Within the context of capital budgeting, money capital rather than labor or material is usually the scarce resource. In the presence of what is termed capital rationing projects must be ranked in terms of their net benefits compared to the costs of investment. Even if funds are plentiful, the actual projects may be mutually exclusive. The acceptance of one precludes others, an obvious example being the most profitable use of a single piece of land. To assess investment decisions, the following methodologies are commonly used:
Payback; Accounting Rate of Return); Present Value (based on the time value of money).
Payback (PB) is the time required for a stream of cash flows to cover an investment's cost. The project criterion is liquidity: the sooner the better because of less uncertainty regarding its worth. Assuming annual cash flows are constant, the basic PB formula is given in years by:
PB = payback period
I0 = capital investment at time period 0
Ct = constant net annual cash inflow defined by t = 1
Management's objective is to accept projects that satisfy their preferred, predetermined PB.
Activity 1
Shorttermism is a criticism of management today, motivated by liquidity, rather than profitability, particularly if promotion, bonus and share options are determined by next year's cash flow (think subprime mortgages). But such criticism can also relate to the corporate investment model. For example, could you choose from the following using PB?
Cash flows (£000s)
Year 0
Year 1
Year 2
Year 3
Project A
(1000)
900
100

Project B
(1000)
100
900
100
The PB of both is two years, so rank equally. Rationally, however, you might prefer Project B because it delivers a return in excess of cost. Intuitively, I might prefer Project A (though it only breaks even) because it recoups much of its finance in the first year, creating a greater opportunity for speedy reinvestment. So, whose choice is correct?
Unfortunately, PB cannot provide an answer, even in its most sophisticated forms. Apart from risk attitudes, concerning the time periods involved and the size of monetary gains relative to losses, payback always emphasizes liquidity at the expense of profitability.
Accounting rate of return (ARR) therefore, is frequently used with PB to assess investment profitability. As its name implies, this ratio relates annual accounting profit (net of depreciation) to the cost of the investment. Both numerator and denominator are determined by accrual methods of financial accounting, rather than cash flow data. A simple formula based on the average undercoated cost of an investment is given by:
ARR = average accounting rate of return (expressed as a percentage)
Pt = annual posttax profits before depreciation
Dt = annual depreciation
I0 = original investment at cost
S0 = scrap or residual value
Tie ARR is then compared with an investment cutoff rate predetermined by management.
Activity 2
If management desire a 15% ARR based on straightline depreciation, should the following five year project with a zero scrap value be accepted?
I0 = £1,200,000 Pt = £400,000
Using Equation (2) the project should be accepted since (£000s):
The advantages of ARR are its alleged simplicity and utility. Unlike payback based on cash flow, the emphasis on accounting profitability can be calculated using the same procedures for preparing published accounts. Unfortunately, by relying on accrual methods developed for historical cost stewardship reports, the ARR not only ignores a project's real cash flows but also any true change in economic value over time. There are also other defects:
 Two firms considering an identical investment proposal could produce a different ARR simply because specific aspects of their accounting methodologies differ, (for example depreciation, inventory valuation or the treatment of R and D).
 Irrespective of any data weakness, the use of percentage returns like ARR as investment or performance criteria, rather than absolute profits, raises the question of whether a large return on a small asset base is preferable to a smaller return on a larger amount?
Unless capital is fixed, the arithmetic defect of any rate of return is that it may be increased by reducing the denominator, as well as by increasing the numerator and vice versa. For example, would you prefer a £50 return on £100 to £100 on £500 and should a firm maximise ARR by restricting investment to the smallest richest project? Of course not, since this conflicts with our normative objective of wealth maximisation. And let us see why.
Activity 3
Based on either return or wealth maximisation criteria, which of the following projects are acceptable given a 14 percent cutoff investment rate and the following assumptions:
Capital is limited to £100k or £200k. Capital is variable. Projects cannot be replicated.
