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You may follow the methods shown in the mp4 on Decision Analysis for a way to do part (b) of this question if you wish.

(a) Discuss how a utility function can be assessed. What is a standard gamble and how is it used in determining utility values?

(b) Alan Barnes invests primarily in the share market. Recently he has become concerned about the share market as a good investment. During the next year he must decide whether to invest \$10,000 in the share market or in a government bond at an interest rate of 9%.

Alan expects the share market to be good, fair or bad, giving a return of 14%, 8% or 0% respectively on his money.

1.Develop a decision matrix showing the two possible strategies, the three states of the share market and the monetary gains or losses under the six possible action-state scenarios.

Answer the following questions. Each answer must be supported with appropriate calculations and/or a table of figures, and you must state for questions 2 to 5 which alternative would be selected.

1. Which alternative would an optimist choose?
2. Which alternative would a pessimist choose?
3. Which alternative is indicated by the criterion of regret?
4. Assuming probability of a good market = 0.4, a fair market = 0.4 and a bad market = 0.2, using expected monetary values what is the optimum action?
5. What is the expected value of perfect information?

## Assessing the Utility Function

An individual usually enjoys the consumption of certain units of a commodity. This provides them with satisfaction which is known as the utility of the commodity. The utility of the commodity can thus be expressed in the form of a function. The function is designed by describing the decision problem. There are several characteristics associated with the utility function which is important to be identified as well. Various quantitative restrictions can be used to specify the function of utility which will be of help in making decisions by comparison of different strategies or strategies (Garriga, 2014).

One of the common ways with which utility can be assessed is the standard gamble. When the utility of a commodity is 1, that is the best possible outcome of the gamble. Similarly, when the utility of a commodity is 0, that is the worst possible gamble. In between the best and the worst cases, there is also a case in which intermediary outcomes are available. From these intermediary cases, the best and the worst outcomes are considered by the decision makers in order to gamble. Probabilities are associated with the intermediary outcomes and gambles, with the help of which the decision maker makes the decision (Hutchins et al., 2015).

• Investment Amount = \$10,000

The amount is to be invested in Share Market or Government Bond.

Government Bond interest rate = 9%

Share Market interest rate is 14%, 8% and 0% when the market is good, fair and bad respectively.

1. Considering the condition stated above, the decision matrix that can be obtained is given in table 1.

Table 1: Decision Matrix with all possible Strategies

 Strategy Market Condition Good Fair Bad Share Market 11400 10800 10000 Government Bond 19000 19000 19000
1. The optimist approach in choosing the best strategy is to invest in the share market. The results are shown in table 2.
 Table 2: Optimist Approach in Selecting Best Strategy Strategy Market Condition Best Profit Good Fair Bad Share Market 11400 10800 10000 11400 Government Bond 10900 10900 10900 10900
1. The approach of the pessimist in selecting the best strategy is to select the investment in the Government Bonds. The results are shown in table 3.
 Table 3: Pessimist Approach in Selecting the Best Strategy Strategy Market Condition Least Profit Good Fair Bad Share Market 11400 10800 10000 10000 Government Bond 10900 10900 10900 10900
1. Table 4 shows the criterion of regret.
 Table 4: Criterion of Regret in Selecting Strategies Strategy Market Condition Maximum Good Fair Bad Share Market 0 100 900 900 Government Bond 500 0 0 500
1. Probability that the market is good = 0.4

Probability that the market is fair = 0.4

Probability that the market is bad = 0.2

Thus, the expected monetary value is given by the following table 5. It can be seen from the table that the expected monetary value is high in case of investing in the share market.

 Table 5: Expected Monetary Value (EMV) from different Strategies Strategy Market Condition Expected Profit Good Fair Bad Share Market 456 432 200 456 Government Bond 436 436 218 436
1. Expected value given perfect information (

Thus, Expected Value of Perfect Information (EVPI) = EV|PI – EMV = \$(1110 – 456) = \$654.

Returns from Market:

Favorable = \$100,000

Unfavorable = –\$60,000.

Probability of favorable market = 0.5

Probability of unfavorable market = 0.5

• Expected Return = \$((100,000 * 0.5) – (60,000 * 0.5)) = \$20,000.

Thus, it can be seen that the business will earn a profit. Thus, Jim can start the business of producing men’s razor.

• Probability of favorable market depending on the track record of his friend = (0.5 * 0.7) + (0.5 * 0.3) = 0.5. Therefore, the probability of unfavorable market = (1 – 0.5) = 0.5.
• The required posterior probability = (0.3 * 0.5) = 0.15.
• Expected net gain / loss = \$((100,000 * 0.5) + (– 60,000 * 0.5)) – \$5,000 = \$15,000.

Engagement of friend has decreased the expected profit margin but there is still profit from the business. Engaging the friend is thus a better choice as his engagement will provide the market conditions more accurately.

• The formulas used in the construction of the simulation is shown with the help of the following excel sheet given in figure 1. shows the results of the simulation from the formulas listed in figure 1.
• Over the 12-month period, Tully tyres have a monthly average profit of \$21,584.16.
• By increasing the selling price from \$200 to \$220 and increasing the profit margin from 22% to 32%, the sales is unaffected, and the average profit has been obtained as \$28,811.12. The revised simulation results are given in figure 3.

