1.      Je zadaná tabulka s portfolii CP A a B na třech trzích. Trh CP ri riziko "korelaceA,B" výnosnost k riziku riziko k výnosnosti (= variační koeficient) I A 0.22 0.3 0.15 I výnosnost 0.261589242 1.114761376 B 0.31 0.32 riziko 0.234659406 0.897052967 II A 0.26 0.29 -0.06 II výnosnost 0.295148765 1.397146151 B 0.34 0.33 riziko 0.211251174 0.715744734 III A 0.18 0.2 0.09 III výnosnost 0.224674321 1.226445067 B 0.41 0.38 riziko 0.183191507 0.815364689 a) Pro každý trh určete portfolio s minimálním rizikem. b) Vypočítejte pro tato portfolia očekávanou výnosnost rp. "c) Určete, na kterém trhu je nejvýhodnější investovat" I A 0.22 0.3 0.15 B 0.31 0.32 kovar mat 0.09 0.0144 0.0144 0.1024 mat soustavy vekt prav stran inverzni mat váhy 0.18 0.0288 1 0 3.056234719 -3.056234719 0.537897311 XA 0.537897311 0.0288 0.2048 1 0 -3.056234719 3.056234719 0.462102689 XB 0.462102689 1 1 0 1 0.537897311 0.462102689 -0.110130073 lambda -0.110130073 výnosnost portfolia riziko portfolia 0.261589242 0.234659406 0.261589242 II A 0.26 0.29 -0.06 B 0.34 0.33 kovar mat 0.0841 -0.005742 -0.005742 0.1089 mat soustavy vekt prav stran inverzni mat váhy 0.1682 -0.011484 1 0 2.445179085 -2.445179085 0.560640441 XA 0.560640441 -0.011484 0.2178 1 0 -2.445179085 2.445179085 0.439359559 XB 0.439359559 1 1 0 1 0.560640441 0.439359559 -0.089254117 lambda -0.089254117 výnosnost portfolia riziko portfolia 0.295148765 0.211251174 0.295148765 III A 0.18 0.2 0.09 B 0.41 0.38 kovar mat 0.04 0.00684 0.00684 0.1444 mat soustavy vekt prav stran inverzni mat váhy 0.08 0.01368 1 0 2.928772259 -2.928772259 0.805763824 XA 0.805763824 0.01368 0.2888 1 0 -2.928772259 2.928772259 0.194236176 XB 0.194236176 1 1 0 1 0.805763824 0.194236176 -0.067118257 lambda -0.067118257 výnosnost portfolia riziko portfolia 0.224674321 0.183191507 0.224674321 ##### Sheet/List 2 ##### 1.      Je zadaná tabulka investičních možností: Firma 1 Firma 2 Firma 3 Kovariance m 0.8 0.3 0.6 "s1,2" -0.1 s 1.2 0.8 1.1 "s1,3" -0.5 "s2,3" 0.3 a) Formulujte a řešte zadanou úlohu s prodejem CP nakrátko Lagrangeovou metodou. b) Řešte předchozí model s předem určenou výnosností 15%. c) Vždy spočítejte výnosnost a riziko sestaveného portfolia. a) kovar mat 1.44 -0.1 -0.5 -0.1 0.64 0.3 -0.5 0.3 1.21 mat soustavy vekt prav stran inverzni matice váhy 2.88 -0.2 -1 1 0 0.219575604 -0.230115233 0.010539629 0.332138842 Firma 1 0.332138842 -0.2 1.28 0.6 1 0 -0.230115233 0.641160764 -0.411045531 0.379918494 Firma 2 0.379918494 -1 0.6 2.42 1 0 0.010539629 -0.411045531 0.400505902 0.287942664 Firma 3 0.287942664 1 1 1 0 1 0.332138842 0.379918494 0.287942664 -0.592633502 -0.592633502 výnosnost portfolia riziko portfolia 0.55245222 0.544349842 0.55245222 b) mat soustavy vekt prav stran inverzni matice váhy 2.88 -0.2 -1 1 0.8 0 0.08400224 0.056001493 -0.140003733 -0.330968826 1.200298675 Firma 1 -0.150924025 -0.2 1.28 0.6 1 0.3 0 0.056001493 0.037334329 -0.093335822 1.779354116 -2.533134217 Firma 2 1.399383984 -1 0.6 2.42 1 0.6 0 -0.140003733 -0.093335822 0.233339556 -0.44838529 1.332835542 Firma 3 -0.248459959 1 1 1 0 0 1 -0.330968826 1.779354116 -0.44838529 -3.835982826 5.870823222 -2.955359343 0.8 0.3 0.6 0 0 0.15 1.200298675 -2.533134217 1.332835542 5.870823222 -10.62684338 4.276796715 výnosnost portfolia riziko portfolia 0.15 1.075602119 0.15 ##### Sheet/List 3 ##### 1.      