£0005
A
B
C
D
Investment
(100)
(100)
(100)
(100)
Return
10
15
20
25
We can summarise our results as follows:
Capital Rationing
Variable Capital
Capital
(£100,00)
(£200,000)
Investment criteria
ARR
Wealth
ARR
Wealth
ARR
Wealth
Project acceptance
D
D
D
C,D
D
B,C,D
Return %
25%
25%
25%
22.5%
25%
20%
Profit (£000s)
25
25
25
45
25
60
When capital is fixed at £100,000, ARR and wealth maximisation equate. At £200,000 they diverge. Similarly, with access to variable funds the two conflict. ARR still restricts us to project D, because the acceptance of others reduces the return percentage, despite absolute profit increases. But isn't wealth maximised by accepting any project, however profitable?
Present Value (PV) based on the time value of money concept reveals the most important weakness of ARR (even if the accounting methodology was cash based and capital was fixed). By averaging periodic profits and investment regardless of how far into the future they are realized, ARR ignores their timing and size. Explained simply, would you prefer money now or later (a "bird in the hand" philosophy)?
Because PB in its most sophisticated forms also ignores returns after the cutoff date, there is an academic consensus that discounted cash flow (DCF) analysis based upon the time value of money and the mathematical technique of compound interest is preferable to either PB and ARR. DCF identifies that finance is a scarce economic commodity. When you require more money you borrow. Conversely, surplus funds may be invested. In either case, the financial cost is a function of three variables:
 the amount borrowed (or invested),
 the rate of interest (the lender's rate of return),
 the borrowing (or lending) period.
For example, if you borrow £10,000 today at ten percent for one year your total repayment will be £11,000 including £1,000 interest. Similarly, the cash return to the lender is £1,000. We can therefore define the present value (PV) of the lender's investment as the current value of monetary sums to be received (or repaid) at future dates. Intuitively, the PV of a ten percent investment which produces £11,000 one year hence is £10,000.
Note this disparity has nothing to do with inflation, which is a separate phenomenon. The value of money has changed simply because of what we can do with it. The concept acknowledges that, even in a certain world of constant prices, cash amounts received or paid at various future dates possess different present values. The link is a rate of interest.
Expressed mathematically, the future value (FV) of a cash receipt is equivalent to the present value (PV) of a sum invested today at a compound interest rate over a number of periods:
FVn = future value at time period n
PV = present value at time period zero (now)
r = periodic rate of interest (expressed as a proportion)
n = number of time periods (t = 1, 2, ..n).
Conversely, the PV of a future cash receipt is determined by discounting (reducing) this amount to a present value over the appropriate number of periods by reference to a uniform rate of interest (or return). We simply rearrange Equation (3) as follows:
If variable sums are received periodically, Equation (4) expands. PV is now equivalent to an amount invested at a rate (r) to yield cash receipts at the time periods specified.
PVn = present value of future cash flows
r = periodic rate of interest
n = number of future time periods (t = 1, 2 ...n)
Ct = cash inflow receivable at future time period t.
When equal amounts are received at annual intervals (note the annuity subscript A) the future value of Ct per period for n periods is given by:
Rearranging terms, the present value of an annuity of Ct per period is:
If these equations seem daunting, it is always possible formulae tables based on corresponding future and present values for £1, $1 and other currencies, available in most financial texts.
Activity 4
Your bankers agree to provide £10 million today to finance a new project. In return they require a 12 per cent annual compound rate of interest on their investment, repayable in three year's time. How much cash must the project generate to breakeven?
Using Equation (3) or compound interest tables for the future value of £1.00 invested at 12 percent over three years, your eventual break even repayment including interest is (£000s):
To confirm the £10k bank loan, we can reverse its logic and calculate the PV of £14,049 paid in three years. From Equation (4) or the appropriate DCF table:
Activity 5
The PV of a current investment is worth progressively less as its returns becomes more remote and/or the discount rate rises (and vice versa). Play about with Activity 4 data to confirm this.
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