Fall 2018, Test 3

General comments:

- you are allowed up to 5 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;

1. Use Lagrange multiplier method to solve the following
constraint optimization problem: find the point of the line
2x_{1} − x_{2} = 1 which is the closest to
0, i.e., in precise terms, minimize the sum
x_{1}^{2} + x_{2}^{2} under the
above constraint.

2-3.

2. Suppose that we have two investments, one with expected return 10 and variance 20, another with expected return 20 and variance 10, and we want to have a return of 13. Assuming that these two investments are independent, use the general formulas that we had in class to find the optimal portfolio.

3. Same as in Problem 2, but this time, the two investments are not independent: the covariance is -0.5. Describe the optimal portfolio for this case.

4-7. Assume that we have ten estimates for the a company's worth:
three estimate of 2 Billion dollars, five estimates of 3 Billions,
and two outliers: an over-pessimistic estimate of 0 Billion, and
an over-optimistic estimate of 10 Billions.
8-10. An investor placed her money into two hedge funds. The first
one led to annual returns of 10%, 5%, 5%, 5%, and 10%. The second
one lead to annual returns of 9%, 0%, 9%, 9%, and 9%.

4. What will be the combined estimate if we use the standard least
squares methods (i.e., l^{2}).

5. What will be the combined estimate if we use the l^{1}
method? Explain in what sense this method is more robust.

6. What is the general class of robust techniques that includes
both l^{2} and l^{1} as particular cases? What is
the justification for using methods from this class?

7. Describe the first few steps of an algorithm for providing the
l^{p}-estimate for p = 1.5. (No need to actually perform
the computations).

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

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

10. What is the justification for the peak-end rule?