Portfolio optimization University of Oulu - Department of Finance Fall 2015
Portfolio optimization Minimum variance portfolio variance is minimized, no requirement regarding the level of expected return Required return portfolio variance is minimized, subject to a given level of the expected return Tangent portfolio the ratio of expected excess return to volatility is maximized, no other requirements Optimal solutions closed-form solutions are available for the standard problems, non-negativity constraints require the applying of numerical techniques
Portfolio optimization numerical example r = 40% E(R m) = 100% µ = 008220 013097 010622 e µ = 004220 009097 006622 Σ = 0083702 0035427 0013336 0035427 0194149 0063565 0013336 0063565 0163351 3 3 3 µ p = w i µ i = µ w σ 2 p = w i w j σ ij = w Σw i=1 i=1 j=1
Minimum variance portfolio Optimization problem: min w Σw min st 1 w = 1 st n n w i w j σ ij i=1 j=1 n w i = 1 i=1 Optimal solution: w = Σ 1 1 1 Σ 1 1
Minimum variance portfolio scalar presentation min 3 3 w i w j σ ij i=1 j=1 = w 1 w 1 σ 11 + w 1 w 2 σ 12 + w 1 w 3 σ 13 + w 2 w 1 σ 21 + w 2 w 2 σ 22 + w 2 w 3 σ 23 + w 3 w 1 σ 31 + w 3 w 2 σ 32 + w 3 w 3 σ 33 st 3 w i = w 1 + w 2 + w 3 = 1 i=1
Minimum variance portfolio matrix presentation min w Σw = [ ] w 1 w 2 w 3 σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33 w 1 w 2 w 3 = [ ] w 1 σ 11 + w 2 σ 21 + w 3 σ 31 w 1 σ 12 + w 2 σ 22 + w 3 σ 32 w 1 σ 13 + w 2 σ 23 + w 3 σ 33 w 1 w 2 w 3 = (w 1 σ 11 + w 2 σ 21 + w 3 σ 31 )w 1 + (w 1 σ 12 + w 2 σ 22 + w 3 σ 32 )w 2 + (w 1 σ 13 + w 2 σ 23 + w 3 σ 33 )w 3 = w 1 w 1 σ 11 + w 1 w 2 σ 21 + w 1 w 3 σ 31 + w 2 w 1 σ 12 + w 2 w 2 σ 22 + w 2 w 3 σ 32 + w 1 w 3 σ 13 + w 2 w 3 σ 23 + w 3 w 3 σ 33 st 1 w = [ 1 1 1 ] w 1 w 2 w 3 = w 1 + w 2 + w 3 = 1
Minimum variance portfolio numerical example min w Σw = [ w 1 w 2 w 3 ] 0083702 0035427 0013336 0035427 0194149 0063565 0013336 0063565 0163351 st 1 w = [ 1 1 1 ] w 1 w 2 w 3 = 1 w 1 w 2 w 3 w = Σ 1 1 1 Σ 1 1
Minimum variance portfolio inverse of covariance matrix Σ = 0083702 0035427 0013336 0035427 0194149 0063565 0013336 0063565 0163351 Σ 1 = 12951 2311 0158 2311 6315 2269 0158 2269 7018 ΣΣ 1 = 0083702 0035427 0013336 0035427 0194149 0063565 0013336 0063565 0163351 12951 2311 0158 2311 6315 2269 0158 2269 7018 0 1 0 1 0 0 0 0 1 = I
Minimum variance portfolio optimal weights w = Σ 1 1 1 Σ 1 1 = [ 1 1 1 ] 12951 2311 0158 2311 6315 2269 0158 2269 7018 1 1 1 12951 2311 0158 2311 6315 2269 0158 2269 7018 1 1 1 12951 2311 0158 2311 + 6315 2269 0158 2269 + 7018 = [ 12951 2311 0158 2311 + 6315 2269 0158 2269 + 7018 ] 1 1 1 10482 1735 4591 = [ ] 10482 1735 4591 1 1 1 10482 1735 10482 1735 4591 4591 = [ ] = = 0624 0103 10482 + 1735 + 4591 16808 0273
Minimum variance portfolio optimal solution w Σw = [ 0624 0103 0273 ] 0083702 0035427 0013336 0035427 0194149 0063565 0013336 0063565 0163351 0624 0103 0273 = 00595 1 w = [ 1 1 1 ] 0624 0103 0273 = 1 σ p = 00595 244% µ p = µ w = [ 008220 013097 010622 ] 0624 0103 0273 = 94%
Minimum variance portfolio µ 15% 10% 938% Minimum variance portfolio 5% r = 4% Risk-free asset 2439% 10% 20% 30% 40% 50% σ
