Methods and Models for Appraising Investment Projects Under Uncertainty

Abstract

Investment decision-making under uncertainty requires a lot of additional considerations. This chapter picks this up and describes decision theory, the risk-adjusted analysis, the sensitivity analysis, the decision-tree method and options pricing models. All these cover a wider range within uncertain environment situations for single investment projects.

Assessment Material

1.1 Exercise 8.1 (Sensitivity Analysis)

A company plans to purchase a machine. The price is €50,000. A volume of 1,000 units of product X can be manufactured and sold per period using the machine. Cash outflows dependent on the production volume total €40 per unit, and a sales price of €100 per unit can be achieved. The economic life of the machine is expected to be three periods. Cash outflows independent of the production volume total €25,000 in the first period and rise by 10 % in each subsequent period. The interest rate is assumed to be 9 %.

The investment decision should be made using the NPV method, and a sensitivity analysis should provide additional information.

• (a) Calculate the NPV of the investment project using the formula

  $$ \mathrm{N}\mathrm{P}\mathrm{V}=-{\mathrm{I}}_0+{\displaystyle \sum_{\mathrm{t}=1}^{\mathrm{T}}\left(\kern0.1em \left(\mathrm{p}-{\mathrm{cof}}_{\mathrm{v}}\right)\cdot \mathrm{x}\hbox{-} {\mathrm{COF}}_{\mathrm{ft}}\right)\kern0.3em \cdot {\mathrm{q}}^{-\mathrm{t}}} $$

  Parameters:

  • NPV = Net present value

  • x = Expected annual sales and production volume

  • p = Sales price of the product X

  • cof_v = Cash outflows dependent on the production volume (per unit)

  • COF_ft = Cash outflows independent of the production volume at time t

  • I₀ = Initial investment outlay

  • ${\mathrm{q}}^{-\mathrm{t}}=\frac{1}{{\mathrm{q}}^{\mathrm{t}}}=\frac{1}{{\left(1+\mathrm{i}\right)}^{\mathrm{t}}}$ = Discounting factor in t

  • t = Time index

  • T = Economic life of the project

• (b) Determine NPVs assuming that sales prices of €60, €80, €120 or €140 (each per unit) can be achieved.

• (c) Use sensitivity analysis to determine critical values for the:

  • Initial investment outlay

  • Sales price

  • Sales and production volumes

  • Production volume-dependent cash outflows

  • Production volume-independent cash outflows

  • Liquidation value

  • Economic life

  • Uniform discount rate

1.2 Exercise 8.2 (Sensitivity Analysis)

A company plans to acquire a new machine to manufacture a product. The forecasted data are:

Table 8

Table 8.8 Production volume and output-independent cash outflows

Assume that production and sales volumes are always identical. Tax and transfer payments can be ignored. The initial investment outlay occurs at t = 0, the liquidation value at the end of the economic life, and current cash flows at the end of each period. The expected NPV of the investment project is €10,899.53.

Use sensitivity analysis to find the following critical values at which NPV = 0:

• (a) The critical value of the initial investment outlay.

• (b) The critical value of the economic life.

• (c) The critical value of the liquidation value.

• (d) The critical value of the uniform discount rate.

• (e) The critical value of the sales price.

• (f) The critical value of the production volume-dependent cash outflows.

• (g) The critical level of the sales and production volumes.

• (h) The critical level of the production volume-independent cash outflows.

• (i) The critical values of the sales and production volumes at t = 1.

1.3 Exercise 8.3 (Sensitivity Analysis)

In the following exercise, the investment problem in Exercise 5.5 should be reconsidered using a sensitivity analysis.

• (a) Assume that investment projects A and B are pursued until the end of their technical lives (t = 4 or t = 3). Applying the NPV criterion, determine for each project in isolation (without considering subsequent projects):

  • (a1) The critical liquidation value.

  • (a2) The critical level of the annual cash flow surpluses which result in a change in absolute profitability.

• (b) Based on Exercise 5.5, consider the possibility of prolonging the economic lives of projects A and B up to the end of their technical lives. Determine, by what percentage the liquidation values of both projects (i.e. A & B jointly) must increase at the end of the technical life so that the technical and optimum economic lives are identical, assuming:

  • (b1) A single substitution of machine A by machine B (Exercise 5.5, a).

  • (b2) An unlimited chain of A and B, one after another (Exercise 5.5, b).

