■ Theoretical case for NPV
We have suggested that managers should base investment decisions on the net pres- ent value criterion: accept all projects that offer a positive net present value. We will now justify this claim by presenting the theoretical case for the NPV rule.
There are three fundamental financial decisions facing individuals and shareholders:
1 Consumption decisions: how much of my available resources should I spend on imme- diate consumption?
2 Investment decisions: how much of the resources available should I forgo now in the expectation of increased resources at some time in the future? How should such decisions be made?
3 Financing decisions: how much cash should I borrow or lend to enable me to carry out these investment and consumption decisions?
Clearly, these decisions are interrelated and should not be viewed in isolation.
Individuals are faced with the choice of how much of their wealth should be con- sumed immediately, and how much should be invested for consumption at a later date. This applies equally to young children with their pocket money, undergraduates with their grants, professionals with their capital, and shareholders with their invest- ment portfolios. All these cases involve a trade-off between immediate and delayed consumption.
We are primarily concerned with how managers should reach investment decisions.
Cash generated from business operations can be utilised in two ways: it can be dis- tributed to the shareholders in the form of a dividend, or reinvested within the busi- ness. Periodically, the directors decide how much of the shareholders’ wealth to distribute in the form of dividends and how much to withhold for investment pur- poses, such as building up stock levels or purchasing new equipment. The sharehold- ers will only be willing to forgo a higher present level of consumption (in the form of dividends) if they expect an even greater future level of consumption. This willingness to give up consumption now in order to increase future consumption characterises investment decisions.
■ Graphical example
Derek Platt is the sole proprietor of Platt Enterprises, a new business with just one asset of £4 million in cash. He has a number of interesting investment ideas (all lasting just a year), but before investing his capital within the business, he asks the following key questions:
1 What return could I earn by investing my capital (or some part of it) in the capital market?
2 How much should I invest within the business?
3 What is the net present value of the business?
Before addressing these issues and in order to present a conceptual framework for the NPV rule, it is first necessary to make certain simplifying assumptions that allow us to portray in two-dimensional form the essential features of the investment–
consumption decision model. The basic assumptions are as follows:
1 Investors are wealth-maximisers.
2 Only two periods are considered – the present period and the next period This two-period model implies that investments involve an immediate cash outlay in in return for a cash benefit in the following period,
3 All information for decision-making is known with certainty.
4 Investment projects are entirely independent of each other and are divisible.
These assumptions are clearly unrealistic in the setting within which investment decisions are taken in practice. Nonetheless, they serve as a useful guide to the rele- vance and limitations of the net value approach.
Let us assume that Platt Enterprises has £4 million available for investment but there are only two possible projects, each costing £2 million and having a one-year life.
Figure 3.4 illustrates the investment opportunities line for the two projects, showing the cost of investing this year and the payoffs arising next year. Platt could invest
£4 million, in both projects, producing a £4 million payoff next year. But he would probably prefer to invest only in project A, costing £2 million, but giving a payoff of
£3 million. Project B is unprofitable, offering only £1 million from £2 million invest- ment. If there are no opportunities to invest surplus cash externally, say by putting it
t1. t0
1t12. 1t02
on deposit with a bank or investing in short-term securities, Platt would have to pay a dividend to shareholders of the £2 million unused cash.
In this example, the choice was fairly straightforward. But if Platt had hundreds of potential projects, it would be far harder to know where the cut-off point for investment should be drawn. He requires a criterion for judging between cash today and cash receivable next year. In effect, he requires a rate of exchange for the transfer of wealth across time. Suppose he requires a minimum of £115 receivable next year to induce him to give up £100 now, the rate of exchange would be This rep- resents a premium for delayed consumption of one year of
This exchange rate between today’s money and tomorrow’s money varies with the level of present consumption sacrificed. Platt may be willing to forgo the first £100 of potential dividend in return for an additional 15 per cent next year, but to persuade him to delay the consumption of a further £100 will probably require something in excess of 15 per cent. This variable exchange rate for the transfer of wealth across time at various levels of investment is termed the marginal rate of time preference, and dif- fers from individual to individual.
The investment opportunities line is concave to the origin rather than a straight line. This indicates the decreasing returns to scale of each subsequent investment opportunity. As a wealth-maximiser, Platt will first select those investment projects offering the greatest return and work down towards those offering the least return.
Point C represents the marginal project beyond which it ceases to be worthwhile to invest – the marginal return from the next £1 in investment would not be sufficient to compensate for the sacrifice involved in giving up a further £1 in dividends. For Platt, C represents the point where the marginal return on investment equals his marginal rate of time preference.
■ Borrowing and lending opportunities
So far, under our highly simplistic assumptions, our owner-manager, Platt, is given only two decisions – consumption and investment. The more he invests, the less he can consume now, and vice versa. This ignores the third choice open to him, namely the
£115
£10010.15, or 15%
£115t0:£100t1 or £1.15:£1.
Period 0 (£m) A 4
B 2 0
A 3 B
4
Investment outlay Investment opportunities Investment line
payoff Period 1 (£m)
C
Figure 3.4
Investment opportuni- ties for Platt Enterprises
financing decision. Where capital markets exist, individuals and firms can buy and sell not only real assets (i.e. fixed and current), but also financial assets. As we saw earlier in this chapter, when perfect capital markets are introduced (i.e. no borrower can influence the interest rate, all traders have equal and costless access to information, no transaction costs or taxes), there will be a single market rate of interest for both borrowing and lending.