Required return portfolio Optimization problem: Optimal solution: min w Σw min st µ w = µ st 1 w = 1 n n w i w j σ ij i=1 j=1 n w i µ i = µ i=1 n w i = 1 i=1 w = Σ 1 µ(1 Σ 1 1µ µ Σ 1 1) + 1(µ Σ 1 µ µ Σ 1 1µ ) µ Σ 1 µ1 Σ 1 1 (µ Σ 1 1) 2
Required return portfolio scalar presentation min 3 3 w i w j σ ij i=1 j=1 = w 1 w 1 σ 11 + w 1 w 2 σ 12 + w 1 w 3 σ 13 + w 2 w 1 σ 21 + w 2 w 2 σ 22 + w 2 w 3 σ 23 + w 3 w 1 σ 31 + w 3 w 2 σ 32 + w 3 w 3 σ 33 st 3 w i µ i = w 1 µ 1 + w 2 µ 2 + w 3 µ 3 = µ i=1 3 w i = w 1 + w 2 + w 3 = 1 i=1
Required return portfolio matrix presentation min w Σw = [ ] w 1 w 2 w 3 σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33 w 1 w 2 w 3 = w 1 w 1 σ 11 + w 1 w 2 σ 21 + w 1 w 3 σ 31 + w 2 w 1 σ 12 + w 2 w 2 σ 22 + w 2 w 3 σ 32 + w 1 w 3 σ 13 + w 2 w 3 σ 23 + w 3 w 3 σ 33 st µ w = [ µ 1 µ 2 µ 3 ] w 1 w 2 w 3 1 w = [ 1 1 1 ] w 1 w 2 w 3 = w 1 µ 1 + w 2 µ 2 + w 3 µ 3 = µ = w 1 + w 2 + w 3 = 1
Required return portfolio numerical example min w Σw = [ w 1 w 2 w 3 ] 0083702 0035427 0013336 0035427 0194149 0063565 0013336 0063565 0163351 st µ w = [ 008220 013097 010622 ] w 1 w 2 w 3 1 w = [ 1 1 1 ] w 1 w 2 w 3 = 1 = 009 w 1 w 2 w 3 w = Σ 1 µ(1 Σ 1 1µ µ Σ 1 1) + 1(µ Σ 1 µ µ Σ 1 1µ ) µ Σ 1 µ1 Σ 1 1 (µ Σ 1 1) 2
Required return portfolio optimal weights w = Σ 1 µ(1 Σ 1 1µ µ Σ 1 1) + 1(µ Σ 1 µ µ Σ 1 1µ ) µ Σ 1 µ1 Σ 1 1 (µ Σ 1 1) 2 1 Σ 1 1 = [ 1 1 1 ] 12951 2311 0158 2311 6315 2269 0158 2269 7018 1 1 1 = 168080 µ Σ 1 1 = [ 008220 013097 010622 ] 12951 2311 0158 2311 6315 2269 0158 2269 7018 1 1 1 = 157651 µ Σ 1 µ = [ 008220 013097 010622 ] 12951 2311 0158 2311 6315 2269 0158 2269 7018 008220 013097 010622 = 015936
Required return portfolio optimal weights 1 Σ 1 1 = 168080 µ Σ 1 1 = 157651 µ Σ 1 µ = 015936 = = 12951 2311 0158 2311 6315 2269 0158 2269 7018 12951 2311 0158 2311 6315 2269 0158 2269 7018 w = Σ 1 µ(1 Σ 1 1µ µ Σ 1 1) + 1(µ Σ 1 µ µ Σ 1 1µ ) µ Σ 1 µ1 Σ 1 1 (µ Σ 1 1) 2 = = 008220 013097 010622 008220 013097 010622 074512 039610 043529 004753 002527 002777 (168080 009 157651) + 1 1 (015936 157651 009) 1 ( 006379) + 015936 168080 157651 2 12951 2311 0158 2311 6315 2269 0158 2269 7018 0193 ( 006379) + 10482 1735 (001747) 4591 0193 + 003031 018312 008020 0193 0193 0136 0005 0052 0705 0026 0269 1 1 1 (001747)
Required return portfolio optimal solution w Σw = [ 0705 0026 0269 ] 0083702 0035427 0013336 0035427 0194149 0063565 0013336 0063565 0163351 1 w = [ 1 1 1 ] 0705 0026 0269 = 1 0705 0026 0269 = 00608 σ p = 00608 246% µ p = µ w = [ 008220 013097 010622 ] 0705 0026 0269 = 90%
Required return portfolio µ 15% 10% 900% Required return portfolio 5% r = 4% Risk-free asset 2465% 10% 20% 30% 40% 50% σ
Tangent portfolio Portfolio optimization Optimization problem: max w µ r w Σw max st 1 w = 1 st [ n ]/ n n w i µ i r w i w j σ ij i=1 i=1 j=1 n w i = 1 i=1 Optimal solution: w = Σ 1e µ 1 Σ 1e µ
Tangent portfolio numerical example max w µ r w Σw = [ ] w 1 w 2 w 3 [ ] w1 w 2 w 3 008220 013097 010622 r 0083702 0035427 0013336 0035427 0194149 0063565 0013336 0063565 0163351 w 1 w 2 w 3 st 1 w = [ 1 1 1 ] w 1 w 2 w 3 = 1 w = Σ 1e µ 1 Σ 1e µ
