mathematical finance, 8(4):325-347, 1998.web.cs.iastate.edu/~honavar/portfolio-selection.pdf ·...

23
Mathematical Finance, 8(4):325-347, 1998.

Upload: others

Post on 01-Aug-2020

19 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Mathematical Finance, 8(4):325-347, 1998.web.cs.iastate.edu/~honavar/portfolio-selection.pdf · Mathematical Finance, 8(4):325-347, 1998. ! #" %$'&)( * +% -,. */0 1 #+324,. " 5%687

Mathematical Finance, 8(4):325-347, 1998.

����������� ������������������������������! #"����%$'&)(��*����+%����-,.���*/01 #+324,.���"

5%687�9;:=<?>A@CBEDGFIHAJ�DK:L JMFONQPQRSBET�6VUA:XW*UZY[JMT\F=6VR\9]J�UX^�_�9GBEUA_�BV`

aCUb9G7�BETc`�9GRd�JVY L 6VDG9eY[JMT\UQ9K6^f6gU�Rh6 L T\PQi�j L0k l�mMn�ogp:ZNQq�r�_s`�B�>tPA_E`\_M>uBV:ZP

v�J�HABETSR�w�>�^�_hqA6VNQ9GTSBk�x�y4x1z 6gHA`

{E| n <}6gT\~ k 7�BsUZPQB�jZv�J�J�F k��f�8l�?DGJ�TSqA6gF�<�6gTS~�jA��� nf�8l��M�`S_�q�6gNQ9]T\B�r�T\B8`�BV6VTh_hq->�6VR\RE>�_�J�F

��J�Tc6gF�^�9GUQ�MBETk�x�y�x1z 6gH�`

{s| n <�6gTS~ k 7�BEU�PQB�j�v�J�J�F k�������.DGJMT\qA6VF�<�6VT\~�jA��� n��VlM���`�9GUb��BET�rCTSBV`�B86gTh_hq->�6gR\Rs>�_�JMF

��6gU�Y[T\BV:=��>Q��6VT\FIPQR\qL JMFONQPQRSBET�6VUA:XW*UZY[JMT\F=6VR\9]J�UX^�_�9GBEUA_�BV`

aCUQ9]7�BETh`9GRd3JVY L 6gD]9eY[J�TSUQ9;6^�6VUZRc6 L TSPQi�j L0k lMm�nMogpF�6gUZY[TSBV:�r%_E`�B�>uPA_E`S_�>uBs:ZP

���A��\�h�M����� ¢¡V£[ ¤¥ ¦E§0¨h¦�©�¦Eª;«�¬�¦g �¬�¦s­� ¤[§¥®� ¦E§�¨h«�¯�©�£[¬�§[°V®²±-°V¬�³*°´¨h³*°V¬� ­� ¤�¨h«�®�©�¤[§-§¥°V 0¤*¨h®C �±� \¨h«�§[°´¨�¤�§[°V 

µ  ¤¥§0³�©�¦V¤¥§¶¨h¦E§·ªK£¥  µ ¨h«¸¨�¦V³ ¹�¡�©�£¥§;º]©�«�¬�©�¹V §¥ £[®C¬�¦V ¹�¬�¦´°g¬�¦g¹V¤[¬�¯�°E§.º]£¥©�®»§¥°V �¨h³�§[¼M¨�«�®�¨h£[½� §?©�¼V§[³�©�®� �¤S¾¿-°V À¨�«�¯�©�£¥¬�§[°V®² ®�¡g«�©hÁE¤�¨�®�¼g«�§¥¬�¡V«�¬t³S¨�§[¬�­� .¼V¡M¹�¨h§[ �£¥¼V«� 0¹g £[¬�­� �¹�¼V¤[¬�¦V¯4¨ÂºG£¶¨h®C ±�©�£[½�¬�¦E§[£¥©s¹g¼V³ ¹ µ ÁÃ}¬�­E¬�¦V ¦4¨h¦g¹%�I¨�£[®�¼V§¥°�¾�Ä.¼V£�¨h«�¯�©�£[¬�§[°g®#¬�¤-­� £¥Á%¤¥¬�®�¡V«� ?§¥©�¬�®�¡g«� �®� ¦E§�¨�¦V¹�£[ ÅE¼V¬�£[ �¤}©�¦V«�Á�³©�¦V¤[§*¨h¦E§¤[§[©�£¶¨�¯� ?¨h¦g¹�³©�®C¡V¼V§[¬�¦V¯�§¥¬�®C �¡� £�¤[§¥©s³*½Â¬�¦� S¨h³*°�§¥£¶¨�¹V¬�¦V¯�¡� £[¬�©E¹�¾Z�� -§[ ¤¥§[ ¹�§[°g ?¡� £;º]©�£[®�¨h¦V³ -©�º�©�¼V£¨h«�¯�©�£[¬�§[°V®Æ©�¦C£[ S¨�«�¤[§[©E³*½�¹�¨h§¶¨Çº]£¥©�®�§[°V �È. ±OÉ�©�£[½�ÊE§¥©s³*½�ËQÌE³*°M¨�¦V¯� �¨h³�³¼V®�¼V«¸¨h§¥ ¹�¹V¼V£¥¬�¦V¯À¨ÂÍ�ÍcªKÁ� S¨h£¡M £¥¬�©s¹Z¾AÄ?¦C§[°V¬�¤�¹M¨h§*¨VÎE©�¼V£Ï¨�«�¯�©�£¥¬�§[°V®�³«� S¨�£[«�Á0©�¼V§[¡� £·º]©�£¥®�¤�§[°V  µ  ¤[§-¤¥¬�¦g¯�«� ?¤¥§[©E³*½%¨h¤�±� «�«�¨�¤ÏÐ�©h­� £SÑ ¤¼V¦V¬�­� £¥¤¶¨�«A¡M©�£[§·º]©�«�¬�©�¤¥ «� ³§[¬�©�¦I¨h«�¯�©�£[¬�§[°V®3¾��� �¨h«�¤[©�¡V£¥ ¤[ �¦s§Â£[ �¤[¼V«�§[¤¢º]©�£}§¥°V �¤[¬�§¥¼M¨h§¥¬�©�¦´¬�¦3±-°V¬�³*°�§[°V ¬�¦E­� ¤[§¥©�£-°M¨�¤?¨h³³ �¤[¤?§¥©�¨h¹V¹g¬�§¥¬�©�¦M¨�«�Ò¥¤[¬�¹V �¬�¦Eº]©�£¥®%¨�§[¬�©�¦�¾ Ó

Page 2: Mathematical Finance, 8(4):325-347, 1998.web.cs.iastate.edu/~honavar/portfolio-selection.pdf · Mathematical Finance, 8(4):325-347, 1998. ! #" %$'&)( * +% -,. */0 1 #+324,. " 5%687

� ������������ ��������������������� �"!$#%�'&(�")+*-,.�/�0,.�%1"����!�23� �"!4#5*-6%&(��,-!�7�298:7/,<;�7=#5;�7/,-��1%���$#5*.23&%�>!�!?7/�$��#"23�$8@� #5*-!�7A#5�3!�7/�0BC����!;�&(�/��!?#%�%!D)E��� B�#5*.#%�/;���F0��&/��!�GH&%*-,<&I,.�%1"����!�23� �"!@��!���#5!>��6%J%K�LM7/�#5*-6%&(��,-!�7�2N� 24�/*-&OJ/��#423P/*-!�,.�/*-,-; #5!�,-1%�4P���FQ#5!>���P/*-�4F�� ��,-1%��FAP/��,.�/6R#SG+��#%23��8@&(��TI,.�"!���&/FQP/;���FUB"JRV:,-1(,.�/� �=#%�/FW�U#"��23P/!�7�XZY"[�\�K�]^P��^#5*-6%&/��,-!�7�2_,-�^1%� ��J��,.24�/*-�3!�&4,.24�/*-� 23� �"!`#"�/F0,-!��:!�,.23�#%�/F0�>!�&(��#56"�`����a�P/,.��� 23� �"!��:6(��&58b*<,.�/� #%�D*<Jc,.�0!?7/��(P�23B�� �d&5G���!�&/;eT"� Kf�g���� �D,h2i� �%!��c&(�j��� #k*lM��8_m�&(�>Ton/!>&(;�Tpfqg(;�7�#%�/6%��FQ#k!�#A,.�/F�,<; #k!��U!?7�#5!�&(P���#5*-6%&(�D,<!?7�2r&(P/!��C� �sGH&(��23�t &51%� ��uv�`Xxw�\�P��/,<1"� ����#5*y�C&(��!�GH&%*-,-&�#k*<6"&(��,-!�7�2�K

Ld7/�^GH&%*-*<&58M,.�/6���,.24�/*-�3��g�#%24�/*-�3F�� 23&(�/��!��>#5!����M!�7/�4�C&O8@� �@&kGz;�&(�/��!�#%�"!�)E��� B�#k*h#"�/;���FR��&/��!�GH&%*-,<&��>!���#5!��?)6%,-��� K'{����?P�23��!�7�#5!|!E8@&W,.�%1"����!�23� �"!��4#%����#}1O#5,-*.#%B/*-�%K=LM7/�I~�����!3,-�$#W��,-��Tk)HG������%�@�/&5)+6(�D&O8d!�7W,.�%1"����!�23� �"!��!�&/;�T08:7/&%���415#5*.P/�0�/��1%� �^;�7�#%�/6%��� K�LM7/�I����;�&(�/F',h�"1%����!?23� �%!3,-��#c7%J���&"!�7/��!�,-; #5*M7/,-6(7/*-JU1%&"*h#k!�,-*<�I��!�&/;�TQK]^�'��1%� ��FQ#}J/� ��!?7/�41O#5*.P/��&5GM!�7/,-�|��!�&/;eTcF�&(P�B/*-���$#%�/F'&(�W&/F�FUFQ#}J/�,-!��31O#5*.P/�I,-�i7�#k*<1"��F�K0LM7/�$����*.#5!>,<1"�����!�P��>�/�4&5G^!�7/�I~�����!4�>!�&(;�T'; #%�pBC�0F�����; ��,.BC��F�B"JA!�7/������a�P/� �/;����"� �%�}�%� ������#%�/F�&5G�!�7/�c����;�&(�/FbB%JA!�7/�����a�P/� �/;��A�� ��Y��"�� �?Y�� �}����K�ld��,<!?7/� �:,h�"1%����!?23� �%!^#5*-&(�/�3; #%�c,.�/; ��� #5���3,.�c1O#5*.P/�4B"JR23&(�D�^!�7�#%�W#3G+#5;�!�&/�@&5G�Y��B�P/!M#^��!?��#5!���6%J|;�&(23B/,.�/,h�/6S!�7/��!�8@&^,.�%1"����!�23� �"!��z; #"�46(�D&O8=��g���&/�/� �%!�,.#5*-*-J%KM]^�/�:��P/;e7S��!��>#5!���6%J|���/*-,<!>�M!�7/�,.�%1"����!�&(��uv��!�&%!�#5*"8�� #k*<!?73��1%� �/*-J4B���!�8@��� �3!�7/�M!�8@&�,.�"1%����!�2i� �%!��}�%#%�/F$24#5,.�"!�#5,.�/�z!?7/,<����1%� �|���/*-,-!�#k!�!�7/�@� �/F&5Gy� #k;e7IFQ#}J%Kz]^�I&/F�F�FQ#}J(�@!�7/�����*.#5!>,<1"�^8�� #k*<!?7IF���; ��� #5������B"J�#^G+#5;�!�&/�z&kG ��|� ��� ��^� ��^�N�� K��M&58@��1%� ���&(����1%� ��FQ#}J/�M!�7/�����*.#5!�,-1%�38@� #5*-!�7I6(�D&O8d��B%J �� � �z� �� � Y ���� K�LM7/P/� �Q#OG�!�� ��!E8@&3;�&(�/����; P/!>,<1"��!?��#5F�,.�/6FQ#}J(�d!�7/�,.�"1%����!�&(��u��@8�� #k*<!?7�6(��&58M�MB"J�#�G�#5;�!>&(�@&5G ��3�p���N�� Kz�H!:!�#%Tk����&/�/*<J�!�8@��*-1%�^FQ#}J(�d!�&4F�&(P�B/*-�3!�7/�8@� #5*-!�7��Q#%�/F�&51%� �dY%�c!���#5F�,.�/64FQ#}J/�M!�7/�^8@� #5*-!�7I6(��&58M�MB"J�#�G�#5;�!>&(��&5Gd� ��k�D� K�s�%1"����!�23� �"!M��!��>#5!���6%,-���M8�7/,-;�7W24#5,.�%!?#5,.�0#|~Qg/��FcG���#k;�!�,-&(�I&5Gz!�7/�|!�&%!�#5*�8@� #5*-!�7�,h��� #5;�7�&5G�!�7/��P��/F�� �E)*-J(,.�/6�,.�"1%����!�23� �"!�� � *-,.T5�4!�7/�|&(�/�F�����; �D,hBC��F�#"B�&51%�%�Q#%�D�; #5*-*-��Fp¡?¢%£�¤�¥H¦%£Q¥¨§�©�ªO«>¦%¬-¦%£Q¡?ª�­`®¯¢%©?¥ °e¢"¬<±¨¢�¤�¥H©�¦%¥Hªs²(±¨ª�¤�K³z�D��1(,-&(P/��*-J%� t &51%� �|XZw}\zF�����; �D,hBC��F�#��C&(��!�GH&%*-,-&5)+����*-��;�!�,-&(�A#5*-6%&(��,-!�7�2´!�7�#5!����&515#%B/*-Jµ�C� �sGH&(��23��¶�#k*h2i&%��!^#5�8@��*-*<·W#5�!?7/��«>ª�¤�¥M;�&/�/��!�#%�"!�)E��� B�#5*.#%�/;���F=��&/��!�GH&%*-,<& K4�+�'!�7/,-����#%�C� ����8@�3F�����; ��,.B��0#��/��8¸#5*-6%&(��,-!�7�2´8M,-!�7��,.23,-*.#%������&(�C� ��!�,-��� Kº¹�,.T5�W!�7/��������P/*-!��SG�&(� t &51%� ��u��$#k*<6"&(��,-!�7�2��@!�7/,-�0�C� �sG�&/��24#%�/;��W����&(��� �D!EJ=,-������&51%� �8M,-!�7/&(P/!:24#%T",h�/6�#%�"J4��!�#k!�,-��!�,-; #5*y#5�>��P�24�/!�,-&(�/�M&/�I!�7/����#5!?P����^&5G�!�7/�^��!�&/;eTI24#%�>T5��! K

Ld7/��!�7/��&(����!>,<; #k*�B�&/P��/F»8@�U����&51%�'&(�b!�7/�U�C� �sGH&(��24#%�/;��'&5G4&(P��0#5*-6%&/��,-!�7�2¼����*.#5!>,<1"�U!�&=!?7/��BC����!;�&(�/��!?#%�%!D)E��� B�#5*.#%�/;���F���&(�D!�G�&"*<,-&�,-��/&%!#5�:��!��D&(�/6�#k��!�7/�4BC&(P��/FU����&51%��FWB%J t &51%� �M#%�/F�]^��F�� �"!�*-,-;e7�X.�%�>\EK�d&O8@��1%� ����!�7/�!>,h2i��#%�/FW����#5;��3����a�P/,h�D��F0GH&(�@&/P���#5*-6%&/��,-!�7�2�,-�^*-,.�/� #%�:,.�c!�7/���/P�23B�� �d&5Gz��!�&/;eT"�M8�7/� �D� #5�t &51%� ��uv�y#5*-6%&(��,-!�7�2º,-����g���&(�/� �"!�,.#5*�,.�|!�7/�M�(P�23BC� ��&5GQ��!�&/;eT"� K¾½R&/����&51%� ���}8���!�����!���F3&(P���#5*-6%&(�D,<!?7�2º��g¯�C� ��,x)23� �"!�#5*-*-J3&(�7/,-��!�&/��,-; #5*�FQ#5!�#�G��D&(2º!�7/��ld��8=m�&(��T|n/!�&/;eT|f�g/;e7�#%�/6"�`�Hldm`n�f � #5;�; P�23P/*.#5!���F3&51%� �y#^Y"Y})+J%� #%���� �D,<&/F���#%�/F�GH&(P��/F�!�7�#5!d&(P��@#5*-6%&(��,-!�7�2¿;�*-� #%��*-J�&(P/!��C� �sG�&/��23�M!�7/�#k*<6"&(��,-!�7�2¸&5G t &51%� �@#%�/F0]^��F�� �"!�*-,-;e7�K

