Investment Decisions with Sampling

Investment Decisions with SamplingAuthor(s): Harold Bierman, Jr. and Vithala R. RaoSource: Financial Management, Vol. 7, No. 3 (Autumn, 1978), pp. 19-24Published by: Wiley on behalf of the Financial Management Association InternationalStable URL: http://www.jstor.org/stable/3665006 .

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Investment Decisions with Sampling

Harold Bierman, Jr., and Vithala R. Rao

Harold Bierman, Jr., is Nicholas H. Noyes Professor of Business Administration, and Vithala R. Rao is Professor of Marketing and Quantitative Analysis, both at the Graduate School of Business and Public Administration, Cornell University.

* Conventional capital budgeting techniques, such as those used by Bierman and Smidt [1] or Merrett and Sykes [3] lead to a rejection of investments with a negative net present value (leaving out portfolio effects). This paper argues that one should not always reject an investment with a negative net present value in situations of uncertainty and when there is a possibility of replicating the investment.

Multi-plant firms have an opportunity to innovate sequentially that is frequently not available to firms with single plants (unless the single plant has multiple production lines). Consider the development of a new type of equipment in a multi-plant company. The analysis for a single unit of equipment 'indicates a negative net present value. But there is some probability that the equipment would be successful and would have a positive present value in any subsequent use. In other words, there is uncertainty about the outcome but there is some probability that it would be a desirable investment. In such a situation, the possibility that the firm may miss out on a techno- logical break-through may be sufficient motivation for trying the equipment as a sample investment.

If one unit of the equipment has a positive net pres-

i 1978 Financial Management Association

ent value, then some may argue that all the units should be acquired for the entire firm. On an expected value basis this is true. Under conditions of uncertainty and risk aversion, however, trying the investment on a small scale may help to determine if the fore- casted good result will actually occur. If the result is good, then the remainder of the units can be purchased. The cost of this policy is delay of the investments, however, which may be a disadvantage.

The objective of this paper is to emphasize that an apparently poor individual investment might be good when considered in the broader context of subsequent investments. We also want to show explicitly why the greater the uncertainty the better it is to try an investment if the investment can be replicated (although we are not sure the investment should be undertaken until the calculations are made). Some managers will prefer to apply intuition, making the decision without the type of calculations that will be illustrated. The model, however, does incorporate the expert judgment of the decision-maker into the decision process.1 The decision will depend on the decision-maker's evaluation of 'The methodology of decision calculus is useful in quantifying the experience and judgment of managers. See Little [2].

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FINANCIAL MANAGEMENT/AUTUMN 1978

the probabilities of events rather than the intuitive judgment of the effect of the combination of probabilities and outcomes. Most important, the model allows the decision-maker to focus on the relevant variables.

The Basic Model

The examples show the investment sampling process where the investment, although initially not desirable, may appear acceptable after considering the consequent opportunity of obtaining additional information and the possibility of sequential decisions. The chance of an improvement of the initially undesirable investment follows logically.

We shall assume initially that undertaking one investment would allow perfect information about what could happen if all the identical investments were undertaken. In a subsequent section, the analysis in- cludes imperfect information. This complicates the analysis but does not change the basic logic of undertaking an undesirable investment because of the information that can be obtained.

Exhibit 1 shows the basic model with the net present value of the profits lost by not undertaking the investment. Vo is the random variable "net present value" with mean Vo for one unit of equipment, and Vb is the break-even present value. Vb is the value when Vo is equal to zero, therefore, Vb is defined to be zero. C is the slope of the net present value curve for all units of investment.

Vo is to the left of Vb, and the correct decision seems to be to reject the investment. If the investment is rejected the present value is zero. But Vo is a random variable with a probability density function. This is a "prior betting distribution." If we are certain that the net present value of the investment is Vo (the variance of the distribution is zero), then the investment would be rejected. If there is some probability that Vo > Vb then further analysis is required.

Exhibit 1. Net Present Value Lost by not Under- taking Investment

Net Present Value

The value of C is crucially important because it defines the relationship of the net present value potential of all the equipment for different values of the random variable. C depends on the number of units of equipment in which the firm can feasibly invest. The more units of equipment the steeper the slope.

The profit potential for multiple investments, given an undesirable single investment, is a function of the slope of the net present value line (which in turn depends on the number of units of equipment), the variance of the probability density function, and the distance between Vo and Vb.

Mathematically, if the value of Vo is positive, the expected profits are: EVPI = 'o C(Vo)f(Vo)dVo, where f(Vo) is the probability density function of Vo, and EVPI gives the expected value of perfect information. The expected cost of obtaining this information is the expected loss that will occur if the true value of Vo is less than Vb. The expected loss of undertaking one unit of investment if Vo is less than Vb is: E(Lo) = f'°o0 (- Vo)f(Vo)dVo.

If EVPI > E(Lo), the investment ought to be tried.2 Exhibit 2 shows the net present value if one investment is undertaken and then all investments are undertaken if Vo > Vb.

2The basic expected value of information models are well known. See Schlaifer [4].