Tangent portfolio optimal weights w = Σ 1e µ 1 Σ 1e µ = [ 1 1 1 ] 12951 2311 0158 2311 6315 2269 0158 2269 7018 12951 2311 0158 2311 6315 2269 0158 2269 7018 008220 004 013097 004 010622 004 008220 004 013097 004 010622 004 032584 032670 025165 = [ ] 10482 1735 4591 004220 009097 006622 = 032584 032670 025165 090419 = 03603 03614 02783
Tangent portfolio optimal solution w Σw = [ 03603 03614 02783 ] 0083702 0035427 0013336 0035427 0194149 0063565 0013336 0063565 0163351 1 w = [ 1 1 1 ] 03603 03614 02783 = 1 03603 03614 02783 = 00736 σ p = 00736 271% µ p = µ w = [ 008220 013097 010622 ] 03603 03614 02783 = 107%
Tangent portfolio Portfolio optimization µ 15% 10% 1065% Tangent portfolio 5% r = 4% Risk-free asset 2712% 10% 20% 30% 40% 50% σ
The three portfolios Portfolio optimization µ 30% 20% 10% Risk-free asset r = 4% 10% 20% 30% 40% 50% σ
Minimum variance portfolio The SAS System Obs Company ISIN W BETA ER STDR ERP STDRP 1 Norvestia FI0009000160 021165 058064 007484 020909 006273 011317 2 Kesko Corporation FI0009000202 009935 070333 008220 028931 3 Stockmann A FI0009000236 004581 107083 010425 030315 4 Stockmann B FI0009000251-004742 112749 010765 031422 5 Tieto Corporation FI0009000277-000504 151621 013097 044062 6 Metsa Board A FI0009000640-000680 132231 011934 045143 7 Metsa Board B FI0009000665-003004 153087 013185 052172 8 Nokia Corporation FI0009000681 002840 123446 011407 043157 9 Uponor FI0009002158 003071 089957 009397 029894 10 Interavanti FI0009002349 014327 014741 004884 034182 11 Citycon FI0009002471-001787 087742 009264 028716 12 Raisio FI0009002943 004992 064405 007864 031095 13 Rautaruukki FI0009003552-012822 130037 011802 037081 14 Wartsila FI0009003727-008447 136215 012173 036483 15 Viking Line FI0009005250 021455 038934 006336 022893 16 Nokian Tyres FI0009005318-000814 135390 012123 042357 17 Sponda FI0009006829 003959 099136 009948 034661 18 Ramirent FI0009007066-000726 120026 011202 038748 19 Fortum FI0009007132 014474 069419 008165 025577 20 Rapala FI0009007355 008026 038842 006331 029279 21 Keskisuomalainen FI0009007546 008894 060424 007625 028345 22 Sanoma FI0009007694 009840 085523 009131 026706 23 Teleste FI0009007728 002947 145004 012700 051571 24 Elisa FI0009007884-001234 110367 010622 040417 25 Aspo FI0009008072 004254 077206 008632 030626
Minimum variance portfolio µ 15% 10% 627% 5% r = 4% Risk-free asset 1132% 10% 20% 30% 40% 50% σ
Required return portfolio The SAS System Obs Company ISIN W BETA ER STDR ERP STDRP 1 Norvestia FI0009000160 008848 058064 007484 020909 009000 015860 2 Kesko Corporation FI0009000202 004000 070333 008220 028931 3 Stockmann A FI0009000236 003184 107083 010425 030315 4 Stockmann B FI0009000251 001818 112749 010765 031422 5 Tieto Corporation FI0009000277 005671 151621 013097 044062 6 Metsa Board A FI0009000640 001792 132231 011934 045143 7 Metsa Board B FI0009000665 002031 153087 013185 052172 8 Nokia Corporation FI0009000681 006313 123446 011407 043157 9 Uponor FI0009002158 004397 089957 009397 029894 10 Interavanti FI0009002349 