A person wants to invest in one of three alternative investment plans: stocks, bonds or a savings accountIt is assumed that the person wishes to invest all of the funds in one plan. The conditional Pay-off of theinvestments are based on three potential economic conditions: accelerated, normal or slow growth. ThePay-off matrix is given below:Alternative InvestmentEconomic ConditionsAccelerated GrowthNormal GrowthSlow Growth10,6008,0006,5006,0005,0004,6001,0005,000StockBondsSaving5,000Determine the best investment plan using each of the following criteria:(i) Laplace(ii) Maximin(iii) Miximax(iv) Hurwicz with coefficient of optimism a = 0.6.

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Answered: A person wants to invest in one of…

Practical Management Science

6th Edition

ISBN:9781337406659

Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter7: Nonlinear Optimization Models

Section: Chapter Questions

Problem 63P

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Suppose you begin year 1 with 5000. At the beginning of each year, you put half of your money under a mattress and invest the other half in Whitewater stock. During each year, there is a 40% chance that the Whitewater stock will double, and there is a 60% chance that you will lose half of your investment. To illustrate, if the stock doubles during the first year, you will have 3750 under the mattress and 3750 invested in Whitewater during year 2. You want to estimate your annual return over a 30-year period. If you end with F dollars, your annual return is (F/5000)1/30 1. For example, if you end with 100,000, your annual return is 201/30 1 = 0.105, or 10.5%. Run 1000 replications of an appropriate simulation. Based on the results, you can be 95% certain that your annual return will be between which two values?
A European put option allows an investor to sell a share of stock at the exercise price on the exercise data. For example, if the exercise price is 48, and the stock price is 45 on the exercise date, the investor can sell the stock for 48 and then immediately buy it back (that is, cover his position) for 45, making 3 profit. But if the stock price on the exercise date is greater than the exercise price, the option is worthless at that date. So for a put, the investor is hoping that the price of the stock decreases. Using the same parameters as in Example 11.7, find a fair price for a European put option. (Note: As discussed in the text, an actual put option is usually for 100 shares.)
In August of the current year, a car dealer is trying to determine how many cars of the next model year to order. Each car ordered in August costs 20,000. The demand for the dealers next year models has the probability distribution shown in the file P10_12.xlsx. Each car sells for 25,000. If demand for next years cars exceeds the number of cars ordered in August, the dealer must reorder at a cost of 22,000 per car. Excess cars can be disposed of at 17,000 per car. Use simulation to determine how many cars to order in August. For your optimal order quantity, find a 95% confidence interval for the expected profit.
Based on Grossman and Hart (1983). A salesperson for Fuller Brush has three options: (1) quit, (2) put forth a low level of effort, or (3) put forth a high level of effort. Suppose for simplicity that each salesperson will sell 0, 5000, or 50,000 worth of brushes. The probability of each sales amount depends on the effort level as described in the file P07_71.xlsx. If a salesperson is paid w dollars, he or she regards this as a benefit of w1/2 units. In addition, low effort costs the salesperson 0 benefit units, whereas high effort costs 50 benefit units. If a salesperson were to quit Fuller and work elsewhere, he or she could earn a benefit of 20 units. Fuller wants all salespeople to put forth a high level of effort. The question is how to minimize the cost of encouraging them to do so. The company cannot observe the level of effort put forth by a salesperson, but it can observe the size of his or her sales. Thus, the wage paid to the salesperson is completely determined by the size of the sale. This means that Fuller must determine w0, the wage paid for sales of 0; w5000, the wage paid for sales of 5000; and w50,000, the wage paid for sales of 50,000. These wages must be set so that the salespeople value the expected benefit from high effort more than quitting and more than low effort. Determine how to minimize the expected cost of ensuring that all salespeople put forth high effort. (This problem is an example of agency theory.)
An automobile manufacturer is considering whether to introduce a new model called the Racer. The profitability of the Racer depends on the following factors: The fixed cost of developing the Racer is triangularly distributed with parameters 3, 4, and 5, all in billions. Year 1 sales are normally distributed with mean 200,000 and standard deviation 50,000. Year 2 sales are normally distributed with mean equal to actual year 1 sales and standard deviation 50,000. Year 3 sales are normally distributed with mean equal to actual year 2 sales and standard deviation 50,000. The selling price in year 1 is 25,000. The year 2 selling price will be 1.05[year 1 price + 50 (% diff1)] where % diff1 is the number of percentage points by which actual year 1 sales differ from expected year 1 sales. The 1.05 factor accounts for inflation. For example, if the year 1 sales figure is 180,000, which is 10 percentage points below the expected year 1 sales, then the year 2 price will be 1.05[25,000 + 50( 10)] = 25,725. Similarly, the year 3 price will be 1.05[year 2 price + 50(% diff2)] where % diff2 is the percentage by which actual year 2 sales differ from expected year 2 sales. The variable cost in year 1 is triangularly distributed with parameters 10,000, 12,000, and 15,000, and it is assumed to increase by 5% each year. Your goal is to estimate the NPV of the new car during its first three years. Assume that the company is able to produce exactly as many cars as it can sell. Also, assume that cash flows are discounted at 10%. Simulate 1000 trials to estimate the mean and standard deviation of the NPV for the first three years of sales. Also, determine an interval such that you are 95% certain that the NPV of the Racer during its first three years of operation will be within this interval.