Je dána kovarianční matice a vektor očekávaných výnosností: Emise CP1 CP2 CP3 CP4 CP5 CP6 CP7 ri (v %) CP1 80.5 82.7 85.3 85.1 123.9 22 3.5 1.9 CP2 82.7 184.7 131.5 69.4 49.5 58 -9.9 6.1 CP3 85.3 131.5 374.2 384.5 366.5 103.8 343.5 2.9 CP4 85.1 69.4 384.5 684.8 599.1 51.6 502.7 4 CP5 123.9 49.5 366.5 599.1 871.4 -21.2 520.4 5.7 CP6 22 58 103.8 51.6 -21.2 89.7 74.4 3.4 CP7 3.5 -9.9 343.5 502.7 520.4 74.4 574.6 4.9 "a) Vypočítejte podíly cenných papírů v portfoliu, je-li povolen sell short, při minimalizaci rizika" b) Očekávaná výnosnost portfolia nechť je 5%. a) matice soustavy vektor pravych stran CP1 161 165.4 170.6 170.2 247.8 44 7 1 0 CP2 165.4 369.4 263 138.8 99 116 -19.8 1 0 CP3 170.6 263 748.4 769 733 207.6 687 1 0 CP4 170.2 138.8 769 1369.6 1198.2 103.2 1005.4 1 0 CP5 247.8 99 733 1198.2 1742.8 -42.4 1040.8 1 0 CP6 44 116 207.6 103.2 -42.4 179.4 148.8 1 0 CP7 7 -19.8 687 1005.4 1040.8 148.8 1149.2 1 0 1 1 1 1 1 1 1 0 1 2.137199771 0.04443472 -0.290668798 -0.230272848 -0.773325871 -1.12240341 1.235036437 inverzni matice váhy výnosnost 0.077541972 -0.006186729 0.008788437 -0.009709852 -0.037687458 -0.077748274 0.045001904 2.137199771 CP1 2.137199771 0.395249927 2.137199771 367.6936402 7.85367765 -52.98983461 -41.8810356 -204.7759573 -52.77360681 9.238318563 -0.006186729 0.007803096 -0.003616667 0.000167497 0.001056212 -0.000477668 0.001254258 0.04443472 CP2 0.04443472 0.04443472 7.85367765 0.364679863 -1.698425925 -0.710108395 -1.700944652 -2.892673482 -0.543297127 0.008788437 -0.003616667 0.007543944 -0.002286279 -0.004259006 -0.008793788 0.00262336 -0.290668798 CP3 -0.290668798 riziko -0.290668798 -52.98983461 -1.698425925 31.61554051 25.73578923 82.38249347 33.86450601 -123.311882 -0.009709852 0.000167497 -0.002286279 0.004278044 0.004071768 0.010660407 -0.007181586 -0.230272848 CP4 -0.230272848 3.891496121 -0.230272848 -41.8810356 -0.710108395 25.73578923 36.31192041 106.6853023 13.33648596 -142.9655466 -0.037687458 0.001056212 -0.004259006 0.004071768 0.02058011 0.040174616 -0.023936242 -0.773325871 CP5 -0.773325871 -0.773325871 -204.7759573 -1.700944652 82.38249347 106.6853023 521.1258721 -18.40125221 -497.0265612 -0.077748274 -0.000477668 -0.008793788 0.010660407 0.040174616 0.086382111 -0.050197403 -1.12240341 CP6 -1.12240341 -1.12240341 -52.77360681 -2.892673482 33.86450601 13.33648596 -18.40125221 113.0031104 -103.1339576 0.045001904 0.001254258 0.00262336 -0.007181586 -0.023936242 -0.050197403 0.032435709 1.235036437 CP7 1.235036437 1.235036437 9.238318563 -0.543297127 -123.311882 -142.9655466 -497.0265612 -103.1339576 876.4459992 2.137199771 0.04443472 -0.290668798 -0.230272848 -0.773325871 -1.12240341 1.235036437 -30.28748412 -30.28748412 15.14374206 b) matice soustavy vektor pravych stran CP1 161 165.4 170.6 170.2 247.8 44 7 1 1.9 0 CP2 165.4 369.4 263 138.8 99 116 -19.8 