003404 014741 004884 034182 11 Citycon FI0009002471 001368 087742 009264 028716 12 Raisio FI0009002943 003318 064405 007864 031095 13 Rautaruukki FI0009003552 000551 130037 011802 037081 14 Wartsila FI0009003727 000536 136215 012173 036483 15 Viking Line FI0009005250 008268 038934 006336 022893 16 Nokian Tyres FI0009005318 001094 135390 012123 042357 17 Sponda FI0009006829 001400 099136 009948 034661 18 Ramirent FI0009007066 000752 120026 011202 038748 19 Fortum FI0009007132 009028 069419 008165 025577 20 Rapala FI0009007355 007662 038842 006331 029279 21 Keskisuomalainen FI0009007546 004277 060424 007625 028345 22 Sanoma FI0009007694 008925 085523 009131 026706 23 Teleste FI0009007728 004337 145004 012700 051571 24 Elisa FI0009007884 001954 110367 010622 040417 25 Aspo FI0009008072 005073 077206 008632 030626
Required return portfolio µ 15% 10% 900% 5% r = 4% Risk-free asset 1586% 10% 20% 30% 40% 50% σ
Tangent portfolio Portfolio optimization The SAS System Obs Company ISIN W BETA ER STDR ERP STDRP 1 Norvestia FI0009000160 005832 058064 007484 020909 009668 017872 2 Kesko Corporation FI0009000202 002546 070333 008220 028931 3 Stockmann A FI0009000236 002842 107083 010425 030315 4 Stockmann B FI0009000251 003424 112749 010765 031422 5 Tieto Corporation FI0009000277 007183 151621 013097 044062 6 Metsa Board A FI0009000640 002397 132231 011934 045143 7 Metsa Board B FI0009000665 003264 153087 013185 052172 8 Nokia Corporation FI0009000681 007163 123446 011407 043157 9 Uponor FI0009002158 004721 089957 009397 029894 10 Interavanti FI0009002349 000729 014741 004884 034182 11 Citycon FI0009002471 002141 087742 009264 028716 12 Raisio FI0009002943 002908 064405 007864 031095 13 Rautaruukki FI0009003552 003826 130037 011802 037081 14 Wartsila FI0009003727 002736 136215 012173 036483 15 Viking Line FI0009005250 005039 038934 006336 022893 16 Nokian Tyres FI0009005318 001561 135390 012123 042357 17 Sponda FI0009006829 000774 099136 009948 034661 18 Ramirent FI0009007066 001114 120026 011202 038748 19 Fortum FI0009007132 007694 069419 008165 025577 20 Rapala FI0009007355 007573 038842 006331 029279 21 Keskisuomalainen FI0009007546 003146 060424 007625 028345 22 Sanoma FI0009007694 008700 085523 009131 026706 23 Teleste FI0009007728 004677 145004 012700 051571 24 Elisa FI0009007884 002735 110367 010622 040417 25 Aspo FI0009008072 005274 077206 008632 030626
Tangent portfolio Portfolio optimization µ 15% 10% 967% 5% r = 4% Risk-free asset 1787% 10% 20% 30% 40% 50% σ
Constrained minimum variance portfolio n n min w Σw min w i w j σ ij i=1 j=1 st 1 w = 1 st n w i = 1 i=1 w 0 w i 0 i w 1 w i 1 i