1.4 Exercise 8.4 (Decision-Tree Method)

A company is, at t = 0, considering an investment to extend production capacity. This investment requires an initial outlay of €40,000 and raises capacity by 5,000–20,000 product units, with variable cash outflows unchanged at €12 per unit. The sales price is constant and independent of the sales volume, at €20 per unit. The planning period consists of two periods and the uniform discount rate is set at 10 %.

The sales volume will be 20,000 units at time t = 1 if there is a favourable development (H: high demand), which has an expected probability (p) of 0.5. In the case of an unfavourable development (L: low demand) (probability p = 0.5), the sales volume will be 17,000 units.

At the end of period 1 the company may execute the same extension investment with an initial investment outlay of €30,000, if it has not invested at t = 0. The variable cash outflows and the sales price remain unchanged.

If there is high demand in period 1, the probability p of having further demand growth (sales volume: 20,000 units) in period 2 is p = 0.75. If there is low demand in period 1, this probability reduces to p = 0.25. Further low demand in period 2 would result in sales of only 17,000 units in that period.

• (a) Illustrate the decision problem by means of a decision-tree.

• (b) Determine the optimum decision sequence, assuming that the company wants to maximise its expected NPV.

• (c) What risk attitude does an investor who maximises the expected NPV have?

1.5 Exercise 8.5 (Decision-Tree Method)

A company is planning for an investment decision under uncertainty with a 2-year planning period.

At t = 0, three alternatives exist:

• I: Big investment (initial investment outlay: €22,000/maximum attainable cash inflow surplus: €100,000)

• II: Small investment (€12,000/€80,000)

• III: Refrain alternative (no investment) (€0/€60,000)

Then at t = 1 the following possibilities exist:

Provided that I was executed:

• I a: No subsequent investment opportunity (€0/€100,000)

Provided that II was undertaken:

• II a: Extension investment (€13,000/€100,000)

• II b: No subsequent investment (€0/€80,000)

Provided that III was undertaken:

• III a, III b, III c according to the options at t = 0

Demand in the second period will be either high (H: maximum attainable cash flow surplus €100,000) or low (L: maximum attainable surplus €60,000).

In the first period, the expected probabilities are: H: p = 0.1 and L: p = 0.9. In the second period, the probability of H is 0.8, provided that period 1 also had high demand, or p = 0.4 otherwise.

• (a) Illustrate the decision problem with the help of a decision-tree.

• (b) Determine the optimum decision sequence for the investor using a uniform discount rate of 10 %.

1.6 Exercise 8.6 (Decision-Tree Method)

• (a) A company must decide between two mutually exclusive strategic investment projects, A and B, at t = 0. This company uses the scenario method, and three scenarios (optimistic, most likely, pessimistic) are formulated. The occurrence of the optimistic scenario (opt) has a probability of 0.3, the most likely scenario (mlike) has a probability of 0.5, and the pessimistic scenario (pess) has a probability of 0.2. The cash flows from the investment projects are dependent on the occurrence of these scenarios, with the following data forecasted:

  Table 8.9 Data for the strategies A and B

  Table 8.9 (continued)

  The planning period spans 5 periods, the uniform discount rate is set at 10 %. Prepare a suitable form of investment appraisal to help with this decision. Discuss briefly which project is preferable.

• (b) It is further assumed that project B may be extended at time t = 2. The cash flows associated with it are forecasted as follows (in €):

  Table 8.10 Cash flows of the three scenarios

  For the extended investment project, an economic life of 3 periods is assumed. If the extension project goes ahead, the market share will increase by 5 % of the market size under all three scenarios. Apart from that, the remaining data are unchanged.

  • (b1) Illustrate the decision problem in graphical form.

  • (b2) What decisions should be taken at t = 2?

    How does this change the decision situation at t = 0?

1.7 Exercise 8.7 (Economic Life and Replacement Time Decisions Using the Decision-Tree Method)

• (a) A company must determine the optimum economic life of a new machine A, characterised by the following data (€’000):

  Table 8.11 Data for the new machine A

  The uniform discount rate is 10 % and the initial investment outlay is €550,000. Determine the optimum economic life of machine A, and the NPV that can be achieved when:

  • (a1) There is no replacement.

  • (a2) There is one identical replacement.

  • (a3) The machines are replaced twice by identical machines.

  • (a4) The machines are replaced an infinite number of times by identical machines.

• (b) Now, 5 years after starting to use machine A, the company is discussing its replacement. There is a rumour that the machine manufacturer may introduce a technically improved machine B, which serves the same function, onto the market within the next few years.

