Portfolio Management. Summer Term Exercise 2: The Capital Asset Pricing Model (CAPM) Prof. Dr. Hans-Peter Burghof / Katharina Nau

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1 Unversty of Hohenhem Char of ankng and Fnancal Servces Portfolo anagement Summer Term 0 Eercse : The Catal sset Prcng odel (CP) Prof. Dr. Hans-Peter urghof / Katharna Nau Sldes: c/o aron Schulz/ Robert Härtl CP Eercse

2 Queston It s assumed that the assumtons of the CP hold, n artcular the formal relatons of the securty market lne. In the table below several arameter values are gven for stocks,, C, and D as well as for the market ortfolo and the rsk-free asset: sset Eected return Standard devaton n % Correlaton wth the market ortfolo eta Stock 9,8 0,64 Stock 40 0,70,4 Stock C 0,6 4 0,9 Stock D 0 arket ortfolo Rsk-free asset 0,0 5 a) Calculate the mssng values f ossble and show your workng. b) Decde whether the statements below are rght or wrong and gve reasons b) The sloe of the catal market lne equals the beta factor of the market ortfolo b) Dversfcaton averts systematc rsks b3) Observed returns are always hgher for securtes wth a hgh beta than for securtes wth a low beta c) Determne the relevant erformance measures for stock gven a realzed return of 0.%, a realzed standard devaton of 6%, a beta of 0.7 and the market ortfolo as benchmark CP: Eercse

3 Queston On the catal market two securtes wth the followng rsk-return characterstcs est: Stock : = 0% = 0% Stock : = 5% = 5% a) ssume that you own a ortfolo whch conssts of 35% stock and 65% stock. For whch correlaton between the stocks, s the standard devaton of the ortfolo return mamal and mnmal resectvely. Comute the resectve standard devaton of the ortfolo return. b) ssume that,=0. Is t ossble to realze a ortfolo wth =4% f the rsk should mamally be =0%. For ths, determne the rsk for an effcent ortfolo wth =4%. CP: Eercse

4 Soluton Queston Part a) 35% 0% 65% 5% 0,35,,,, * * *,,, a[ ], 0,35 n[ ], 0,095 CP: Eercse 3

5 Soluton Queston Part b) 4% *0. (- )* , 0 0,046 0,046 0,040, 0,0 return of 4% cannot be realzed wth a rsk of merely 0% CP: Eercse 4

6 Queston 3 a) In what way dffer the assumtons of the Zero-eta-CP and the classcal CP? b) Prove that c) The eected market return s 4% and the rsk-free rate s 6%. What s the beta of a securty wth an eected return of 0% n CP- equlbrum. What share of an nvestor s wealth needs to be nvested n the market ortfolo n order to realze a return of 0%? d) You eect a stock rce of 0 n the net erod for the stock n art c). How much are you wllng to ay f your decson s based on the CP. e) securty has an eected return of µ =5.5% and a covarance wth the market return of σ, =0.35. The rsk-free rate s r f =7% and the varance of the market ortfolo s σ =0.35. What s the eected return of a securty wth a covarance of σ, = f the assumtons of the CP hold? f) You are the manager of an nvestment fund and calculate wth the followng numbers: σ =0.; µ =6% and r f =8%. Would you recommend to buy stocks wth µ =% and σ, =0.0? CP: Eercse 5

7 Queston 4 The market ortfolo conssts of the followng securtes: Number of shares Prce er share 0, Eected stock rce Standard devaton of stock returns Stock % Stock % The covarance n rces for the stocks s σ, =35. ssume that the assumtons of the CP hold. a) Calculate the eected return and the standard devaton of the return of the market ortfolo. b) Calculate the betas of the two stocks. c) What s the rsk-free rate? CP: Eercse 6

8 Soluton Queston 4 Part a) Eected return of the market ortfolo: Calculaton of the stock weghts: , 0, 5% % arket return: 3% Calculaton of the standard devaton: CP: Eercse cov( ~ r,r ~ cov( ~ r, ~ r ) ~ E [( E ~ 0, ) E( ~ ) ( 0, E( ~ ) ~ ] [ ~ )] [( E( ~ ) ] 0, ) E( ~ ( [ 0, 0, 0, 0, 0, 0, 0, 0, ) cov( ~, ~ 0,008 0, 09 ) , E ( ~ 0,0045 ) )] E( ~ ))( ~ E( ~ )) 7

9 Soluton Queston 4 Part b) The beta of a securty s defned as:, Frstly, the covarances of the stock returns wth the market return need to be determned n order to calculate the beta factors., E ( ~ r )( ~ r ) E ( ~ r ) [ ( ~ r )] E[( ~ r )( ~ r )],,,,,, 0,005 0,0069 0,005 0,008 0,0069 0,008,963 0,859 CP: Eercse 8

10 Soluton Queston 4 Part c) Securty market lne: r r ( r ) f f r f r f 6,5% CP: Eercse 9

11 Queston 5 There est three securtes and the followng nformaton s avalable:, Stock 0% 0,339 0,477 9% Stock,356 0,953 Stock 3 5% 0,794 arket,9% ssume that the assumtons of the CP hold. Identfy the mssng values n the table above. CP: Eercse 0

12 Soluton Queston 5 Soluton:, Stock 0% 0,339 0,477 9% Stock 0,0%,356 0,953 5% Stock 3 5% 0,8465 0,794 % arket 4,07%,9% CP: Eercse