Constrained minimum variance portfolio The SAS System Obs Company ISIN W BETA ER STDR ERP STDRP 1 Norvestia FI0009000160 016634 058064 007484 020909 007132 012496 2 Kesko Corporation FI0009000202 007281 070333 008220 028931 3 Stockmann A FI0009000236 000000 107083 010425 030315 4 Stockmann B FI0009000251 000000 112749 010765 031422 5 Tieto Corporation FI0009000277 000000 151621 013097 044062 6 Metsa Board A FI0009000640 000000 132231 011934 045143 7 Metsa Board B FI0009000665 000000 153087 013185 052172 8 Nokia Corporation FI0009000681 000344 123446 011407 043157 9 Uponor FI0009002158-000000 089957 009397 029894 10 Interavanti FI0009002349 012518 014741 004884 034182 11 Citycon FI0009002471 000000 087742 009264 028716 12 Raisio FI0009002943 002515 064405 007864 031095 13 Rautaruukki FI0009003552 000000 130037 011802 037081 14 Wartsila FI0009003727 000000 136215 012173 036483 15 Viking Line FI0009005250 023997 038934 006336 022893 16 Nokian Tyres FI0009005318 000000 135390 012123 042357 17 Sponda FI0009006829-000000 099136 009948 034661 18 Ramirent FI0009007066 000000 120026 011202 038748 19 Fortum FI0009007132 005600 069419 008165 025577 20 Rapala FI0009007355 010754 038842 006331 029279 21 Keskisuomalainen FI0009007546 005952 060424 007625 028345 22 Sanoma FI0009007694 009234 085523 009131 026706 23 Teleste FI0009007728 000941 145004 012700 051571 24 Elisa FI0009007884 000000 110367 010622 040417 25 Aspo FI0009008072 004231 077206 008632 030626
Constrained minimum variance portfolio µ 15% 10% 5% r = 4% Risk-free asset 713% 1250% 10% 20% 30% 40% 50% σ
Constrained required return portfolio n n min w Σw min w i w j σ ij i=1 j=1 st µ w = µ st n w i µ i = µ i=1 1 w = 1 n w i = 1 i=1 w 0 w i 0 i w 1 w i 1 i
Constrained required return portfolio The SAS System Obs Company ISIN W BETA ER STDR ERP STDRP 1 Norvestia FI0009000160 008840 058064 007484 020909 009000 015860 2 Kesko Corporation FI0009000202 004000 070333 008220 028931 3 Stockmann A FI0009000236 003139 107083 010425 030315 4 Stockmann B FI0009000251 001862 112749 010765 031422 5 Tieto Corporation FI0009000277 005674 151621 013097 044062 6 Metsa Board A FI0009000640 001791 132231 011934 045143 7 Metsa Board B FI0009000665 002030 153087 013185 052172 8 Nokia Corporation FI0009000681 006315 123446 011407 043157 9 Uponor FI0009002158 004403 089957 009397 029894 10 Interavanti FI0009002349 003405 014741 004884 034182 11 Citycon FI0009002471 001366 087742 009264 028716 12 Raisio FI0009002943 003321 064405 007864 031095 13 Rautaruukki FI0009003552 000550 130037 011802 037081 14 Wartsila FI0009003727 000534 136215 012173 036483 15 Viking Line FI0009005250 008277 038934 006336 022893 16 Nokian Tyres FI0009005318 001096 135390 012123 042357 17 Sponda FI0009006829 001403 099136 009948 034661 18 Ramirent FI0009007066 000744 120026 011202 038748 19 Fortum FI0009007132 009026 069419 008165 025577 20 Rapala FI0009007355 007664 038842 006331 029279 21 Keskisuomalainen FI0009007546 004274 060424 007625 028345 22 Sanoma FI0009007694 008919 085523 009131 026706 23 Teleste FI0009007728 004341 145004 012700 051571 24 Elisa FI0009007884 001949 110367 010622 040417 25 Aspo FI0009008072 005076 077206 008632 030626
Constrained tangent portfolio max w µ r w Σw max st 1 w = 1 st [ n ]/ n n w i µ i r w i w j σ ij i=1 i=1 j=1 n w i = 1 i=1 w 0 w i 0 i w 1 w i 1 i