À &%*-*-&O8d,h�/6 t &O1"� �4#%�/Fj]^��F�� �"!�*-,<;�7ÁX.�"��\��@8��W#5*-��&p��������� �"!�������P/*-!��IGH&(�4!?7/�0��,-!�P�#5!>,<&/�p,.�b8:7/,<;�7�!�7/�,.�%1"����!�&(��7�#5�z��&/23�@~��/,-!��I¶���,-F���,.�"GH&(��24#k!�,-&(����·3��P/;�74#5�z!?7/�M; P������ �"!@,h�"!�� �����>!���#5!>�%KÂn/,-F��:,.�"G�&(�>24#5!�,-&(�I24#}J����&51/,<F��7/,.�"!��@!�&!?7/�^,h�"1%����!>&(��!�7�#k!z&(�/�^&(��#|����!@&5G���!�&/;eT"�M#%���M*-,.T5��*-J�!�&&/P/!���� �sGH&(��2Ã!�7/�:&%!�7/� ����!>&(;�T%��,.�!�7/�^��&(�D!�G�&"*<,-&QK�½W&(����&51%� ���5!�7/���>,<F��:,.�"G�&/��24#5!�,-&(�I24#}J�BC��F�� �C� �/F�� �%!d&(�4!?7/����#5��!@#%�/F4G+P/!�P����:B�� 7�#}1(,-&(��&5G!�7/�^24#%��Tk��! K�{M!z!�7/�BC��6%,.���/,.�/6$&5G�� #5;e7S!���#kF�,h�/63FQ#}J%�/!�7/�^��,-F��^,.�"G�&/��24#5!�,-&(�S,-����������� �"!���FI!�&3!�7/�:,h�"1%����!>&(�#5�4#W��,.�/6%*-����; #k*h#"�4��� ��������� �"!�,.�/6�!�7/�=¶>��!�#5!���·c&5G^!�7/�4~��/,-!��c��,-F���,.�"G�&(�>24#5!�,-&(��Ä@!�7/���>,<6/�/,Z~Q; #"�/;��0&5G^!�7/,-�,.�"G�&/��24#5!�,-&(��23P/��!:B��*-� #%�>�/��FRB"J$!�7/�^,.�"1%����!�&/��K

Å Æp��ÇdÈ���ÉÊ�?��ËM����Ç@Ìt &(�/�>,<F�� �3#c��&(��!DG�&%*-,-&0;�&/�%!�#k,h�/,.�/6�ÍÎ��!�&/;eT"� K�fz#5;�7W!��>#5F�,.�/60FQ#}J%� � !�7/�I��� �sGH&(��24#"�/;��4&5GM!�7/�4��!>&(;�T%�|; #%�B��IF�����; ��,.BC��FpB%J�#I1%��;�!�&/�^&5G�®�©�±¨¡?ª�©Eª}¬-¦%¥H±ÐÏ}ª�¤���F�� �/&%!���F=B%JWÑ � �HÒ � ��Ò � � �}� ����ÒCÓ � 8:7/� ���$Ò�Ô@,-�3!�7/�$�/��g/!FQ#}J�uv�z&(�C� �/,h�/64����,-;��:&5G�!�7/�^ÕH!�7|��!�&/;�TF�,-1/,-F���FRB"J�,-!���&(��� �/,.�/6$���D,<;��:&(�3!?7/��; P����D� �%!�FQ# J%K�LM7/P/�z!?7/��15#5*.P/�

ÖH×�ØOÙ�Ú}Û ÜZÝ�Þ�ßQݨÜZà@Ù^á¨â ã�ä�åMæ�ã?ç�èHØ5Þ�çHÙsÛã>éHê ÜZݨé+ã>éHÜ�ëZä5ì�æ�Ù�è�Þ�Ú}ëZâÙ�í?ÚOã>ë�ëZä^æ�Ùsë�ë(Ú5çHÙ�àzÜ�Û�Ú5ÝHÙ�ç�î5ØOÞ>Ú é+ç�î%æ�Ù�Ù+ï ç�î}ÙEÝHè?ð5ã�ç¾Ý¨ØOÙ�ݨÜZà@Ùê%Ù�ÝÐæ�Ù�ÙsÛ`ã?è�ݨÜZÞ�Û5ç�ð

Page 3: Mathematical Finance, 8(4):325-347, 1998.web.cs.iastate.edu/~honavar/portfolio-selection.pdf · Mathematical Finance, 8(4):325-347, 1998. ! #" %$'&)( * +% -,. */0 1 #+324,. " 5%687

&5G�#%��,.�%1%���>!�23� �"!^,h�c��!>&(;�T�Õ�,h�/; �D� #5�����$�¨&(��G+#5*-*<� � !�&�ÒCÔ�!�,.23���^,-!��������1/,-&(P/�^1O#5*.P/�3G+��&(2N&(�/�423&(���/,.�/6�!�&!�7/���/��g(!}K${ ��&(��!DG�&%*-,-&0,-�3F��?~��/��FpB"JU#�8@��,-6(7"!1%��;�!�&/� � � ��� � ��� � � �}� �����@Ó � ��P/;�7�!?7�#5!��@Ô�� [c#%�/F� ÓÔ � � Ô � �%K�LM7/�MÕ�!�73� �"!���J&kG�#^��&/��!�GH&%*-,<& � ,-�@!�7/�M����&(�C&(��!�,-&(�3&kG�!�7/�M!>&%!�#5*���&(�D!�G�&"*<,-&|1O#5*.P/�:,h�"1%����!>��F4,.�!�7/�3ÕH!�7S��!�&/;eT K��^,-1%� ��#I��&(��!DG�&%*-,-& � #%�/F0!?7/�����,-;��i�D��*h#k!�,-1%����Ñ��(,.�"1%����!�&/����P/��,.�/6I!�7/,-�`�C&(��!�GH&%*-,-&$,.�/; ��� #5�>��¨&(��F���; ��� #k��� � !?7/��,h��8�� #k*<!?7IG+��&(2Ã&(�/�23&(�>�/,h�/6I!�&3!�7/�3�/��g(!�B"J�#|G�#k;�!�&(��&5G

�� Ñ �Ó�Ô� �

� Ô Ò Ô �

����� ����������� �"!$#&%('*)�#&����#,+-�.���./0'1��#&��+��!�7/,-�Â��#"��� ���58@�z#%����,.�"!�� ������!>��F,.�|&(�")+*-,h�/�M�C&(��!�GH&%*-,-&�����*-��;�!�,-&(�|��!?��#5!���6%,-��� K¾{M!�!�7/���>!�#%��!�&5GQ� #5;�7:FQ#}J32?�5!�7/���&/��!�GH&%*-,<&3����*-��;�!�,-&(����!���#k!���6%J36%��!���!�7/�`���D��1(,-&(P/�����D,<;���D��*h#k!�,-1%���M&5Gy!?7/����!>&(;�T324#%��T5��!�Ñ � � �}� ���>Ñ&4�5 � K�À���&(2!�7/,-�,.�"GH&(��24#k!�,-&(����!�7/�4��!��>#5!���6%Jc,h2S23��F�,.#5!���*-JU�>��*<��;�!>�,-!��4��&(�D!�G�&"*<,-& � 4 GH&(�^!�7/�SFQ# J%KS{M!!?7/�$B���6%,.���/,.�/6&5G�!?7/���/��g/!3FQ# Jp�¨FQ#}J"2��Á� � �y!?7/������,-;��0����*.#5!>,<1"���GH&(�|FQ#}J62�#%���S&(B/��� ��1%��F=#%�/F=!�7/�I,.�%1"����!�&(��uv�8@� #5*-!�7,.�/; ��� #5�����^B"J�#|G+#5;�!�&/�z&kG � 4 Ñ 4 K]:1"� �I!�,.23�%�^#A����a¯P/� �/;��'&5G�FQ#5,-*-Jj����,-;����D��*h#k!�,-1%����Ñ � ��Ñ � � �}� ���>Ñ&7»,-��&(B/�>� ��1%��F»#%�/Fj#A����a�P/� �/;���&5G��&/��!�GH&%*-,<&"� � � � � � �}� �}��� � 7�,-�����*-��;�!���F�KSÀ���&/2�!�7/�4BC��6%,.���/,h�/6'&5G�FQ#}JU�3!?7���&(P/6(7c!�7/�4B���6",h���/,.�/6W&5G@FQ# J8U�b�%�C!�7/�^8@� #5*-!�7I8M,-*-*Â7�#}1%�^,.�/; ��� #5�>��FRB"J0#^G+#5;�!�&/�@&5G

9 7 �;: � 4�< �=: Ñ 4�< �?>�@BA�7C4 � �

4 Ñ 4 �n/,.�/;��3,.�W#4!�J��/,<; #k*Â24#"��T5��!M!�7/�|8@� #5*-!�7�6(�D&O8d�M��g¯�C&(�/� �"!�,.#5*-*<JcG+#5��! � !�7/��G�&(�>24#5*y#%��#k*<J/��,-�:&5Gz&(P��:#5*-6%&5)

��,-!�7�2´8M,-*<*@B��$���D����� �%!>��F�,.�W!�� ��23�^&kG�!�7/�4�/&(��2S#5*-,ED���FA*-&%6(#%�D,<!?7�2 &5G�!�7/�38@� #5*-!�7�#k;e7/,-��1%��F�K3���SF�� �/&%!��!�7/,-�^�/&(��24#5*-,FD���F0*-&%6/#%��,-!�7�2¸&5G�!�7/�^8@� #5*-!�7cB%J

G 9 7 ��: � 4�< ��:}Ñ 4�< �H>�@�A� �8

7�4 �

*-&%6JI � 4 Ñ 4LK �

���B� MN#&�O+*'1PQ�R'(�S%T�.U PQ��PQ�O/V�XWY!Z#&%1'[)�#&����#&+�b,-!�7�!?7/��BC� �/�?~Q!�&kG`7/,.�/F���,-6(7%!}��&(�'� #5;e7'FQ#}JW&(�/�$; #"��,.�"1%����!�#k*<*@&5GM&(�/�%uv��8@� #5*-!�7µ,h�A!?7/�$��,.�/6%*-�0BC����!�)��� �EG�&(�>23,.�/6���!�&/;eTµGH&(�|!�7�#5!3FQ#}J%K���!4,-�4;�� ��!�#k,h�/*-Jp#%B/�?P���F=!�&�7/&(���c!�&���� �sGH&(��2Î#5�38���*-*�#k�$#W��������;�,-� �%!#56%� �"!:8M,-!�7c!�7/,-�^*<��1"��*z&5Gz,.�"GH&(��24#5!>,<&/�R#"B�&(P/!^!�7/��G�P/!?P����%K�+�/�>!�� #5F���,.�W!�7/,-���#%�C� ����8@�;�&/24����!��4#56/#5,.�/��!#I23&(���3������!���,-;�!���Fc;�*.#5���^&5G�,.�%1%���>!�23� �"!^��!���#5!>��6%,-���M; #5*-*-��F�;�&(�/��!�#"�%!�)E�D� B�#5*.#%�/;���F���&(��!DG�&%*-,-&%� Kd{��^�/&%!���Fc,.�!�7/�^,.�"!���&/FQP/;�!�,-&(����#i;�&(�/��!�#%�"!�)E��� B�#k*h#"�/;���F0��&(��!DG�&%*-,-&4,-�^��� B�#5*.#%�/;���F0� #k;e7IFQ#}JI��&4!�7�#5!M#|~Qg/��FIG+��#5;�!�,-&(�I&5G!�7/�M8@� #5*-!�7|,-��7/��*-FI,.�4� #5;�7|&5G�!�7/�MP��/F�� ��*-J(,.�/64,.�"1%����!�2i� �%!��}KÂLd7/� ���?G�&/���%�%#^;�&(�/�>!�#%�"!�)E��� B�#5*.#%�/;���F$�C&(��!�GH&%*-,-&��!��>#5!���6%J4� 2S�/*<&5J/��!?7/���#%2i�^,h�"1%����!?23� �%!d1%��;�!�&(� � &(�I� #5;�7�!?��#5F�,.�/64FQ#}J�#"�/F0!�7/�3������P/*-!�,.�/6$8@� #5*-!�7c#%�/F�/&(��2S#5*-,ED���Fc*-&%6(#%��,-!�7�2i,<;|8@� #5*-!�70#}GH!�� �O8j!��>#5F�,.�/64FQ#}J(��#"���

9 7 � � �&>�@BA� 9 7 � � ��: Ñ 4 < � �7C4 � �\ Ñ 4 �"G 9 7 � � �R>�@�A� G 9 7 � � ��:}Ñ 4 < � � �

87�4 �

*-&%6]I �^ Ñ 4 K �

ld&%!��4!�7�#k!^��P/;e7A#��>!���#5!���6"J023,-6(7%!3����a�P/,h�D�$15#5��!�#"23&(P��"!��&5G@!���#5F�,.�/6Q����,.�/;���#5!!?7/�$B���6%,.���/,.�/6�&5GM� #5;�7FQ#}J"2^!�7/�I,.�%1%���>!�23� �"!i���D&(��&(�D!�,-&(�/�4#%���$��� B�#k*h#"�/;���FoB�#5;�Tc!�&W!�7/�$1"��;�!�&(� � KW�+�'!�7/,-�$��#%��� �|8@�I,<6/�/&(���;�&(242i,<�>��,-&(��;�&%�>!����H7/&58@��1%� ���"�����!�7/�^F�,-��; P/�>��,-&(�c,h�cn/��;�!�,-&(��_ � K

Y

Page 4: Mathematical Finance, 8(4):325-347, 1998.web.cs.iastate.edu/~honavar/portfolio-selection.pdf · Mathematical Finance, 8(4):325-347, 1998. ! #" %$'&)( * +% -,. */0 1 #+324,. " 5%687

��,-1%� ��#��>��a¯P/� �/;���&5GMFQ#5,-*-JU����,-;��0����*.#5!>,<1"����Ñ � �>Ñ � �}� �}�E�>Ñ&7=8@�4; #%�'F��?~��/�%��,.�U�D��!���&%���C��;�! ��!?7/��BC����!��� B�#5*.#%�/;���Fp�C&(��!�GH&%*-,-&01%��;�!>&(�^8�7/,-;�7�8@&(P/*-F=7�#}1%�$#5;�7/,-��1%��FU!?7/�$24#5g/,.2�P�2 8@� #5*-!�7 9 7 ��#%�/F=7/� �/;���#5*-��&!�7/�24#kg(,.23P�2¸*-&%6(#%�D,<!?7�23,-;8@� #5*-!�7��XG 9 7 K����F�� �/&"!��^!�7/,-�`�C&(��!�GH&%*-,-&$B"J � � � � � �L8 � KÂLd7�#5!M,-� �

