Revision basic techniques Compounding Discounting 1 FV = PV (1 + r ) n PV = FV (1 + r ) n FV = Future Value PV = Present Value r = rate n = number of periods
ExampleJohn deposits £200 in an account that pays 6% per year. At the end of 6 years what will he have?
AnnuityAn equal sum of money (AMT) received or paid over a fixed period of time. Compounding Discounting 1 (1 + r ) − 1 n 1− FV = AMT (1 + r ) n PV = AMT r r
Example: AnnuityJill has been faithfully depositing $2,000 at the end of each year for the past 10 years into an account that pays 8% per year. How much money will she have accumulated in the account?
Example: AnnuityJohn wants to make sure that he has saved up enough money for college expenses: $40,000 per year for 4 years. How much money will John need to have accumulated in an account that earns 7% per year?
Perpetuity A perpetuity is an equal sum of money (AMT), received or paid over an infinite period (will never cease). AMT PV = rEg: You are considering the purchase of abond that pays $60 per year forever, and therate of interest you want to earn is 10% pa, whatis the present value of the bond?
Capital Budgeting Decisions• Long-term decisions - go or no-go decision on a product, service, facility, or activity of the firm. – Involve longer time horizons – cost larger sums of money – require a lot more information to be collected e.g. estimates of the timing and amount of cash flow – Models used have a pre-determined accept or reject criterion.
Methods1. Net Present Value (NPV)2. Internal Rate of Return (IRR)3. Profitability Index (PI)4. Payback Period
1. Net Present Value (NPV)• Discounts all the cash flows from a project back to time 0 using an appropriate discount rate, r: CF1 CF2 CF3 CFnNPV = − CF0 + + + + ... + (1 + r ) (1 + r ) (1 + r ) 1 2 3 (1 + r ) nA +NPV implies that the project is adding value to the firm’s bottom line, therefore, the higher the NPV the better.RULE: Accept if NPV>0 reject if NPV<0
Example 1• The owner of Perfect Images Salon is considering the purchase of a new tanning bed.• It costs $10,000 and is likely to bring in after-tax cash inflows of $4,000 in the first year, $4,500 in the second year, $10,000 in the third year, and $8,000 in the fourth year.• The company’s cost of capital is 10%.• Calculate the NPV for the new tanning bed.
2. Internal Rate of Return (IRR)• The IRR is the discount rate which when applied to the cash flows gives a nil (zero) NPV: CF1 CF2 CF3 CFn £0 = −CF0 + + + + ... + (1 + r ) (1 + r ) (1 + r ) 1 2 3 (1 + r ) n RULE: Accept if IRR > hurdle rate Reject if IRR < hurdle rateHurdle rate = company’s cost of capital (WACC)• Note that the IRR is measured as a percent, while the NPV is measured in money terms.
IRR Method1. Find the NPV(a) at one discount rate (a%)2. Find the NPV(b) at a second discount rate (b%)3. Interpolate between the two NPV’s to find the approximate IRR NPVa IRR ≈ a % + ( b% − a % ) ( NPVa − NPVb )
Problems with the IRR• In most cases, NPV decision = IRR decision – i.e. a project with a positive NPV will have an IRR that exceeds its hurdle rate - similarly, the highest NPV project will also have the highest IRR.• There are some cases where the IRR method is problematic - Multiple IRR’s -Projects that have non- normal cash flows i.e. --multiple sign changes, often end up with multiple IRR’s. (normal = negative followed by positive cash flows)• IRR ignores the relative size of investments RULE: select the project if it has a positive NPV
Multiple Internal Rates of Return
3. Profitability Index (PI)• If faced with a constrained budget, we should choose projects that makes the best use of the scarce resource• The PI can be used to calculate the ratio of the NPV the cost of a project as follows: NPV PI = cos t• In essence, how NPV we are getting per pound invested.
Example 3: PI Using the cash flows and NPVs @ 10%, calculate the PI of each project. Which one should be accepted, if they are mutually exclusive? Why? Year A£ B£ 0 -10,000 -7,000• 5,000 9,000• 7,000 5,000• 9,000 2,000 NPV £7,092.41 £6,816.68
Capital RationingOften associated with general problems of cash shortage: i. the business has become loss making implying that there will be insufficient cash with which to replace these assets when necessary. ii. High inflation may mean that even though the business is profitable it is still failing to generate sufficient funds to replace assets. iii.Growing businesses may face a shortage of working capital with which to finance expansion, result in a period of capital rationing. iv. Seasonal or cyclical businesses may face times of cash shortage causing periodic capital rationing. v. A large one-off item of expenditure e.g. property, may mean a temporary shortage of cash for further investment. vi. Investment opportunities – there are more investment opportunities available than the funds allocated to the capital budget permit. This means that projects must be ranked for investment.
