S183 M1 S09 (Solutions) - University of California Los...

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Unformatted text preview: University of California, Los Angeles Department of Statistics Statistics 0183/ C283 Instructor: Nicolas Christou Exam 1 04 May 2009 got/{Tim} Problem 1 (20 points) Answer the following questions: Name: a. Suppose the average beta. for a group of stocks for the years 2004-08 (this is the historical period with the subscript 1) is ,81 = 1.0 and their variance is 051 = 0.25. The estimate of beta of stock A obtained from the regression of the returns of stock A on the market for the same period is 5,; = 1.183 and its variance is of?“ = 0.22. What it your best forecast of beta for stock A using the Vasicek’s technique? ‘1" _ 0.17 40-11. 0« “+0511 g 0.11— I . arr Mg) "2. [.0‘73- v b. You are given that the variance of the return of the market is a?“ = 0.2091 and the covariance between the returns of stock A and the market is cum = 0.2474. Find the beta of stock A. gtfggh: 9.25533- : m8} 4 ‘L (7-7109! C. Portfolios A and B were constructed using the single index model. The beta of portfolio A is fipA .= 0.9 while the beta of portfolio B is HPB = 1.2. If the variance of the returns of the market is a?" = 0.30 find the covariance between portfolio A and portfolio B. (525 2 ((5-7) 0'30 1- 0319’} GEN/[ampli- tpfifia-tzgtia, GdP64"QFG’R‘Lfl—F‘Y‘fil Si d. Assume that the single index model holds. The characteristics of two stocks A and B are the following: Stock 0: fl 0'? A -0.0022 1.06 0.01 B 0.0084 0.81 0.05 If 63,, = 0.002 find the correlation coefficient between stocks A and B. a 0 K pm : Eflfl‘ : (I.06)(o-&t)o.oaL p in 59 Problem 2 (20 points) Answer the following questions: a. Suppose stocks A and B are chemical stocks and stocks 6‘ and D are utility stocks. Let Ia and If; the indexes for the chemical and utility stocks respectively, and let Rm the market index with variance 0,2“. The following regression models hold: RA = 0,4 +fiAIcfl-i- (:34 Re = as +1331cu+ 6.3 RC = ac + £30111 + EC R9 = CED + 331:! + CD and I3", = To + bcifim + do“. In = Wu+buh+6u Assumptions: All the error terms are independent. and all the indexes are independent from all the error terms. Also, uar(e,4) = 63A, var{53} = 0'33, var-{cc} = 035., var(eD) = 03D, var(60) = agc, and uni-(6U) = 0'30. 1. Write down the covariance between stocks A and B. I 07953 : CGVE’XA +6kIsn+ $4 ) Kg {—figIchd—{g'l :: GJCQA Ice; Q“ as] ‘- GAQE 0;: = GAQBEE5314' .. 2_ Write do the covariance between stocks A and C‘. ._ . 6 I a all") (Sic -: chxnrmicfi is, «Medea: .. waA cu; : QRQC GI (1111,10) 1: Gaga bcglm 07.1“- b. Suppose the multi group model holds, and we have data for three industries with 10 stocks in each industry. The 3 x 3 correlation matrix using the assumption of the multi group model is given below: r ‘L 3 I 0.4073 0.0403 0.0722 fi = 0.0403 0.4633 0.1275 ’> 0.0722 0.1275 0.0960 1. Suppose the first stock in the first industry has 01 = 0.15 and the last stock in the third industry has :56: 0.05. Compute the covariance between these two stocks. ‘ -:_ .oaaf 07,9“ ': (Train PB 2 (El-(5)6311; 0'0311 a 2 2. Suppose short sales are allowed. Using the multi group model the optimum portfolio (point of tangency) can be found by solving '1) = A'IC. Once we obtained the elements of the vector (I! we can compute the z,‘s and from there the mi’s. Write down the elements of the matrix A and the vector C. " Z M HM?“ PM?!" “if” m C’TO‘FN) 1- 1» I- 1. I491” "L _,. '43:")! IM, "1. P33 ) 2’ v ‘ "2E _——- '1' :r- (It-(93%) Problem 3 (20 points) _ Suppose that the single index model holds, R,— = 0.002, of“ = 0002548013, and Rm ——~ 0003602158. 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: > tablel stock Ratio coll c012 c013 c014 c015 [1,] 3 0.0124333171 0.5865097? 