� � >�@�A� #"��6�24#kg� 9 7 � � � � #%��6z2S#5g� G 9 7 � � � �8�7/� ���@!�7/�@24#5g/,.2�P�2 ,-��!�#%T5� ��&51%� ��#5*-*Q��&"����,.B/*-����&(�D!�G�&"*<,-&:1%��;�!�&(�D���¨,HK��%K � 1%��;�!>&(����,h��� Ó 8M,-!�7|�/&(�")E�/��6(#5!>,<1"�;�&(24�C&(�/� �"!���!?7�#5!��?P�2 !�&4&(�/� � K^��!�� ��#5!�,-1%�323��!�7/&/F��:G�&/�@~��/F�,.�/60!?7/,<�:1%��;�!�&/��P/��,.�/6�!�7/�3� �"!�,.���3����a�P/� �/;��&5G�����,-;��4����*.#5!�,-1%���:Ñ � �}� � ���>Ñ&7U#%���^F�,-��; P/�>����F�,.�c&(P��M� #"��*-,<� �^��#"��� �|X.����\Â8:7/,<;�7c6%,-1%���:����1%� ��#5*�P���FQ#5!����:GH&(���&%*-1/,.�/6i#�6%� �/� ��#5*Q2i,<g/!�P����:����!�,.24#k!�,-&(�4����&(B/*-� 2��/,.�/;�*.P/F�,.�/6�23P/*-!�,.�/*-,<; #k!�,-1%�P���FQ#5!>����*-,.T5�:!�7/&%���dF�����; ��,.B���F,.�W!�7/,-����#"��� ��K3���SF�� �/&%!��4!�7/�3*-&%6(#"��,-!�7�23,-;$8@� #5*-!�7W#5;�7/,-��1%��F=P/��,.�/60!�7/�3&(�/!�,.24#k*�;�&(�/��!�#%�"!�)E��� B�#k*h#"�/;���F��&/��!�GH&%*-,<& � � B"JNG 9��7 ��:}Ñ 4 < � K���7/� �/��1"� �z,-!@,-�M;�*-� #%�qG��D&(2Ã!�7/��;�&/�%!���g/! �"8��d8M,-*<*¾&(23,-!@!�7/��F�� �C� �/F�� �/;�J�&(�!�7/�����D,<;��:����*.#5!�,-1%����#"�/F4��,.24�/*-J3F�� �/&%!��M!�7/�M#%BC&51%�MB%J G 9 � K t *-� #%�D*<J%� � � F�� �C� �/F���&/�!?7/�M� �%!>,h�D�M����a�P/� �/;��&5Gz����,-;�������*.#5!>,<1"��� : Ñ 4 < #"�/F024#}J�BC�^FQ��#%24#5!>,<; #k*<*-J�F�,���� ��� �"!MGH&(��F�,���� �D� �%!^24#%��Tk��!�BC� 7�#}1(,-&(���}K

]^B"1/,<&/P/��*-J%�^!�7/�W&(�/!�,.24#5*:1%��;�!�&/� � � ; #%��&(�/*-J�BC��;�&(24��P/!>��F�#OG�!�� �S!�7/�W� �"!�,.��������a�P/� �/;���&5G4����,-;������*.#5!�,-1%���^,-�3T(�/&58��A�H#5!:8�7/,-;e7��C&%,.�%!}��,<!^,-�3�/&$*-&(�/6%� ��&5Gz1O#k*hP/� � K��d&58���1"� ���Q!?7/��#5*-6%&(�D,<!?7�2 F�����; ��,.B���FA,.�!�7/,-�4��#%��� �I�H#5�^8@��*-*M#5� t &51%� ��u��4Xxw�\M#5*-6%&(��,-!�7�2 � ��� �EG�&(�>23�#5*.23&%��!3#5�8@��*-*�#k� � � 8:7/,<*-��P/��,.�/6W&(�/*-J�!�7/�������1/,-&(P/��*-J0&(B/�>� ��1%��F07/,-��!�&(�DJ$&5Gz����,-;����D��*h#k!�,-1%����!>&i2S#%T5�^� #5;�7IFQ#}J�u��d,h�"1%����!?23� �%!dF���;�,<�>,<&/��K��� �"�O� ���.%T+(PQ�Z!Z#&%('*)�#&����#&+t &51%� �dXZw�\�,.�%!?��&(FQP/;���F�!?7/�`�/&%!>,<&/�3&5GÂP��/,-1%� ���?#5*��C&(��!�GH&%*-,-&QK�{^�4&(�")+*-,.�/���&/��!�GH&%*-,<&3����*-��;�!�,-&(��#k*<6"&(��,-!�7�2Ã!�7�#5!������P/*-!��^,.�c!�7/�3����a�P/� �/;�� : � 4 < ,-�^��#5,-FW!�&�B����(£ ±ÐÏ}ª ©>¤e¦"¬��H����*.#5!�,-1%�3!�&I!�7/�|����!^&5G�#5*-*�;�&(�/��!�#%�"!�)E��� B�#k*h#"�/;���F��&/��!�GH&%*-,<&"� � ,xG

*<,.27���� 24#5g������ � G 9 � ��: Ñ 4 < ��� G 9 ��: � 4 < �=: Ñ 4 < ��� � [Á�LM7�#5!I,-� �^#=P��/,-1%� ����#5*�C&(��!�GH&%*-,-&=����*-��;�!�,-&(�º#5*-6%&(��,-!�7�2 ��g�7/,.B/,<!>�0!�7/���?#%23�R#k��J¯2S�/!�&%!�,-;W6(��&58M!�7���#5!��c,.��/&(��2S#5*-,ED���FW*-&%6(#"��,-!�7�23,-;48@� #5*-!�70#k��!�7/�3B�����!^��� B�#5*.#%�/;���FA��&(��!DG�&%*-,-&SG�&(�|¦%£ �$�>��a¯P/� �/;��3&5G�����,-;��4����*.#5!�,-1%���: Ñ 4 < K

�s��n/��;�!�,-&(��!38@�^#5FQ#%�/!M#|G+��#%23��8@&(��T|F���1%��*-&(�C��F�GH&(����P���� �D1(,-����Fc*<� #"���/,.�/6�#%�/F�6",<1"�#�>,h2S�/*<�3P���FQ#5!>���P/*-��!�7�#5!3����*-��;�!��I#R�/��8��C&(��!�GH&%*-,-&�1%��;�!�&(�`G��D&(2 !�7/�����D��1(,-&(P/�4&/�/�%KW�������D&O1"�$!�7�#k!�!?7/,<�4#5*-6%&/��,-!�7�2 ,<�P��/,-1%� ����#k*�,.�0n/��;�!�,-&(�#"QK���$ %���W � ��� )�#&%'& P.'T��#&��+�W��� #k*<,-!�J%��!?7/�,.�%1"����!�&(�^23,-6(7"!^7�# 1"�i2i&(���,.�"GH&(��24#k!�,-&(�c!�7�#%�)(DP/�>!�!�7/�4����,-;��$�D��*h#k!�,-1%���^&(B/��� ��1"��FR�>&4G+#%��Kn/,-F���,.�"GH&(��24#k!�,-&(�3��P/;�74#5�@������1O#5,-*-,.�/64,.�%!>� ������!��>#5!�����&(��;�&(�/�?P�23� �s)+;�&(�"~QF�� �/;��:~Q6(P��D���z; #%�3,.�/F�,-; #5!��:8�7/,-;�7��!�&/;�T%�y#"����*-,hTk��*<J^!>&M&(P/!��C� �sG�&/��2b!�7/��&%!?7/� ����!�&/;eT"��,.�^!�7/�@��&(��!DG�&%*-,-&QK¾ÀQ&%*-*-&58M,.�/6 t &51%� ��#%�/F3]^��F�� �"!�*-,<;�7IX.�%�>\E�8@�^F�� �/&%!��|!�7/���,-F��3,.�"G�&/��24#5!�,-&(�cB%J�#%��,h�"!���6%� �+*iG���&/2�#3~��/,-!��|����! :(�%��YC� � �����-, < KzLM7/P/� �Q!?7/�B�� 7�#}1(,-&(�&5G�!�7/�24#%�>T5��!@,h�/;�*.P/F�,.�/60!?7/�^��,-F���,.�"GH&(��24#k!�,-&(�I,-�`�/&58bF�� �/&%!���FWB%JI!�7/��>��a¯P/� �/;��J: Ñ 4 �.* 4 < K

À &%*-*-&O8d,h�/6 t &51%� �M#"�/FR]^��F�� �"!�*-,-;e7AX.�%�>\E�C8��3#5*-*-&O8º!�7/�|;�&(�/��!�#%�"!�)E��� B�#k*h#"�/;���FR��&/��!�GH&%*-,<&I!�&4��g��/*-&%,-!^!�7/���,-F���,.�"G�&/��24#5!�,-&(�=B"J���g���#%�/F�,.�/6U!?7/�$��,.�/6%*-�W��&(��!DG�&%*-,-&W,.�%!�&�#c����!4&kGd�C&(��!�GH&%*-,-&%� ��&(�/�SG�&(�|� #5;�7A��&%���>,hB/*-�1O#k*hP/�|&5Gy!�7/�^�>,<F��^,.�"GH&(��24#k!�,-&(��KzLM7/P/� ��#3;�&(�/��!?#%�%!D)E��� B�#5*.#%�/;���F0��&/��!�GH&%*-,<&S8M,-!�74�>,<F��|,.�"G�&(�>24#5!�,-&(�I;�&(�/�>,<�>!��&5G^!�7/��1%��;�!�&/��� � ��� � � � �HY � ��� ����� � � , � #%�/FpP/�����$�C&(��!�GH&%*-,-&U1%��;�!>&(� � � * 4 � &/�AFQ# J 2�K�LM7/��8@� #5*-!�7=#%�/F�/&(��2S#5*-,ED���F=*-&%6/#%��,-!�7�23,-;I8�� #k*<!?7�������P/*-!�,.�/6cG��D&(2 P/��,.�/6�#��>��!&5G@;�&(�/��!�#%�"!�)E��� B�#k*h#"�/;���F��C&(��!�GH&%*-,-&%�4B�#5����F&(�I��,-F��|,h�"GH&(��2S#5!�,-&(�0#"���%�

9 7 � � � � ��: Ñ 4 �-* 4 < �?>�@BA� 7C4 � �

� * 4 � Ñ 4 � G 9 7 � � � � ��:}Ñ 4 �.* 4 < �?>�@BA� �8

7�4 �

*-&%6 I � � * 4 � Ñ 4 K �

!

Page 5: Mathematical Finance, 8(4):325-347, 1998.web.cs.iastate.edu/~honavar/portfolio-selection.pdf · Mathematical Finance, 8(4):325-347, 1998. ! #" %$'&)( * +% -,. */0 1 #+324,. " 5%687

�(P/��!�*-,.T5�M!?7/��F��?~��/,-!�,-&(�S&5G�!�7/�MB�����!�;�&(�/�>!�#%�"!�)E��� B�#5*.#%�/;���F4��&/��!�GH&%*-,<& �%8@�@F��?~��/�M!�7/�MB�����!��>,<F��d,h�"GH&(��2S#5!�,-&(�F�� ��� �/F�� �"!^��&(��!DG�&%*-,-&4����! � � � � #5�@!�7/�24#kg(,.23,FD�� �@&5G 9 7 � � � � �=: Ñ&4¨�.* 4 < � KÂld&%!��^!?7�#5!@!�7/�^F�,.23� �/��,-&(��&kGÂ#��,-F��3,.�"G�&(�>24#5!�,-&(�cF�� ��� �/F�� �"!��&/��!�GH&%*-,<&��>��*<��;�!>,<&/�R���D&(B/*-� 2 ,<� , !�,.23���^*.#%��6%� �d!�7�#%��!�7/���,.�/6%*-�$�C&(��!�GH&%*-,-&����*-��;�!�,-&(�W����&(B/*-� 2�K

Ld7/���>��a¯P/� �/;��4&kG���,-F��I,h�"GH&(��2S#5!�,-&(� : * 4 < ;�&(P/*-F=B��423� #%�/,.�/6%*-�����4��#"�/F�&(2_�/&%,-���%�y�/��,-!�7/� �|#SG�P��/;�!>,<&/�&5GM!�7/����#5��!324#%��T5��!�/&/�#0���D��F�,<;�!>&(�|&5G@G�P/!�P��D��24#"��T5��!�� KI]^��!?7/�$&%!?7/� �7�#%�/F���,-!423,-6(7%!3B���#W��� �sGH��;�!,.�/F�,-; #5!�&(�3&kG�!?7/�0B�����!3,.�%1"����!�23� �"! KWf�g/!�� �/F�,.�/6�!�7/�I!E8@&5)+,.�"1%����!�2i� �%!|��g¯#%2S�/*<��6",<1"� ��,.�=n/��;�!�,-&(�b�%��8@�23,-6(7"!7�#}1%�3��,-F��4,.�"G�&(�>24#5!�,-&(� * � �3&(�c&(F�F'FQ# J/�4�¨8�7/� �W!�7/�31%&"*h#k!�,-*<�S��!�&/;eT�*<&"������7�#5*xG@,<!>�1O#k*hP/� � #%�/F* � Y:&(�|��1%� �3FQ#}J/���¨8�7/� �|!�7/�M1"&%*.#5!�,-*-�M��!�&/;eT�F�&(P�B/*-��� � KzLd7/,<����,-F��M,.�"G�&/��24#5!�,-&(�i; #%�3B��:��g¯�/*-&%,-!���F�B%J|!�7/�;�&(�/��!?#%�%!D)E��� B�#5*.#%�/;���F���&(�D!�G�&"*<,-&I����! � ��� � � ���%��[ � #%�/F � �HY � � �H[��}� � !�&4F�&(P�B/*-�3,-!��:8�� #k*<!?7���1%� ��J�&"!�7/� �!���#kF�,h�/6�FQ#}J%K�d&58@��1%� ����!�7/�3&(�/*-J0�>,<F��S,h�"GH&(��2S#5!�,-&(�W;�&(2S2�P��/,-; #5!>��FR!>&�!�7/�3,h�"1%����!>&(�4�H#5!:!�7/�4B���6%,.���/,.�/6&5G�FQ#}JN2 � ,-�:!�7/���,.�/6%*-�31O#k*hP/� * 4�8M,-!�70�/&iG�P���!?7/� �4¶���g��/*.#%��#5!�,-&(�/�}��·�#"�/F�!�7/�����a�P/� �/;��]: * 4 < 24#}J$&/�M24#}J�/&%!@;�&(�"!�#5,.��#%�"J�P/�>�?G�P/*�,.�"GH&(��24#k!�,-&(��Kz�d� �/;��%��!�7/�^,.24��&/��!�#%�/;��|&5Gy� #5;�7I��,-F��,.�"GH&(��24#5!>,<&/�I1O#5*.P/��23P/��!MBC�*-� #%���/��F�G���&/2¸!�7/���� �EG�&(�>24#%�/;��^&5G�!�7/�24#%�>T5��!MFQP���,.�/6�������1(,-&(P/�:!���#kF�,h�/6SFQ# J/� K

{^�W&(�")+*-,.�/�0,.�"1%����!�2i� �%!3#5*-6%&(��,-!�7�2 ,.��!?7/,<�3����!>!�,.�/6µ7�#k��#5;�;������3&(�µFQ# J 2B�&%!�7µ!�&0!�7/����#5��!37/,-��!�&(��J&5GM����,-;�������*.#5!>,<1"������#5�B��?GH&(��� � #%�/F�!�&�!�7/�I��#5��!#%�/F�; P��>��� �%!|��,-F��$,.�"GH&(��24#k!�,-&(�W1O#k*hP/��� * � �}� �}���.* 4 KILM7/�6%&(#k*Q&5Gy!?7/�`#5*-6%&/��,-!�7�2¸�/&O8b,-�M!�&|,.�%1%���>!M,.��#�2S#%���/� ��;�&(24����!>,<!>,<1"�^8M,-!�7 9 7 � � � � � ��: ÑR4��-*T4 < � �/!�7/�:8@� #5*-!�7&5Gz!�7/�3B�����!^;�&/�/��!�#%�"!�)E��� B�#5*.#%�/;���FW��&(�D!�G�&"*<,-&�8M,-!�7c��,-F��3,.�"G�&/��24#5!�,-&(��K:]^�/�3; #%��� #5�>,<*-JWF��?~��/�4#$�/&"!�,-&(��&5GP��/,-1%� ����#k*<,-!�J0#%��#5*-&%6"&(P/�M!�&4!?7/�^F��?~��/,<!>,<&/�06",<1"� ��,.�0n/��;�!�,-&(�cY�K !�K

