Decision-making under conditions of risk and uncertainty

Transcription

1 Decision-making under conditions of risk and uncertainty Solutions to Chapter 12 questions (a) Profit and Loss Statement for Period Ending 31 May 2000 Revenue ( journeys): 0 3 miles ( ) ) 4 5 miles ( ) ) Over 5 miles ( ) ) Juvenile fares ( ) ) Senior citizen fares ( ) ) ) Advertising revenue ) ) Less: Variable costs (20 routes 4 buses 150 miles 330 days 0.75) ( ) Less: Fixed costs ( ) Net profit ) Question (b) Assuming the same passenger mix as 2000 the weighted average fare per passenger for year ending 31 May 2001 is ( )/ The break-even point is where: Total revenue from fares Advertising revenue Total cost Let x number of passenger journeys Break-even point: x ( ) x x Maximum capacity utilization passenger journeys ( /0.6) Break-even capacity utilization / % (c) (i) Expected value and probability estimates for 2001 Capacity Revenue Inflation Costs Combined Net Expected Utilization Fares Adverts probability profit value % (Probability) ( 000) ( 000) (%) (Probability) ( 000) ( 000) ( 000) a b b b a a DECISION-MAKING UNDER CONDITIONS OF UNCERTAINTY 95

2 Notes a Fare revenues at 60% capacity for 2000 were Assuming 5% inflation fare revenues for 2001 at 60% capacity will be ( ). At 70% and 50% capacity utilization fare revenues will be as follows: 70% 70/ % 50/ b Variable costs vary with bus miles which are assumed to remain unchanged. Predicted costs at the different inflation levels are as follows: 8% ( ) % ( ) % ( ) (c) (ii) The answer to this question requires the preparation of a cumulative probability distribution that measures the cumulative probability of profits/ (losses) being greater than specified levels. Cumulative probability distribution Losses greater than probability Probability of a loss occurring 0.40 Profits greater than Profits greater than Profits greater than Profits greater than (d) The following factors have not been incorporated into the analysis: (i) Change in the passenger mix. (ii) Changes in the number of routes and the number of days operation per year. (iii) Changes in fare structure such as off-peak travel or further concessions for juveniles and senior citizens. (iv) Changes in cost levels due to factors other than inflation (e.g. more efficient operating methods). Question (a) For each selling price there are three possible outcomes for sales demand, unit variable cost and fixed costs. Consequently, there are 27 possible outcomes. In order to present probability distributions for the two possible selling prices, it would be necessary to compute profits for 54 outcomes. Clearly, there would be insufficient time to perform these calculations within the examination time that can be allocated to this question. It is therefore assumed that the examiner requires the calculations to be based on an expected value approach. The expected value calculations are as follows: (i) Variable cost (ii) Fixed costs ( 10 10%) 10/ ( 10 6/ ( 10 5%) 4/ (iii) 17 selling price (iv) 18 selling price (units) (units) units units units units units units DECISION-MAKING UNDER CONDITIONS OF UNCERTAINTY

3 Expected contribution 17 selling price ( ) selling price ( ) The existing selling price is 16, and if demand continues at units per annum then the total contribution will be [( ) units]. Using the expected value approach, a selling price of 18 is recommended. (b) Expected profit fixed costs Break-even point fixed costs ( )/contribution per unit ( 7.60) units Margin of safety expected demand ( units) units 6003 units % margin of safety 6003/ % of sales Note that the most pessimistic estimate is above the break-even point. (c) An expected value approach has been used. The answer should draw attention to the limitations of basing the decision solely on expected values. In particular, it should be stressed that risk is ignored and the range of possible outcomes is not considered. The decision ought to be based on a comparison of the probability distributions for the proposed selling prices. For a more detailed answer see Probability distributions and expected value and Measuring the amount of uncertainty in Chapter 12. (d) Computer assistance would enable a more complex analysis to be undertaken. In particular, different scenarios could be considered, based on different combinations of assumptions regarding variable cost, fixed cost, selling prices and demand. Using computers would also enable Monte Carlo simulation (see Chapter 14) to be used for more complex decisions. (a) Possible Alternative outcomes Probability types of (level of of Payoff machine hire orders) outcomes ( 000) High High [( ) 2300] Medium [( ) 2300] Low [( ) 2300] Medium High ( ) Medium [( ) 1500] Low [( ) 1500] Low High ( ) Medium ( ) Low [( ) 1000] Question (b) Expected values: High hire level ( ) ( ) ( ) Medium hire level ( ) ( ) ( ) Low hire level ( ) ( ) ( ) (c) Using the expected value decision rule, the medium hire contract should be entered into. Managers may be risk-averse, risk-neutral or risk-seeking. A risk-averse manager might adopt a maximin approach and focus on the worst possible outcome for DECISION-MAKING UNDER CONDITIONS OF UNCERTAINTY 97