The existence of a capital market permits owners to transfer wealth across time in a manner different from the investment–consumption pattern of the firm. This is shown by the interest rate line in Figure 3.5, which represents the exchange rate between cur- rent and future cash flows under perfect capital market conditions. Its slope is where idenotes the single period rate of interest.
In our example, the interest rate is found by relating present wealth to next year’s wealth at any point on the graph. At the extremes, this is
The interest rate is therefore 20 per cent.
With the introduction of financing opportunities afforded by the capital market, Platt can now identify the appropriate level of corporate investment. He should con- tinue to invest until project M – where the interest rate line is tangential to the invest- ment opportunities line. At this point, all investments offering a return at least as high as the market rate of interest are accepted, since they all offer positive net pres- ent values. Reading off the graph in Figure 3.5, we find that investment as far as M would mean a dividend of £3 million today and an investment of £1 million (i.e.
). It is not worth investing further as the projects offer negative NPVs. It would be more beneficial for Platt to withdraw the £3 million from the busi- ness and to invest it in the capital market at 20 per cent p.a.
What then is the net present value of the £1 million investment programme envisaged by Platt? Reading off the investment opportunities curve, we find that the capital outlay will produce cash flows of £2.4 million next year. The NPV is therefore £1 million:
The new value of the business becomes £5 million (starting capital of £4 million plus NPV of investment programme).
We suggested earlier that the £3 million not invested by the firm would be paid out as a dividend. An alternative would be for the firm to invest all or part of it on behalf of the owners in the capital market until such time as investment opportunities
NPV£2.4 m
1.2 £1 m£2 m£1 m£1 m
£4 million£3 million
£6 million>£5 million1.20.
11i2, Owner’s consumption
requirement
Interest rate Investment opportunities
Today (£ million) Invest
Dividend
0 1.0 1.5 2.0 3.0 4.0 5.0 6.0
Next year (£ million)
M 5.0
4.2 4.0 3.0 2.4 2.0 1.0
Figure 3.5
Investment and financ- ing opportunities for Platt Enterprises
offering positive NPVs arise. Suppose Platt is only looking for a dividend of £1.5 mil- lion. The extra £1.5 million can be invested in the capital market to earn £1.8 million next year (i.e. or ). Platt’s cash flow next year will then be the £2.4 million from capital investments plus the £1.8 million from financial investments.
■ Separating ownership from management
Most firms are characterised by a large number of shareholders (owners), few of whom are actively involved in the management of the firm. It is obviously impossi- ble for managers to evaluate investment decisions on the basis of the personal investment–consumption preferences of all the shareholders. Happily, the existence of capital markets renders any such attempt unnecessary. Managers do not need to select an investment programme whose cash flows exactly match shareholders’ preferred time patterns of consumption. The task of the manager is to maximise present value by accepting all investment proposals offering a return at least as good as the market rate of interest.
This criterion maximises the current wealth of the shareholders, who can then transform that wealth into whatever time pattern of consumption they require. They can do this by lending or borrowing on the capital market until their marginal rate of time preference equals the capital market rate of interest. This Separation Theorem, as it is usually termed, leads to the following decision rules:
1 Corporate management should invest in projects offering positive net present val- ues when discounted at the capital market rate.
2 Shareholders should borrow or lend on the capital market to produce the wealth dis- tribution which best meets their personal time pattern of consumption requirements.
■ Capital market imperfections
Based on the assumptions laid down at the start of the chapter, managers should undertake investments up to the point at which the marginal return on investment is equal to the rate of return in the capital market. You will recall that two important assumptions were the existence of perfect capital markets and the absence of risk.
When these assumptions are relaxed, the argument in favour of the net present value rule becomes weaker. For one thing, there is no longer a unique rate of interest in the capital market, but a range of interest rates varying with the status of the borrower, the amount required and the perceived riskiness of the investment. A detailed analysis of investment under risk is the subject of subsequent chapters, but at this stage we can say that a project’s return should be compared with the rate of return on investments in the capital market of equivalent risk – the greater the investment risk, the higher the required rate of return.
A major concern involves the particular capital market imperfections where the borrowing rate is substantially higher than the lending rate. In this case, the two-period investment model will resemble Figure 3.6. The steeper line represents the interest rate for the borrower and the flatter line represents the lending rate. The existence of two different interest rates gives rise to two different points on the investment opportunities line CD. Prospective borrowers, having to pay a higher rate of interest for funds, would prefer the company to invest only BD this year (i.e. up to project Y).
However, prospective lenders will require the company to discount at the lower lend- ing rate, leading to a much greater investment of AD, with investment Xbeing the marginal project.
£1.5 m1.20
£4.2 m£2.4 m,
There is no simple solution to the investment–consumption decision when capital market imperfections prevail. Fortunately, in the UK, USA, Japan and much of Western Europe, capital markets are highly competitive and function fairly well, so that differ- ences between lending and borrowing rates are minimised, but significant differentials can be found in emerging capital markets such as that in Turkey.
■ Formula for the present value of a perpetuity
This formula derives from the present value formula:
Let and We now have:
(i)
Multiplying both sides by bgives us:
(ii)
Subtracting (ii) from (i) we have:
Substituting for aand b,
Multiplying both sides by and rearranging, we have:
PVX i
11i2
PV¢1 1
1i≤ X 1i PV11b2a
PVba1bb2b3p2 PVa11bb2p2
1>11i2 b.
X>11i2a
PV X
1i X
11i22 X
11i23p
Lending rate
Borrowing rate
Period 0 (£)
Period 1 (£)
C X
Y
0 A B D
Figure 3.6
Investment decisions in imperfect capital markets
APPENDIX III