Exhibit 2. Net Present Value if Investment is Under- taken

Net Present Value for All Investments

Probability Density Function

NPV for all units

-·__C

0 Net present value for one unit: V0

Vo 0

Loss for One Investment I

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Vo



BIERMAN AND RAO/INVESTMENT DECISIONS WITH SAMPLING

Numerical Example The example shows an investment that costs

$1,000,000. The outcome can either be e1 or e2 in all future years.

Probability Outcome Event of Events (a perpetuity) Present Value

e, .4 $150,000 $1,500,000 e2 .6 40,000 400,000

The time value factor (required return) is 10%. The expected present value of the benefits is $840,000 ($1,500,000 x .4 + $400,000 X .6). Since the investment costs $1,000,000 and the expected benefits are only $840,000, the investment has a negative net present value of $160,000; it should be rejected on an expected present value basis.

However, the firm may have 11 of these investments. There is a .4 probability that each unit of equipment will perform to produce a net present value of $500,000 or $5,500,000 in total. There is a .6 probability of eleven units losing $600,000 per unit or $6,600,000 in total. The expected value is negative ($5,500,000 X .4 - $6,600,000 X .6 = -$1,760,000). We can eliminate the uncertainty by buying one item of equipment for a cost of $1,000,000 and a negative expected value. Exhibit 3 shows the decision tree that evolves.

The firm invests in the ten additional units of equipment only if event e1 occurs and the process proves to be feasible. If the process is feasible, each investment adds $500,000 of net present value. Multiplying the $500,000 by the 11 machines, we find the upper path leading to $5,500,000 of present value. The expected value of perfect information for eleven units is: .4 ($5,500,000) = $2,200,000.

The expected cost (net present value) of obtaining this information is equal to the $600,000 net loss associated with one piece of equipment that will occur

Exhibit 3. Decision Tree: Perfect Information

- $1,000,000

No further 1,500/ 000 investm ent

Invest in 10 more, each with a net present

400,000 value of $500,000

if the event is e2. This is an expected cost of $360,000. Because the expected value is $2,200,000 and the expected cost is $360,000, we would advocate undertaking the single investment in the hope that we will find out that e, is the true state of the world.

Imperfect Information

Now assume that after the one investment is made, and the results observed, we still cannot be sure of the desirability of the investment; that is, the information obtained is imperfect. We will now assign a set of probabilities reflecting the reliability of the information. The conditional probabilities exist as shown in Exhibit 4.

Exhibit 4. Conditional Probabilities

The actual state is: Event el Event e2 (High (Low

Present Present Observed Event Value) Value)

Investment seems to be profitable: G .9 .2

Investment seems to be not profitable: B .1 .8

If the investment actually is good, there is still a .1 probability that it may not appear profitable. If the investment actually is not good, there is a .2 probability that it may appear profitable. Extending the example to reflect an initial profitable probability of .4 and a .6 not-profitable probability, we will observe either profitable or not-profitable operations. The problem that has to be solved is whether or not it is desirable to go ahead and make the initial investment. The com- putations of the relevant revised probabilities are shown in the appendix to this paper. Exhibit 5 shows the probabilities and the outcomes if purchase of one unit is undertaken to obtain information, and then the decision is whether or not to undertake the additional units.

In Exhibit 5, we have crossed out two inferior paths to simplify the presentation. The expectation of the G path and investing in ten more units is: Value of G path = .75 ($5,500,000) - .25 ($4,400,000) = $3,025,- 000. The value of the B path and not investing in ten more units is: Value of B path = .077 ($500,000) - .923 ($400,000) = -$330,700. There is .48 probability of G and .52 probability of B; thus the expected value of the process is $1,280,000 (.48 X $3,025,000 - .52 x $330,700).

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Where we once found that undertaking purchase of one or eleven units with negative expected values was undesirable, now that we can sample, with perfect information, we find that the purchase of one unit becomes desirable. Even with imperfect information it is desirable to try one unit. The expected present value is $1,280,000.

Sample Investment: Normal Probability Assumption

Assume now that the prior betting distribution of outcomes is normally distributed. (This assumption is not essential and is made only to simplify the arithmetic as we shift from the discrete to the con- tinuous model.) Assume also that once the trial invest-

Exhibit 5. Imperfect Information Net Present Value

5,500,000

Invest in 10 more units

Investment seems to be profitable: G

Do not invest

Invest in 10 more units

Investment seems to be not profitable: B

Do not invest

400.000

4,400,000

500,000

-400,000

5,500,000

4,400,000

500,000

22



BIERMAN AND RAO/INVESTMENT DECISIONS WITH SAMPLING

ment is undertaken we will know how the machine be-

ing considered will operate in the other environments of the company.

The piece of equipment being considered has a normal distribution of net present values. The prior distribution of outcomes (V0) has a mean of (Vo =

$-2,000) and a standard deviation of (a, = $8,000). Since the distribution is for net present values, the break-even value (Vb) is equal to zero.

The expected net present value is a negative $2,000. If purchase of only one piece of equipment can be undertaken, the project would be rejected. However, assume that the firm uses 500 of these machines. For every dollar increase in net present value of one machine, the total net present value increases by $500. This is C.