{:�$�/&%!>,<;���FbB%J t &51%� �4#"�/Fp]^��F�� �%!>*<,-;�7ºX.�%�>\E��!�7/�0,.�"1%����!�&(�i; #%�=��#%��!�,-!�,-&(� !�7/��!���#kF�,h�/6AFQ#}J(�IB�#5����F&(�A!�7/�I��,-F��0,.�"GH&(��24#k!�,-&(����#"�/F=!���� #5!|� #5;e7=��#"��!�,-!�,-&(�A��� ��#%�>#5!���*-J%K=f�g��/*<&",<!>,h�/6A!�7/����,-F��c,h�"GH&(��2S#5!�,-&(�',<�!�7/� ���?GH&(���3�/&�2i&(���F�,��$; P/*-!^!�7�#"�R�>P����/,h�/6 , ;�&(�/,-���^&5Gz&(P��:#5*-6%&(��,-!�7�2�� &(�/��G�&(�d� #5;�70�C&%����,.B/*-�41O#k*hP/�3&5G!�7/�^��,-F��|,h�"GH&(��2S#5!�,-&(��Kzn/,.�/;��!?7/�^*<&"6(#%��,-!�7�2¸&kGÂ!�7/�^8@� #5*-!�7I,-��#kF�F�,<!>,<1"�%�Q!�7/�^*-&%6/#%��,-!�7�2¸&5G�!�7/�^8@� #5*-!�7I&(�!�7/�M� �"!�,.���M�>��a¯P/� �/;��^8M,-!�7i��,-F���,.�"GH&(��24#5!>,<&/�3,-� (�P/��!z!�7/�d��P�2Á&5G�!�7/�d*<&"6(#%��,-!�7�23��&kG�!?7/�M8@� #5*-!�7/�z6%� �/� �>#5!���FB"J$!�7/� , ;�&(�/,-���^&5G�!�7/��#5*-6%&/��,-!�7�2�K����� �6�X��P.'1�.W��N#&%�^,-��!���,.B�P/!�,-&(��#k*�2i��!�7/&(F��M#%�D������&(B�#%B/*-J4!?7/�`23&"��!z;�&(2S23&(�4#%�����D&(#5;�73!�&�#kFQ#%�/!�,-1%�:,h�"1%����!?23� �%!���!���#k!���6%,-���GH&(�3��� B�#5*.#%�/;���Fo�C&(��!�GH&%*-,-&%� K=V:��*-*-J»X.� w�\�#5���?P�23��F=!�7/����g/,<�>!�� �/;��0&5G^#%�=P��/F�� ��*-J(,.�/6=F�,-��!��D,hB�P/!>,<&/�=&5G^!�7/�����,-;��4����*.#5!�,-1%���#%�/F�P/����F��@#}J%���:F���;�,<�>,<&/�W!�7/��&(��J�!>&$���C��;�,xG�Jc!�7/�4�/��g(!^��&(�D!�G�&"*<,-&�1%��;�!>&(��K� :�/F�� �d1O#%��,-&(P/�;�&(�/F�,-!�,-&(�/� ��,-!^8M#5�^F�� 23&(�/�>!���#5!���Fp�¨�%Kv6QKyX��(���C�&_���"���Y}\ � !�7�#5!:8M,-!�7����D&(B�#%B/,-*-,<!�JW&(�/�4!?7/���@#}J%���^F���;�,-��,-&(�#%������&/#5;e7A#5;�7/,-��1%���4!�7/����#"23�$6/��&58M!�7���#k!��$&kG�!�7/��8@� #5*-!�7A#5�3!�7/�0B�����!I��� B�#5*.#%�/;���Fp��&/��!�GH&%*-,<& KU�+� !�7/,-�#%������&/#5;e7��C!�7/�4����,-;��i�D��*h#k!�,-1%�����a�P/� �/;����^; #%�WB��|FQ��#}8��IG+��&(2N&(�/�&5G�#IT/�/&O8:�I����!�&kG���&%���>,hB/*-�3F�,-��!���,.B�P")!�,-&(�/� K|LM7/,-�#%������&(#5;�7�8M#5�^P/����F�B"J0{:*<6"&(��!4X.�>\z8�7/&I;�&(�/��,-F�� ����F'!�7/�3����!^&5G�#5*-*�� ��6%&/F�,<;I#%�/FW��!�#5!�,-&(��#"��JF�,-��!���,.B�P/!�,-&(�/�S&(��,.�"~��/,-!��c����a¯P/� �/;����}�z#%�/FA����!�,.24#5!���FA!�7/��P��/F�� ��*-J(,.�/6�F�,-��!?��,.B�P/!�,-&(�=,.�'&(��F�� �|!�&0;�7/&/&%���!�7/�^�/��g(!@��&(�D!�G�&"*<,-&31%��;�!�&/��K t &51%� ��#%�/F �^*.P/����X � [}\�;�&/�/��,-F�� ����F$!?7/�`�����>!���,-;�!���FI; #5���M8�7/� �D��!�7/�:����!�&5Gy����,-;������*.#5!�,-1%���:,-����£ ±Ð¥Hª:#%�/F�6(#}1%�^#%�I,.�%1"����!�23� �"!M��;�7/� 23�8d,<!?70P��/,-1%� ����#k*y����&(�C� ��!�,-��� K

Ld7/�`23&"��!z;�*-&%����*-J�����*.#5!���F�������1/,<&/P/��������P/*-!��M#%���^B"J t &51%� �MXxw�\y#%�/F t &51%� ��#%�/F�]^�DF�� �%!�*-,-;�7�X.�%�>\EK�LM7/��J����&51%�|!�7�#5!M;�� ��!?#5,.�0,.�"1%����!�2i� �%!M�>!���#5!���6",<���^#%���3P��/,-1%� ����#5*�8M,-!�7/&(P/!^24#%T",.�/60#%�"J$��!?#5!�,-��!�,-; #5*�#5����P�2S�/!�,-&(�/�&(�A!�7/�0��#k!�P�����&5G^!�7/����!�&/;�T�24#%��Tk��! K t &51%� �IXZw�\`����&51%��F=!�7�#5!3!�7/��8@� #5*-!�7=#k;e7/,-��1%��FbB%J=7/,-��P��/,-1%� ���?#5*��&/��!�GH&%*-,<&4#5*-6%&/��,-!�7�2Ã,<�S¶�#5*.23&%��!@#5�@*.#%��6%��·4#5��!�7/�B�����!@;�&(�/�>!�#%�"!�)E��� B�#5*.#%�/;���F0�C&(��!�GH&%*-,-&QK��M,-��#%��#k*<J/��,-�M8M#5�,.24����&51%��F=B%J t &51%� �|#%�/F=]^��F�� �%!>*<,-;�7bX.�%�?\@8�7/&R#k*<�>&R,.�"!���&/FQP/;���F=!�7/���/&%!�,-&(�W&5G:��,-F��$,.�"GH&(��24#5!>,<&/���z#%�/F6%� �/� ��#k*<,FD���F t &51%� ��u��#k*<6"&(��,-!�7�2 !�&0P/���S!�7/��^,.��,-;e7/*-��!O�D���%Y��}� � �>� ���%Y � #%�/FW!�7/��^,h�D,<;�7/*-��!O���%�}� �}���}� � ���D,<&/���&51%� ��!�7/��>��!M&5GÂ#5*-*y�C&%����,.B/*-�4��&(�D!�G�&"*<,-&I1%��;�!�&(�D� K

t &O1"� ��#%�/F=]^��F�� �"!�*-,-;e7�u��3,.�"1%����!�23� �"!�>!���#5!���6",<���P/�>�$#%��#}1%� ��#k6%,.�/6023��!�7/&/F�!�&R�/,-;�Tc!�7/��,.���C&(��!�GH&%*-,-&1%��;�!�&/��� KyLd7/����&(�D!�G�&"*<,-&^1%��;�!>&(��P/����F4&(�iFQ# J 2�,-�z!�7/�d8���,-6(7"!���F4#}1%� ��#56"��&51%� ��¦%¬¨¬�GH� #5��,.B/*-�^��&(��!DG�&%*-,-&1"��;�!�&(���

"

Page 6: Mathematical Finance, 8(4):325-347, 1998.web.cs.iastate.edu/~honavar/portfolio-selection.pdf · Mathematical Finance, 8(4):325-347, 1998. ! #" %$'&)( * +% -,. */0 1 #+324,. " 5%687

�H#5*-*��)+F�,.23� �/�>,<&/��#5*�1%��;�!�&(����8M,-!�7I�/&(�")E�/��6(#5!�,-1%�d;�&(24��&/�/� �%!��@!�7�#k!z��P�2 !�&$� � �%8�7/� ���d!�7/��8@��,-6(7"!z&5G�� #5;�7��&"����,.B/*-�i�C&(��!�GH&%*-,-&41%��;�!�&/�@,-��F���!�� �>23,.�/��FRB"J$,-!��^��� �sGH&(��2S#%�/;��^,.��!�7/���#5�>! KzLM7�#5!@,-� �

� 4 �� � 9 4B5 � � � ����� � � �� 9 4�5 � � � ����� � � �

� ��� �8�7/� ��� ��� ,-�4&(�/�I&5G�!�7/� ^,.��,-;e7/*-��!IF�,-��!���,.B�P/!�,-&(�/�$2i� �%!�,-&(�/��Fp#"B�&51%�%KWlM&%!>�$!�7�#k!!�7/����&(�D!�G�&"*<,-&�1"��;�!�&(���#%���I8@��,-6(7%!>��F�#5;�;�&/��F�,.�/6�!�&0!?7/��,h�S��#5��!4��� �sGH&(��2S#%�/;��%� 9 4�5 � � � � ��#5�38���*-*M#5�3!�7/������,-&(� � � � � K ^,<�>; ����!��#%������&5g/,.24#5!�,-&(��XZw�\�&(�3����; P��D��,-1%�$�>� ��,-������g���#%�/��,-&(�bX �%�>\�#%����P/����FA!�&0��15#5*.P�#5!���!�7/�I#%B�&51%�I,.�%!���6/��#5*-� K0�s�B�&"!�7W; #5�����}��7/&O8@��1%� ����!�7/�4!�,.23�$#"�/F�����#5;��4����a�P/,.����F�GH&(�d~��/F�,.�/6W!�7/�4�/��8 �C&(��!�GH&%*-,-&01%��;�!�&/�^#%����� #%�D�!�&6(��&58���g���&/�/� �%!�,.#5*-*-J4,.�3!�7/�@�(P�23BC� ��&5GQ��!�&/;eT"� K��j7/,-*<�@!�7/�MBC&(P��/F���#5;�7/,-��1%��F4B%J|!�7/�z6"� �/� ��#5*-,FD���F�P��/,-1%� ���?#5*��&/��!�GH&%*-,<&#k*<6"&(��,-!�7�2º&5G t &51%� ��#%�/F$]^�DF�� �%!�*-,-;�74#%���@��!��D&(�/6%� �¾!�7�#%�|&(P���� �}8@�@��7/&58U!�7�#k!�&(�7/,-��!�&/��,-; #5*���!�&/;eTFQ#5!�#i&(P��@#5*-6%&(��,-!�7�2 ��� �sGH&(��2i��BC��!�!�� �d8�7/,-*-������a�P/,.��,.�/6�!�,.23�#%�/F0�?��#5;��^*-,h�/� #"�M,.��!?7/���/P�2�BC� �@&5G���!�&/;eT"� K

� � �4È����IÈ��� �Ë��y���^Ç ��:����� ��È����9Ì�ÇMÈ�Ç@ q�y����� Ë:È�:�:�������4É Ì]^P���G+��#%2i��8�&/��T:GH&(��P���FQ#k!�,.�/6i#|��&(�D!�G�&"*<,-&|1%��;�!�&(��,-�M#%��#5*-&%6%&/P/��!�&!�7/��G���#"23��8@&(��T:F���1%��*-&(�C��F0B%J4V:,-1(,.�/� �#%�/Fj�A#%��23P/!�7ºXZY%[�\^GH&(�I&(�")+*-,.�/�p����6(�������>,<&/��K �+��!�7/,-�0&(�")+*-,.�/�AG��>#%23��8@&(��TA!�7/�A��&(��!DG�&%*-,-&p1"��;�!�&(�I,-!�����*xG� �/; #%�/��P/*.#5!>����!�7/�^�/��;�������#"��J$,.�"GH&(��24#k!�,-&(�SG��D&(2Ã!�7/�`���D��1(,-&(P/�^����,-;�������*.#5!>,<1"��� KzLM7/P/� ��#5!@!�7/�^��!�#"��!z&5G�FQ# J2�� !�7/�#5*-6%&/��,-!�7�2�;�&/24��P/!����M,-!��^�/��8 �C&(��!�GH&%*-,-&$1"��;�!�&(� � 4�� � #5��#|G+P��/;�!�,-&(��&5G � 4z#%�/Fc!�7/� (�P/��!M&(B/�>� ��1%��F����,-;�������*.#5!�,-1%���^Ñ 4 K3�+�'!�7/�3*-,h�/� #"���D��6(��������,-&(�c����!>!�,.�/6R#%��#5*-J0D���FUB"JµV:,-1/,.�/� �U#%�/F'�U#%��23P/!�7���!�7/��JW��7/&58!�7�#5!@6%&/&/F��C� �sG�&/��24#%�/;��:; #%�IB��#5;�7/,-��1%��F0B%JI;�7/&(&%�>,h�/64#|1%��;�!�&/� � 4�� � !�7�#k!�,-�3¶�;�*-&%����·S!�& � 4�K����`#kFQ#%�/!!�7/��,.�:23��!�7/&/F0#%�/F�~��/F0#4�/��8º1%��;�!�&/� � 4�� � !�7�#5!�H#"������&5g(,.24#5!>��*<J � 24#kg(,.23,FD����:!�7/�:G�&"*<*-&58M,.�/6SG�P��/;�!�,-&(���

�4� � 4�� � � ��� *-&%6�� � 4�� � Ñ 4 � ��� � � 4�� � � � 4 � � �HY �8�7/� ��� ��� [3,-�:��&(23�3��#%��#%2i��!�� ��; #5*-*<��F'!�7/�^*-� #%���/,.�/60��#5!��#"�/F � ,<�^#3F�,-��!�#"�/;��i2i� #5��P����^!�7�#k!M��� ��1%���:#5�#���� ��#k*<!�J�!>� ��2�K3LM7/,-����� ��#k*<!�J�!>� ��2�� ��� � � 4�� � � � 4 � ��!�� �/F��!>&0T5��� � � 4�� � ;�*-&%�>�4!�& � 4EK4LM7/�I��P�����&%�>�&5G�!�7/�^~��D��!�!>� ��2¸,-�^!�&$24#kg(,.23,FD��3!�7/�*-&%6/#%��,-!�7�23,-;38@� #5*-!�7�,xGz!�7/�|; P������ �"!`����,-;��4����*.#5!�,-1%�|Ñ 4 ,-���� ��� #k!���F�KLM7/�4*-� #%�>�/,h�/6���#k!�� � ;�&/�%!��D&%*-�!�7/�$�D��*h#k!�,-1%�4,.24��&/��!�#%�/;��$BC��!�8���� �µ!�7/�4!�8@&�!�� ��23� K4�s�%!?P/,<!>,<1"��*<J%�q,ZG � 4@,<�G+#%��G+��&(2 !�7/�`BC����!@;�&(�/��!�#"�%!�)E�D� B�#5*.#%�/;���F��C&(��!�GH&%*-,-& � � !?7/� �$#|��24#k*<* *-� #%���/,.�/6$��#5!>��23� #%�/�@!�7�#5! � 4�� � 8M,-*<*23&51%�:&(�/*-J$�>*<&58M*-JI!�&58M#%��F � � K�]^�I!�7/��&"!�7/� �M7�#%�/F��/,xG � 4�,<�:#5*.��� #5F�JI;�*<&"���^!�& � � !�7/� ��#|*.#%��6%�^*-� #%�>�/,h�/6��#5!>�24#}J4; #%P/���|!�7/�#5*-6%&(��,-!�7�2N!�&$B��2i,<�>*<��F�B"J$FQ#}J")+!�&5)+FQ#}J���P/;�!�P�#k!�,-&(�/� K

