S183 M1 S10 (Solutions)

# S183 M1 S10 (Solutions) - Statistics 0183 C283 Name Que l 0...

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Unformatted text preview: Statistics 0183 / C283 Name: Que l 0 _ Problem 1 (20 points) The betas for 10 stocks in two historical periods 2000-2004 and 2005-2009 are as follows: [1.] [2.] [3.] [4.] [5.] [6.3 [7.] [8,] [9.] [10.] hotel .9072828 .0374136 .9871119 .0084073 .7606293 .8047901 .9533157 .8036708 .0867607 .0315184 HHOOOOHOHO University of California, Los Angeles Department of Statistics Instructor: Nicolas Christou 223’ ’1. sex/.6“ was A oz? g3” Exam 1 07 May 2010 \_ beta2 0.7333501 1.0096043 .1143143 ‘0 .1011334 6 W F59“ .7711333 iiﬁgl .7334646 .9914738 - .103334043. .9524100. .3759303 OOHOOOHH 3.. Explain how you can obtain an estimate for the beta of stock 8 for the period 2010-2014 using the fl(?C* Blume's technique. 116%?5 \rs «in! \$.é7ﬁari _ (F1g7r 7*}fél F=r7*7’41t> <Lry\/E; more? 83 Fun 101wa . A 5% , 51 F?LAK‘\ 'TTfi rancﬁ’i‘rw a? ‘3 067341.. 47 «1%: mi. (W 5874/, b. Suppose that for the second period 2005~2009 the variance of the return of the 5&P500 index is a?“ = 0.00217. Assume that the single index model holds. Find the covariance between stocks 1 and 3 during the same period. (1 is ::: (5‘ Ci“; (likiL' 33‘ (Iirli:S£3 a. aciliL;i F- .-— )(I‘IW),MUC} =2) 0"]; c. Explain how you can obtain an estimate for the beta of stock 8 for the period 2010-2014 using the MQEmLi + W (iii Vacicek’s technique. m1 ( MIL, magi ML{BH4'1:)+ Wm») .. tram, (.103 + ('5 d. Suppose that the correlation coefﬁcient between stock A and S&P500 during the period 2005-2009 is 0.20. The variance of the return of stock A during the some period is 0.0143 and the variance of the return of the 8851350 ' dex was a?" = 0.00217. Find the beta of stock A. O.'L. ila"”‘F} :9) ximoozi4 G ‘04;— 0.0016 EH 0'.er .69 Problem 2 (20 points) : Use the following for questions (a) and (b) below: 0 I O O a 00 60 28 -- 0. 0C! Stock R a \ “.3 “#3 3132 f 25 _: [1m 43/] :06 l 2;? +7! (gr/.323 204'“ It is also given that pAB = 0.1. a. What expected return on stock B would result in an optimum portfolio of %A and éB? Assume short sales are allowed and that f; = 0.04. 2‘3 firth {sq-rm =L —~=) 9+1”: 3 0-0. ~D.Ot( : 9,09 234. + 0.0016 2‘}? “" .- 0410th 2r? r-u.oLr .— “-°0!6"34+ as; 0.99 1.) %+::rm)"/ (9-02, ‘1:— 0.0‘Ffé % =9 2:4 “z: "z; ~ r b. What expected return on stock B would mean that stock B would not be held? Assume short sales are allowed and that R f = 0.04. l: SIMILL was Hm 2) K8740 T) %"’ ° —— moq'ﬁq 4.0.ddltf a? 1 BW (25: 04'1- ~orou _. £3 .— 0.0% :. moomaq «’- maﬁa—(2;; 2A .5) gr; =1- 1-... «' 04m "2;; a Mot—r- amalé ("Ll 4,—- c. Suppose X and Y represent the returns of two stocks. Show that these two random variables X and Y cannot possibly have the following properties: E(X) = 0.3,E(Y) = 0.2,E(X2) = D.1,E(Y2) = 0.29, and E(XY) = 0. Reminder: ox}: = E(X - px)(Y — py) = EXY — (EXXEY). memzoJ—m‘ : MI =5 S‘brﬁ :10"! W6!) 3: 0.29.. 0.7.1 3 ("Mr :1) 5"» (if) :5.)"’ Q'Wﬂ) :: 5xv_{e><)[€vl : o-(MW‘W) 5) (EVE—0'0" f): ,LLLQ : 4.1. 0-I of alvzp é / Data for problem 3: #Create the ticker vector for the two stocks plus the S&P500: > ticker <- C("ihm", “xom”, "‘GSPC") > data <- getReturns(ticker, start="2005-01-3P', end="2009+12-31") #Get the summary statistics: > summary(data\$ﬂ) ibm xom 'GSPC Min. :-0 . 205144 Min. :-0 . 116543 Min. :-0 . 1694245 181: On. :-0.013633 151'. 0a. :-0.028956 1.51: llu. :-0.0184670 Median : 0.009613 Median : 0.003498 Median : 0.0099800 Mean (0.033026) Mean :% Mean : 0.0001331 3rd 0m: . 3rd ﬂu.: 0.045 3rd Qu.: 0.0277094 Max. : 0.129405 Max. : 0.233054 Max. : 0.0939251 #Get the variance covariance matrix: _ > _cov(-data\$ﬂ) .-—.. \\ ' ibm xom (1%!) 