Exhibit 6 shows the "betting distribution" on the net present value. If ao equaled zero, and if management were sure that a net present value of $-2,000 would occur, then the project would be rejected. There is some probability, however, that the net present value will be positive: EVPI = Cao N(D), where N(D)

= I + (Z-D) 0 (Z)dZ,3 0 (Z) is the unit normal density function for the standardized random variable,

Z -0 Vo,andD Vo Z = o - o , and D = .For example, we O'o 2'o

-$2,000 have D = .25, and from a table of $8,000

normal loss integrals N(D) = .2863. The expected value of perfect information for 500 pieces of equipment is: EVPI = 500($8,000) (.2863) = $1,145,000.

Define N(D*) = D + N(D), where D* results from integrating more than '/2 the density function. The expected cost of undertaking the investment to find out

3See Schlaifer, [4]. Tabled values of N(D) can be found in Schlaifer.

Exhibit 6. Probability Density Function: Betting Distribution of Example

Probability Density

II

0, = 8,000

-2000 0

o0 V

b

the true value of V0 is: E(Lo) = j-°c(-Vo)f(Vo)dVo =

aoN(D*). For the example, we have N(D*) = .25 + .2863 = .5363 and E(Lo) = $8,000 (.5363) = $4,290. Since the expected value of the perfect information is $1,145,000 and the expected cost is only $4,290, the single investment with a negative expected value of $2,000 (considering only one machine) is acceptable.

If the firm could only undertake one unit of equipment, C = $1 and EVPI = $8,000 (.2863) = $2,290. Now the expected cost is $4,290 and the expected value is $2,290. Thus, the expected net present value is a negative $2,000 (equal to the present value of one unit of investment) and the investment would be re-

$4,290 jected. There must be $4,290 1.87 units of

$2,290 investment for the firm to want to sample the investment of this example.

Delaying Other Investments

The sampling procedure (trying one investment before proceeding with the remainder) will result in the delay of other investments, adversely affecting their present value if they have positive present values. We are focusing, however, on investments that would- otherwise be rejected. If the alternative were to make all the investments now or do nothing, the firm would do nothing. The firm wants to obtain information as cheaply as possible, which is accomplished by undertaking a minimum-sized investment. While the expected value of this investment, taken by itself, is negative, the information that can be gathered justifies undertaking the investment. The fact that the other investments will be delayed is unfortunate, but it does not affect the basic sampling strategy. Delaying the investments decreases their net present value, but it also enables the firm to avoid investing funds in undesirable investments. On balance, it is a desirable strategy if the investment would otherwise be rejected.

Conclusion

An investment with a negative net present value traditionally has been rejected. This paper suggests a technique to evaluate investments where it may be good to undertake one "undesirable" investment in order to obtain information. As long as there is uncertainty about the outcome, and there is some probability that the investment will turn out to have a positive net present value, one must do the com- putations to determine whether an investment, undesirable by itself, may become worthwhile because of the additional investments that would be undertaken upon the information attached to its success.

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Appendix. Calculations to Revise Probabilities

From Exhibit 4 we obtain:

P(G I e,) = .9 P(B | e) = .8

P(G e,) = .2

probabilities are: =P(G I e,) P(e) = = P(G | e2) P(e) =

P(G) =

P(B, e) = P(B I e1) P(e1) P(B, e2) = P(B I e2) P(e2)

P(B)

P(B I e1)

.9 X .4 = .36

.2 X .6 = .12

.48

.1 x .4 = .04

.8 X .6 = .48

.52

.1

.36 P(e, G) .48

.04 P(e, B) .52

.75

.077

P(e, G) .25

P(e, | B) = .923

References

1. Harold Bierman, Jr., and Seymour Smidt, The Capital Budgeting Decision, New York, The Macmillan Co., 1975.

2. John D. C. Little, "Models and Managers: The Concept of Decision Calculus," Management Science (April 1970), pp. B466-85.

3. A. J. Merrett and A. Sykes, The Finance and Analysis of Capital Projects, New York, John Wiley & Sons, Inc., 1963.

4. R. Schlaifer, Analysis of Decisions Under Uncertainty, New York, McGraw-Hill Book Co., 1969.

FINANCIAL MANA GEMEN T ASSOCIA TION 1978 ANNUAL MEETING

Dates: October 12-13, 1978

Place. Radisson South Hotel Minneapolis, Minnesota

Program Participation:

Meeting Arrangements.

Placement Information:

Professor Robert F. Vandell Darden Grad. Sch. of Business Administration University of Virginia P. O. Box 6550 Charlottesville, Virginia 22906 Telephone: (804) 924-7417

Professor Peter Rosko Department of Finance and Insurance

College of Business Administration University of Minnesota 271 Nineteenth Avenue South Minneapolis, Minnesota 55455 Telephone: (612) 373-0358

Professor Donald J. Puglisi College of Business & Economics University of Delaware Newark, Delaware 19711 Telephone: (302) 738-2556

The joint P(G, e) =

P(G, e2) =

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Investment Decisions with Sampling