4 each alternative and then select the alternative with the largest payoff. This approach would lead to the selection of the low initial hire level. A risk-seeking manager might adopt a maximax approach and focus on the best possible outcomes. This approach would lead to choosing the high initial hire contract, since this has the largest payoff when only the most optimistic outcomes are considered. (d) With perfect information, the company would select the advance plant and machinery hire alternative that would maximize the payoff. The probabilities of the consultants predicting high, medium and low demand are respectively 0.25, 0.45 and The expected value calculation with the consultant s information would be: Advance Expected hire Payoff value level ( 000) Probability ( 000) High market high Medium market medium Low market low Expected value with consultant s information Expected value without consultant s information Maximum amount payable to consultant Question (a) Selling price Maximum demand Maximum revenue Total variable cost Fixed costs R & D cost Estimated profit ( 000) Totalrevenue Total cost 1000 At 80 selling price Output Optimal output level Figure Q DECISION-MAKING UNDER CONDITIONS OF UNCERTAINTY

5 The above analysis is based on the maximum sales demand. On this basis, the analysis indicates that profits are maximized at an output level of units when the selling price is 80. It is preferable to use the most likely demand level and to incorporate uncertainty around the most likely demand into the analysis. (b) For a selling price of 90 there are three different demand levels, and for each demand level there are three different outcomes for actual unit variable cost. Therefore there are nine possible outcomes. The contribution and probability of each outcome is presented in the following schedule: (1) (2) (3) (4) (5) (6) (7) (8) Weighted Unit Total Joint outcome Demand variable Unit contribution probability (6 7) (000) Probability cost Probability contribution ( 000) (2 4) ( 000) Expected total contribution Fixed costs Expected profit (c) To compare the three selling prices, it is necessary to summarize the information in part (b) for a 90 selling price in the same way as part (c) of the question. Note that fixed costs are deducted from the total contribution column in the schedule presented in (b) to produce the following statement: Prices under review Probability of a loss Greater than or equal to Probability of a profit Greater than or equal to Expected profit Loss ( ) The following items should be included in the memorandum: (i) The 90 selling price has the largest expected profit, but there is also a 0.34 probability of not making a profit. (ii) Selling price of 80 may be preferable, because there is only a 0.10 probability of not making a profit. A selling price of 80 is least risky, and the expected value is only slightly lower than the 90 selling price. DECISION-MAKING UNDER CONDITIONS OF UNCERTAINTY 99

6 (iii) Subjective probability distributions provide details of the uncertainty surrounding the estimates and enable the decision-maker to select the course of action that is related to his personal risk/profit trade-off (see Chapter 12 for an explanation of this). (iv) Subjective probabilities are subject to all the disadvantages of any subjective estimate (e.g. bias). (v) Calculations are based on discrete probabilities. For example, this implies that there is a 0.7 probability that demand will be exactly A more realistic interpretation is that represents the mid-point of demand falling within a certain range. (d) If the increase in fixed costs represents an additional cost resulting from an increase in volume then this incremental cost is relevant to the pricing decision. If the fixed costs represent an apportionment then it is not relevant. Nevertheless, we noted in Chapter 11 that selling prices should be sufficient to cover the common and unavoidable long-run fixed costs. The research and development expenditure is a sunk cost, and is not a relevant cost as far as the pricing decision is concerned. However, the pricing policy of the company may be to recover the research and development expenditure in the selling price. The amount recovered per unit sold should be a policy decision. Note that the decision to write off research and development in one year instead of three will affect the reported profits. Question (a) The calculations of the product variable costs per unit are: Newone Newtwo Labour and materials Variable overheads 6 (6 hrs 1) 2 (2 hrs 1) Unit variable cost Low-price alternative: The contributions per unit are 32 for Newone ( ) and 14 ( 60 46) for Newtwo. The probability distributions are as follows: Newone Newtwo Demand Probability Contribution Demand Probability Contribution a a Note a Machine capacity restricts outputs to 1000 units of Newone and 3000 units of Newtwo. Note that estimates indicate with 100% certainty that Newone will yield a contribution of and Newtwo will yield a contribution of Higher price alternative: The contributions per unit are 42 for Newone ( ) and 24 ( 70 46) for Newtwo. The probability distributions are as follows: Newone Newtwo Demand Probability Contribution Demand Probability Contribution a a Expected value Expected value DECISION-MAKING UNDER CONDITIONS OF UNCERTAINTY