^,���� ��� �"!F�,-��!�#"�/;��IG�P��/;�!�,-&(�/�3*-� #5F=!�&cF�,���� �D� �%!4P���FQ#k!����>P/*<���}K�]^�/�$&5G:!�7/��2S#5,.��;�&(�"!���,.B�P/!�,-&(�/�3&5G!�7/,-�:*<,.�/�|&5GÂ8@&(�>T4,-�M!�7/�P/�>�&5G�!�7/�����*.#5!�,-1%�|� �%!?��&(�"J$#5��#3F�,-��!?#%�/;��^G+P��/;�!�,-&(�IGH&(�M2i&%!�,-1O#5!>,h�/6IP���FQ#5!���� �

"!�#��%$'&(& ) �6>�@�A��� � *

Ô}*<&"6 * Ô+ Ô �

½�#%�"J�&"!�7/� ��&(�")+*-,.�/�#5*-6%&(�D,<!?7�23�@8M,-!�7$23P/*-!�,.�/*-,-; #5!�,-1%�8@��,-6(7"!�P��CFQ#5!����^XZY��5��!C�(Y%[/�Q� "5\�#%�D��#5*-��&423&%!�,-15#5!���FB"J0!�7/,-�F�,-��!?#%�/;��4G+P��/;�!�,-&(��#"�/FU#%���3!�7/P/���&/&%!���FW,.��!�7/�423,.�/,.23P�29����*.#5!�,-1%�4� �"!���&(�"JR���D,h�/;�,.�/*-��&5GMV^P/*<*x)B�#5;�TIX.����� ��!�\EK�L�&|F�� ��,-1%�^*-� #%���/,.�/6$��P/*-���MP/��,.�/64����*.#5!�,-1%�:� �%!��D&(�%J%�/8@�M����! � � � 4�� � � � 4 � � !,# � � 4�� � &-& � 4 � K��!�,-�Â7�#"��F^!�&�24#kg(,.23,FD����=��,.�/;��@B�&%!�7�!�� ��23��F�� �C� �/F$�/&(�")+*-,.�/� #%��*-J&(� � 4�� � Ky]^�/�@��&%�>��,.B/*<�M#%�����D&(#5;�7:,<�!�&P/���:#%�3,-!�� ��#5!>,<1"��&/�/!�,.23,FD #5!�,-&(�I#5*-6%&(��,-!�7�2��"��P/;�74#5��6(��#kF�,<� �"!z����& (E��;�!�,-&(���"!�&:~��/F4!�7/�:24#5g/,h23P�2 1%��;�!�&(�� 4�� � !�7�#k!�2S#5g(,.23,FD����.�ÁP��/F�� ��!�7/�^;�&(�/��!��>#5,.�%! � ÓÔ � � 4�� �Ô � �4�¨�����%�/GH&(�q,h�/�>!�#%�/;��%� À�*<��!>;e7/� �:X.� Y�\ � KÂLd7/,<�#%������&/#5;e7c,-�^!�,.23�4;�&/�/��P�23,.�/6R#5�^,-!����a�P/,.������>&%*-1(,.�/6R#IF�,���� ��� �"!�/&(�")+*-,.�/� #%�|��a¯P�#5!>,<&/�W&(�c� #5;�70!?��#5F�,.�/6��� �D,<&/F�K^À�P���!�7/� ��23&/���%��#k��8@�3*.#5!�� �dF�� 23&(�/��!��>#5!��,.�Wn/��;�!�,-&(� �C�Q!�7/�4�C&(��!�GH&%*-,-&�#5*-6%&/��,-!�7�2 !�7�#5!d~��/F��^!�7/���g�#5;�!$�>&%*.P/!�,-&(��!�&�fqa¯P�#k!�,-&(�»�HY � ,.�b����#5;�!�,-;��WF�&/�����/&%!IJ/,<��*-F»BC��!�!�� �������?P/*<!>�$!�7�#"�p!?7/�R#5*-6%&(�D,<!?7�2Ê8@�

Page 7: Mathematical Finance, 8(4):325-347, 1998.web.cs.iastate.edu/~honavar/portfolio-selection.pdf · Mathematical Finance, 8(4):325-347, 1998. ! #" %$'&)( * +% -,. */0 1 #+324,. " 5%687

�/&58 ���D����� �%!^8:7/,<;�7W,-�B�#5����FW&(�c!�7/�|GH&%*-*-&O8d,h�/6�� �I;�,<� �"!�#"������&5g(,.24#5!>,<&/��K�+�/��!>� #5FW&5G�~��/F�,.�/60!�7/�3��g�#5;�!24#5g/,.23,FD�� ��&5G �3�(8@���D� �/*h#k;��^!�7/�@~�����!�!�� ��2 8M,-!�74,-!���~����>!�)+&(��F�� ��Ly#}J/*<&/�z��&"*<J��/&(23,.#5*�#%�D&(P��/F � 4�� � � � 4�KLM7/,-�z#%������&5g/,.24#5!�,-&(��,<����� #5�>&(��#%B/*-�@,ZG,����#k!�,-��~Q���Â#:¹�,.�/��;�7/,-! DM;�&(�/F�,-!�,-&(�3#%�/F|!�7/�z1%��;�!>&(� � 4�� � ,-�z����*.#5!�,-1%��*-J;�*-&%���I!�& � 4 KS���$#5*-��&WP/����#�¹¾#56(��#%�/6"�$23P/*<!>,h�/*-,-� �3!�&07�#"�/F�*<��!?7/�4;�&(�/��!��>#5,.�%!^!�7�#k!^!�7/�4;�&(24�C&(�/� �"!��&5G� 4�� � 2�P/�>!��?P�2¸!�&4&(�/�%K�LM7/,-��*-� #5F��^P/�M!�&$2S#5g(,.23,FD�� ��º,h�/�>!�� #5F0&kG ���

��$� � 4�� � � � � � �� *-&%6�� � 4 Ñ 4 � � Ñ 4 � � 4�� � � � 4 �� 4 Ñ 4 � � � � � 4�� � � � 4 � � � � Ó�

Ô �� 4�� �Ô � � � �

LM7/,-�:,<�dF�&(�/��B"J$�>��!�!�,.�/64!�7/�3Í ��#%��!>,h#k*�F�� ��,-1O#5!>,<1"����!�&JD�� ��&R� GH&(�:���pÕ��pÍ � �� ��I� � 4�� � � � �� � 4�� �Ô � �ÒV4Ô

� 4 Ñ 4 �� � � � 4�� � � � 4 �� � 4�� �Ô � � � [j� � ! �

�¨G�!�7/�3����*.#5!�,-1%�^� �"!���&(�"J$,-�^P/����FR#k��!?7/�F�,-��!�#%�/;���G�P��/;�!>,<&/��!�7/� ��fqa¯P�#5!>,<&/�U� ! � BC��;�&(23���

�Ò 4Ô

� 4 Ñ 4 � �¨*-&%6 � 4�� �Ô� 4Ô �b� � � � � [ &(� � 4�� �Ô � � 4Ô ��g�� ��

Ò 4Ô� 4 Ñ 4 � � � � � �

fz�"GH&(��;�,.�/6=!�7/��#5F�F�,-!�,-&(��#k*;�&(�/��!?��#5,.�%! � ÓÔ � � 4�� �Ô � �W6%,-1%���0#A��&/��!�GH&%*-,<&bP���FQ#5!>�R8:7/,<;�7b8@�W; #5*-*^!�7/�ª�>®¯¢%£ ª £ ¥H±Ð¦%¥Hª?­|²(©�¦%­%±¨ª £ ¥Â�HfO�$� � �s� P���FQ#k!����

� 4�� �Ô ��$4Ô ��g¯�� � Ò 4Ô � � 4 Ñ&4 �� Ó� � � 4� ��g��]I � Ò 4� � � 4 Ñ 4 K � �" �

{º��,.23,-*h#"�^P���FQ#5!��:GH&(�@!?7/�^; #5���^&5G�*-,h�/� #"�^����6(�������>,<&/�I8�#k�z~��D��!M6%,-1%� �cB%J�V:,-1(,.�/� �W#%�/F0�A#%��23P/!�7cXZY%[�\EK�s�4#5F�F�,-!�,-&(�S!�&!?7/��P���FQ#5!>��� �%8@��#5*-��&3�/����F$!>&^;e7/&/&%����#"�3,.�/,<!>,h#k*��C&(��!�GH&%*-,-&1%��;�!>&(� � � K��j7/� �4�/&3����,-&(�

,.�"G�&/��24#5!�,-&(�c,-�:6%,-1%� ����#$�D� #5��&(��#%B/*-�3;�7/&%,-;���8@&(P/*-F�B��3!�&I��!�#"��!M8M,-!�7W#%����a¯P�#5*�8@��,-6(7%!^#5���>,<6/�/��FR!>&$� #5;�7&5Gz!�7/�3��!>&(;�T%�:,.�W!�7/�3��&(��!DG�&%*-,-&Q��!�7�#k!�,-� � � � � ������Íc� �}� ���}����Í � K@�j7/� �0�>,<F��S,h�"GH&(��2S#5!�,-&(�c,-���������� �"!���F��8@��� 24�/*-&5Jp#0�>��!4&5G��&(��!DG�&%*-,-&A1%��;�!�&(���}KU�'�0P/���0!�7/�0fO�$� � � P���FQ#5!>�0!�&�;�7�#%�/6%��!?7/�0��&(��!DG�&%*-,-&A1%��;�!�&(�,.�/F���g(��F=B"JR!?7/���>,<F��I,.�"G�&/��24#5!�,-&(��K4�d� �/;��%��!�7/�4����&(B/*-� 2 &5G��C&(��!�GH&%*-,-&�����*-��;�!�,-&(�'8M,-!�7W��,-F��I,h�"GH&(��2S#5!�,-&(���,.24�/*-J=����FQP/;����3!�&R#W��#%��#k*<*-��*@����*-��;�!�,-&(�A&5G , F�, ��� ��� �"!4��&(�D!�G�&"*<,-&%�}K0�¨G�!?7/�$��,-F��I,.�"G�&(�>24#5!�,-&(�',-��,.�/F�����F,.�"G�&/��24#5!�,-1%�%��!�7/������!3&5G��&/��!�GH&%*-,<&"�48M,-*-*`#5;�7/,-��1%��*.#%��6%� �i8�� #k*<!?7A!�7�#%�A#W����a�P/� �/;��0&5G��&(�D!�G�&"*<,-&�1"��;�!�&(���������P/*-!�,.�/6SG��D&(2¸!�7/�^� �"!�,.����>��a¯P/� �/;��%K����^F�� 23&(�/�>!���#5!��:!�7/,-�:,h��!�7/�^��g���� ��,.23� �"!�#5*�����;�!�,-&(��!�7�#5!�G�&"*<*-&58M� K

�s�4!?7/�`�/��g/!@����;�!�,-&(�I8@�`#"��#5*-J D��:&(P�������� � � )EP��CFQ#5!��MB�#5����F0�C&(��!�GH&%*-,-&3����*-��;�!�,-&(��#5*-6%&(��,-!�7�2�K����:;�&(2|)��#%����!�7/���C� �sGH&(��24#%�/;��@&5G�!�7/�Mf �4� � � )EP��CFQ#5!��M#5��8@��*-*�#5�y!?7/��f�g�#5;�!�fO�$� � � )EP���FQ#5!>����8�7/,-;e7324#5g/,.23,FD�����!�7/��b,.�If�a�P�#5!�,-&(�4Y|��#5!�7/� �q!�7�#%�|!�7/��#%�����D&Og/,.24#5!�,-&(� �� � 8d,<!?73&%!�7/� �q&(�")+*-,h�/�^��&/��!�GH&%*-,<&3����*-��;�!�,-&(�I#5*-6%&(�D,<!?7�23�GH&(�@F�,���� ��� �%!d����!�!�,.�/6%�M,.�cn/��;�!�,-&(� ��K� �¿��Ë:È��3̾��Ì�+�S!�7/,-�z�>��;�!�,-&(���/8��M#%��#k*<J0D��:!�7/�M*-&%6(#%�D,<!?7�23,-;�8@� #5*-!�73&/B/!�#5,.�/��F$B"J�!?7/��f �I� � � �C&(��!�GH&%*-,-&�P��CFQ#5!��^��P/*-�%K��������&51%�|8�&/����!�)+; #5�>��B�&/P��/F��&(�c!�7/�4P���FQ#k!��38�7/,-;e7W,.24�/*-JW!�7�#5!^!?7/�ifO�4� � � P���FQ#5!��3,-��#k*h2i&%��!^#5�^6%&/&(FW#5�!�7/�4B����>!^;�&(�/��!�#%�"!�)E��� B�#k*h#"�/;���Fµ�C&(��!�GH&%*-,-&�8�7/� �c;�� ��!�#5,.�W#5���?P�24�/!�,-&(�/�7/&%*-FW&(�c!�7/�4����*.#5!�,-1%�31%&%*.#5!>,<*-,-!�J0&5G!�7/�M�>!�&(;�T"�z,.�4!?7/����&(�D!�G�&"*<,-&QK����:#5*-��&��������>� �%!@#^1O#"��,.#%�%!�&5Gyf �4� � � 8:7/,<;�7I����a¯P/,.�����M�/&|��P/;e7S#5����P�24�/!>,<&/�/� K

{:*-!�7/&(P/6(7A!?7/�0#%��#5*-J(�>,<�S,<�I��������� �"!���FAG�&/��#µ��,.�/6%*-�R��&/��!�GH&%*-,<&'1%��;�!�&(����,-!4; #%�=BC��6%� �/� ��#5*-,FD���Fb!�&W!�7/�23P/*-!�,.�/*<�I1%��;�!�&/���`Tk� �/!^8�7/� �W��,-F��4,.�"GH&(��24#5!>,<&/�0,-�3��������� �"!B%JW��#%��!�,-!�,-&(�/,.�/60!?7/�4!���#5F�,.�/6�FQ#}J/�B�#5����FW&(�!�7/�^��,-F��|,h�"GH&(��2S#5!�,-&(�0#"�/F�!���� #5!�,.�/64� #k;e7���#%�D!�,-!�,-&(����� ��#"��#5!���*-J%K

_

Page 8: Mathematical Finance, 8(4):325-347, 1998.web.cs.iastate.edu/~honavar/portfolio-selection.pdf · Mathematical Finance, 8(4):325-347, 1998. ! #" %$'&)( * +% -,. */0 1 #+324,. " 5%687

n/,.�/;��UÒ 4Ô ��� ��������� �"!��R���D,<;��A����*.#5!�,-1%��� �:8@��7�# 1"�R!?7�#5!0ÒV4Ô �Ê['G�&/��#5*-*Õ4#%�/F 2�K¸À�P���!�7/� ��23&/���%�M8@�#5���?P�23�3!�7�#5!24#5g/ÔOÒ 4Ô � �|GH&(�^#5*-*�2�K3���3; #%�'24#%T5�3!�7/,-�#5���?P�24�/!�,-&(�W8M,-!�7/&/P/!^*<&"���&5G@6%� �/� ��#5*-,-!�JW��,.�/;��23P/*-!�,.�/*<J/,.�/63!�7/������,-;��M����*.#5!�,-1%����Ñ 4 B"J#M;�&(�/�>!�#%�"! � ��,.24�/*-J4#5F�F���*-&%6 � !�&�!?7/�z*-&%6(#%�D,<!?7�23,-;�8@� #5*-!�7��5*-� #}1/,h�/6!�7/�:F�,���� �D� �/;��`BC��!E8@��� �S!�7/�:*<&"6(#%��,-!�7�23,-;:8@� #5*-!�7$#k;e7/,-��1%��F�B%JS!�7/��f �I� � � )EP���FQ#5!>�`#%�/FI!�7/��BC����!M#5;�7/,<��1"��F*-&%6(#%�D,<!?7�23,-;$8@� #5*-!�7"G 9 � P��/;�7�#%�/6%��F�K0³�P/!�#%�/&%!?7/� �^8M#}J%��!�7/�$#5���?P�23��FU*-&58@� �^B�&(P��/F��S&(�AÒ 4Ô P/����FA,.�LM7/��&(�D� 2 �4�HBC��*-&O8 � ; #%��BC��1/,-��8���F0#k��#�*-&O8@� �@B�&/P��/F�&(�4!?7/�`��#k!�,-&&5Gy!?7/��8@&(���>!z!�&3B�����!M����,-;������*.#5!�,-1%���GH&(�@!���#kF�,h�/6SFQ# J]2�K

