See file attached
FIN 322 – Summer 1998 PAGE 2 FIN 205 - Efficient Frontier (Risk and Return) Professor A. Spieler The following table (Table 1) represents CLOSING prices for America Online (AOL), Anheuser Busch (BUD), Coca-Cola (KO), and TransWorld Airlines (TWA). Assume the return on a Treasury Bill is 4% per year (.33% per month). Clearly indicate your answers by underlining or putting them in a box. The data in Tables 1 and 2 can be downloaded in spreadsheet form from the class Blackboard site. AUTONUM \* Arabic (a) Calculate the MONTHLY returns for each asset (AOL, BUD, KO, TWA). (b) Calculate the variance and standard deviation of RETURNS for each asset (AOL, BUD, KO, TWA) over the 36-month period from 7/96-6/99. (c) Comment on the difference between expected return and realized return. (d) Find the total return over the 36-month period for each stock. Which stock had the highest and lowest returns? Was this expected? Are these results consistent with the general risk-return relationship? (e) Are these the actual closing prices or prices adjusted for changes in capitalization, i.e. stock splits, etc.? How do you know? AUTONUM \* Arabic (a) Calculate the COVARIANCE and CORRELATION matrix for all pairs. Display your answers in a 4x4 table. (b) Verify that covariance between the asset and itself is equal to its variance (there may be some rounding error). (c) Verify that the correlation between all pairs of assets falls in the range [-1,1]. Verify that the correlation with an asset and itself is 1. (d) Do all assets move together? Interpret your results. AUTONUM \* Arabic Graph the Capital Allocation Line for BUD and the risk-free asset. Extend the line past the 100% investment in BUD, i.e. short the risk-free asset. How do you interpret a weight in the risk-free < 0%? autonum \* arabic (a) graph the efficient frontier for only 2 assets, aol and twa. use the formula var(w1x + w2y) = w12var(x) + w22var(y) +2w1w2cov(x,y). (b) what is the minimum variance portfolio? identify by weight in each asset to the nearest 0.1% autonum \* arabic verify that the portfolio weights (in table 2) sum to 1 for each of the 50 portfolios. autonum \* arabic calculate the expected return for each of the 50 portfolios. autonum \* arabic calculate the variance and standard deviation for each portfolio. since the assets are not independent, i.e. cov(0, you must use following formula: var (w1x + w2y + w3z + w4w) = w12var(x) + w22var(y) + w32var(z) + w42var(w) +2 w1 w2cov(x,y) + 2 w1 w3cov(x,z) + 2 w1 w4cov(x,w) + 2 w2 w3cov(y,z) + 2 w2w4cov(y,w) +2 w3w4cov(z,w), where w1, w2, w3, and w4 are arbitrary constants. note: w1 + w2 + w3 + w4 = 1. autonum \* arabic plot all the portfolios and identify the efficient frontier. list all portfolios that lie on the efficient frontier. autonum \* arabic identify the following portfolios by number: (a) equally-weighted portfolio (b) the single-security portfolios (#1- #4) (c) the minimum variance portfolio (d) the “market” portfolio for the 4 securities (ignore the single security portfolios #1-#4). hint: to identify the market portfolio, use the slope, i.e. sharpe