Simultaneous Decision-Making Models

Abstract

Decisions about investment programmes often involve simultaneous choices about types and numbers of investment projects. Additionally, models used for simultaneous decision-making might need to accommodate choices within a range of company areas such as financing, production, sales, human resources and tax policy. In this chapter, the finance and production areas—because of their relevance and close connections with investment decisions—are selected to illustrate ways of supporting investment decision-making in a broader sense than has been discussed previously. In the sections of this chapter some models are presented in detail, their practical relevance is discussed, and problems with their practical application are analysed. A static and two multi-tier models of simultaneous investment and financing/production decisions are thoroughly described in this chapter, illustrated by examples.

Appendices

Assessment Material

1.1 Exercise 7.1 (Dean Model for Simultaneous Financing and Investment Decisions)

The choice is between the investment and finance projects below, each with their given cash flows, a_jt or d_it (in €’000):

Table 7.7 Cash flows for the investment projects

Table 7.8 Cash flows of for financing projects

• (a) For each project, calculate the internal rate of return (IRR) or the effective rate of interest. From this, deduce the capital supply and capital demand curves and draw these on a graph. Determine the optimum investment and financing programme as well as the endogenous rate of interest. What is the maximum compound value?

• (b) Take another look at the choice of investment and financing projects in part a) of the exercise. Assume all the investment projects must be realised in full (i.e. they are indivisible). Ascertain the optimum investment and financing programme and calculate the maximum compound value.

• (c) State the assumptions made by the Dean model.

1.2 Exercise 7.2 (Multi-tier Model for Simultaneous Financing and Investment Decisions)

A company faces the task of planning its investment and financing programme. It must choose between three investment projects (x₁, x₂, x₃). At any point in time, excess funds may be invested in the short term (x_4t). Interest on such short-term investments is 5 %. The investment projects are characterised by the following net cash flows (€’000):

Table 7.9 Net cash flows for the investment projects

There are also two financing projects available to the company (y₁, y₂) with the following net cash flows (€’000):

Table 7.10 Net cash flows for the financing projects

Each loan can be drawn down for up to €600,000 and divided up at will. Each investment project may be undertaken up to five times, but must be realised in full each time (i.e. the projects are indivisible).

The company invests internal funds as follows: €200,000 at the beginning of the first period and €100,000 each at the beginning of the second and third periods.

Formulate a multi-tier model for the simultaneous planning of an investment and financing programme appropriate to the problem described above.

1.3 Exercise 7.3 (Multi-tier Model for Simultaneous Financing and Investment Decisions)

A company is faced with two investment projects (x₁, x₂) and two forms of long-term financial investment (x₃, x₄) plus one short-term financial investment (x_5t) in each period. The company may take up two loans (y₁, y₂) of up to €1,000,000 each.

For the available investment projects and loans, the following monetary consequences are expected (€’000):

Table 7.11 Net cash outflows per unit of the variables (projects)

There are no internal funds available.

• (a) Formulate a multi-tier model for maximising the compound value of the investment and financing programme.

• (b) The following programmes are proposed:

  • (i) x₁ = 1.5; x₂ = 1; x₃ = 1; y₁ = 1; y₂ = 1

  • (ii) x₁ = 1; x₂ = 1; y₁ = 1; y₂ = 1

  (The values of the variables x_5t are not given here but may be deduced from the other variables.)

  Are the programmes feasible and, if so, optimal? Briefly outline the reasons for this.

• (c) How does the model change if additional cash inflows in the amount of €10,000 are expected for each unit of investment project 1 at each of the times t = 4 and t = 5, and 10 % is the rate of interest for calculation purposes?

• (d) In optimising a Hax and Weingartner model, the following endogenous compounding factors q ^*_t were determined for the times t:

  $$ {\mathrm{q}}_0^{*} = 1.93908;\;{\mathrm{q}}_1^{*} = 1.4916;\;{\mathrm{q}}_2^{*} = 1.243;\;{\mathrm{q}}_3^{*} = 1.1;\;{\mathrm{q}}_4^{*} = 1 $$

  Determine the endogenous rates of interest for periods 1–4, and assess the profitability of an additional project with the following cash flow profile:

  Table 7.12 Cash flow profile of the additional project

1.4 Exercise 7.4 (Static and Multi-tier Models for Simultaneous Financing and Investment Decisions)

• (a) A choice must be made between the investment and finance projects below, each with their forecasted cash flows, a_jt or d_it (in €’000).

  Table 7.13 Cash flows of the investment projects

  Table 7.14 Cash flows of the financing projects

  • (a1) Determine the optimum investment and financing programme when the investment and finance projects may all be divided at will.

    What is the maximum compound value?

  • (a2) Ascertain the optimum investment and finance programme assuming the investment projects cannot be divided.

    What is the maximum compound value?

  • (a3) Which programme is optimal if neither the finance projects nor the investment projects can be divided?

    What is the maximum compound value?