L¾&��� 23,.�/FI!�7/����� #kF�� ���(#^�C&(��!�GH&%*-,-&�1"��;�!�&(��,-��#�1%��;�!�&(��&5G��/&(�")E�/��6(#5!>,<1"�`�/P�23B�� ����!�7�#5!���P�2 !�&4�%K�LM7/�f �I� � � �C&(��!�GH&%*-,-&�P��CFQ#5!��3#5*-6%&(��,-!�7�2�P/�>���M!�7/�^GH&%*-*-&58M,.�/6$��P/*-���

� 4�� �Ô ��Z4Ô ��g��]I � � ��� ��� � � K� 4

8�7/� ��� ��� [i,<�:!�7/�^*-� #%���/,.�/6���#5!��%� #%�/F�� 4 ,-�:!�7/��/&(��24#k*<,FD #5!>,<&/�� 4 � �

� Ô Ó� 4Ô ��g¯� �

�Ò 4Ô

� 4 Ñ 4� �

LM7/�:GH&%*-*-&O8d,h�/6I!�7/��&(�D� 2¸;e7�#"��#5;�!�� ��,FD����^#36%� �/� ��#k*����D&(��� ��!�JI&5G�!�7/�f �I� � � )EP��CFQ#5!��%K� ��������������� ª}¥'$�� � Ó «�ªI¦3®�¢%©�¥ °�¢%¬-±¨¢0Ï ª?¡ ¥H¢"©��¦"£Q­0¬-ª}¥�Ñ � � � �}����ÑR7�«>ª$¦�¤eª����/ª £ ¡�ªS¢�°:®�©�±¨¡?ª$©�ª ¬-¦%¥H±¨Ï ª�¤� ±¨¥! �Ò 4Ô �"� � [|°�¢%©3¦%¬¨¬�Õ>� 2¦%£ ­424#5g Ô Ò 4Ô � �^°�¢%©¦%¬¨¬Q2$#&%�¢%© � � [W¥! (ªI¬-¢�²(¦"©�±¨¥! ('3±Ð¡ � ª�¦%¬-¥! W­'�(ªI¥H¢0¥! /ª®¯¢"©�¥ °�¢%¬-±Ð¢�Ï}ª�¡}¥H¢%©>¤z®�©�¢}­'�(¡?ª?­0« �4¥! (ª*),+�� � � § ��®¯­%¦%¥HªS±.¤�«>¢'�/£Q­%ª?­M°�©�¢-'_«�ª}¬<¢ � ¦5¤�°e¢"¬Ð¬-¢ � ¤/.

7�4 �

*-&%6�� � 4 Ñ 4 � �7�4 �

*-&%6��%$ Ñ 4 ��� 0 !�#�� $ &(& � � �� � � 8

�-� � �% �(©?¥! (ª}©1'3¢%©�ª2�:± ° � � ±.¤4¡3 (¢5¤�ª £A¥H¢R«>ª$¥! (ª��/£Q± °�¢%©1'N®�©�¢?®¯¢"©�¥H±¨¢%£AÏ ª?¡ ¥H¢%©4��¦%£ ­ � ªI¤�ª ¥ �c� Y-��5 Yq*-&%6�Í �(8¥! (ª}£ � ª* (¦"Ï ª 7�

4 �*-&%6�� � 4 Ñ 4 � �

7�4 �

*-&%6��%$ Ñ 4 ���"6 Y(8�*-&%6MÍY-� �

7 ������8 ¹���!:9 4 � "!�#��%$'&(& � 4�� � � � "!�#��%$'&(& � 4 � K�LM7/� �9 4 � � � Ô * Ô *<&"6���� 4�� �Ô �T� 4Ô �

� � � Ô * ÔD� � Ò 4Ô � � 4 Ñ 4 � *-&%6;� 4 �� � � $ Ñ&4

� 4 Ñ 4 �A*<&"6<� 4 � � � �L¾&4B�&(P��/FW*-&%6;� 4 ��8@�^P/���!�7/�:G+#5;�!M!?7�#5!�G�&(�d#5*-*>=?��XZ[��}�?\�#%�/F0Ò@� ���

*-&%6��D� � =��D� �?A � ��� ��=�Ò��Ò � � ��� ��_ �LM7/,-�zB�&/P��/F�8M#5��1%� ��,x~Q��F4B"J��g�#%23,.�/,.�/6^!�7/��~�����!�!E8@&MF�� �D,<15#5!�,-1%����&5G�B��HÒ � � =�Ò��4Ò � � � � *h���D� � =@��� �*A � ����¨�����4X ����\ � K

C

Page 9: Mathematical Finance, 8(4):325-347, 1998.web.cs.iastate.edu/~honavar/portfolio-selection.pdf · Mathematical Finance, 8(4):325-347, 1998. ! #" %$'&)( * +% -,. */0 1 #+324,. " 5%687

n/,.�/;��3ÒV4Ô ��Xx[�� �?\�#%�/F���,.�/;�� � � �j� � ��� � � � ÒIGH&(� � � [3#%�/FRÒ �'XZ[�� �e\E�%8@�7�#}1%���� 4 � �

Ô � 4Ô A�� � ���� � � � � �� �Ô � 4Ô ��� � ��� � A�� � � � � � � � Ò 4Ô �

� � � ��� � A�� � � � � � � � � 4 Ñ 4 �ld&O8|�¯#"���/*<J/,.�/6�,.�/��a¯P�#5*-,-!�J _��/8@�7�#}1%�

*<&"6 � 4 � � � � ��Q� � 4 Ñ 4 � � �

t &(23B/,.�/,.�/6$8d,<!?70f�a�P�#5!�,-&(��� � � 6%,-1%��� �9 4 � �

� � � $ Ñ 4� 4 Ñ 4� � � �

� � � 4 Ñ 4 � �� � � *-&%6��%$ Ñ 4 � � 4 Ñ 4 � � � ��Q� � 4 Ñ 4 � �

��,.�/;��$� � A � � � Ò�GH&(�@#5*-*ÂÒ¾Kn/,.�/;��3ÒV4Ô ���/��#%�/Fc��P�2423,.�/64&51%� �M#k*<*R2���8@�7�#}1%�

� "!,#�� $ &(& � � � � "!�#��%$'&(& � 7,� � � � !,#�� $ &(& � � �� �7�4 �

�¨*-&%6�� � 4 Ñ 4 � � *-&%6��%$ Ñ 4 ��� � � � 8�-� � �

8�7/,-;�7A,h2S�/*<,-����!�7/�S~�����!4BC&(P��/F=��!�#5!>��FU,.�A!�7/�I!�7/��&(��� 2�K�LM7/������;�&(�/F=B�&(P��/F�&5G�!�7/�I!�7/��&(�D� 2_GH&%*-*-&58M�B"J0��!��>#5,-6(7%!DG�&(�D8�#"��F0#5*-6%� B���#����/&"!�,.�/60!�7�#5! "!�#��%$'&(& � � � �j*-&%6MÍ98:7/� � � � ,-�!�7/�4P��/,xG�&/��2 ����&/B�#%B/,-*<,-!�J1%��;�!�&/��K

n/,.�/;��G 9 7 �

�8

7�4 �

*-&%6 I � 4 Ñ 4�K �LM7/��&(�D� 2_�,.242i��F�,h#k!���*-J06%,-1%��� G 9��7 � G 9 7 � 5 *<&"6�Í��Q��Y-� � 8 � �HP��/F�� �M!?7/�^;�&(�/F�,-!�,-&(�/�^&5G�LM7/��&(�D� 2_� � KLM7/P/� ��GH&(�3#%�A#%������&(����,.#5!>�$;�7/&%,-;���&5G � ��8�7/� �A!�7/���/P�2�BC� �3&5G�FQ#}J/� 8 BC��;�&(23���3*.#%��6%�%�q!�7/��F�,���� �D� �/;��B���!�8@��� �'!�7/�$*-&%6/#%��,-!�7�23,-;I8�� #k*<!?7U#5;�7/,-��1%��F=B%JUfO�4� � � ,-�36(P�#%��#%�"!�����F�!>&0;�&(�"1%� ��6%�I!�&�!�7/�I*-&%6(#%�D,<!?7�23,-;8@� #5*-!�7A&5G^!�7/�0B����>!$;�&(�/�>!�#%�"!�)E��� B�#5*.#%�/;���Fp�C&(��!�GH&%*-,-&QKp�d&O8@��1%� ���zLM7/��&(��� 2 ��,-���/&%!S��!���&(�/6W� �/&(P/6/7=!�&��7/&58b!�7�#5!Mf �4� � � ,-��#3P��/,-1%� ���?#5*y��&/��!�GH&%*-,<&4#5*-6%&/��,-!�7�2�KzLM7/,-�M,-��BC��; #%P/���|;e7/&/&%��,.�/64!�7/����D&(��� � � ����a¯P/,.�����T/�/&O8d*<��F�6"�&5GzB�&%!?7I!�7/��(P�23BC� �M&5G�!���#kF�,h�/6SFQ# J/��#%�/F�!?7/���#5!�,-& �|,.�0#5F�15#%�/;��%K����^8M,-*-*yF�� #5*¾8M,-!�70BC&%!�7&5G�!�7/�����^F�,��$; P/*-!�,-��� ����!?#%��!�,.�/648M,-!�7I!?7/�F�� ��� �/F�� �/;��3&5G � &(���/K

�j7/� �W�/&0*-&58@� �^B�&(P��/F �S&(�WÒV4Ô ,-��T/�/&58�����8@�4; #%�WP/���4!?7/�3G�&"*<*-&58M,.�/6R�C&(��!�GH&%*-,-&RP��CFQ#5!��$#5*-6%&/��,-!�7�28�7/,-;�7�,-�`��#"��#%23��!�� �D,ED���F0B"J0#3��� #5*y�/P�23B�� �,= ��Xx[�� �?\�Ky¹���!

�Ñ 4 � ��� � = ��Í � Ñ 4 �º��= ��Í � �8�7/� ��� � ,<�:!�7/�#5*-*@�%u��M1"��;�!�&(��K�{:�^B��?GH&(���%� 8��2S#5,.�%!�#k,h�c#4��&/��!�GH&%*-,<&I1%��;�!�&/� � 4 8:7/,<;�7c,-�`P��CFQ#5!���FRP/�>,h�/6�Ñ 4 ��#k!�7/� ��!�7�#%�IÑ 4 �

� 4�� �Ô ��Z4Ô ��g��y� � �Ò 4Ô � � 4 �Ñ&4 �� Ô � 4Ô ��g��y� � �Ò 4Ô � � 4 �Ñ 4 � �

Page 10: Mathematical Finance, 8(4):325-347, 1998.web.cs.iastate.edu/~honavar/portfolio-selection.pdf · Mathematical Finance, 8(4):325-347, 1998. ! #" %$'&)( * +% -,. */0 1 #+324,. " 5%687

À�P��D!�7/� ���5!�7/�M��&/��!�GH&%*-,<&^1"��;�!�&(��!�7�#5!�8@�z,.�"1%����!�8M,-!�73,-�z#5*-��&^�>*<,-6(7"!�*-J423&/F�,Z~Q��F�K�n�����;�,x~Q; #5*-*.J%��!�7/�M#5*-6%&/��,-!�7�2P/�����M!?7/����&/��!�GH&%*-,<&S1%��;�!�&(�

�� 4 � ��� � = � � 4 �º��= ��Í � �#%�/F���&S!�7/�^*-&%6(#%��,-!�7�2i,<;38@� #5*-!�7�#5;�7/,-��1%��F0,-�M*-&%6�� �� 4 Ñ&4 � K�'�^; #5*-*�!�7/,-�`2i&(F�,x~Q��FR#k*<6"&(��,-!�7�2 �f �4��=�� � � K� ������������� � ª}¥'$�� � Ó «�ªI¦3®�¢%©�¥ °�¢%¬-±¨¢0Ï ª?¡ ¥H¢"©��¦"£Q­0¬-ª}¥�Ñ � � � �}����Ñ 7 «>ª$¦�¤eª����/ª £ ¡�ªS¢�°:®�©�±¨¡?ª$©�ª ¬-¦%¥H±¨Ï ª�¤� ±¨¥! iÒ 4Ô �p[:°�¢%©^¦%¬¨¬QÕ�� 2�¦%£ ­�2S#5g(Ô5Ò 4Ô � �@°e¢"©^¦%¬Ð¬ 2$# %�¢%© =?���H[�� � �%Y}\z¦%£Q­ � � [��z¥! (ª3¬-¢�²(¦"©�±¨¥! ('3±Ð¡ � ª?¦%¬-¥! ­'�/ª4¥H¢I¥! (ª:®¯¢"©�¥ °�¢%¬-±Ð¢�Ï}ª�¡}¥H¢%©>¤z®�©�¢}­'�(¡?ª?­0« �4¥! (ª �),+���=z� � � §���®�­%¦%¥HªI±.¤«�¢ �(£ ­%ª�­d°�©E¢ ' «>ª ¬-¢ � ¦5¤�°e¢"¬Ð¬-¢ � ¤/.

7�4 �

*-&%6�� �� 4 Ñ 4 � �7�4 �

*<&"6��%$ Ñ 4 ��� Y-=R8 � 0 !�# �%$ &-& � � �� � � 8

�Q� = ��Í � ��

% �(©?¥! (ª}©1'3¢%©�ª2�4± ° � � ±.¤0¡3 (¢5¤�ª £b¥H¢p«>ªR¥! /ª �/£Q± °�¢%©1'´®�©�¢?®�¢%©�¥H±¨¢%£jÏ}ª�¡}¥H¢%©4� 8 �_Y%Í � *-&%6�Í �3¦%£Q­ � ªc¤�ª ¥= � HÍ � *-&%6@Í��Q� �(8 � � � � � ¦"£Q­ ��� 5 �-= � *<&"6�Í �Q�HÍ � 8 � ¥! /ª £ � ª* (¦%Ï ª7�4 �

*-&%6�� �� 4 Ñ 4 � �7�4 �

*-&%6��%$ Ñ 4 ��� YQ�HY%Í � *<&"6�Í � � � � 8 � � � � � C �7 ������8 À���&/2¸&(P��M#5�>��P�24�/!�,-&(�I!�7�#k!�24#5g%Ô%Ò 4Ô � �%��8@�7�#}1%�

�� 4 Ñ 4� 4 �Ñ 4 �

��� � = � � 4 Ñ 4 � = ��Í��� � = ��Í � � 4 Ñ 4 � = ��Í �

LM7/����,-6(7"!�7�#%�/FA��,-F���&5GM!�7/,-�3,h�/��a�P�#5*-,-!�J=,<�3F���; ��� #k��,.�/6µ#k��#�G�P��/;�!>,<&/��&5G � 4 Ñ 4 #%�/FA��&0,-�423,.�/,.23,FD���F8�7/� � � 4 Ñ&4 � �%K�LM7/P/� �

�� 4 ÑR4� 4 �Ñ 4 �º��� � = � � = ��ÍA�

&(�@��a�P/,-1O#5*-� �"!�*-J%�*-&%6�� �� 4 Ñ 4 � � *-&%6�� � 4 �Ñ 4 � �=*-&%6���� � =�� = ��Í �� *-&%6�� � 4 �Ñ 4 ��� Y =z� � � �

LM7/�^*.#5��!:,.�/��a¯P�#k*<,-!�J0P/�����:*-&%6��D� � =$� = ��Í � �=*-&%6���� � = � � � Y-=µGH&(� =?�U�H[��}���%Y}\�KÀ���&(2¸LM7/��&(�D� 2_�#%���/*-,-��FW!�&4!�7/�����,-;��3����*.#5!�,-1%�,.�/��!?#%�/;���� �ÑR4E��8@�7�#}1%�^!�7�#k!