ratio autonum \* arabic plot the cal using the treasury bill. for this question, assume the annual t-bill rate is 12% to make the graph easier to read. presentation answers the questions briefly and to the point. summarize your answers briefly in word file and email your actual spreadsheet,. remember to circle or underline your final answers. table 1 monthly closing prices for the aol, bud, ko, and twa from july 1996 through june 1999. date aol bud ko twa 6/1/99 115.375 71.125 61.563 4.938 5/3/99 119.250 73.063 68.500 5.188 4/1/99 142.750 73.125 68.063 5.438 3/1/99 147.000 76.125 61.375 5.188 2/1/99 88.938 76.688 63.875 5.938 1/4/99 87.875 70.688 65.313 4.750 12/1/98 77.563 65.625 67.000 4.875 11/2/98 43.781 60.625 70.063 5.125 10/1/98 31.844 59.500 67.563 5.000 9/1/98 27.906 54.000 57.625 5.688 8/3/98 20.484 46.750 65.125 6.625 7/1/98 29.281 51.813 80.500 8.313 6/1/98 26.281 47.188 85.500 10.375 5/1/98 20.828 45.938 78.375 10.375 4/1/98 19.984 45.813 75.875 9.688 3/2/98 17.078 46.250 77.438 12.313 2/2/98 15.172 46.875 68.625 13.063 1/2/98 11.953 44.938 64.750 11.625 12/1/97 11.313 44.000 66.688 10.063 11/3/97 9.375 43.188 62.500 7.563 10/1/97 9.625 39.938 56.625 7.438 9/2/97 9.430 45.125 61.000 7.875 8/1/97 8.063 42.625 57.313 7.313 7/1/97 8.438 42.938 69.125 6.938 6/2/97 6.953 41.938 68.000 8.563 5/1/97 6.891 42.875 68.500 8.688 4/1/97 5.641 42.875 63.625 7.250 3/3/97 5.313 41.250 55.750 6.938 2/3/97 4.688 44.500 61.000 5.875 1/2/97 4.625 42.500 57.875 6.438 12/2/96 4.156 40.000 52.625 6.563 11/1/96 4.406 42.375 51.125 7.750 10/1/96 3.391 38.500 50.500 8.000 9/3/96 4.438 37.750 50.875 9.625 8/1/96 3.781 37.875 50.000 13.250 7/1/96 3.813 37.375 46.875 10.750 table 2 portfolio weights for 50 random portfolios portfolio number aol bud ko twa 1 1.00 0.00 0.00 0.00 2 0.00 1.00 0.00 0.00 3 0.00 0.00 1.00 0.00 4 0.00 0.00 0.00 1.00 5 0.25 0.25 0.25 0.25 6 0.50 0.50 0.00 0.00 7 0.50 0.00 0.50 0.00 8 0.50 0.00 0.00 0.50 9 0.00 0.50 0.50 0.00 10 0.00 0.50 0.00 0.50 11 0.00 0.00 0.50 0.50 12 0.16 0.27 0.49 0.08 13 0.12 0.47 0.33 0.08 14 0.21 0.30 0.18 0.31 15 0.46 0.35 0.45 -0.26 16 0.39 0.24 0.02 0.35 17 0.36 0.20 0.18 0.26 18 0.27 0.44 0.24 0.05 19 0.20 0.33 0.49 -0.02 20 0.28 0.50 0.00 0.22 21 0.06 0.19 0.35 0.40 22 0.36 0.32 0.12 0.20 23 0.23 0.46 0.24 0.07 24 0.48 0.19 0.37 -0.04 25 0.00 0.34 0.25 0.40 26 0.38 0.25 0.03 0.34 27 0.21 0.34 0.42 0.02 28 -0.55 0.80 0.36 0.39 29 0.37 -0.02 0.41 0.24 30 0.22 0.25 0.21 0.33 31 0.47 -0.10 0.28 0.36 32 -0.14 0.49 0.17 0.48 33 0.45 0.01 0.29 0.25 34 0.04 0.21 0.44 0.31 35 0.18 0.23 0.41 0.19 36 0.35 0.10 0.49 0.06 37 0.02 0.16 0.42 0.40 38 0.33 0.33 0.12 0.22 39 0.35 0.37 0.34 -0.06 40 0.26 0.14 0.35 0.26 41 0.18 0.13 0.27 0.42 42 0.48 0.07 0.16 0.28 43 0.45 0.17 0.24 0.14 44 0.41 -0.23 0.50 0.32 45 0.43 0.04 0.03 0.50 46 -0.01 0.44 0.36 0.21 47 0.07 0.65 0.21 0.07 48 0.22 0.02 0.31 0.46 49 0.38 0.41 -0.15 0.36 50 0.08 0.40 0.32 0.20 0%?="" autonum="" \*="" arabic="" (a)="" graph="" the="" efficient="" frontier="" for="" only="" 2="" assets,="" aol="" and="" twa.="" use="" the="" formula="" var(w1x="" +="" w2y)="w12VAR(X)" +="" w22var(y)="" +2w1w2cov(x,y).