 2 machines. Assume both have the same annual cash inflows. The aim is to buy the machine that results in the lower present value of outflows. Assume a RROR of 13% p.a. Year 0 1 2 3 4
Machine A $100,000 $170,000 $180,000 $200,000 $220,000
Machine B $150,000 $200,000 $220,000
𝑃𝑉 (𝐶𝑜𝑠𝑡𝑠 𝑜𝑓 𝐴) = $664,949 𝑃𝑉(𝐶𝑜𝑠𝑡𝑠 𝑜𝑓 𝐵) = $499,283 Machine B appears to be cheaper but the comparison is flawed. Consider replicating machine B in years 3 and 4 Year 0 1 2 3 4
Machine A $100,000 $170,000 $180,000 $200,000 $220,000
Machine B $150,000 $200,000 $220,000 + $150,000 = $370,000 $200,000 $220,000
𝑃𝑉(𝐶𝑜𝑠𝑡𝑠 𝑜𝑓 𝐵 𝑡𝑤𝑖𝑐𝑒) = $890,296 𝑃𝑉(𝐶𝑜𝑠𝑡𝑠 𝑜𝑓 𝐴) = $664,949 Therefore, the correct decision is to buy machine A due to lower cash outflow 14
6.7 Decision Tree Analysis
Used to evaluate investment options involving a series of decisions over a period of time Decision Tree Analysis involves taking account of the probability of various events occurring and the effect of those decisions on the NPV of the project Decision tree analysis then involves calculating the NPV of each decision working backwards through the tree. (Assume a discount rate of 10% pa)
Example: The management of ELEC P/L are considering investing in a 1-year research project that will attempt to develop an electric mop. The research program will initially cost $500,000 and if successful will yield a cash flow of $150,000 in perpetuity. Management also believes there is only a 33% chance of successfully developing an electric mop. Assuming a discount rate of 12% pa, should ELEC P/L proceed with the project?
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑝𝑎𝑦𝑜𝑓𝑓 (𝑎𝑡 𝑡 = 1) = 1,250,000 × 0.33 + 0 × 0.67 = $412,500 𝑁𝑃𝑉 (𝑎𝑡 𝑡 = 0) = −500,000 +
412,500 = −131,696.43 1.12
∴ 𝐷𝑜𝑛′ 𝑡 𝑝𝑟𝑜𝑐𝑒𝑒𝑑
Example: More complex problem
Management of a silicon chip manufacturing firm are faced with the decision of investing in two alternative chip producing machines, both of which have an operational life of 8 years. Machine A costs $4m and machine B costs $3m. If machine B is chosen, management may after 3 years choose to upgrade it for a cost of $3m so that it has the same capacity as machine A. However, this will only occur if demand is high. Management also believes that the probability of high demand for chips in the first 3 years will be 0.7. However, if demand is high in the first 3 years management believes there is a 0.8 probability that demand will remain high in the following 5 years. However, if demand is low, management believes that there is a 0.6 probability that it will remain low. Discount rate is 10%
Let H3 = event of high demand in first 3 years
L3 = event of low demand in first 3 years,
H5 = event of high demand in next 5 years
L5 = event of low demand in next 5 years
P(H3)=0.7
P(L3)=0.3
P(H5|H3)=0.8
P(L5|H3)=0.2
P(L5|L3)=0.6
P(H5|L3)=0.4
Management has also estimated the following annual cash flows from the decisions Machine A A B B Upgrade B Upgrade B
Demand H L H L H L 15