7�4 �

*<&"6�� � 4 �Ñ 4 � �7�4 �

*<&"6��%$ �Ñ 4 ��� !�# �%$ &-& � � �� � � 8

�Q� = ��Í � ��Hw �

8�7/� ���^8@�P/����Fc!�7/�:G+#5;�!M!�7�#5! �Ò 4Ô ��= ��ÍUKld&%!��:!�7�#5! $ �ÑR4 � ��� � = ��Í � $ Ñ&4�� = ��Í � $ ÑR4EK t &(23B/,h�/��Fc8M,-!�7�f�a�P�#5!�,-&(�/�3� � � #%�/FU��w � ��#%�/F��P�242i,h�/6c&51%� ��#k*<*O2��¾!�7/,-�^6%,-1%���!�7/�|~����>!B�&(P��/FA&5G�!?7/��!?7/��&(��� 2�K3LM7/�4����;�&/�/FUB�&(P��/F'&5G�!�7/�3!�7/��&/��� 2

GH&%*-*<&58M��G���&/2¸!�7/�:G+#5;�!M!�7�#5! "!�#M�%$ &-& � � � �=*-&%6@Í 8:7/� � � � ,-�:!�7/�P��/,xG�&(�>2�����&(B�#%B/,-*-,-!�J01%��;�!�&/��K^,-1/,-F�,h�/6^,.�/��a�P�#5*-,-!EJ�� C � &kGCLM7/��&/��� 2»YdB%J:!�7/���/P�23B�� �¾&5G�!���#5F�,.�/6:FQ# J/�&8A��7/&58M��!�7�#k!�!?7/�z*-&%6(#%�D,<!?7�23,-;

8@� #5*-!�74#5;�7/,-��1%��F�B%J|!�7/� �f �I��=�� � � )EP��CFQ#5!��M;�&/�%1%� �D6%���z!�&^!?7�#5!�&5G�!?7/��B�����!�;�&(�/��!�#"�%!�)E�D� B�#5*.#%�/;���F��C&(��!�GH&%*-,-&� GH&(�z#"�$#%�����D&(����,.#5!��:;e7/&",<;��^&kG � F�� ��� �/F�� �"!:&(� 8 � K��M&58@��1%� ���"8��:��!�,-*-*�7�#}1%��!?7/��,-����P/�^!?7�#5!@!�7/��*-� #%�>�/,h�/6

w

Page 11: Mathematical Finance, 8(4):325-347, 1998.web.cs.iastate.edu/~honavar/portfolio-selection.pdf · Mathematical Finance, 8(4):325-347, 1998. ! #" %$'&)( * +% -,. */0 1 #+324,. " 5%687

��#5!>�i23P/��!^B��S;e7/&%�>� �W,.��#5F�1O#"�/;��$#5�#iG�P��/;�!�,-&(�'&5G 8K3LM7/�3GH&%*-*-&58M,.�/6R#5*-6%&(�D,<!?7�2_#%�/FW;�&(��&%*-*.#%��JW��7/&58M�7/&58»#3F�&(P�B/*-,.�/6I!���,-;eTI; #"��B���P/�>��F0!�&3&(B/!�#k,h�c#3P��/,<1"� ����#5*���&(�D!�G�&"*<,-&$#k*<6"&(��,-!�7�2�K

Ld7/���>!�#56%��F �f �4� =z� � � )EP���FQ#k!��$��P��/�|,h�µ��!�#56%���:8�7/,-;�7�#%���4�/P�2�BC� ����FWG+��&(2_[�K|LM7/�$�/P�2�BC� �^&5GzFQ#}J/�,.�b��!�#k6%��[A,<�WY%Í � *-&%6MÍA��#%�/Fb!�7/�A�(P�23B�� ��&5G�FQ#}J/��,.�b� #5;�7p�>!�#56%��Õ � [A,-�RY Ô Í � *<&"6�ÍAK Ld7(P/��,xG8 � Y%Í � *-&%6�Í ,-�@!�7/�M!>&%!�#5*Q�/P�23B�� ��&kG�FQ#}J/� �%!?7/��*.#5��!���!�#56"��� �"!�� ����FI,-�M�(P�23B�� �D��F � *-&%6 � � 7� Ó��������(Ó � K�{M!!�7/�^��!?#%��!@&5G�� #5;e7I�>!�#56%�^!�7/��C&(��!�GH&%*-,-&41%��;�!�&/�@,-�`���?)+,.�/,-!�,.#5*-,FD���F�!�&4!?7/�P��/,ZGH&(��2 ����&(�C&(��!�,-&(�I1"��;�!�&(�@#%�/F =#%�/F � #%�D�^����!�#5�@,.�0LM7/��&/��� 2�Y4P/��,.�/6I!�7/���/P�23B�� �@&kGÂFQ#}J/�M,.��!�7/�^��!?#56%�^#5�M!�7/�^15#5*.P/�GH&(�O8K� ���/� ������������� (ª|¤e¥H¦�²/ª�­ �) +�� =z� � � § ��®¯­"¦%¥Hª$± ¤^¦��(£ ±ÐÏ}ª ©>¤e¦%¬"®�¢%©�¥ °�¢%¬-±¨¢I¤eª}¬<ª?¡ ¥H±¨¢%£c¦%¬ ²(¢%©?±Ð¥! �'&#7 ������8 ���M~��D��!�BC&(P��/F�!�7/�MF�,���� ��� �/;�� � 74 � *-&%6��%$ Ñ 4 ��� � 74 � *<&"6�� �� 4 Ñ 4 � G�&(��#%�"J $Â�/#%�"J4����a�P/� �/;���&5G����,-;������*.#5!�,-1%����: ÑR4 < #%�/FI!�7/� �� 4�;�&(24��P/!���F�B"J4!�7/�:��!�#56%��F �f �$� =z� � � )EP���FQ#k!��%Kz¹���!�� � � *-&%6 � � 7� Ó � ����(Ó �B��3#4B�&/P��/F0&(�I!�7/�^*.#5�>!M��!�#56"��(P�23B�� ��K�À���&/2 Ld7/��&(��� 2 Y8@�^&(B/!?#5,.�

7�4 �

*<&"6��%$ Ñ 4 ��� 7�4 �

*-&%6�� �� 4 Ñ 4 � � "(Í � *-&%6@Í � ��Ô� �Y � � � �HY Ô � � � � Y%Í � *-&%6MÍ

� Y�� � � Í � *-&%6@Í��D�Â� ��Ô��� �HY � � � �Ô �

� Y � � � Í � *-&%6@Í��D�Â� �HY � � � � � � � � �Y � � � � � �� _%Í � *-&%6@Í��D�Â�º�HY � � � � � �� _%Í � *-&%6@Í��D�Â�º� 8

Y%Í � *-&%6@Í � � � � � �ld&O8|�%����!�!>,h�/6 $0!�&^!�7/�^B����>!�;�&/�/��!�#%�"!�)E��� B�#5*.#%�/;���F���&/��!�GH&%*-,<&3#%�/FIF�,<1/,-F�,.�/6�B"J 8o#5*-*-&58M�MP/��!>&����8���,-!��:!�7/�������1/,-&(P/��*-,.�/�3,.��!�� ��2i��&kGÂ!�7/�3�/&(��24#k*<,FD���Fc*<&"6(#%��,-!�7�23�d&5GÂ!?7/�^8�� #k*<!?7�#5;e7/,-��1%��F

G 9 � �;: Ñ 4 < � � G 9 ��: �� 4 < �=: Ñ 4 < � � _%Í � *-&%6MÍ=�����j� 7� Ó � ����(Ó � � � � �8 �{:�Z8�� �b�C!�7/�#%B�&51%�BC&(P��/F06"&(���M!>&$[��/;�&(24�/*-��!�,.�/6�!�7/�����&/&5G�K

�s�3��P�2��/!�7/�dF�, ��� ��� �/;��^B���!�8@��� �S!�7/��#}1%� ��#56"��FQ#k,<*-J4*-&%6/#%��,-!�7�23,-;:,h�/; �D� #5���:,h�S8@� #5*-!�73&5G�!�7/� �f �$� =z� � � )P���FQ#k!���#%�/Fc!�7/��BC����!:;�&(�/��!�#%�"!�)E��� B�#k*h#"�/;���FR��&/��!�GH&%*-,<&IFQ��&(�/�d!�&ND�� �D&$#5!M!�7/�3��#5!��"!������HÍ � *<&"6�Í � �(8 � � � � �GH&(� 8 ��Y%Í � *-&%6MÍAKW�j7/� �A!�7/���>#5!�,-&�B���!�8@��� �A!�7/�0B�����!4#%�/F=8@&(�D��!��!>&(;�TW&(�A� #5;�7'FQ# JA,-�$B�&/P��/F���F#%�/FW����*.#5!�,-1%��*-J���24#5*-*���#5�M; #%��&5G�!>� ��B����g�����;�!>��F0,.�W����#5;�!�,-;�� � ��!?7/�f �$� � � )EP��CFQ#5!��|; #%��B��3P/����F0,.�/��!�� #kF6%,-1/,h�/60#i;�&(�%1"� ��6%� �/;��4��#5!��^!>& D�� ��&I&5G#!�� 5 �¨*-&%6�Í � �18 � KM�s�0;�&/24��#%��,-��&(��� !�7/�3B�&(P��/F��^����&51%��FRB"J t &51%� �#%�/FU]^�DF�� �%!�*-,-;�7pX.�"��\�GH&(�d!�7/��,.�#5*-6%&/��,-!�7�2_;�&(�"1%� ��6"��!>&ND�� ��&W#5!:!�7/�$��#5!>�$!����HÍb*-&%6O8 � �(8 � K��s�W!�� ��2i��&5G!�7/�3�(P�23B�� ��&5Gz!���#5F�,.�/64FQ#}J/�Z8��!?7/��,h�:B�&/P��/F��#%���23P/;e7I�?P���� ��,-&(��� �����C��;�,h#k*<*-JW;�&(24��#"����F0!�&S&(P��MB�&/P��/FGH&(� �f �$� =z� � � KzLM7/�^&/�/*<J�; #k���^,.��8�7/,-;�7�&(P��@B�&/P��/F��`7�#}1%�#%��#kF�1O#%�"!�#56%�:,-��8:7/� �I!�7/���/P�23B�� �@&5G���!�&/;�T%�Ír,.�/;�*.P/F���Fp,.�A!�7/����&/��!�GH&%*-,<&',<�4����*.#5!>,<1"��*<J=*.#%�D6%��#"�/FU!�7/��24#%�>T5��!�7�#5�4BC&(P��/F���Fo����*.#5!>,<1"��1%&%*.#5!�,-*-,-!EJ'��&!�7�#5!If �4� � � ; #%��B��RP/�>��F�K ^�����/,-!��c!�7/�0;�&/24��#%��#5!>,<1"��,h�"GH� ��,-&(��,-!�J=&5G^&(P��3!�7/��&/����!�,-; #5*^B�&(P��/F�� �@,.� &(P����g���� ��,.23� �"!�� �¾8��|GH&(P��/FW!�7�#5!:&(P��^#5*-6%&(�D,<!?7�2 F�,<FAB���!�!�� ������1%� �c!�7/&(P/6(7c!�7/�4�/P�2�BC� �:&5G�!?��#5F�,.�/6�FQ#}J(��88M#5�@*.#%��6%���¨&51%� � ���Z[%[%[ � #%�/F�!�7/�3��&(�D!�G�&"*<,-&%�d,h�/;�*.P/F���F�&/�/*<J0#�G���8j�>!�&(;�T"� K

% &('��Ç��Â��É Çd����Ì*)¸�����,+.-0/1& 4Ë@��Ë���4!>����!���FA&(P��P��CFQ#5!��$��P/*-���3&(�A7/,-��!�&(��,-; #5*���!�&/;eTW24#"��T5��!^FQ#5!�#�G��D&(2 !�7/��lM��8Ãm�&(��TWn/!�&/;eTWf�g/;�7�#%�/6%�#5;�; P�23P/*.#5!���F�&51%� �^#�Y"Y})+J%� #%�:��� ��,-&/F�K4À &(�:� #5;�70��g���� �D,h2i� �%! ��8@�4������!��D,<;�!>��F�&(P��^#5!�!>� �%!�,-&(�c!�&0#I�?P�B/����!

� [

Page 12: Mathematical Finance, 8(4):325-347, 1998.web.cs.iastate.edu/~honavar/portfolio-selection.pdf · Mathematical Finance, 8(4):325-347, 1998. ! #" %$'&)( * +% -,. */0 1 #+324,. " 5%687

&5GM!�7/��!T_��>!�&(;�T"�GH&(��8�7/,-;e7'8@��7�#}1%�4FQ#k!�#�� � #"�/FU;�&(2S��#%����FA!�7/��fO�$� � � )EP���FQ#5!>��8M,-!�7�!?7/��BC����!3��,.�/6%*-���!�&/;�T4,.�c!�7/��?P�B/����!`#k��8@��*-*Â#k�M!�7/��BC����!:;�&(�/��!�#%�"!�)E��� B�#k*h#"�/;���Fµ�C&(��!�GH&%*-,-&4GH&(�@!?7/���P�B/����!}KM����G�&(P��/Fc!�7/�B����>!$;�&(�/�>!�#%�"!�)E��� B�#5*.#%�/;���Fb��&(��!DG�&%*-,-&=B%J=#%���/*-J(,.�/6p#AB�#5!�;�7=24#5g/,.2�P�2�)+*<,.T5��*-,.7/&/&(FÁ23,-g(!?P����0����!>,h2S#5!�,-&(�����&/;���FQP�����#5�^F�����; ��,.B���F=,.�W&/P��:� #%��*-,-� ����#"��� �4X ����\EK3{MGH!�� ��F���!�� ��23,.�/,.�/6W!�7/�4B�����!|;�&(�/��!�#%�"!�)E��� B�#k*h#"�/;���F��&/��!�GH&%*-,<& �z8@�I!�7/� �A;�&(24��P/!���F=,-!��I��� �sGH&(��24#"�/;��$&(�A!?7/�0����,-;��0����*.#5!�,-1%������a�P/� �/;��%K=����#5*-��&�;�&(2S��#%����F!�7/�0�C� �sG�&/��24#%�/;��c&5G�&(P��4P��CFQ#5!��W��P/*-���$!�&W!�7�#k!4&5G t &51%� ��u��3P��/,<1"� ����#5*^��&(�D!�G�&"*<,-&U#k*<6"&(��,-!�7�2Ê#%�/F=!>&U#%�P���FQ#k!��W!�7�#5!4,-�0B�#5����Fb&(� ��g¯#k;�!�24#kg(,.23,FD #5!�,-&(��&5G�fqa¯P�#k!�,-&(�»�HY � Kb���0!>� ��2Ê!?7/�R*.#5!>!�� �423��!�7/&/Fb!�7/�f�g�#5;�!�fO�4� � � )EP���FQ#5!>�%Kz���|;�&(24��#%�D��F�!�7/�������?P/*<!>�MGH&(�M#5*-*��?P�B/����!��:&5GÂ�>!�&(;�T"�M;�&(�/��,-F�� ����F�B"J t &51%� ��XZw�\y,.�7/,-����g���� �D,h2i� �%!��}K

n�P�2S24#%��,FD�,.�/6�&/P��y�����?P/*<!>� �58@��G�&(P��/F|!�7�#5! �5��� �>7�#%�/�y�?P�������,-��,.�/6%*-J%�%!�7/��8@� #5*-!�7#5;�7/,-��1%��F�B"J^!�7/�@f �4� � � )P���FQ#k!��$8M#5�|*.#%��6%� �|!�7�#%�'!�7/�I8@� #5*-!�7U#k;e7/,-��1%��F=B%J'!�7/��P��/,-1%� �D��#5*M��&(�D!�G�&"*<,-&�#5*-6%&(�D,<!?7�2�KRLd7/,<�3&(P/!>;�&(23�,-�4;�&(�"!���#%�DJR!>&0!�7/����P��C� ��,-&(�|8@&(����!�)+; #k����BC&(P��/F��$����&51%��FµG�&(�|!�7/��P��/,-1%� ����#k*��C&(��!�GH&%*-,-&U#5*-6%&(�D,<!?7�2�K�LM7/�F�,���� ��� �/;��,.�0�C� �sGH&(��24#%�/;��|8M#5�@*h#"��6%����!@GH&(�M�C&(��!�GH&%*-,-&%�M;�&(2S��&%����Fc&5Gy1%&"*h#k!�,-*<�|��!�&/;eT"� K