1 6.1 0 CP3 170.6 263 748.4 769 733 207.6 687 1 2.9 0 CP4 170.2 138.8 769 1369.6 1198.2 103.2 1005.4 1 4 0 CP5 247.8 99 733 1198.2 1742.8 -42.4 1040.8 1 5.7 0 CP6 44 116 207.6 103.2 -42.4 179.4 148.8 1 3.4 0 CP7 7 -19.8 687 1005.4 1040.8 148.8 1149.2 1 4.9 0 1 1 1 1 1 1 1 0 0 1 1.9 6.1 2.9 4 5.7 3.4 4.9 0 0 5 0.393500391 0.437029695 -0.655179483 -0.044328174 0.036014728 0.415333712 0.417629131 inverzni matice váhy výnosnost 0.016039416 0.007660609 -0.004068331 -0.00315134 -0.009140957 -0.023510276 0.01617088 2.286870663 -0.378674055 CP1 0.393500391 5 0.393500391 12.46482586 14.22203112 -21.99148151 -1.484412384 1.755887217 3.595547509 0.575180292 0.007660609 0.00468536 -0.000721958 -0.001309155 -0.00537105 -0.012689387 0.007745581 0.010736227 0.085258694 CP2 0.437029695 0.437029695 14.22203112 35.27676809 -37.65277498 -1.344467347 0.779105534 10.52776359 -1.806911685 -0.004068331 -0.000721958 0.004856308 -0.000915259 0.001708481 0.002544365 -0.003403606 -0.259380927 -0.079159711 CP3 -0.655179483 riziko -0.655179483 -21.99148151 -37.65277498 160.6291498 11.16699892 -8.647974707 -28.24586151 -93.98917008 -0.00315134 -0.001309155 -0.000915259 0.003578658 0.001027626 0.004876574 -0.004107102 -0.246233458 0.040381057 CP4 -0.044328174 6.313645966 -0.044328174 -1.484412384 -1.344467347 11.16699892 1.3456231 -0.956443464 -0.950006826 -9.306352769 -0.009140957 -0.00537105 0.001708481 0.001027626 0.00733021 0.01499997 -0.01055428 -0.842795834 0.175762112 CP5 0.036014728 0.036014728 1.755887217 0.779105534 -8.647974707 -0.956443464 1.130258656 -0.317112373 7.827232162 -0.023510276 -0.012689387 0.002544365 0.004876574 0.01499997 0.038550595 -0.024771841 -1.254395478 0.333945838 CP6 0.415333712 0.415333712 3.595547509 10.52776359 -28.24586151 -0.950006826 -0.317112373 15.47343764 12.90508601 0.01617088 0.007745581 -0.003403606 -0.004107102 -0.01055428 -0.024771841 0.018920369 1.305198807 -0.177513935 CP7 0.417629131 0.417629131 0.575180292 -1.806911685 -93.98917008 -9.306352769 7.827232162 12.90508601 100.2183368 2.286870663 0.010736227 -0.259380927 -0.246233458 -0.842795834 -1.254395478 1.305198807 -30.65171899 0.921530536 -26.04406632 -0.378674055 0.085258694 -0.079159711 0.040381057 0.175762112 0.333945838 -0.177513935 0.921530536 -2.331513485 -10.73603689 39.86212538 ##### Sheet/List 4 ##### 1.      Mějme bezrizikové aktivum s výnosností " a portfolia umístněná na efektivní množině. Sestrojte graf umístnění jednotlivých portfolií, jestliže budeme měnit podíly investování do bezrizikového aktiva a rizikového portfolia." Riziková portfolia A B C D 6.20% 4% 7.50% 8.40% rf 14.50% 9.70% 17% 20% 0.035 U všech portfolií budeme volit podíly (váhy) takto: 1 2 3 4 5 0.2 0.4 0.5 0.6 0.8 Portfolio 0.8 0.6 0.5 0.4 0.2 kombinace rf s A výnosnost 0.0566 0.0512 0.0485 0.0458 0.0404 riziko 0.116 0.087 0.0725 0.058 0.029 B výnosnost 0.039 0.038 0.0375 0.037 0.036 riziko 0.0776 0.0582 0.0485 0.0388 0.0194 C výnosnost 0.067 0.059 0.055 0.051 0.043 riziko 0.136 0.102 0.085 0.068 0.034 D výnosnost 0.0742 0.0644 0.0595 0.0546 0.0448 riziko 0.16 0.12 0.1 0.08 0.04