11:01 0.0039985797 0.0004087865 0.0017214346 xom 0.0004087865 0.0035546519 0.0 P 328 :GS 0.0017214346 0.000987832 0.002172_6_2_9_5 #Run the regression of the returns'fof IBM on the returns of 3&P500 #and obtain alpha, beta, mse: > regi <- 1m(data\$R[,1]' " data\$R[,3]) > summary(reg1)\$coef [1] [1] 0.008920891 —" < > summary(rag1)\$coef[2] ——) e [1] 0.7923277 > summary(regl)\$sigma“2 __’ [1] 0.002580861 0:5. L #Run the regression of the returns of EXXON-M811. on the returns of 88513500 #and obtain alpha, beta, use: > reg2 <- lm(data\$R[,2] " dara\$R[,3]) > summary(reg2)\$coef [1] [1] 0.008046374 > eummaryCreg2)\$coe:E [2] [1] 0.4546716 > summary(reg2)\$sigma"2 [1] 0.003159995 Problem 3 (20 points) Using the package atockPox-tfolio we have obtained the returns of IBM, Exxon-Mobil, and the 3&P500 index for the period 2005d01—31 to 2009-12—31. The summary statistics, variance-covariance matrix of the returns, and the regressions of the returns of IBM and Exxon-Mobil on the index are shown on the previous page. :1. Using the single index model compute the variance of the returns of IBM. 1 1. ’L "L ‘1 L g 0“ = 0.00 (Xe-A: Gus/x (rm + a: :2 0.??7, 6.00'Lf?)+ 0.0016 :9) m4 4: W" 0310.»: - a 00367 b. What is the beta of a portfolio that consists of 80% IBM and 20% Eamon-Mobil? QP: .: Q(Uraﬁl)+0rl(d-4‘rr) ’5) .—.__- c. Using the historical variance-covariance matrix of the returns and assuming Rf = 0.008 we get the following: I. > z [.1] ibm 0.256622417 xom 0.000559498 1. Explain how these 2 values were computed. No calculations, but please be very speciﬁc! -f W_64VﬁHL Mil-r3436: £210; 2:“ a ear-ex a! w M __ 2. Compute the proportion of the investor' v. ealth that goes into each stock. Rum : “£0346 1— =) 6r1f(€t¥4.000ffrf —— _ be point G on the graph below. C: = 0.963%? Oswald) + o: Panel: sanded mm 0011 0.000 0.006 000 0.05 0.10 015 '.' 2L: Pontollo mandate devlaﬂon Problem 4 (20 points) Using the single index model three stocks x, y, 2 were ranked based on the excess return to beta ratio as follows: —————-———. a ' ‘ 7 Stock 1' Thin —r’—(R‘:.f2 m 23:1 —"—IJ—‘1'(R :2 m gi- }=1 gg- Ci 1:; 0.0395 13.7170 13.7170 342.2115 347.2115 01 a: 0.0080 4.3307 18.0477 538.4952 835.7067 Cg 2 0.0067 1.9733 20.0210 294.4627 11301694 03 Assume R; = 2% and that the variance of the returns of the market is a?“ = 0.0023. 9.. Find 01.02103. C: .2“ f—ﬁ—ra‘ 007—? x '1'?’.?ﬂ § C' :1 4‘ if- 0-0013F 3" ‘ '5 M1- e) 6 20¢!qu 3 +duao73 x ((90.169L/ b. What is the composition of the optimal portfolio when short sales are not allowed? C“’::C.=O‘.ol?r‘/ ., on S‘th l- a? [MA {N yam (3: c. Show the optimal portfolio of part (b) on the graph below. On this graph 1:, y,z are the three stocks. 0.10 0.06 008 '3? .229“ % diomo oo oo o 00 ° °o 0.04 Porﬂolio expected return “38° 0 'b O O D O O 0.00 0.02 -0.02 Portiolio standard deviation Problem 5 (20 points) Part A: A portfolio manager wants to present to his clients the efﬁcient frontier using 25 stocks. Explain clearly and in detail how you would help this portfolio manager to trace out the eﬂicient frontier when short sales are aIIOWed but no riskless lending and borrowing exists. You must show a graph, the inputs you are using, the vectors and matrices you are multiplying, etc. One should be able to follow step by step your procedure and be able to trace out the efﬁcient frontier. 43):. {j 6 Two RF. VALULﬂ "0 RA‘ ' R3 6? “‘6 a New 5M3 (MC? aw? “WP PUMFoLw MPH/Lift?) Cum/é _ 1 M W (I W C W l (TL %/L c . o L f" . x_ u L'ﬁ' 0 '0] 0‘ Part B: I " Suppose short sales are allowed and three stocks X,Y,Z are used to construct the efﬁcient frontier. Let A and B be two portfolios on the efﬁcient ﬁ'ontier with: RA = 0.006.031 = 0.1,]?3 = 001,05 = 0.2 and 0,43 = 0.02. The composition of portfolio A is 0.53X, —0.50Y, 0.972. The composition of portfolio B is 0.53X, -1.30Y, 2.27Z. Find the composition of the minimum risk portfolio in terms of the two portfolios and in terms of the three stocks X, Y, Z. 0n the previous page draw the graph of the expected return against standard deviation and show approximately the portfolio possibilities curve, identify the efﬁcient frontier, and place the two portfolios A, B, and the minimum risk portfolio on the graph. m *rcw cir— AJ)’ ,- ~— g x 6‘40 an ~— "0 X4} 2 . ME; 01".. 1 (max) ...
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