="" (b)="" what="" is="" the="" minimum="" variance="" portfolio?="" identify="" by="" weight="" in="" each="" asset="" to="" the="" nearest="" 0.1%="" autonum="" \*="" arabic="" verify="" that="" the="" portfolio="" weights="" (in="" table="" 2)="" sum="" to="" 1="" for="" each="" of="" the="" 50="" portfolios.="" autonum="" \*="" arabic="" calculate="" the="" expected="" return="" for="" each="" of="" the="" 50="" portfolios.="" autonum="" \*="" arabic="" calculate="" the="" variance="" and="" standard="" deviation="" for="" each="" portfolio.="" since="" the="" assets="" are="" not="" independent,="" i.e.="" cov(0,="" you="" must="" use="" following="" formula:="" var="" (w1x="" +="" w2y="" +="" w3z="" +="" w4w)="w12VAR(X)" +="" w22var(y)="" +="" w32var(z)="" +="" w42var(w)="" +2="" w1="" w2cov(x,y)="" +="" 2="" w1="" w3cov(x,z)="" +="" 2="" w1="" w4cov(x,w)="" +="" 2="" w2="" w3cov(y,z)="" +="" 2="" w2w4cov(y,w)="" +2="" w3w4cov(z,w),="" where="" w1,="" w2,="" w3,="" and="" w4="" are="" arbitrary="" constants.="" note:="" w1="" +="" w2="" +="" w3="" +="" w4="1." autonum="" \*="" arabic="" plot="" all="" the="" portfolios="" and="" identify="" the="" efficient="" frontier.="" list="" all="" portfolios="" that="" lie="" on="" the="" efficient="" frontier.="" autonum="" \*="" arabic="" identify="" the="" following="" portfolios="" by="" number:="" (a)="" equally-weighted="" portfolio="" (b)="" the="" single-security="" portfolios="" (#1-="" #4)="" (c)="" the="" minimum="" variance="" portfolio="" (d)="" the="" “market”="" portfolio="" for="" the="" 4="" securities="" (ignore="" the="" single="" security="" portfolios="" #1-#4).="" hint:="" to="" identify="" the="" market="" portfolio,="" use="" the="" slope,="" i.e.="" sharpe="" ratio="" autonum="" \*="" arabic="" plot="" the="" cal="" using="" the="" treasury="" bill.="" for="" this="" question,="" assume="" the="" annual="" t-bill="" rate="" is="" 12%="" to="" make="" the="" graph="" easier="" to="" read.="" presentation="" answers="" the="" questions="" briefly="" and="" to="" the="" point.="" summarize="" your="" answers="" briefly="" in="" word="" file="" and="" email="" your="" actual="" spreadsheet,.="" remember="" to="" circle="" or="" underline="" your="" final="" answers.="" table="" 1="" monthly="" closing="" prices="" for="" the="" aol,="" bud,="" ko,="" and="" twa="" from="" july="" 1996="" through="" june="" 1999.="" date="" aol="" bud="" ko="" twa="" 6/1/99="" 115.375="" 71.125="" 61.563="" 4.938="" 5/3/99="" 119.250="" 73.063="" 68.500="" 5.188="" 4/1/99="" 142.750="" 73.125="" 68.063="" 5.438="" 3/1/99="" 147.000="" 76.125="" 61.375="" 5.188="" 2/1/99="" 88.938="" 76.688="" 63.875="" 5.938="" 1/4/99="" 87.875="" 70.688="" 65.313="" 4.750="" 12/1/98="" 77.563="" 65.625="" 67.000="" 4.875="" 11/2/98="" 43.781="" 60.625="" 70.063="" 5.125="" 10/1/98="" 31.844="" 59.500="" 67.563="" 5.000="" 9/1/98="" 27.906="" 54.000="" 57.625="" 5.688="" 8/3/98="" 20.484="" 46.750="" 65.125="" 6.625="" 7/1/98="" 29.281="" 51.813="" 80.500="" 8.313="" 6/1/98="" 26.281="" 47.188="" 85.500="" 10.375="" 5/1/98="" 