41. Practice exam.

# 41 Practice exam. - University of California Los Angeles Department of Statistics Statistics C183/C283 Instructor Nicolas Christou Exam 1 04 May

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Unformatted text preview: University of California, Los Angeles Department of Statistics Statistics C183/C283 Instructor: Nicolas Christou Exam 1 04 May 2009 Name: Problem 1 (20 points) Answer the following questions: a. Suppose the average beta for a group of stocks for the years 2004-08 (this is the historical period with 2 ¯ the subscript 1) is β1 = 1.0 and their variance is σβ 1 = 0.25. The estimate of beta of stock A obtained ˆ from the regression of the returns of stock A on the market for the same period is βA = 1.183 and 2 its variance is σβ = 0.22. What it your best forecast of beta for stock A using the Vasicek’s technique? ˆ A1 2 b. You are given that the variance of the return of the market is σm = 0.2091 and the covariance between the returns of stock A and the market is σAm = 0.2474. Find the beta of stock A. c. Portfolios A and B were constructed using the single index model. The beta of portfolio A is βP A = 0.9 2 while the beta of portfolio B is βP B = 1.2. If the variance of the returns of the market is σm = 0.30 ﬁnd the covariance between portfolio A and portfolio B . d. Assume that the single index model holds. The characteristics of two stocks A and B are the following: Stock A B α -0.0022 0.0084 β 1.06 0.81 σ2 0.01 0.05 2 If σm = 0.002 ﬁnd the correlation coeﬃcient between stocks A and B . Problem 2 (20 points) Answer the following questions: a. Suppose stocks A and B are chemical stocks and stocks C and D are utility stocks. Let IC and IU the indexes for the chemical and utility stocks respectively, and let Rm the market index with variance 2 σm . The following regression models hold: RA = α A + βA I C + A RB = αB + βB IC + B RC = αC + βC IU + C RD = αD + βD IU + D and IC = γC + bC Rm + δC IU = γ U + bU R m + δ U Assumptions: All the error terms are independent, and all the indexes are independent from all the 2 error terms. Also, var( A ) = σ 2A , var( B ) = σ 2B , var( C ) = σ 2C , var( D ) = σ 2D , var(δC ) = σδC , and 2 var(δU ) = σδU . 1. Write down the covariance between stocks A and B . 2. Write down the covariance between stocks A and C . b. Suppose the multi group model holds, and we have data for three industries with 10 stocks in each industry. The 3 × 3 correlation matrix using the assumption of the multi group model is given below: 0.4073 ρ = 0.0403 ¯ 0.0722 0.0403 0.4633 0.1275 0.0722 0.1275 0.0960 1. Suppose the ﬁrst stock in the ﬁrst industry has σ1 = 0.15 and the last stock in the third industry has σ = 0.05. Compute the covariance between these two stocks. 2. Suppose short sales are allowed. Using the multi group model the optimum portfolio (point of tangency) can be found by solving Φ = A−1 C . Once we obtained the elements of the vector Φ we can compute the zi ’s and from there the xi ’s. Write down the elements of the matrix A and the vector C . Problem 3 (20 points) 2 ¯ Suppose that the single index model holds, Rf = 0.002, σm = 0.002548013, and Rm = 0.003602158. Using the single index model we obtain the following variance covariance matrix for three stocks 1, 2, and 3. > var_covar [,1] [,2] [,3] [1,] 0.013422089 0.005415671 0.002142743 [2,] 0.005415671 0.024634485 0.002914967 [3,] 0.002142743 0.002914967 0.010748691 Also, the mean returns of the three stocks are: > R_bar R1 0.005274547 R2 0.001527333 R3 0.010364922 The table below show the ranking of the stocks based on the excess return to beta ratio: > table1 stock Ratio col1 col2 col3 col4 col5 [1,] 3 0.0124333171 0.58650977 0.5865098 47.17243 47.17243 0.001334083 [2,] 1 0.0026197367 0.43353212 1.0200419 165.48690 ????????? ??????????? [3,] 2 -0.0002779702 -0.04654724 0.9734946 167.45407 380.11340 0.001260063 a. Find the two missing values in the table1 above. b. The last column in table1 are the Ci ’s. What is the value of C ∗ when short sales are allowed, and when short sales are not allowed? c. When short sales are allowed the composition of the optimum portfolio is: > short_sales_composition stock x_short [1,] 3 0.9648372 [2,] 1 0.2217000 [3,] 2 -0.1865372 ¯ This portfolio has RG = 0.01088493 and σG = 0.1046062. Write the expression that computes these two numbers. No calculations! d. When short sales are not allowed the values of the zi ’s are: > no_short_sales stock z_no_short [1,] 3 0.7535748 [2,] 1 0.1236640 Find the composition of the optimum portfolio and compute its expected return and standard deviation. e. On the previous page you see the plot of the expected return against the standard deviation of many portfolios of these three stocks when short sales are allowed and when short sales are not allowed. Indicate the position of the optimum portfolio on this graph when short sales are allowed and when short sales are not allowed. f. Consider the case when short sales are not