According to the initial information provided, it has been observed that the average profit is around \$21,000. But with the increase in the selling price by \$40 and increasing the profit margin by 2%, it is known that there will be no effect in the sales of the product. From the increased credentials, the results have shown that there is an increase in the average profit, which is around \$28,000 now. Thus, the increased rates must be used for the business.

Thanks and Regards

Name and Signature

• The table 6 shows the necessary calculations to estimate the overhead cost using high low method.

Table 6

 Machine Hours (x) Overhead Cost (y) Highest Activity 3,800 \$48,000 Lowest Activity 1,800 \$46,000

Marginal Cost = \$(48,000 – 46,000) / (3800 – 1800) = \$1

Fixed Cost = \$(48,000 – 1 * 3800) = \$(46,000 – 1 * 1800) = \$44,200

Equation to estimate overhead cost:

Overhead Cost (y)= 44,200 + Machine Hours (x)

When, machine hours = 3000, overhead cost = \$(44,200 + 3,000) = \$47,200

• Table 7 gives the regression results to estimate the overhead cost with the help of machine hours.
 Table 7: Overhead Cost against Machine Hours Regression Statistics Multiple R 0.104 R Square 0.011 Adjusted R Square -0.113 Standard Error 15447.614 Observations 10 ANOVA df SS MS F Significance F Regression 1 2.1E+07 2.1E+07 0.088 0.774 Residual 8 1.9E+09 2.4E+08 Total 9 1.9E+09 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 59198.785 21473.783 2.757 0.025 9680.152 108717.417 MH -2.304 7.774 -0.296 0.774 -20.230 15.621

The model developed is insignificant as the significance F is more than the level of significance (0.05). Thus, Machine hours is not a good predictor of overhead cost.

Table 8 gives the regression results to estimate the overhead cost with the help of batches.

 Table 8: Overhead Costs against Batches Regression Statistics Multiple R 0.91 R Square 0.83 Adjusted R Square 0.81 Standard Error 6379.22 Observations 10 ANOVA df SS MS F Significance F Regression 1 1.06E+09 1.06E+09 39.427 0.000 Residual 8 3.26E+08 4.07E+07 Total 9 1.93E+09 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 6555.556 7666.868 0.855 0.417 -11124.274 24235.385 Batches 234.568 37.357 6.279 0.000 148.422 320.714

The model developed is significant as the significance F is less than the level of significance (0.05). Thus, batches is a good predictor of overhead cost. Moreover, batches can predict 83% accurately the variations in the overhead cost.

Overhead Cost = 6555.556 + 234.568 * Batches

Table 9 gives the regression results to estimate the overhead cost with the help of batches and machine hours.

 Table 9: Overhead Costs against Machine hours and Batches Regression Statistics Multiple R 0.913 R Square 0.833 Adjusted R Square 0.785 Standard Error 6783.922 Observations 10 ANOVA df SS MS F Significance F Regression 2 1.6E+09 8.0E+08 17.468 0.002 Residual 7 3.2E+08 4.6E+07 Total 9 1.9E+09 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 9205.658 12704.918 0.725 0.492 -20836.700 39248.016 MH -0.931 3.422 -0.272 0.793 -9.022 7.161 Batches 233.827 39.820 5.872 0.001 139.667 327.987

The model developed is significant as the significance F is less than the level of significance (0.05). Moreover, the model can predict 83% accurately the variations in the overhead cost.

Overhead Cost = 9205.658 + 233.827 * Batches – 0.931 * MH

• Model developed in table 8 and 9 has shown the same value of R Square but the adjusted R Square is higher in the model in table 8. Thus, the better will be considered as the model which involves only batches to predict overhead cost.
• Overhead Cost = 6555.556 + 234.568 * 150 = \$41,740.74
• Contribution margin for Product A = \$(12 – 8) = \$4

Contribution margin for Product B = \$(15 – 10) = \$5

• For Product B, break even = (5000 / 5) = 1000 units
• For Product A, break even = (5000 / 4) = 1250 units
• Ratio of manufacture of A and B is 3:1.
• Profit of \$3,500 will be earned if 1333 units of product A and 667 units of product B are sold. The necessary calculations are shown in table 10.

Table 10

 Product A B Total Sales Unit 3 1 4 Sales price \$36 \$15 \$51 Variable cost \$24 \$10 \$34 Total fixed costs \$5,000 Total Contribution \$12 \$5 \$17 Weighted average Contribution \$4 Targeted Profit (Before Tax) \$3,500 Targeted Sales Volume 1333 667 2000
• Profit of \$8,400 will be earned after tax, if 855 units of product A and 428 units of product B are sold. The necessary calculations are shown in table 11.

Table 11

 Product A B Total Sales Unit 3 1 4 Sales price \$72 \$15 \$87 Variable cost \$24 \$10 \$34 Total fixed costs \$5,000 Total Contribution \$48 \$5 \$53 Weighted average Contribution \$13 Targeted Profit (After Tax) \$8,400 Tax Rate 30% Targeted Profit (Before Tax) 12000 Targeted Sales Volume 855 428 1283

References

Garriga, E. (2014). Beyond stakeholder utility function: Stakeholder capability in the value creation process. Journal of Business Ethics, 120(4), 489-507.

Hutchins, R., Viera, A. J., Sheridan, S. L., & Pignone, M. P. (2015). Quantifying the utility of taking pills for cardiovascular prevention. Circulation: Cardiovascular Quality and Outcomes, 8(2), 155-163.

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