Constrained tangent portfolio The SAS System Obs Company ISIN W BETA ER STDR ERP STDRP 1 Norvestia FI0009000160 005831 058064 007484 020909 009668 017872 2 Kesko Corporation FI0009000202 002545 070333 008220 028931 3 Stockmann A FI0009000236 002838 107083 010425 030315 4 Stockmann B FI0009000251 003430 112749 010765 031422 5 Tieto Corporation FI0009000277 007183 151621 013097 044062 6 Metsa Board A FI0009000640 002400 132231 011934 045143 7 Metsa Board B FI0009000665 003262 153087 013185 052172 8 Nokia Corporation FI0009000681 007163 123446 011407 043157 9 Uponor FI0009002158 004721 089957 009397 029894 10 Interavanti FI0009002349 000729 014741 004884 034182 11 Citycon FI0009002471 002139 087742 009264 028716 12 Raisio FI0009002943 002908 064405 007864 031095 13 Rautaruukki FI0009003552 003828 130037 011802 037081 14 Wartsila FI0009003727 002734 136215 012173 036483 15 Viking Line FI0009005250 005042 038934 006336 022893 16 Nokian Tyres FI0009005318 001561 135390 012123 042357 17 Sponda FI0009006829 000776 099136 009948 034661 18 Ramirent FI0009007066 001114 120026 011202 038748 19 Fortum FI0009007132 007695 069419 008165 025577 20 Rapala FI0009007355 007574 038842 006331 029279 21 Keskisuomalainen FI0009007546 003143 060424 007625 028345 22 Sanoma FI0009007694 008698 085523 009131 026706 23 Teleste FI0009007728 004678 145004 012700 051571 24 Elisa FI0009007884 002734 110367 010622 040417 25 Aspo FI0009008072 005276 077206 008632 030626
Constrained portfolios Portfolio optimization µ 30% 20% 10% Risk-free asset r = 4% 10% 20% 30% 40% 50% σ
Beta-neutral minimum variance portfolio min w Σw min st β w = 0 st 1 w = 1 n n w i w j σ ij i=1 j=1 n w i β i = 0 i=1 n w i = 1 i=1
Beta-neutral minimum variance portfolio The SAS System Obs Company ISIN W BETA ER STDR ERP STDRP 1 Norvestia FI0009000160 031437 058064 007484 020909 004000 014622 2 Kesko Corporation FI0009000202 014880 070333 008220 028931 3 Stockmann A FI0009000236 005737 107083 010425 030315 4 Stockmann B FI0009000251-010199 112749 010765 031422 5 Tieto Corporation FI0009000277-005648 151621 013097 044062 6 Metsa Board A FI0009000640-002737 132231 011934 045143 7 Metsa Board B FI0009000665-007199 153087 013185 052172 8 Nokia Corporation FI0009000681-000052 123446 011407 043157 9 Uponor FI0009002158 001966 089957 009397 029894 10 Interavanti FI0009002349 023427 014741 004884 034182 11 Citycon FI0009002471-004416 087742 009264 028716 12 Raisio FI0009002943 006388 064405 007864 031095 13 Rautaruukki FI0009003552-023966 130037 011802 037081 14 Wartsila FI0009003727-015935 136215 012173 036483 15 Viking Line FI0009005250 032442 038934 006336 022893 16 Nokian Tyres FI0009005318-002405 135390 012123 042357 17 Sponda FI0009006829 006091 099136 009948 034661 18 Ramirent FI0009007066-001960 120026 011202 038748 19 Fortum FI0009007132 019004 069419 008165 025577 20 Rapala FI0009007355 008332 038842 006331 029279 21 Keskisuomalainen FI0009007546 012740 060424 007625 028345 22 Sanoma FI0009007694 010606 085523 009131 026706 23 Teleste FI0009007728 001789 145004 012700 051571 24 Elisa FI0009007884-003892 110367 010622 040417 25 Aspo FI0009008072 003570 077206 008632 030626
Beta-neutral minimum variance portfolio µ 15% 10% 5% r = 4% Risk-free asset 1462% 10% 20% 30% 40% 50% σ