Part 1. True / False Questions

(50pts/5pts each)

Read each problem carefully and circle your final answer. No partial credit will be given for the problems in Part 1.

1. [ True / False ] Assume that there are two risky assets, P and Q, and a risk-free asset F in an economy as shown below. In this economy, short sales of the risky assets and the risk-free asset are not allowed. In such an economy, the portfolio R as in the figure below can be constructed.

P R

Q

The curve represents the investment opportunity set created by P and Q when short sales are allowed.

F

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2. [ True / False ] The graph below shows the indifference curves for investor A and B. The graph reveals that the investor A would choose P over Q while the investor B would choose Q over P.

Investor B

Q P

3. [ True / False ] When all individual asset returns follow normal distributions, the returns of any portfolio of these individual assets would also follow a normal distribution.

4. [ True / False ] In the CAPM, the beta of a risky asset represents the sensitivity of the return of the risky asset to the market portfolio return.

5. [ True / False ] In the Markowitz Portfolio Selection Model, all the portfolios optimally chosen by investors would have the same Sharpe ratio.

6. [ True / False ] Under the CAPM, the beta of the market portfolio is zero. 7. [ True / False ] Any asset without the default risk can be a risk-free asset.

()

Investor A

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8. [ True / False ] Assume that an investor's utility function is defined as follows

1 =() 2

2

where A represents investor's risk aversion. The investor would strictly prefer an investment opportunity X to Y if and only if

11

() = () 2 > () 2 = ()

9. [ True / False ] Is the following situation possible when the CAPM holds in the financial market? Mark "True" if possible, and mark "False" if not possible:

Portfolio Risk-free Market

Expected Return 5

10

Standard Deviation 0

20

22

A 15 30

10. [ True / False ] The distribution below is negatively skewed and it means we would have more extreme positive outcomes under the distribution.

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Part 2. Multiple Choice & Short Answer Questions

(70pts)

To earn the full points for each of the following problems, show me ALL YOUR WORK. Even if your choice of the answer is correct, you may not get the full points without any supporting work in the exam paper. Also, some partial credit may be given for a problem with wrong answer depending on your work on the exam paper.

1. [20pts] Suppose that the linear regression model of the following specification has been estimated for a fund titled "PanAgora":

= + +

where is the excess return of the fund (=PanAgora) over the risk-free rate and is the

excess return of the market portfolio (S&P500 index fund) over the risk-free rate. The samples used are of monthly frequency, and the sample period is from 2011 to 2020. The regression results are shown in the Appendix. Assume that the market risk premium (() ) and the risk-free rate () are 0.0063 and 0.0002, respectively.

Throughout the problem 1, provide the answers rounded to the nearest ten-thousandth (i.e. ).

(1-a) [2pts] What is the estimated alpha for the PanAgora fund?

(1-b) [3pts] Is the estimated alpha for the PanAgora fund statistically significant when the level of significance is set at 5%? How did you determine its statistical significance?

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(1-c) [5pts] When S&P500 index goes up 10% in a month, is the return of the PanAgora fund expected to be higher or lower than 10% for the month? Briefly explain.

(1-d) [5pts] What is the expected return of the PanAgora fund? Recall the following

() = + ()

(1-e) [5pts] Draw the Security Market Line (SML) and mark the PanAgora fund. Also, show the alpha and beta of the PanAgora fund.

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2. [30pts] There are two risky asset, A and B, and one risk-free asset in the market. The risk- free rate is 2% and the risky asset A has the following return distribution:

Bear Market Probability 0.4

Risky Asset Return - 4%

Bull Market 0.6

+ 11%

Throughout the problem 2, provide the answers rounded to the nearest ten-thousandth (i.e. ).

(2-a) [2pts] What is the expected return of the risky asset A?

(2-b) [3pts] What is the standard deviation of returns on the risky asset A?

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(2-c) [5pts] The risky asset B has the following statistical properties: () = 0.20 and = 0.40. Correlation between the returns on the two risky assets is 0. Draw the investment opportunity set that can be created by the risky assets A and B in the expected return - standard deviation space below.

(2-d) [5pts] Write down the optimization problem to find the Optimal Risky Portfolio (ORP).

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(2-e) [5pts] The Optimal Risky Portfolio (ORP) constructed from the risky asset A and B has the following statistical properties:

() = 0.0753 and = 0.0910

Draw the Capital Allocation Line made from the risk-free asset and the ORP in the graph in (2-c). Also, mark the risk-free asset and the ORP in the same graph. Calculate the slope of the Capital Allocation Line made from the risk-free asset and the ORP.

(2-f) [5pts] An investor has the following utility function with A = 10:

1 =() 2

2

What would be the best portfolio choice for this investor when there are two risky asset, A and B, and the risk-free asset? Answer this by calculating mean return of the best portfolio.

(2-g) [5pts] Mark the best portfolio chosen in (2-f) in the graph in (2-c). Show this with the Capital Allocation Line and the indifference curve of the investor.

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3. [20pts] This problem is an estimation and hypothesis testing exercise using the Law of Large Numbers (LLN) and Central Limit Theorem (CLT). As we discussed in class, the LLN and CLT normally requires a large enough sample for a reasonably precise estimate. However, you are supposed to use a small dataset just for computational convenience here. Assume that the conditions for the LLN and the CLT are satisfied.

Provide all your answers rounded to the nearest ten-thousandth (i.e., )

Annual Returns on BioStar (BS):

Year Return 2017 - 0.08 2018 - 0.05 2019 + 0.10 2020 + 0.15

(3-a) [3pts] To apply the LLN and CLT for estimation, a set of regularity conditions is required for the samples. Briefly explain these conditions. Also, briefly explain pros and cons for adopting large sample (i.e., What is the benefit of having large sample? And, what is a major concern when you try to adopt large sample?)

(3-b) [2pts] Estimate the mean return of the BS.

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(3-c) [3pts] Estimate the variance of the returns on the BS.

(3-d) [4pts] Show 95% confidence interval for the true expected return () of BS. Even though we have very small sample, assume that the t-stat follows not a Student t-distribution but a normal distribution.

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(3-e) [4pts] Test the following hypothesis at the significance level of 5%.

0: = 0

(3-f) [4pts] Consider another stock: ITStar (ITS). Based on our discussion of the LLN in class, suggest a way to estimate covariance between BS and ITS. Judging only from looking at the return data, would the covariance be positive or negative? Why?

Annual Returns on ITStar (ITS):

Year 2017 2018 2019 2020

Return

+ 0.12 + 0.08 - 0.05 - 0.20

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Appendix: PanAgora Fund

SUMMARY OUTPUT

Regression Statistics Multiple R

R Square

Adjusted R Square Standard Error Observations

ANOVA

0.48 0.23 0.22 0.03

119

df Regression 1 Residual 117 Total 118

Coefficients Intercept 0.01 S&P500(e) 0.44

SS MS

0.0425 0.0425 34.7468 0.1432 0.0012

0.1857

Standard Error 0.00

0.07

t Stat 3.20 5.89

F

P-value 0.00 0.00

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Significance F 3.71E-08

Lower 95% 0.00 0.29

Upper 95% 0.02 0.59