Cashflow p.a. 1m 0.5m 0.6m 0.2m 1m 0.5m
Decision 2
Upgrade B 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑝𝑎𝑦𝑜𝑓𝑓 𝑝. 𝑎. = 1𝑚 × 0.8 + 0.5 × 0.2 = 0.9𝑚 0.9𝑚 1 𝑁𝑃𝑉(𝑎𝑡 𝑡 = 3) = −3 + (1 − ) = 0.4117𝑚 0.1 1.15 Do not upgrade B 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑝𝑎𝑦𝑜𝑓𝑓 𝑝. 𝑎. = 0.6𝑚 × 0.8 + 0.2𝑚 × 0.2 = 0.52𝑚 0.52𝑚 1 𝑁𝑃𝑉(𝑎𝑡 𝑡 = 3) = (1 − ) = 1.971𝑚 0.1 1.15 ∴ 𝐷𝑜 𝑛𝑜𝑡 𝑢𝑝𝑔𝑟𝑎𝑑𝑒 𝑖𝑓 𝑑𝑒𝑚𝑎𝑛𝑑 𝑖𝑠 ℎ𝑖𝑔ℎ 𝑖𝑛 𝑓𝑖𝑟𝑠𝑡 3 𝑦𝑒𝑎𝑟𝑠
Decision 1
Buy machine A Assume high demand in first 3 years 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑝𝑎𝑦𝑜𝑓𝑓 𝑝. 𝑎. = 1𝑚 × 0.8 + 0.5 × 0.2 = 0.9𝑚 0.9𝑚 1 𝑁𝑃𝑉(𝑎𝑡 𝑡 = 3) = (1 − ) = 3.412𝑚 0.1 1.15 Assume low demand in first 3 years 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑝𝑎𝑦𝑜𝑓𝑓 𝑝. 𝑎. = 1𝑚 × 0.4 + 0.5 × 0.6 = 0.7𝑚 𝑁𝑃𝑉(𝑎𝑡 𝑡 = 3) = 𝑁𝑃𝑉(𝑎𝑡 𝑡 = 0) = −4𝑚 + 0.7 (
3.142𝑚 1.13
1𝑚
0.7𝑚 1 (1 − ) = 2.654𝑚 0.1 1.15 1
2.654𝑚 1.13
+ 0.1 (1 − 1.13 )) + 0.3 (
+
0.5𝑚 1 (1 − 1.13 )) 0.1
= $0.5065𝑚
Buy machine B Assume high demand in first 3 years 0.52𝑚 1 𝑁𝑃𝑉(𝑎𝑡 𝑡 = 3) = (1 − ) = 1.971𝑚 [𝑓𝑟𝑜𝑚 𝑑𝑒𝑐𝑖𝑠𝑖𝑜𝑛 2] 0.1 1.15
16
Assume low demand in first 3 years 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑝𝑎𝑦𝑜𝑓𝑓 𝑝. 𝑎. = 0.6𝑚 × 0.4 + 0.2 × 0.6 = 0.36𝑚 0.36𝑚 1 𝑁𝑃𝑉(𝑎𝑡 𝑡 = 3) = (1 − ) = 1.365𝑚 0.1 1.15 N𝑃𝑉(𝑎𝑡 𝑡 = 0) = −3𝑚 + 0.7 (
1.971𝑚 1.13
+
0.6𝑚 1 (1 − 1.13 )) + 0.1
1.365𝑚 1.13
0.3 (
+
0.2𝑚 1 (1 − 1.13 )) 0.1
= −$0.4062𝑚
∴ 𝐵𝑢𝑦 𝑚𝑎𝑐ℎ𝑖𝑛𝑒 𝐴
Topic 7: Risk, Return & Investment I 7.1 Realised Returns 𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑟𝑒𝑡𝑢𝑟𝑛 =
𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 + 𝐸𝑛𝑑𝑖𝑛𝑔 𝑚𝑎𝑟𝑘𝑒𝑡 𝑣𝑎𝑙𝑢𝑒 − 𝐵𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝑚𝑎𝑟𝑘𝑒𝑡 𝑣𝑎𝑙𝑢𝑒 𝐵𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝑚𝑎𝑟𝑘𝑒𝑡 𝑣𝑎𝑙𝑢𝑒 = 𝑑𝑖𝑑𝑖𝑑𝑒𝑛𝑑 𝑦𝑖𝑒𝑙𝑑 + 𝑐𝑎𝑝𝑖𝑡𝑎𝑙 𝑔𝑎𝑖𝑛𝑠 𝑦𝑖𝑒𝑙𝑑 𝑅𝑻 =
𝐶𝐹1 + 𝑃1 − 𝑃0 𝑃0
Example: Suppose you bought 100 shares of ANZ one year ago at $34.99. At the end of the year, you sell the stock for $30.02. How did you do assuming you had 100 shares? 𝑅𝑒𝑡𝑢𝑟𝑛 𝑖𝑛 % =
(30.02 × 100) − (34.99 × 100) = −0.14204 = −14.204% 34.99 × 100
7.2 Average Returns (𝑅1 + ⋯ + 𝑅𝑇 ) 𝑅̅ = 𝑇
∑𝑛𝑖=1 𝑅𝑖 𝑜𝑟 𝐸(𝑅𝑎𝑠𝑠𝑒𝑡 ) = 𝐸(𝑅) = 𝑛
𝑛
𝑜𝑟 𝐸(𝑅) = 𝑅̅ = ∑ 𝑝𝑖 𝑅𝑖 𝑖=1
𝑤ℎ𝑒𝑟𝑒 𝑝𝑖 =
1 𝑛
Note: pi can also be a probability How to use the calculator: 1. Clear the statistics memory using “-m/C STAT” key 2. Enter the first value and press “∑+” 3. Continue entering the values and pressing “∑+” 4. Find mean by pressing “𝑥̅ , 𝑦̅” 5. Find standard deviation by press “𝑆𝑥 , 𝑆𝑦 ”
7.3 Holding Period returns
Also known as the compounded realised returns 𝐻𝑃𝑅 = (1 + 𝑅1 ) × (1 + 𝑅2 ) × … × (1 + 𝑅𝑇 ) − 1
7.4 What is risk?
Risk can be defined as the uncertainty of the future outcomes or the probability of an adverse outcome Alternatively, it is a chance of a financial loss or the variability of returns associated with an asset A common measure of risk is the variance or standard deviation of expected returns
7.5 Variance and Standard Deviation 𝑉𝑎𝑟(𝑅) =
1 [(𝑅1 − 𝑅̅ )2 + (𝑅2 − 𝑅̅ )2 + ⋯ + (𝑅𝑇 − 𝑅̅ )2 ] 𝑇−1 17
