(Relevant to AAT Examination Paper 4: Business Economics and Financial Mathematics)

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2 1 The annuity offers \$110,000 at the end of each year forever, for an initial cash outflow of \$1,000,000 now 2 Adam has an another choice to invest elsewhere to earn a return of 10% per annum 3 If Adam did invest at 10% per annum, how much would he need to invest now, at 10%, to obtain \$110,000 at the end of each year forever? 4 Is it worth investing \$1,000,000 in the annuity, or should he invest elsewhere at 10%, in order to earn these future cash flows? If Adam invested elsewhere at 10% per annum, the amount required to earn \$110,000 at the end of each year forever would be as follows Year Cash flow Discount factor Present value \$ 10% \$ 1 110, , , ,860 n = 110,000 Adam s case is a perpetuity, or an annuity whose payments are paid forever, ie the term of the annuity is not finite The present value of a perpetuity = The present value of an annuity = \$1,100,000 The alternative is to invest \$1,000,000 in the annuity, or \$1,100,000 elsewhere at 10%, in order to earn the future cash flows We then obtain the following conclusion - It is worth investing in the annuity, by \$100,000 (\$1,100,000 - \$1,000,000) - The annuity gives a return of more than 10% per annum The net present value is the difference between the present value of cash inflows from the annuity (\$1,100,000) and the present value of cash outflows (in this example, \$1,000,000): since net present value is positive, the annuity is acceptable The NPV method can also be used in comparing two or more investment projects and the decision rule is to choose the project with the highest NPV Expected value It is important to realize that there are a lot of uncertainties in the future such as over cash flows, interest rates and so on Consider the following examples of decision-making problems Amy must decide whether to purchase an apartment to live in as an investment, or whether to rent it If sales of apartments continue to increase, it will be more profitable to purchase the - 2-2

3 apartment If sales levels decline, it will be more profitable to rent the apartment Which decision should she make? AA Company is considering three options regarding the prices of shoes for next year AA Company could (option 1) increase the prices by 4 percent, (option 2) increase the prices by 2 percent, or (option 3) keep the prices as they are The final decision will be determined by estimated demand In the above cases, the decisions are characterized by several options and factors not controlled by the decision maker For example, AA Company has no control over the sales volume These cases tell us a lot about the nature of decision making We can list the decision options, determine the possible future events and even assign probabilities, but we still face uncertainty in our decision making The three components in any decision process are as follows: The options or alternatives are the choices available to the decision maker and which are under the control of the decision maker AA company can decide to increase its prices by 4 percent, or by 2 percent The economic environment determines the outcome of the decision that actually happens This is not under the control of the decision maker AA Company does not know whether demand for its shoes will be high after it has increased prices By the same token, Amy cannot know whether the real estate market will be either good or poor over the next year A payoff is required to compare each combination of decision options and the economic environment Amy thinks that if the economic climate is good and she decides to buy an apartment, the payoff will be an estimated \$100,000 On the other hand, if Amy decides to buy the apartment and the economic climate is poor, the payoff is estimated to be -\$90,000 The expected value approach is used to take this uncertainty into account We now introduce the application of probability to business decision making under conditions of uncertainty We first introduce expected value as an appropriate criterion for decision making and extend its usage to investment appraisal The expected value is the weighted average of each value multiplied by their corresponding probabilities Expected value = Where x i = the value of the payoffs p i = the probability of the state of nature Over the past decade, in six years the economic climate has been good and in four years it has been poor We can therefore project that the probability of a good economic climate is 06 and the probability of a poor economic climate

4 The expected value of Amy s decision to buy an apartment is: EV (purchase apartment) = \$100, (\$90,000) 04 Expected net present value method = \$60,000 \$36,000 = \$24,000 Here we extend the expected value idea in evaluating an investment project The expected net present value (ENPV) method uses the idea of expected value through uncertain cash flows, an unknown future economic environment, and so on Example 2 Becky Company is considering whether to invest \$340,000 in a project which has only a 20% chance of being successful and receiving cash inflows of \$210,000 at the end of year one and year two However, there is 80% chance that the project will fail, in which case the cash inflows will only be \$150,000, received at the end of each of year one and year two Is it worth Becky Company investing in the project if the cost of capital is 5%? The solution for the Becky Company s problem is summarized as follows: 1 Construct a table to lay down the outcomes for each event and each year 2 Assign probabilities to the events Discount factor Success (02) Failure (08) Year 5% Cash flows, \$ PV Cash flow, \$ PV (340,000) (340,000) (340,000) (340,000) , , , , , , , ,050 NPV = 50,390 NPV = (61,150) 1 Calculate the ENPV by weighting the net present values by the assigned probabilities and summing the weighted values ENPV = (02 \$50,390) + (08 - \$61,150) = \$-38,842 2 If the ENPV is positive, the investment is acceptable If it is negative, the investment is unacceptable If it is zero, the company should be indifferent to the investment Since the ENPV is \$-38,842, Becky Company should not invest in the project - 4-4

5 Limitations of using expected net present value method in decision making Although the ENPV method is a good way of considering an investment appraisal, it may not always be applicable The probabilities, future discount factors and cash flows are also estimated, ie they are not known with certainty Here we consider the limitations of the NPV method Suppose Mr Chan has the chance to invest a new product Assume that if the new product is invested in and is successful, there will be an estimated net profit of \$2,000,000 However, if the product is not successful, there will be an estimated cost of \$100,000 The payoff figures for this decision, including probabilities, are given in table 1 Table 1: Project payoffs State Success Failure Options Invest \$2,000,000 -\$100,000 Do not invest 0 0 When Mr Chan assigns a subjective probability of 09 to success, the investment makes a profit of \$1,790,000 (\$2,000, \$100, = \$1,790,000) Because the expected value is \$1,790,000, Mr Chan should invest the new product Now consider another situation If the probability of failure now is 04 and the estimated loss is \$500,000 if the product is not successful, then the expected profit for the investment now is \$1,000,000 (\$2,000, \$500, = \$1,000,000), as compared with an expected value of \$0 for the option Mr Chan may still chose to invest in the new product based on the positive expected value However, Mr Chan may be unwilling to accept a 04 probability of losing \$500,000 regardless of the higher expected profit, since a loss of \$500,000 would result in his bankruptcy Why would some decision makers reject the higher expected profit in this type of situation? They may do so because the probability assessment is subjective and difficult to estimate If the decision exposes the investor to a high risk, the expected value method may not be a good guide to use - 5-5

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