Disregard the "R Excercises." I would need all answers and through explanation for each answer.

Review Questions 1. The followings are descriptive statistics of daily cc returns on S&P 500 index (Jan 3, 2000 - Feb 21, 2014: T=3,555). (a) Describe the stylized facts on the return series based on the above information. (b) (ARCH(1) Model) Assuming that we use the following model: rt = t et ; et iid N (0; 1); t = 1; :::; T 2 2 0: t = ! + 1 rt 1 ; ! > 0 and 1 Show that this model can generate the stylized facts you expained in (a). (c) (GARCH(1,1) Model) Now we turn to the following model: rt = t et ; et iid N (0; 1); t = 1; :::; T 2 2 2 0 and t = ! + 1 rt 1 + 1 t 1 ; ! > 0; 1 1 0: Show that this model can be interpreted as ARMA (1,1) model for squared returns. 2 Compare the sample covariance to the single index covariance. Are they similar? Review Questions 1. (Single Index Model) SI model for asset returns has the form Rit = i + i RM t + "it ; "it i = 1; :::; N assets, t = 1; :::; T time periods, iid N 0; 2 "i where Rit denotes the cc return on asset i at time t, and RM t denotes the return on a market index portfolio at time t. 1.1. In the SI model, what are the interpretations of i and "i ? 1.2. SI model with large portfolios can be constructed as N X 1 Rit = Rp;t = N i=1 where N 1 X = N i=1 i, N 1 X = N i=1 + R M t + "t ; i N 1 X and "t = "it : N i=1 In a large "well diversi...ed" portfolio, we could diversify away all nonmarket variance, hence Rp;t + RM t . Explain how this is possible. From the earlier discussion, we may use the S&P 500 index as the "market return" for our SI model. The following represents R linear regression output from estimating the SI model for the four Northwest stocks (Boeing, Microsoft, Nordstrom and Starbucks) using monthly continuously compounded 3 return data over the the period November 1998 -October 2003. > summary(boeing.fit) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.00216 0.01366 0.16 0.875 sp500 0.63862 0.27354 2.33 0.023 * Residual standard error: 0.106 on 58 degrees of freedom Multiple R-squared: 0.0859, Adjusted R-squared: 0.0701 F-statistic: 5.45 on 1 and 58 DF, p-value: 0.023 > summary(nord.fit) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.00432 0.01414 0.31 0.76 sp500 1.50799 0.28312 5.33 1.70E-06 *** Residual standard error: 0.11 on 58 degrees of freedom Multiple R-squared: 0.328, Adjusted R-squared: 0.317 F-statistic: 28.4 on 1 and 58 DF, p-value: 1.7e-06 4 > summary(msft.fit) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.0012 0.014 0.09 0.93 sp500 1.6971 0.2808 6.04 1.20E-07 *** Residual standard error: 0.109 on 58 degrees of freedom Multiple R-squared: 0.386, Adjusted R-squared: 0.376 F-statistic: 36.5 on 1 and 58 DF, p-value: 1.16e-07 > summary(sbux.fit) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.0183 0.0171 1.07 0.288 sp500 0.6666 0.3418 1.95 0.056 . Residual standard error: 0.132 on 58 degrees of freedom Multiple R-squared: 0.0615, Adjusted R-squared: 0.0454 F-statistic: 3.8 on 1 and 58 DF, p-value: 0.056 Signif. codes: 0 ? **?0.001 ? *?0.01 ? ?0.05 ??0.1 ??1 1.3. Make a table showing the estimated values of i , its estimated standard error, the estimate of 2"i , and the Ri2 values from the four regression equations. Asset ^ i SE ^ i ^ 2"i Ri2 (1) 5 2 2 1.4. Characterize the relation among i = iM 2 , Ri = M correlation coe cient iM . Can j i j > 1 or jRi j > 1? = ( ) ) iM = ( iM i M and the (Ri ) : (Ri ) : i j i j > 1? possible not possible jRi j > 1? possible not possible ( ( ( ( ) ) ) ) 1.5. From (1), which asset appears to be most correlated to "Market"? Which asset has the highest ...rm speci...c risk? What does it mean in terms of risk diversi...cation? 