allowed. Suppose a new portfolio is constructed as follows: 70% in the optimum portfolio and 30% in the risk free asset. Find the expected return and standard deviation of this new portfolio and place it on the graph on the previous page. Plot for problem 3: Portfolios with and without short sales for problem 3 0.006 0.004 0.002 0.000 Rp 0.008 0.010 0.012 q q qq qq q qq q qq qq qq q q qq q q qqq qq q q q qq q qq q qqqq q qq q q q q qq q q qq qq qqqqqqq qq qq q q q q q qq q q qq q qqq q q q q q q qq q q q q q qq qq qq q q qq qq q qq q q q qq q qq q q q q q q q q q qq q q qq q q q q q q q qq q qq qqqq q q q qq q q q q qq q qq qq q q q qq qqq q q qq qqq q q q q qq qq q q q q q qq q q qq q qq q q q q q qq qq qq q q qqqq q q q q qq q qqq q q q qq qq q q qq q qq q q q q qqq q qq q q q qqq q q qq q q qq qq q q qq q q q q q qq q qq q q q q qq q qq q q q q q qq q qq qq qq qq q qq q q q qq qq qq qqqq q q qq q q q qq q qqqq q q q q q q q q qq q q q q qq q q q qq q q q qq qq qq qq qq qq qq q qq q q q q q q q qq qqqq q q q q qq q q q q q qq q q q q qq q q qq q qq q q qqq q qq qq qq qq qq q qq q q q qq q q q q qq qq q q qq q q q qq q q qq q qq q qq qq q q q q q q q qq q q qq qq q qq q qq qqq qq q qqq qqq q qq q q qqqq q q q q qqq q q 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qq qqqq q qq q q q qq qqq q q qq q q qq q q qq qqq qqq qq qqqq q q q qq q q qq q q qq q q q q qq q q qqqq qqqqq qq q q qq qq qqq qq q q q qq q q qq q q qqq q q qq q q q q qq qq q qq q q q q q q q qq q qq q q q q q q q qqqq q qq qqq q q qqq q qqq qq qqq q qq q qqqqqq qq qqqqq q qq q q q q qqqqq q qq q q qqq qq q q q q q q qq q q q q q qq qq q qq q q q qq q q q qq qqqq q qq qq q qq qqq q q qq q qq q qq qqq qqqqqq qq q q q q q qqq qqqq q qq q q qqq qqq q q qqqqqq q qq qq q qq qq q q qq q q q q qqqq qqq q q q q q qq q q q q qq qq qq qq qqqq q q qq qq q q qq q q qq q q q q qq q qq qq q q q q qq q q q q qqqqqq q q qqq q q qq qq qq q qq qq qqq qq q qqq q q q q qq q qq q qq q q qq qqq qqq q q qq q q q q q qqq q qq qq q q qqqqq qq qq q qq qqqqq q qqq qqq qqqqq qqq q q q q q qqq q q qq qq q q q qq qq q qq q qq qqq qq qqqq qqqqqq q qq qqq q q qq q qq qqqq q qq qq qqq q qq qq q q q q q qq q q qq q qqq qqq q qqq qq q qqq qq q qq qq q qq qq qq qq q q qq qq q q qq q qqq qqqqq qq q qqq qqq q q qq q qq q qq qqq qqq q qqq qq qq 0.00 0.02 0.04 0.06 σp 0.08 0.10 0.12 Problem 4 (20 points) You are constructing a portfolio from three assets. The ﬁrst two assets are stocks 1 and 2. The third asset is the risk free T-bill which has return 3%. The characteristics of the two stocks are as follows: Stock 1 2 ¯ R 0.15 0.08 σ 0.30 0.15 The correlation coeﬃcient between stocks 1 and 2 is ρ12 = 0.15. a. Find the composition of the minimum risk portfolio (point M on the graph on the previous page). b. Suppose you want to form a portfolio by combining the two stocks and the risk free asset that will give you expected return of 10% (point A on the graph). Find the composition of portfolio A in stock 1, ¯ stock 2, and the risk free asset. Note: For your convenience you are given that RM = 0.09113636. c. Calculate the standard deviation of portfolio A. Note: For your convenience you are given that σM = 0.1414013 d. Suppose you still want a return of 10% but you are only allowed to combine the two stocks. Find the composition of this portfolio and show it on the graph. Plot for problem 4: 0.12 0.14 Portfolio possibilities curve for problem 4 0.10 A q 0.08 0.06 0.04 0.02 Rf = 0.03 0.00 Rp Mq 0.00 0.05 0.10 0.15 σp 0.20 0.25 Problem 5 (20 points) Using the constant correlation model we completed the table below on 6 stocks. Assume Rf = 0.001 and average correlation ρ = 0.2530345. > table1 R4 R3 R5 R1 R6 R2 Rbar 0.015036250 0.010364922 0.009990582 0.005274547 0.003806880 0.001527333 Rbar_Rf 0.014036250 0.009364922 0.008990582 0.004274547 0.002806880 0.000527333 sigma 0.1181161 0.1029387 0.1360799 0.1152052 0.1385122 0.1560782 Ratio 0.118834339 0.090975688 0.066068411 0.037103779 0.020264496 0.003378646 col1 0.2530345 0.2019374 0.1680099 0.1438429 0.1257541 0.1117065 col2 0.1188343 0.2098100 ????????? 0.3129822 0.3332467 0.3366254 col3 0.03006919 0.04236849 ?????????? 0.04502026 0.04190712 0.03760324 a. Find the two missing numbers in the table above. b. Find the cut-oﬀ point C ∗ if short sales are not allowed. c. Find the cut-oﬀ point C ∗ if short sales are allowed. d. Write down the expression in matrix form that computes the variance of the portfolio when short sales are allowed. No calculations. ¯ e. You are given a new stock with R = 0.005, and σ = 0.15. Will anything change when short sales are allowed and when short sales are not allowed in terms of the portfolio allocation. Brieﬂy explain your answer without doing all the calculations. ...
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## This note was uploaded on 06/02/2011 for the course STATS 183 taught by Professor Nicolas during the Spring '11 term at UCLA.

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