Question 1.

A bank manager considers an investment strategy. She has three options: a stock for a big company, a bond and a stock for a start-up company, whose stock returns are denoted by SB, BB, and SS, respectively. It is known that SB and BB follow normal distributions: SB~N(8%,10%) and BB~N(2%,1%) and SB and BB have a correlation of -0.5. SS are independent of both SB and BB, and has discrete distribution:

1. Using @Risk or equivalent software, simulate returns of SB, BB and SS and fill out the following summary statistics of simulated data. Use percentage returns up to 2 decimal points (for instance, 3.99%).

2. The bank manager asks you which investment you recommend among the following four strategies.

a. For each investment strategy, report a histogram of simulated returns and also report a table including summary measures (Minimum, Maximum, Mean, 90% CI, Mode, Median, Std Dev).

b. Among four strategies, which one is the best and the worst strategy in terms of average return?

c. Which one is the safest strategy in terms of standard deviations?

d. Under some bank regulation, the bank manager maintains the Value-at-Risk 5% of the portfolio return being -2.5% or above. Find a portfolio that satisfies the regulation and achieve an average return of 4% or above. Report your portfolio and simulation outcomes (histogram, mean and 5% percentile). 

Questions 2.

Suppose that you were wondering whether to open a café. There are two choices: Strategy #1 is not to open (Not IN) and Strategy #2 is to open (IN). The table below shows unit price and cost per customer and a fixed cost per day in dollars (note: you have to pay the fixed cost, such as a rent every day regardless of the number of customers.)

The number of customers varies according to weather condition and you consider the following probability table.

If you do not open a café, then you can invest your asset into a fixed income security, which generates a return of $380 per day.

1. Fill out a table below regardingrisk profiles for profitsaccording to strategies and weather conditions. [Hint: profit is given by # customers * (unit price - unit cost) - fixed cost.]

2. Obtain Expected Monetary Value (EMV) for two strategies.

3. Draw a decision tree by using PrecisionTree software. [Hint: your tree has four end nodes.]

4. Conduct a one-way sensitivity analysis by varying the probability of being sunny. Use base-value +/-25% with 11 steps. [Hint: your original file has to set up the probability of rain as a formula of (1 - the probability of being sunny), rather than a value of 0.25.] Report a sensitivity graph and a strategy region graph (EMV and variations in the probability) and discuss which strategy is the best in terms of EMV (30 words or less). 

5. Conduct a two-way sensitivity analysis by varying the probability of being sunny (0.75) and the fixed investment return per day ($380). Use a base value +/-25% with 11 steps for both variables. [Hint: To refer a fixed return ($380) for the sensitivity analysis, you can choose only one cell. Thus, before doing this analysis, you should link another cell to the cell that you will use for this analysis. In other words, if you put a value of $380 in two cells, the software does not recognize that they are the fixed cost. Please wait for Week 10 lecture.]Report a figure of strategy region.Also, discuss which strategy is the best in term of EMV when the probability of being sunny is 60% (50 words or less).

Questions 3.

This question asks you to generate X-bar charts, using a file, X-bar-chart-data.xlsx. The file contains 50 averages of subsamples following N(2,3) and the ones following N(2,6). The subsample size is 5 (n=5). 

1. Report a X-bar chart using averages of subsample generated by N(2,3). Here, assume that the standard deviation (σ = 3) is known and use 2-sigma deviation. You report only the chart but you may follow steps below:

Step 1: obtain the average of 50 averages of subsample from N(2,3).

Step 2: Obtain the UCL and LCL with the sample size n =5 and 2-σ deviation.

Step 3: plot 50 subsamples from N(2,3) with values obtained in Step 1-2.

2. Report a X-bar chart, that has averages of subsample generated by N(2,6) with the UCL and LCL obtained in Step 1-2 above.

3. Can the X-bar chart detect a change of distribution in 2?Explain (no more than 50 words).