Hard capital rationing – a firm is prevented from undertaking attractive investments for reasons external to the firm.Soft capital rationing – management places a limit on the amount of capital investment that may be undertaken: it is due to factors internal to the firm, such as:• management may decide to limit funds to retained earnings: – They do not wish to issue further equity to prevent outsiders from gaining control of the business or to avoid earnings dilution – They do not wish to commit the company to meeting large fixed interest payments on additional debt capital.• number of projects undertaken restricted in order to ensure adequate management resources available for them to realise their full potential.
4. Payback Period• The length of time in which an investment pays back its original cost.RULE: Accept if Payback period ≤ the cutoff period• Main focus is on cost recovery or liquidity.• Assumes that all cash outflows occur right at the beginning of the project’s life, followed by a stream of inflows that occur uniformly over the year.
Payback Period Flaws• The payback period method has two major flaws: -It ignores all cash flow after the initial cash outflow has been recovered -It ignores the time value of money – this can be solved by calculating the payback period using discounted cash flows.
Example 4 - Payback PeriodCalculate the payback period for the following cash flows: Yr Cash Flow 0 -470,000 1-5 140,000pa
Overview of Decision Models1. Net present value (NPV) – economically sound – properly ranks projects across various sizes, time horizons, and levels of risk, without exception for all independent projects.2. Internal rate of return (IRR) – provides a single measure (return) – has the potential for errors in ranking projects – can also lead to an incorrect selection when there are two mutually exclusive projects or incorrect acceptance or rejection of a project with more than a single IRR3. Profitability index (PI) – incorporates risk and return – but the NPV-to-cost ratio is actually just another way of expressing the NPV
Overview of Decision Models4. Payback period – simple and fast, but economically unsound – ignores all cash flow after the cutoff date – ignores the time value of money4a. Discounted (Adjusted) payback period – incorporates the time value of money – still ignores cash flow after the cutoff date.
Importance of Cash Flow• Cash flow measures the actual inflow and outflow• While profits represent merely an accounting measure of periodic performance. - A firm can spend its operating cash flow but not its net income. - Some firms have net losses (due to high depreciation write-offs) and yet can pay dividends from cash balances, while others show profits and may not have the cash available. Thus, cash flow is broader than net income.
Relevant Project Cash Flows Incremental Cash FlowRelevant cash flows are: the incremental cashflows of the company as a whole as a result ofaccepting the project.Those cash flows which would have arisen anyway,irrespective of whether or not the project isundertaken, can be ignored, e.g. apportioned costsand money already spent or committed.RULE: We are only concerned with the timing and size of future cash flows
5 important issues1. sunk costs2. opportunity cost3. working capital4. capital expenditures and tax implications5. inflation
1. Sunk Costs• Expenses already incurred, or that will be incurred, regardless of the decision to accept or reject a project. -e.g. marketing research exploring business possibilities in a region would be a sunk cost, since its expenditure has taken place prior to undertaking the project and will have to be paid whether or not the project is taken on.RULE: do not considered as part of the relevant cash flows when evaluating the project
2. Opportunity Costs• Costs that may not be obvious• Result from benefits lost as a result of a project -e.g. a project uses existing equipment rather than selling it, the value that could be realized by selling would be a relevant opportunity costRULE: include opportunity costs as relevant cash flows
3. Working Capital• Additional cash flows arising from changes in current assets (inventory and receivables) and current liabilities (accounts payables) - as a result of a new project.• At the end of the project, these additional cash flows are recovered and must be shown as cash inflows.• Even though the net cash outflows--due to increase in net working capital at the start-- may equal the net cash inflow at the end, the time value of money effects make these costs relevant.RULE: include changes in working capital as relevant cash flows
4. Capital Spending & Tax Implications• Capital expenditures are expensed on an annual basis giving the company tax relief• The portion written off each year is called the depreciation expense; a accumulated total is kept in the balance sheet - Accumulated Depreciation.• The book value of an asset equals its original cost minus its accumulated depreciation and allows the gain or loss at disposal to be calculated.RULE: -include capital expenditure and sales proceeds as relevant cash flows -include tax relief as a relevant cash flow -do not include depreciation as it is not a cash flow
5. Inflation – cash flowsTo find the future cash flow with inflation (nominal): FV = PV × (1 + r ) nWhere:r = inflation rate n = number of periodsFV = future value with inflation – nominal cash flowPV = present value without inflation – real cash flowRULE: future cash flows must include the effects of inflation
5. Inflation – discount rateIf the cash flows include inflation then the discount rate must include inflation: n = (1 + r )(1 + i ) − 1Where:n = nominal discount rater = real discount ratei = inflation rateRULE: discount rate must include the effects of inflation