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**COMPLEX INVESTMENT DECISIONS**

Chapter 11 COMPLEX INVESTMENT DECISIONS

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LEARNING OBJECTIVES Show the application of the NPV rule in the choice between mutually exclusive projects, replacement decisions, projects with different lives, etc. Understand the impact of inflation on mutually exclusive projects with unequal lives Make choice between investments under capital rationing Illustrate the use of linear programming under capital rationing situation

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**Complex Investment Problems**

How shall choice be made between investments with different lives? Should a firm make investment now, or should it wait and invest later? When should an existing asset be replaced? How shall choice be made between investments under capital rationing?

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**Projects with Different Lives**

The choice between projects which have different lives should be made by evaluating them for equal periods of time. Example: A firm has to choose between two projects X and Y, which are designed differently, but perform essentially the same function. Cash flows of projects are in real terms and the real discount rate is 10 per cent. The present value of costs are shown below:

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**Projects with Different Lives**

Project X has 4-year life while Project Y has 2-year life. Project Y will be replicated to compare it with Project X. Project Y costs more than project X.

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**Annual Equivalent Value (AEV) Method**

The method for handling the choice of the mutually exclusive projects with different lives, as discussed in last slide, can become quite cumbersome if the projects’ lives are very long. We can calculate the annual equivalent value (AEV) of cash flows of each project. We shall select the project that has lower annual equivalent cost.

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AEV: Example In the earlier example, the present value of cash flows of X is Rs 215,100. You can divide Rs 215,100 by a 4-year present value factor for an annuity of Re 1 at 10 per cent (3.1699) to obtain AEV. Similarly, AEV for Y can be calculated. Y is more costly.

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AEV for Perpetuities When we assume that projects can be replicated at constant scale indefinitely, we imply that an annuity is paid at the end of every n years starting from the first period. where NPV¥ is the present value of the investment indefinitely, NPVn is the present value of the investment for the original life, n and k is the opportunity cost of capital.

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**Nominal Cash Flows and Annual Equivalent Value**

Continue with earlier example. Let us assume expected inflation of 4% . The real cash flows of X and Y can be converted into nominal cash flow (as shown below) and the real discount rate into nominal discount rate: (1.04) x(1.10) -1= Notice that the ranking of projects changes at higher inflation rate of 15%. Thus, the choice should be based on real AEV. Inflation and Annual Equivalent Value

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**Investment Timing and Duration**

The rule is straightforward: undertake the project at that point of time, which maximizes the NPV.

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**Tree Harvesting Problem**

The maximisation of the investment’s NPV would depend on when we harvest trees. The net future value of trees increases when harvesting is postponed; but the opportunity cost of capital is incurred by not realising the value by harvesting the trees. The NPV will be maximised when the trees are harvested at the point where the percentage increase in value equals the opportunity cost of capital.

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**Tree Harvesting Problem**

Suppose the net future value obtained over the years from harvesting the trees is At and if the opportunity cost of capital is k, then the net present value (NPV) of the net realisable value of trees is given by:

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**Tree Harvesting Problem**

To determine the optimum harvesting time, which maximizes the NPV, we set the derivative of the NPV with respect to t in Equation equal to zero. Land may have value since the trees can be replanted. Therefore, the correct formulation of the problem will be to assume that once the trees are harvested, the land will be replanted. Thus, if we consider a constant replication of the tree-harvesting investment indefinitely, then the NPV will:

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**Replacement of an Existing Asset**

Replacement decisions should be governed by the economics and necessity considerations. An equipment or asset should be replaced whenever a more economic alternative is available.

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Example A company is operating equipment, which is expected to produce net cash inflows of Rs 4,000, Rs 3,000 and Rs 2,000 respectively for next 3 years. A design, which is considered to be a technological improvement and more efficient to operate, has appeared in the market. It is expected that the new machine will cost Rs 12,000 and will provide net cash inflow of Rs 6,000 a year for 5 years. What should the company do? Assume 12 per cent discount rate.

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Example The correct method of analysis is to compare the annual equivalent value (AEV) of the old and new equipments as given below. A chain of new machines is equivalent to an annuity of Rs 9,630 = Rs 2,671 a year for the life of the chain. The existing machine is still capable of providing an annuity of: Rs 7,390 = Rs 3,076. So long as the existing machine generates a cash inflow of more than Rs 2,671 there does not seem to be an economic justification for replacing it.

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**Investment Decisions Under Capital Rationing**

Capital rationing refers to a situation where the firm is constrained for external, or self-imposed, reasons to obtain necessary funds to invest in all investment projects with positive NPV. Under capital rationing, the management has to decide to obtain that combination of the profitable projects which yields highest NPV within the available funds.

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**Why Capital Rationing? There are two types of capital rationing:**

External capital rationing: imposed by capital markets Internal capital rationing: self-imposed by the company internally

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Profitability Index The objective of the NPV rule under capital constraint should be to maximise NPV per rupee of capital rather than to maximise NPV. Projects should be ranked by their profitability index, and top- ranked projects should be undertaken until funds are exhausted. The Profitability Index does not always work. It fails in two situations: Multi-period capital constraints. Project indivisibility.

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**Limitations of Profitability Index**

Multi-period capital constraints Project indivisibility

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**Profitability Index: Example**

The NPV and profitability index of the following four projects are shown. Given the budget constraint of Rs 50, projects M and N will be selected as per PI.

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**Programming Approach to Capital Rationing**

Capital rationing presents a situation of maximising net present value of several projects subject to funds constraint. Hence, programming approach can be used for decision making. Linear Programming (LP) Integer Programming (IP) Dual variable

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Example Let us consider four projects – L, M, N and O, given earlier. The company has budget constraint of Rs 50 each in year 0 and year 1. We need to maximise NPV subject to budget constraints. Since investments will be positive, we will put as constraints.

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**The LP Solution of the problem:**

Example Maximize NPV Subject to: The LP Solution of the problem:

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Integer Programming A large number of projects in practice are indivisible. When projects are not divisible, we can use integer programming (IP) by limiting the X’s to be integers of either 0 to 1. Integer programmes are difficult to solve. It may take unwieldy number of iterations for the model to converge on a solution. Also, other restrictions may prove to be redundant on account of integer restriction.

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Dual Variable Dual variables for the budget constraints may be interpreted as ‘opportunity costs’ or ‘shadow prices.’ In the earlier example, dual variables for the budget constraints in periods 0 and 1 respectively, are 0.344, and The dual variables of for period 0 imply that NPV can be increased by Rs if the budget in period 0 is increased by Re 1. In other words, the opportunity cost of the budget constraint for period 0 is 34.4 per cent, and for period 1 it is 8.6 per cent. Dual variables provide information for deciding the shifting of funds from one period to another.

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**Extensions of Programming Approach**

The use of LP or IP models can be extended to cope with other constraints. A firm may like to provide for the carry over of unspent cash from one period to another. In addition to financial constraints, non-financial constraints can also be included.

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**Limits to the Use of Programming Approach**

First, they are costly to use when large, indivisible projects are involved. Second, these models assume that future investment opportunities are known. The discovery of investment opportunities in practice is an unfolding process.

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**Capital Rationing in Practice**

Capital rationing does not seem to be a serious problem in practice. It may arise due to the internal constraint or the management’s reluctance to raise external funds. When companies face the problem of shortage of funds, they use simple rules of choosing projects rather than the complicated mathematical models

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