20.828="" 45.938="" 78.375="" 10.375="" 4/1/98="" 19.984="" 45.813="" 75.875="" 9.688="" 3/2/98="" 17.078="" 46.250="" 77.438="" 12.313="" 2/2/98="" 15.172="" 46.875="" 68.625="" 13.063="" 1/2/98="" 11.953="" 44.938="" 64.750="" 11.625="" 12/1/97="" 11.313="" 44.000="" 66.688="" 10.063="" 11/3/97="" 9.375="" 43.188="" 62.500="" 7.563="" 10/1/97="" 9.625="" 39.938="" 56.625="" 7.438="" 9/2/97="" 9.430="" 45.125="" 61.000="" 7.875="" 8/1/97="" 8.063="" 42.625="" 57.313="" 7.313="" 7/1/97="" 8.438="" 42.938="" 69.125="" 6.938="" 6/2/97="" 6.953="" 41.938="" 68.000="" 8.563="" 5/1/97="" 6.891="" 42.875="" 68.500="" 8.688="" 4/1/97="" 5.641="" 42.875="" 63.625="" 7.250="" 3/3/97="" 5.313="" 41.250="" 55.750="" 6.938="" 2/3/97="" 4.688="" 44.500="" 61.000="" 5.875="" 1/2/97="" 4.625="" 42.500="" 57.875="" 6.438="" 12/2/96="" 4.156="" 40.000="" 52.625="" 6.563="" 11/1/96="" 4.406="" 42.375="" 51.125="" 7.750="" 10/1/96="" 3.391="" 38.500="" 50.500="" 8.000="" 9/3/96="" 4.438="" 37.750="" 50.875="" 9.625="" 8/1/96="" 3.781="" 37.875="" 50.000="" 13.250="" 7/1/96="" 3.813="" 37.375="" 46.875="" 10.750="" table="" 2="" portfolio="" weights="" for="" 50="" random="" portfolios="" portfolio="" number="" aol="" bud="" ko="" twa="" 1="" 1.00="" 0.00="" 0.00="" 0.00="" 2="" 0.00="" 1.00="" 0.00="" 0.00="" 3="" 0.00="" 0.00="" 1.00="" 0.00="" 4="" 0.00="" 0.00="" 0.00="" 1.00="" 5="" 0.25="" 0.25="" 0.25="" 0.25="" 6="" 0.50="" 0.50="" 0.00="" 0.00="" 7="" 0.50="" 0.00="" 0.50="" 0.00="" 8="" 0.50="" 0.00="" 0.00="" 0.50="" 9="" 0.00="" 0.50="" 0.50="" 0.00="" 10="" 0.00="" 0.50="" 0.00="" 0.50="" 11="" 0.00="" 0.00="" 0.50="" 0.50="" 12="" 0.16="" 0.27="" 0.49="" 0.08="" 13="" 0.12="" 0.47="" 0.33="" 0.08="" 14="" 0.21="" 0.30="" 0.18="" 0.31="" 15="" 0.46="" 0.35="" 0.45="" -0.26="" 16="" 0.39="" 0.24="" 0.02="" 0.35="" 17="" 0.36="" 0.20="" 0.18="" 0.26="" 18="" 0.27="" 0.44="" 0.24="" 0.05="" 19="" 0.20="" 0.33="" 0.49="" -0.02="" 20="" 0.28="" 0.50="" 0.00="" 0.22="" 21="" 0.06="" 0.19="" 0.35="" 0.40="" 22="" 0.36="" 0.32="" 0.12="" 0.20="" 23="" 0.23="" 0.46="" 0.24="" 0.07="" 24="" 0.48="" 0.19="" 0.37="" -0.04="" 25="" 0.00="" 0.34="" 0.25="" 0.40="" 26="" 0.38="" 0.25="" 0.03="" 0.34="" 27="" 0.21="" 0.34="" 0.42="" 0.02="" 28="" -0.55="" 0.80="" 0.36="" 0.39="" 29="" 0.37="" -0.02="" 0.41="" 0.24="" 30="" 0.22="" 0.25="" 0.21="" 0.33="" 31="" 0.47="" -0.10="" 0.28="" 0.36="" 32="" -0.14="" 0.49="" 0.17="" 0.48="" 33="" 0.45="" 0.01="" 0.29="" 0.25="" 34="" 0.04="" 0.21="" 0.44="" 0.31="" 35="" 0.18="" 0.23="" 0.41="" 0.19="" 36="" 0.35="" 0.10="" 0.49="" 0.06="" 37="" 0.02="" 0.16="" 0.42="" 0.40="" 38="" 0.33="" 0.33="" 0.12="" 0.22="" 39="" 0.35="" 0.37="" 0.34="" -0.06="" 40="" 0.26="" 0.14="" 0.35="" 0.26="" 41="" 0.18="" 0.13="" 0.27="" 0.42="" 42="" 0.48="" 0.07="" 0.16="" 0.28="" 43="" 0.45="" 0.17="" 0.24="" 0.14="" 44="" 0.41="" -0.23="" 0.50="" 0.32="" 45="" 0.43="" 0.04="" 0.03="" 0.50="" 46="" -0.01="" 0.44="" 0.36="" 0.21="" 47="" 0.07="" 0.65="" 0.21="" 0.07="" 48="" 0.22="" 0.02="" 0.31="" 0.46="" 49="" 0.38="" 0.41="" -0.15="" 0.36="" 50="" 0.08="" 0.40="" 0.32="">