7 Note a Output is restricted to 1000 units of Newone and 3000 units of Newtwo. Recommendations The above probability distributions indicate that Newtwo is preferable to Newone, irrespective of which price is set. At the higher selling price Newtwo yields a higher expected value. There is only a 0.2 probability that a lower contribution will be earned if the higher price is selected in preference to the lower price. The advantage of the lower price is that the outcome is certain, but, given the high probability (0.8) of earning higher profits with the higherprice alternative, a selling price of 70 is recommended. With the higher-price alternative, there is a 0.70 probability that machine hours will not be utilized. Any unused capacity should be used to sell Newone at 130 selling price. (b) Decision problems require estimates of changes in costs and revenues for choosing alternative courses of action. It is therefore necessary to distinguish between fixed and variable costs. Regression analysis can be used to estimate a cost equation, and tests of reliability can be applied to ascertain how reliable the cost equation is in predicting costs. For a description of regression analysis and tests of reliability you should refer to Chapter 24. A common test of reliability is the coefficient of determination, which can be calculated by squaring the correlation coefficient. The coefficient of determination for the cost equation used in the question is 0.64 (0.8 2 ). Consequently, 36% of the variation in cost is not explained by the cost equation used in the question. It is possible that activity bases other than machine hours might provide a better explanation of the relationship between costs and activities. Alternatively, changes in costs might be a function of more than one variable. In such circumstances, cost equations based on multiple regression techniques should provide more reliable cost estimates. The following is a decision tree relating to the question: Question (1) MD s price (1) CN s price 1.2 (1) KL s price P = 0.6 P = 0.2 P = 0.1 Decision tree 1.1 P = 0.4 P = 0.3 (4) (5) Expected Unit sales (millions) contribution (6) Total contribution ( m) (7) Joint probability (2 3) (8) Expected value (6 7) ( m) P = P = 0.4 P = P = P = 0.3 P = P = 0.7 P = P = 1.0 P = 1.0 P = Figure Q12.22 DECISION-MAKING UNDER CONDITIONS OF UNCERTAINTY 101

8 The variable cost per litre is as follows: Direct materials 0.12 Direct wages 0.24 Indirect wages etc. (16 % 0.24) and the range of contributions are: 0.80 for a selling price of for a selling price of for a selling price of 0 The decision tree indicating the possible outcomes presented in Figure Q12.22 shows that the expected value of the contribution is maximized at a selling price of Fixed costs are common and unavoidable to all alternatives, and are therefore not included in the analysis. However, management might prefer the certain contribution of 1.74 million at a selling price of 0. From columns 6 and 7 of the decision tree it can be seen that there is a 0.60 probability that contribution will be in excess of 1.74 million when a selling price of 1.20 is implemented. The final decision depends on management s attitude towards risk. Question (a) Budgeted net Profit/Loss outcomes for year ending 30 June Client Fee per Variable cost Contribution Total contrib. Days Client day per client day per client day per year ` (b) The maximax rule looks for the largest contribution from all outcomes. In this case the decision maker will choose a client fee of 180 per day where there is a possibility of a contribution of The maximin rule looks for the strategy which will maximize the minimum possible contribution. In this case the decision maker will choose client fee of 200 per day where the lowest contribution is This is better than the worst possible outcome from client fees per day of 180 or 220 which will provide contribution of and respectively. The minimax regret rule requires the choice of the strategy which will minimize the maximum regret from making the wrong decision. Regret represents the opportunity lost from making the wrong decision. The calculations in part (a) are used to list the opportunity losses in the following regret matrix: State of nature Low variable Most likely variable High variable cost of 70 cost of 85 cost of 95 Choose a fee of Choose a fee of Choose a fee of DECISION-MAKING UNDER CONDITIONS OF UNCERTAINTY

9 At a variable cost of 70 the maximum contribution is derived from a fee of 180. Therefore there will be no opportunity loss. At a fee of 200 the opportunity loss is ( ) and at the 220 fee the opportunity loss is ( ). The same approach is used to calculate the opportunity losses at variable costs of 85 and 95. The maximum regrets for each fee are as follows: The minimum regret is and adopting a minimum regret strategy will result in choosing the 200 fee per day alternative. (c) The expected value of variable cost = = For each client fee strategy the expected value of budget contribution for the year is calculated as follows: * fee of 180 : ( ) = * fee of 200 : ( ) = * fee of 220 : ( ) = A client fee of 200 per day is required to give the maximum expected value contribution of Note that there is virtually no difference between this and the contribution where a fee of 180 per day is used. (d) Profit can be increased by making cost savings provided that such actions do not result in a fall in demand and a reduction in revenues. Alternatively, investments may be made that will increase the level of service and thus demand. Profits will increase if the extra revenues exceed the increase in costs. The balanced scorecard approach to performance measurement and the determinants of performance measurement relating to service organizations described in Chapter 23 can be used to identify appropriate performance areas for the health centre. The performance areas identified in Exhibit 23.5 in Chapter 23 include quality of service, flexibility, resource utilization and innovation. Each of these areas is discussed below. (i) Quality of service may be improved by upgrading facilities such as a cafeteria, free daily newspapers and better waiting room facilities. This may increase demand and generate additional revenues which exceed the cost increases. (ii) Flexibility of service may be improved by providing additional sports/exercise facilities that are not currently available. In addition, additional exercise and dietary consultants who can provide services that are not currently available. (iii) Resource utilization may be improved by better scheduling relating to the use of the exercise equipment and staff time and extending the opening hours. The aim should be to provide at least the same level of service with fewer resources. (iv) Innovation may take the form of new services such as an extension of the range of health advice that can be provided and introducing on-line booking systems which can be directly accessed by the clients. DECISION-MAKING UNDER CONDITIONS OF UNCERTAINTY 103