0.5865098 47.17243 47.17243 0.001334083 [2,] 1 0.0026197367 0.43353212 1.0200419 165.48690 ????????? 7????7????? [3.] 2 —0.0002779702 -0.04654724 0.9734946 167.4540? 380.11340 0.001260063 a. Find the two missing values in the table1gh‘of. tittiw‘kfifiti ‘* 1' mom rr- were): More rm ,__ .__"-—————-"—"'”- C1 1 114m»? 0% H‘ ‘ l+on°lF93m (“L'GGI) b. The last column in tablel are the 01’s. What is the value of 0' when short sales are allowed, and when short sales are not allowed? ' Prue-uh) : Ctrflub’l‘wfl NUT ALLth-fi : CV1 .._.._ aaoffigffll— c. When short sales are allowed the composition of the optimum portfolio is: > short_sa1es_composition stock x_short [1,] 3 0.9648372 [2,] 1 0.2217000 [3,] 2 -0.1865372 This portfolio has Ra ——- 0.01088493 and do = 0.1046062. Write the expression that computes these two numbers. No calculations! IF... —- ¢—- —- ‘-— H R [, ~ Plot for problem 3: Portfolios with and without short sales for problem 3 0.004 0.006 0.008 0.010 0.012 0.002 0.000 0.00 0.02 0.04 0.06 0.08 0.10 0.12 d. When short sales are not allowed the values of the zi’s are: > no_short_sales stock z_no_short [1.] 3 0.7535743 [2.] 1 0.1236640 Find the composition of the optimum portfolio and compute its expected return and standard deviation. K7 :: T; 0 ‘ g 5.7” 309’ 5 {3:03am}; oJW‘frlé-WZ'U/ : 32.1- 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. , 670“r\(r/) IGKFL” <73fi¢\r(lLJ“CL\l 1‘. 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. iv = 0"” amid“? + 0”“ (“ml L 0.0093 7’; Go. ._“_‘ 0.}6 z; Plot for problem 4: Portfolio possibilities curve for problem 4 0.04 0.06 0.08 0.10 0.12 0.14 0.02 0.00 0.00 0.05 0.10 0.15 0.20 0.25 ’ m‘KCFEIEa‘ LVE— 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 [2 or 1 0.15 0.30 2 0.03 0.15 The correlation coefficient between stocks 1 and 2 is p12 = 0.15. 3.. Find the composition of the minimum risk portfolio (point M on the graph on the previous page). ._ (mi-F Mme (9'0" : 0\()”70?0 Y” _ c.101‘+0.lil-10"”)fi7fifitd) I ? “Sp )9 .—=— 4. 9943670?! H 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 assfit': Note: For your convenien you are~given that RM = 0.09113636. LC’T )Cr: Myrr— nu / iv—VC‘ W -; o-ro c) K‘: 0‘” “‘0‘” :IJlflQ‘T’gl mock A. 3, Muhqu oJiWfi‘afi) :1 with L: “WW! (ti-WMOW :: M Calculate the standard deviation of portfolio A. Note: For your convenience you are given that 0M = 0.1414013 ' Q: /.zmg,6:, 2: Ammo «J‘f’W‘DL—w 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. :3 far Mml + mm m = we 7“ 7‘“ War— 0-09 0'1954’ / K :_ /—~ 2 l XL : 09H” . Problem 5 (20 points) Using the constant correlation model we completed the table below on 6 stocks. Assume R! = 0.001 and average correlation p = 0.2530345. > tablel X1 54 fiber Rbar_Ri sigma Ratio c011 c012 c013 R4 0.015036250 0.014036250 0.1181161 0.118834339 0.2530345 0.1188343 0.03006919 R3 0.010364922 0.009364922 0.1029387 0.090975658 0.2019374 0.209310°)0.04235349 85 0.009990582 0.008990582 0.1360799-0.066068411 0.1680099 ??7?????? ?????????? R1 0.005274547 0.004274547 0.1152052 0.037103779 0.1438429 0.3129822 0.04502026 RB 0.003806880 0.002806880 0.1385122 0.020264496 0.1257541 0.3332467 0.04190712 H2 0.001527333 0.000527333 0.1560782 0.003378646 0.1117065 0.3366254 0.03760324 3.. Find the two missin numbers in the table above. b. Find the cut-off point C‘ if short sales are not allowed. pF : 0.0063)”; c. Find the cut-off point C" if short sales are allowed. V1 Z C T— 0‘ 07,} [x 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 o‘ = 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. w .0164 ~00 [0:00) “‘5’ f 0 0 f h) / ‘ WWI/5 (W No - f . f“ 04w; 'DoW CALCMLMIM ...
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