Amanda has 30 years to save for her retirement. At the beginning of each year, she puts 5000 into her retirement account. At any point in time, all of Amandas retirement funds are tied up in the stock market. Suppose the annual return on stocks follows a normal distribution with mean 12% and standard deviation 25%. What is the probability that at the end of 30 years, Amanda will have reached her goal of having 1,000,000 for retirement? Assume that if Amanda reaches her goal before 30 years, she will stop investing. (Hint: Each year you should keep track of Amandas beginning cash positionfor year 1, this is 5000and Amandas ending cash position. Of course, Amandas ending cash position for a given year is a function of her beginning cash position and the return on stocks for that year. To estimate the probability that Amanda meets her goal, use an IF statement that returns 1 if she meets her goal and 0 otherwise.)
In Problem 11 from the previous section, we stated that the damage amount is normally distributed. Suppose instead that the damage amount is triangularly distributed with parameters 500, 1500, and 7000. That is, the damage in an accident can be as low as 500 or as high as 7000, the most likely value is 1500, and there is definite skewness to the right. (It turns out, as you can verify in @RISK, that the mean of this distribution is 3000, the same as in Problem 11.) Use @RISK to simulate the amount you pay for damage. Run 5000 iterations. Then answer the following questions. In each case, explain how the indicated event would occur. a. What is the probability that you pay a positive amount but less than 750? b. What is the probability that you pay more than 600? c. What is the probability that you pay exactly 1000 (the deductible)?
A common decision is whether a company should buy equipment and produce a product in house or outsource production to another company. If sales volume is high enough, then by producing in house, the savings on unit costs will cover the fixed cost of the equipment. Suppose a company must make such a decision for a four-year time horizon, given the following data. Use simulation to estimate the probability that producing in house is better than outsourcing. If the company outsources production, it will have to purchase the product from the manufacturer for 25 per unit. This unit cost will remain constant for the next four years. The company will sell the product for 42 per unit. This price will remain constant for the next four years. If the company produces the product in house, it must buy a 500,000 machine that is depreciated on a straight-line basis over four years, and its cost of production will be 9 per unit. This unit cost will remain constant for the next four years. The demand in year 1 has a worst case of 10,000 units, a most likely case of 14,000 units, and a best case of 16,000 units. The average annual growth in demand for years 2-4 has a worst case of 7%, a most likely case of 15%, and a best case of 20%. Whatever this annual growth is, it will be the same in each of the years. The tax rate is 35%. Cash flows are discounted at 8% per year.
If a monopolist produces q units, she can charge 400 4q dollars per unit. The variable cost is 60 per unit. a. How can the monopolist maximize her profit? b. If the monopolist must pay a sales tax of 5% of the selling price per unit, will she increase or decrease production (relative to the situation with no sales tax)? c. Continuing part b, use SolverTable to see how a change in the sales tax affects the optimal solution. Let the sales tax vary from 0% to 8% in increments of 0.5%.
An investor has a certain amount of money available to invest now. Three alternative investmentsare available. The estimated profit in Kwacha of each investment under each economic conditionare indicated in the following payoff table:Event Investment SelectionA B CEconomy declines 500 -2000 -7000No charge 1000 2000 -1000Economy Expand 2000 5000 20,000Based on his own past experience, the investor assigns the following probabilities to eacheconomic condition:( )( )( )Economy declines 0.30No change 0.50Economy expands 0.20PPP===i. Determine the optimal action based on the maximax criterion ii. Determine the optimal action based on the maximin criterion iii. Compute the expected monetary value (EMV) for each investment iv. Compute the expected opportunity loss (EOL) for each investment v. Explain the meaning of the expected value of perfect information (EVPI) in thisproblem vi. Based on the results of (iii) and (iv), which investment would you choose?vii. Compute the coefficient of…
An investor has a certain amount of money available to invest now. Three alternative investments are available. The estimated profit in Kwacha of each investment under each economic condition are indicated in the following payoff table: Event Investment Selection A B C Economy declines 500 -2000 -7000 No charge 1000 2000 -1000 Economy Expand 2000 5000 20,000 Based on his own past experience, the investor assigns the following probabilities to each economic condition: ( ) ( ) ( ) Economy declines 0.30 No change 0.50 Economy expands 0.20PPP=== iv. Compute the expected opportunity loss (EOL) for each investment v. Explain the meaning of the expected value of perfect information (EVPI) in this problem vi. Based on the results of (iii) and (iv), which investment would you choose? Why?
An investor has a certain amount of money available to invest now. Three alternative investments are available. The estimated profit in Kwacha of each investment under each economic condition are indicated in the following payoff table: Event Investment Selection A B C Economy declines 500 -2000 -7000 No charge 1000 2000 -1000 Economy Expand 2000 5000 20,000 Based on his own past experience, the investor assigns the following probabilities to each economic condition: ( ) ( ) ( ) Economy declines 0.30 No change 0.50 Economy expands 0.20PPP=== i. Determine the optimal action based on the maximax criterion ii. Determine the optimal action based on the maximin criterion iii. Compute the expected monetary value (EMV) for each investment

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