• (b) A company wishes to plan its investment and financing programme simultaneously. There are four investment projects to choose from, A-D (investment variables x_A − x_D), with the following net cash flows (€’000):

  Table 7.15 Net cash flows for the four investment projects

  Investment projects A and C may be realised a maximum of three times. Internal funds available at t = 1 amount to €80,000. The investment projects A and B may also be realised at t = 1 (investment variables x_E, x_F), and an upper limit of 3 applies to the realisation of investment project A also at this time.

  The following information on the financing projects 1–3 (financial variables y₁ − y₃) is available:

  • If the first financing project is realised, 60 % of the cash inflows will be received at t = 0 and 40 % at t = 1. At each time, interest at a rate of 10 % is payable on the capital borrowed, which is to be repaid at t = 4.

  • A payment of the full nominal amount of the second financing project will be received at t = 0 if this project is realised. 50 % of the capital is to be repaid at t = 3, and the remaining 50 % at t = 4. At each time, interest at a rate of 9 % is also payable on capital previously received and not yet repaid.

  • The third financing project generates only one positive payment at t = 0. Payments of interest and compound interest, as well as capital repayments, are due at times t = 1 to t = 4. The total amounts payable stay the same and the applicable rate of interest is 6 %.

  • For each of the financing projects the maximum amount is €200,000.

  At each time, a short-term, single-period financial investment may be made, yielding interest at 3 % (investment variables x_Gt, t = 0, 1, 2, 3, 4).

  Also at each time (except t = 4), a short-term, single-period loan may be accessed bearing interest at 7 % (financial variables y_4t, t = 0, 1, 2, 3), while the maximum amount available at each time is, as for the other financing projects, €200,000.

  Formulate a multi-tier model for this problem. Relate the objective function to t = 4 and assume a discount rate of 5 % for period 5.

• (c) The models formulated in (a), and (b) aim to decide simultaneously on an investment and financing programme. Work out the differences between the models and, in so doing, state the differing assumptions involved.

1.5 Exercise 7.5 (Extended Förstner and Henn Model)

The head of a company’s planning department wishes to decide about production and investments simultaneously. The following data are available: The company produces two kinds of products, k (k = 1, 2). For each unit of product, it achieves a price p_k and has to pay variable cash outflows of a_vk resulting from the production process. It can sell maximum amounts of Z_k.

Table 7.16 Data for the two kinds of products

Both products are produced on three machines, j (j = 1, 2, 3). The utilisation of these machines j for each unit of the product k is given below (in units of capacity).

Table 7.17 Utilisation of the machines

At the beginning of the planning period there is an initial stock of machinery with the capacity given below in time units:

Table 7.18 Existing capacity of the machines

Identical machines may be acquired at the beginning of each period. For each type of machine j, I_0j represents the initial investment outlays (in €), and G_j the relevant expansion in capacity (in time units).

Table 7.19 Data for the machines

The liquidation value at the end of the economic life is 20 % of the initial investment outlay for each machine. The decrease in liquidation value occurs evenly throughout all periods of the economic life.

In each case, the total economic life of the existing machine is 2 years and all existing machines have a remaining economic life of 1 year. The cash outflows to acquire these existing machines were equal to those for the machines available for purchase at t = 0.

• (a) Formulate a two-period model with the objective ‘maximising the compound value’. In so doing, assume that the data given here—with the exception of the cash outflows for the aggregates acquired at t = 1 (which rise by 10 % compared with the figures given)—are also valid for the second period. Note that the company must remain liquid at all times. Interest on the short-term financial investment is 10 %. €10,000 of internal funds are available at t = 0 and again at t = 1.

• (b) What problems might be expected in setting up and solving such a model in a real business environment?

1.6 Exercise 7.6 (Extended Förstner and Henn Model)

Prepare a simultaneous investment and production decision using the following underlying data.

A company produces two kinds of product, k (k = 1, 2). It has a monopoly position in the market and achieves prices p_k according to the following formulae. The maximum volumes it can sell, Z_k, and the variable cash outflows per unit, cof_vk, are also given below (with z_k = production amount and sales volume).

Table 7.20 Data for the two products

Both products are manufactured on the machines j (j = 1, 2) and take up the following time units per unit of product on these machines.

Table 7.21 Data for the machines

At the beginning of the planning period, machine 1 has a capacity of 360 time units and a remaining economic life of one period. Its further characteristics are equal to those given below for new type 1 machines.

New type 1 and type 2 machines can be acquired at the beginning of each period. Their economic life is four periods and the liquidation value at the end of the economic life amounts to 20 % of the initial investment outlay. The decrease in their liquidation value occurs linearly throughout all periods of their economic life.

Regardless of the date of acquisition, the cash outflows are €2,000 for the acquisition of machine 1, and €2,500 for machine 2. Each new machine purchased expands capacity by 90 time units (machine 1) and 100 time units (machine 2).

The rate of interest for short-term financial investments is 10 %; there is €40,000 of internal funds available at t = 0.

Given the above, formulate a dynamic two-period model for determining an optimum investment and production programme with the objective of maximising the compound value. In so doing, assume that the data given—with the exception of the variable cash outflows per unit, which rises by 10 %—are valid for both periods. Bear in mind that the company must remain liquid throughout both periods.

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