## CS 5354/CS 4365 Advanced Computational Methods in Economics and Finance Fall 2018, Final Exam

Name: ____________________________

• you are allowed up to 10 pages of handwritten notes;
• if you need extra pages, place your name on each extra page;
• the main goal of most questions is to show that you know the corresponding algorithms; so, if you are running of time, just follow the few first steps of the corresponding algorithm;
Good luck!

1. Traditional utility-based approach to decision making.

1a. What is utility? Give a precise definition.

1b. How do you select the very bad alternative A and the very good alternative A+ that are needed to define utility?

1c. Once you have selected the very bad and the very good alternatives, how do we find the utility of a given alternative A? Explain, in detail.

1d. Which of the two actions is better: B or C? For action B, there are two possible outcomes: a1 with probability 0.3 and utility 100, and a2 with probability 0.7 and utility −20. For action C, there are also two possible outcomes: b1 with probability 0.4 and utility 40, and b2 with probability 0.6 and utility −10.

2. How utility depends on money

2a. By using the fact that utility is proportional to the square root of money, explain why most people prefer \$70 cash to a lottery in which they get \$100 with probability 0.7.

2b. Give an example explaining why the simplified assumption -- that utility is proportional to money amount -- does not lead to a reasonable behavior.

2c. What properties were used to justify the power-law dependence of utility on money?

3. Optimal portfolio: Markowitz theory. Suppose that we have two investments, one with expected return 100 and variance 200, another with expected return 200 and variance 100, and we want to have a return of 140. Assuming that these two investments are independent, find the optimal portfolio.

4. Decision making under interval uncertainty. Suppose that:
• for alternative A, the utility is between 5 and 7,
• for alternative B, it is between 4 and 8, and
• for alternative C, it is between 1 and 9.

4a. Which of these three alternatives will be selected:

• by a perfect optimist, for whom the Hurwicz optimism-pessimism parameter is αH = 1?
• by a perfect pessimist, for whom the Hurwicz optimism-pessimism parameter is αH = 0?
• by a realist, for whom the Hurwicz optimism-pessimism parameter is αH = 0.5?

4b. Explain, in detail, how we can deduce the Hurwicz optimism-pessimism formula from the requirement that decision making under interval uncertainty should not change if we change the starting point (shift-invariance) or if we change the measuring unit (scale-invariance).

5. Probabilistic decision making

5a. Suppose that a decision maker follows McFadden's formula with β = ln(5). If we have three alternatives, with utilities 0, 1, and 2, with what probability will the expert select each of these three alternatives?

5b. What property was used to justify McFadden's formula?

5c. If we have three alternatives, with utilities 0, 1, and 2, with what probability will the expert select each of the three alternatives is this expert uses a power-function modification of McFadden's formula: with the first powers? with the third powers?

5d. What property was used to justify the power-function modification of McFadden's formula?

6. Dealing with outliers: robust techniques. Assume that we have ten estimates for next year's oil prices: three estimate of \$60 per barrel, five estimates of \$50, and two outliers: an over-pessimistic estimate of \$10, and an over-optimistic estimate of \$200.

6a. What will be the combined estimate if we use the standard least squares methods (i.e., l2).

6b. What will be the combined estimate if we use the l1 method? Explain in what sense this method is more robust.

6c. What is the general class of robust techniques that includes both l2 and l1 as particular cases? What is the justification for using methods from this class?

7. Group decision making

7a. Suppose that a group of two people needs to select between two alternatives. For the first alternative, their utilities are 5 and 10; for the second alternative, their utilities are 7 and 8. Which alternative should they select if they use Nash's bargaining solution?

7b. What property was used to justify Nash's bargaining solution?

7c. Give an example that Nash's bargaining solution provides a more fair description of the country's economic situation than the Gross Domestic Product (GDP).

8. How people actually make decisions: peak-end rule.. An investor placed her money into two hedge funds. The first one led to annual returns of 8%, 1%, 1%, 1%, and 8%. The second one lead to annual returns of 7%, 0%, 7%, 7%, and 7%.

8a. Which of the two investments leads to better end results?

8b. Which of the two investments will the investor prefer if he/she follows the peak-end rule?

8c. What is the justification for the peak-end rule?

9a. Let us assume that the trade volume between the two countries is described by the gravity model, with α = 3.

• How will the trade volume change if the GDP of the first country increases by 20%?
• How will the trade volume change if the GDP of the second country increases by 30%?
• How will the trade volume be different if countries with the same GDP were located at a twice larger distance than the given pair?

9b. What properties were used to justify the gravity model?

10. Possible invariances and resulting formulas. Provide the detailed proofs of the three main mathematical results that we had:
• that every shift-invariant function F(x), i.e., a function for which F(x + c) = f(c) * F(x) for some f(c), has the form F(x) = const * exp(β * x);
• that every scale-invariant function F(x), i.e., a function for which F(k * x) = f(k) * F(x) for some f(k), has the form F(x) = const * xα;
• that every additive function, i.e., a function for which F(x + y) = F(x) + F(y), has the form F(x) = const * x.

11. Least Squares (For extra credit) Use the Least Squares method to find the values a and b for which a * xi + b ~ yi, based on the following observations:
• x1 = −1, y1 = 1;
• x2 = 0, y2 = 1;
• x3 = 1, y3 = −1.
12. Generalized Maximum Entropy (For extra credit)

12a. Use the generalized maximum entropy method -- where we maximize the integral of ρ2 -- to find the probability density function in a situation when the only thing we know is the mean value 1 of the random variable located on the interval [0,3].

12b. What is the justification for using the generalized maximum entropy method, of maximizing the integral of ρp?

12c. Explain how we derive the traditional maximum entropy method -- of maximizing the integral of −ρ*ln(ρ) -- as a limit case of the generalized maximum entropy method when p tends to 0.