  For B, an initial investment outlay of €600,000 and the following additional data are forecasted (€’000):

  Table 8.12 Data for machine B

  There is a 60 % probability that B will be available at t = 1, and a 30 % probability it will become available at t = 2. If not available by that time, B will not be offered at all. Technical progress exceeding that achieved by B is not expected for the next few years.

  All data from (a) concerning machine A remain unchanged.

  When should the existing machine A be replaced and how (i.e. by another of type A or by machine B)?

  Assume that, based on market trends, the product generated using the machine can be sold for only another 5 years. Therefore, the planning period will end at t = 5 and the existing machine will be sold at that time. Because of the poor future prospects for the product, the existing machine is to be replaced only once, or not at all.

  Present a graphical illustration of this problem before solving it with appropriate calculations.

1.8 Exercise 8.8 (Decision-Tree Method)

At t = 0, a company has the choice between making an investment or rejecting it (refrain alternative). The investment creates a production capacity of 20,000 units after an initial investment outlay of €350,000.

If the investment is undertaken, no others are possible. If it is rejected at t = 0, then another investment may be undertaken at t = 1 with an initial investment outlay of €300,000 and a capacity of 17,000 units. No other investment projects are possible in later years.

The planning period totals three periods. With regard to future developments, two input measures are assumed to be uncertain. In the first period, maximum sales volume is expected to be either 15,000 units (probability 40 %) or 20,000 units (probability 60 %). The cash outflow per unit for the first period is estimated at either €12 (probability 50 %) or €10 (probability 50 %). It is assumed that the (random) developments that influence the sales volumes and cash outflows per unit are independent.

In all periods, the sales price will be €20 (this is certain). It is further assumed that the per unit cash outflows experienced in the first period will remain unchanged in subsequent periods.

If in the first period the maximum sales volume amounts to of 20,000 units, it will either remain at this level (probability 60 %), or rise to 22,000 units (probability 40 %) in the final two periods.

In the case of a maximum sales volume of 15,000 units in the first period, the maximum sales volume is forecasted to either stay the same (probability 50 %) or to rise to 18,000 units (probability 50 %) in the final two periods.

Other cash outflows need not be considered. The liquidation values at the end of the planning period are either €30,000 with investment at t = 0, or €40,000 with investment at t = 1. The uniform discount rate is 10 %.

• (a) Illustrate the decision problem in the form of a decision-tree.

• (b) Determine the optimum decision sequence and the maximum expected NPV.

1.9 Exercise 8.9 (Decision-Tree Method)

At t = 0, an investor must decide whether to use €510,000 to either make a direct investment in Genetic Engineering Inc., a young genetic technologies company, or to invest in the capital market earning a yield of 8 % per year.

The shares in Genetic Engineering Inc. are traded at the present time (t = 0) on the stock exchange at €500 per share. Their nominal value is €100 per share. The medium-term trend in the price of these shares is influenced by various factors.

During the next year, tests will be done on the newest product developed by Genetic Engineering Inc., the results of which are expected to be available at the end of the year (t = 1). With positive results (probability 40 %), the share price is expected to increase by €100; with negative results, the share price is expected to decline by €50.

The legislative body has announced that a decision on guaranteeing patent protection for gene-technology products will take effect in the second year. Patent protection is expected with 50 % probability. In the case of favourable test results, this patent protection will result in a share price rise at t = 2 of €250; in the case of negative test results, the share price rise will be only €100. A refusal of patent protection is expected to result in a fall in share prices of €100.

Other factors influencing the share price (e.g. general share price index changes and other investor transactions) and tax payments should be ignored.

The investor is considering purchasing the shares at t = 0 or t = 1. After having purchased in t = 0 it will be possible to sell the shares at t = 1 or to hold them until t = 2. His aim is to maximise his expected assets—consisting of shares and/or cash—as at t = 2. The share will be valued at t = 2 at its current market price (future sales expenses are ignored).

The investor assumes that any funds not invested in the shares can be invested elsewhere at 8 % per period. Purchase and sales expenses for any share transaction are estimated at 2 % of the share price. At the end of a period, and independently of how the company develops, he expects to receive a dividend of 10 % on the nominal value of the shares. The dividend payments are made before the purchase or sales transactions. Any dividend the investor receives at t = 1 is reinvested for one period at 8 %.

Illustrate this decision problem with the help of a decision-tree, and determine the optimum investment strategy, taking into account the investor’s compound value maximisation target, and using the rollback procedure.

Further reading: see recommendations at the end of this part.