## Advanced Computational Methods in Economics and Finance Homeworks for the course CS 5354/CS 4365, Fall 2018

General comment: Unless explicitly specified, each homework should be turned in on a sheet of paper. Place it on the instructor's desk in the classroom by the beginning of the class on the day when the homework is due.

1. (Due September 5) With what accuracy can we determine the utility if we only ask 4 questions? Explain your answer.

2. (Due September 5) How many questions do you need to ask a person to determine his/her utility with accuracy 5%? Explain your answer.

3. (Due September 5) Suppose that an alternative A has utility u(A) = 0.6 with respect to the original pair (A, A+), and that with respect to a new pair (A', A'+), we have u'(A) = 0.3 and u'(A+) = 0.8. What is the utility u'(A) of the alternative A with respect to the new pair? Explain your answer.

4. (Due September 17) Suppose that:

• for alternative A, the utility is from 0.4 to 0.6,
• for alternative B, it is from 0.3 and 0.7, and
• for alternative C, it is from 0.0 to 0.8.
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?

5. (Due September 26) Which of the two actions is better: A or B? For action A, there are two possible outcomes: a1 with probability 0.4 and utility 5, and a2 with probability 0.6 and utility −2. For action B, there are also two possible outcomes: b1 with probability 0.3 and utility 4, and b2 with probability 0.7 and utility −1.

6. (Due September 26) To determine the subjective probability of snow in Alaska tomorrow, we asked an expert whether she prefers \$10 with probability 0.6 or \$10 if it snows tomorrow. She selected \$10 with probability 0.6. Based on this answer, what can we conclude about the expert's subjective probability of snowing tomorrow? To be more precise, based on this answer, what is the interval of possible values of the corresponding subjective probability?

7. (Due September 26) Suppose that a decision maker follows McFadden's formula with β = ln(2). If we have three alternatives, with utilities 0, 1, and 4, with what probability will the expert select each of these three alternatives?

8. (Due October 1) In situation as described in Problem 7, 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 second powers?

9. (Due October 1) 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?

10. (Due October 1) 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).

11. (Due October 1) By using the fact that utility is proportional to the square root of money, explain why most people prefer \$500 cash to a lottery in which they get \$1,000 with probability 0.5.

12. (Due October 3) 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.

13. (Due October 3) Let us assume that the trade volume between the two countries is described by the gravity model, with α = 1.

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

14. (Due October 3) Use the Least Squares method to find the values a and b for which a * xi + b ~ yi, based on the following observations:

• x1 = 0, y1 = 0;
• x2 = 1, y2 = 0;
• x3 = 2, y3 = 2.

15. (Due October 15) Write a program that use the Least Squares method to find coefficients a and b of linear regression y = a * x + b based on several pairs (xi, yi). Please turn in a printout of the program, and the printout of the test case -- that you checked by hand -- showing that your program is correct. For extra credit: make your program capable of finding a linear dependence on several variables.

16. (Due October 24) Use Lagrange multiplier method to solve the following constraint optimization problem: find the point of the line x1 − x2 = 1 which is the closest to 0, i.e., in precise terms, minimize the sum x12 + x22 under the above constraint.

16a. (Due November 5; the grade for this h/w will replace the grade for h/w 16 if that grade was not perfect; no need to do this problem if you already got 10/10 on h/w 16) Use Lagrange multiplier method to solve the following constraint optimization problem: minimize the sum 2x12 − x22 + 2x2 under the constraint x1 + x2 = 2.

17. (Due November 5) Suppose that we have two investments, one with expected return 1 and variance 2, another with expected return 2 and variance 1, and we want to have a return of 1.5. Assuming that these two investments are independent, use the general formulas that we had in class to find the optimal portfolio.

18. (Due November 5) Same as in Problem 17, but this time, the two investments are not independent: the covariance is 0.5. Describe the optimal portfolio for this case.

19. (Due November 7) Assume that we have ten estimates for the annual economic growth: four estimate of 2%, four estimates of 3%, and two outliers: an over-pessimistic estimate of −50%, and an over-optimistic estimate of 10%.

• What will be the combined estimate if we use the standard least squares methods (i.e., l2).
• What will be the combined estimate if we use the l1 method? Explain in what sense this method is more robust.
• 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?

20. (Due November 12) Use the algorithm that we learned in class (and which is also described here) to write a program that use the robust lp method to find coefficients a and b of linear regression y = a * x + b based on several pairs (xi, yi). Allow the user to input the value α describing degree of robustness and the value ε > 0 describing the desired accuracy. Please turn in a printout of the program, and the printout of the test case -- that you checked by hand -- showing that your program is correct. For extra credit: make your program capable of finding a linear dependence on several variables.

21. (Due November 12) An investor placed her money into two hedge funds. The first one led to annual returns of 10%, 1%, 1%, 3%, and 9%. The second one lead to annual returns of 8%, 0%, 8%, 8%, and 8%.

• Which of the two investments leads to better end results?
• Which of the two investments will the investor prefer if she follows the peak-end rule?
• What is the justification for the peak-end rule?

22. (Due November 14) Suppose that we have three estimates for the same quantity:

• x1 = 1 with accuracy σ1 = 0.1,
• x2 = 2 with accuracy σ2 = 0.2, and
• x3 = 3 with accuracy σ3 = 0.3.
If we use the weighted least squares approach, what will be the combined estimate? Comment: see this file for details.

23. (Due November 14) 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 2 of the random variable located on the interval [0,3]. What is the justification for using the generalized maximum entropy method, of maximizing the integral of ρp? 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 1.

24. (Due November 26) Write a program that use the weighted Least Squares method to find coefficients a and b of linear regression y = a * x + b based on several triples (xi, yi, σi), where σi is the accuracy with which we measure yi. Reminder: the algorithm is the same as the auxiliary algorithm for the lp-method (see Problem 20), but with wi = (σi)−2. Please turn in a printout of the program, and the printout of the test case -- that you checked by hand -- showing that your program is correct. For extra credit: make your program capable of finding a linear dependence on several variables.

25. (Due November 28) What if the usual scale- and/or shift-invariant approximations are not accurate enough, and we try multi-D approximations to get a better accuracy? Explain what family of functions we need to use:

• if we require shift-invariance,
• if we require scale-invariance, and
• if we require both shift- and scale-invariance.