1.6. The followings are estimated regression lines for each SI model. SI model for SBUX 0.0 -0.4 -0.05 0.0 0.05 0.10 -0.10 -0.05 0.0 0.05 SP 500 returns SP 500 returns SI model for Boeing SI model for NORD 0.10 -0.4 0.1 -0.1 -0.3 -0.2 NORD returns 0.0 0.3 0.2 -0.10 Boeing returns -0.2 SBUX returns 0.0 -0.2 -0.4 MSFT returns 0.2 0.2 SI model for MSFT -0.10 -0.05 0.0 0.05 0.10 SP 500 returns -0.10 -0.05 0.0 0.05 0.10 SP 500 returns For MSFT stock, write down the test statistics for the hypotheses H0 : i = 1 vs. H1 : i 6= 1. Which asset has a (statistically) non-zero intercept (with 5% signi...cance level)? 6 ECON 490 Practice Question: Portfolio Theory No due date Submission This is a set of question for your practice. You need not submit but they could be asked in Quiz 3. We will go over it in the review session. Reading and Program Downloads Please read the course material on compass. For R exercises, the R code (econ490hw8_2016F.R) in Compass will be helpful. Make sure you understand the output rather than just following the hint. Review Questions Consider the constant expected return model for the four Northwest stocks (Boeing, Microsoft, Nordstrom and Starbucks) Rit = i + it t = 1; ;T; i = 1; 2; 3; 4 (Boeing, Msft, Nord and Sbux, respectively); iid N (0; 2i ); cov( it ; jt ) = ij ; cor( it ; jt ) = ij it 1 1. Transfer the model into vector and matrix forms, i.e., de...ne the following vector and matrices, 0 1 0 1 0 1 B C C Rt = B @ A; 0 B =B @ 0 1 B C C x=B @ A; B C C =B @ A; 1 0 t B C C =B @ A C C A 1 1 B 1 C C 1=B @ 1 A 1 where x is a vector of portfolio shares. 2. Write down the optimization problem and give the Lagrangian used to determine the global minimum variance portfolio. Let m denote the vector of portfolio weights in the global minimum variance portfolio. 3. Show that the global minimum variance portfolio m = (m1 ; m2 ; m3 ; m4 )0 that solves the problem in 2 is 1 10 1 : 11 4. Write down the optimization problem used to determine the tangency portfolio when the risk free rate is given by rf . Let t denote the vector of portfolio weights in the tangency portfolio. What does the following ratio represent, in ...nancial economics? t0 rf P 1=2 0 (t t) 2 5. It can be shown (no need to show) that 1 t= 10 ( 1( Explain the economic concept of ( between m in 3 and t. rf 1) : rf 1) rf 1). Discuss the main dierence 6. Denote the tangency portfolio Rtan = t0 R. State Mutual Fund Separation Theorem and draw a portfolio frontier based on the Theorem. Discuss the resulting weights determination (xf for rf and xtan for Rtan ) according to an investor's risk preferences. R Exercises The following questions require R. On Compass is the R script ...les econ490hw8_2016F.R. The ...le contains hints for completing the R exercises. Copy and paste the relevant statistical results and graphs into a MS Word document (or your favorite word processor) while you work, add any comments and answer all questions in this document. Start MS Word and open a blank document. You will save all of your work in this document. Please understand the output. Using the monthly return data on the four Northwest stocks (Boeing, Microsoft, Nordstrom and Starbucks, the same data set from homework 7), you will estimate various portfolio weights you learned from class. 1. For the four Northwest stocks (Boeing, Microsoft, Nordstrom and Starbucks) estimate the CER Model parameters i , 2i , i , ij and ij . Rit = i + "it ; "it iid N (0, 2i ); Cov("it , "jt ) = ij . 1. Compute the global minimum variance portfolio, as in Question 2 from review question. Include the bar chart of this portfolio weights 2. Compute the e cient portfolio (allowing short sales) with target return being to the highest mean return of four asset returns, and explain the dierence with Question 2 above. 3