À�P���!�7/� ��2i&(���%��!�7/��P��/,-1%� ����#5*@��&(�D!�G�&"*<,-&c��!���#5!>��6%JRP/��,.�/6c!�7/� ^,.��,-;�7/*-��!O���%�}� � �>� � � ����,-&(��&(P/!��C� �sG�&/��23��F!�7/� ^,.��,-;�7/*-��!O�����%YC� � �}��� � �%Y � ����,-&(���%F������/,-!��:!�7/��BC��!�!�� ��B�&/P��/F������D&O1"��FiB"J t &51%� ��#%�/F$]^��F�� �"!�*-,-;e7cX.�%�>\QGH&(�!�7/� ^,.��,-;e7/*-��!O�D���%Y��}� � �>� ���%Y � ����,-&(��K���3!�7/� �D�?G�&(�D�iP/�>��F�!�7/� ^,.��,-;e7/*-��!O���"� � �}���}� � ����,-&(�:8�7/� �W6%� �/� �>#5!�,.�/6#5*-*�&5G�!�7/�P��/,-1%� ���?#5*y��&/��!�GH&%*-,<&�������P/*-!��^��� �C&(��!���F�,.��!?7/,<�:����;�!�,-&(��K

�'�$F�,-Fp�/&%!|~��/F=#%�%J'��,-6(�/,x~Q; #%�%!SF�, ��� ��� �/;��c,.��!�7/�0�C� �sGH&(��24#%�/;���#5;�7/,-��1%��FpB%J'!�7/��fO�$� � � )EP���FQ#5!>�#5*-6%&(�D,<!?7�2�#%�/F0!?7/�f�g�#5;�!�f �4� � � #k*<6"&(��,-!�7�2¸8�7/,-*-�3!�7/�^*.#5!�!�� ��8�#k��23P/;�7I��*-&58�� ��K�'��7�#}1%�I!�8�&W��&%�>��,.B/*<�c��g¯�/*.#%��#k!�,-&(�/�$8:7%J�!?7/&%����#5*-6%&(��,-!�7�23�38M,-!�7=BC��!�!�� �3#%��#5*-J/!�,-; #5*�BC&(P��/F��$��� �E)

GH&(��23��Fb8@&(�����c,.�b&(P��I��g���� ��,.23� �"!�� KÃÀ�,.����! �:!�7/��#%��#5*-J/!�,-; #5*B�&/P��/F��0,.�%1"&%*-1%�U#%������&5g/,.24#5!�,-&(�/� �:#%�/Fj#���?~��/��FW#%��#5*-J(�>,<�:23,-6(7%!dJ(,-��*-F�B���!�!�� �@BC&(P��/F���&/�I��&(23�4�¨&(�@#5*-* � &kGy!�7/�#5*-6%&(�D,<!?7�23� Kzn/��;�&(�/F��C!�7/�#%��#5*-J/!�,x); #5*�B�&(P��/F��^#%���|6(P�#%��#"�%!������d!�7�#5!^7/&%*-F�GH&(��#k*<*�����a�P/� �/;����^&5Gz����,-;��i�D��*h#k!�,-1%��� K�Ld7/� ���?G�&/���%�Q!?7/��#5*-6%&(�D,<!?7�23�8M,-!�7IB���!�!>� ��BC&(P��/F��M23,-6(7"!�BC�`7/��F�6",h�/6I#56(#5,.�/��!MP��/P/��P�#5*Q�>��a¯P/� �/;����M&kG¾���D,<;���D��*h#k!�,-1%���M#5!@!�7/�M��g���� �/���^&5G!�7/��,.�:��� �sGH&(��24#"�/;��^&(�I!�7/�^����a�P/� �/;����^&5Gz����,-;�������*.#5!�,-1%���d&(;�; P����D,h�/6I,.�c!�7/�7/,-��!�&(��,-; #5*�24#%��T5��!@FQ#5!?#�K

Ld7/�4~����>!��g�#%24�/*-��6%,-1%� �=B"J t &51%� �^,-�4#W��&(�D!�G�&"*<,-&�B�#5����FA&(�A�+��&/a�P/&%,-� �@��#%�/F��4¹�!�F�K�#%�/F=V:,.�A{`��Tt &(�>��K.�z!�8@&�lMm^n�fÁ�>!�&(;�T"�4;e7/&"��� �AGH&(�3!�7/��,.�S1%&%*.#5!�,-*-,-!�J%K P���,.�/6A!�7/�0Y%Y})+J"� #%�4��� �D,<&/Fb� �/F�,.�/6U,.�j�}w �'����+�D&(a�P/&%,-�$,.�/; ��� #k����Fp,.�b����,-;��RB"J�#RG�#5;�!>&(�3&5G �CKZw%Y���8:7/,<*-��V:,.��{`��T',.�/; ��� #5����F�,.�p����,-;���B%J=#cG+#5;�!�&(�|&5G"QK.� !�KzLM7/�3B�����!:;�&(�/��!�#"�%!�)E�D� B�#5*.#%�/;���FR�C&(��!�GH&%*-,-&�#5;�7/,-��1%���#|G+#5;�!�&(��&5G C !�K C [3#%�/Fc!�7/��P��/,-1%� ���?#5*Â�C&(��!�GH&%*-,-&#38@� #5*-!�7�&5G !"w�KZw C K� d��,.�/6�!�7/�f �I� � � )EP��CFQ#5!��38M,-!�7 ��� [��Z[ �J/,-��*-F��#|G�#k;�!�&(�@&5G C [�K��'���/8�7/,-;e7�,-�`#5*.23&"��!#5�@6%&/&(F�#5�@!�7/�BC����!@;�&(�/��!�#%�"!�)E��� B�#k*h#"�/;���F0��&(��!DG�&%*-,-&QKzLd7/�f�g�#5;�!Mf �$� � � )EP���FQ#k!��^J(,-��*-F��^#�>,h2i,<*.#%��G+#5;�!�&(�&5G C [��ZwC�%K

Ld7/�$������P/*-!��|&5GM!�7/�48@� #5*-!�7�#5;�7/,-��1%��FA&O1"� �:!�7/�$Y%YIJ%� #"����#%�D��6",<1"� �WGH&(�:!�7/,-�4��#5,.�^&5GM��!>&(;�T%�}��#5�8@��*-*#5�4&"!�7/� �4��#5,.��� ��,.�pL�#%B/*-�U�%KA���c&(B/��� ��1"��F=a¯P�#5*-,-!�#5!>,<1"��*<J���,.23,-*.#%�$�D����P/*-!��4GH&(�|!�7/��F�,���� ��� �"!��C&(��!�GH&%*-,-&%�;�&(�/��,-F�� ����F�B"J t &O1"� �^XZw�\ � t &/2423� ��;�,.#5*�½W��!�#5*-��#%�/FWV:,h��{`�>T��¨�����#5*-��&IÀ�,-6(P����$� � � t &(2423� �D;�,h#k*�½W��!�#5*-�#%�/F�½W��,-;�;�& t &(�>��K.�(#%�/F0� ��½ #"�/F t &(; # t &"*h#��¨À�,-6(P��D�Y � K��+��#5*-*�; #5�����}��!�7/�:8�� #k*<!?7�#5;e7/,-��1%��F0B"J�f �4� � �#%�/FW!�7/�4f�g�#5;�!Mf �I� � � ,-�*.#%�D6%� �M!?7�#%�c!�7/�38�� #k*<!?7�&5G@!�7/��P��/,-1%� ���?#5*z��&(�D!�G�&"*<,-&0#5*-6%&/��,-!�7�2�K^½W&(����&51%� ����,.�����1%� �>#5*�; #5�����@!�7/�^8@� #5*-!�7S&5GÂ!?7/�`fO�$� � � )EP���FQ#5!��|,-��#5*.23&%��!M#5�@6%&/&/F0#5�@!�7/��8@� #5*-!�7I&kGy!�7/�B����>!M;�&(�/��!�#"�%!�)��� B�#5*.#%�/;���FW��&(�D!�G�&"*<,-&QK��s��#5*-*�!�7/�:��g¯�C� ��,.23� �%!>��8@�^��� �sGH&(��23��F��/!�7/�^J/,-��*-F���&5G�!�7/�`fO�$� � � )EP���FQ#5!>�#%�/F�!�7/�f�g�#5;�!�fO�4� � � )EP���FQ#5!>�8@� ���^1%� ��JI��,.23,-*.#%����#%�/F0#5*-8M#}J(�@8M,-!�7/,.�c&(�/���� �D;�� �%!:&5G�&(�/�#%�/&%!�7/� ��K

Ld7������3��g¯#%2S�/*<���|;�&(24��#%�D,h�/6�!?7/��FQ#k,<*-JW8@� #5*-!�7W#5;�7/,<��1"��FUB%JWf �4� � � )EP��CFQ#5!��4#%�/F�!�7/�4BC����!^;�&(�/��!�#"�%!��� B�#5*.#%�/;���Fp�C&(��!�GH&%*-,-&�#%���IF�� �/,<;�!>��Fp,.��À�,-6(P��D����� � !�KWlM&%!>�4!�7�#5!38�7/� �'!�7/�I��!�&/;eT"�;�&(�/�>,<F�� �D��Fo#%���I�/&%!1%&%*.#5!>,<*-�3#%�/F���7/&58j#*-&/;eTk)+��!�� �c��� �sGH&(��2S#%�/;��%��#5�@,h��!�7/�^; #5���^&kGÂ� ��½Ê#%�/F t &/; # t &%*.#0�¨�����:!�7/�^1%&"*h#k!�,-*<,-!�J� ×�ØOÙ^çHÙ�ÝMÞ>ßzçHÝHÞ�èHï ç@ß-é+Þ?àÁæ�Ø ÜZè�Ø0æ�Ù:ê Ú Ü�ëZÝ:ݨØOÙ � ã>éHÜZÞ�Ú5ç��%Þ>é+ݨßhÞ>ë�ÜZÞ?ç^èEÞ>ÛOç¨ÜZçHÝHÙ�âcÞ�ßzݨØ5Ù:ßhÞ>ë�ëZÞ�æ�Ü�Û��$çHÝHÞ�è�ï ç���� ��î���ëZè�Þ?ã î

��à�ÙséHÜZè�ã>Û�� é+ã�Û5â ç�î��¾é+èEÞ}î���Þ�è�ã��Þ>ëZã}î���Þ?à@à@Ù+é+èsÜZã>ë �^Ù�ÝHã�ëZç�î���Þ�æ���ØOÙ�à�ÜZèEã>ëZç�î��¾Ú��%Þ�Û Ý�î��Qç��%Ù�ä��^ã>Û�Ú ß.ã?èEݨÚ}éHÜ�Û��}î���� ��Þ>Û(î! ÜZçHèHØ�ê5ã�è�Ø(î ! Þ>é+â%î�"���î�"��3î�"�×���î�"¾Ú ë�ßHî����î�#$���3î�#-Û���Ùsé+çHÞ>ë�ë.î%#-é+Þ�í?Ú5Þ�ÜZç�î'&)(*&}î +�ÜZàqêkÙséHëZä),-��ëZã�éHï%î'+�Ü�Û.�¾éHï%î�+�Þ�â}ã�ï%î�/"Ú ï�ÙsÛOç�î�^ÙsÜZè�èEÞ}î'�^Ùsé+èHï%î��0�0�3î��^Þ>ê Ü�ë.î'�^Þ>éHéHÜZç��:Ü�Û Ü�Û��}î%��12"�î��CÜ�ë�ëZç¨ê Ú}é+ä�î43�è�Ø ë�ÚOà:î%3�ÙEã>é+ç�î%3?Ø5Ùsé+à�ã>Û5SÜ�ë�ë�ÜZã�à@ç�î�ã�Û5â�×�Ù���ã?è�Þ ð

�%�

Page 13: Mathematical Finance, 8(4):325-347, 1998.web.cs.iastate.edu/~honavar/portfolio-selection.pdf · Mathematical Finance, 8(4):325-347, 1998. ! #" %$'&)( * +% -,. */0 1 #+324,. " 5%687

n/!>&(;�T%� �+�D&(a�P/&%,-� t &(2S2�K�½W��! K t &(2S2�K�½W��! K � ��½ �� V:,.�c{`��T � V:,.�c{`��T � ½W��,-;�;�& t &(; # t &%*.#

��&"*h#k!�,-*<,-!�J � n/!�&/;�T0� [�KZ['! " [�KZ[%Y � [�KZ[%Y � [�KZ[�� !��&"*h#k!�,-*<,-!�J � n/!�&/;�T$Y [�KZ['�"[ [�KZ['�"[ [�KZ['!C� [�KZ[�� "t &(������*.#5!�,-&(�I&kGÂ����!�P���� [�KZ[ " � [�KZ[T_ " [�KZ[T_ C [�K ! ��������!:n/!�&(;�T )z8@� #5*-!�7 ��Kxw �%Y�Kx[ �%Y�KZ[ ��!�K "

)z{^³�m � [�K " � w�K C � w�K C � Y�K �� t�� ³ )z8@� #5*-!�7 C !�K C � "'"QKx[ � ['!�Kx[ ����K.�

)z{^³�m Y��%K _ Y'��K�! Y'!�K " ��!�K.�fO�4� � � )z8@� #5*-!�7 ����� � � ��� � w C KZw � "QKZw

)z{^³�m Y��%K " Y'!�K�� Y'!�KZY ��!�K.�) � � t�� ³ [�KZw _ [�K C C [�KZw'� [�KZw%w

fqg¯#k;�!�fO�$� � � )z8@� #5*-!�7 C [�K " �%� [�K � ������ ��� ��)z{^³�m Y��%K�! Y'!�K�� Y'!�KZY ��!�K.�) � � t�� ³ [�KZw _ [�K C _ [�KZw'� [�KZw%w

:�/,-1%� ����#k* )z8@� #5*-!�7 "([�Kx[ C ��K " C "QKZ[ � "QKZY)z{^³�m ����K�! Y��%Kxw Y��%K _ � Y�K �) � � t�� ³ [�K � " [�K � _ [�K C Y [�KZw "

L�#%B/*-�0��� t &(2S��#%��,-��&(�c&5Gz!�7/�38@� #5*-!�7W#5;�7/,-��1%��F�B%Jc!�7/�4f �4� � � )EP���FQ#k!��%��!�7/�3f�g�#5;�!�fO�$� � � )EP���FQ#5!��%��#%�/F!�7/�4P��/,-1%� ����#k*z��&(��!DG�&%*-,-&0#5*-6%&(�D,<!?7�2�K:ÀQ&(�:#5*-*�!�7/�4��&(�D!�G�&"*<,-&%�^;�&/�/��,-F�� ����F���8@�6",<1"�iBC&%!�7c!�7/�3!�&%!?#5*�8@� #5*-!�7#%�/F4!?7/��#}1%� ��#56%�M#%���/P�#5*���� ��;�� �"!zJ/,-��*<FU��{`³qm � GH&(��� #5;�7$�C&(��!�GH&%*-,-&5)+����*-��;�!�,-&(�I#5*-6%&(��,-!�7�2Ã#5�z8@��*-*�#k��GH&(��!�7/�B����>!y;�&(�/�>!�,-!�P/� �%!���!�&/;�T:,h�|!�7/�@��&(�D!�G�&"*<,-&^#%�/F!?7/��B����>!y;�&(�/�>!�#%�"!�)E��� B�#5*.#%�/;���F4��&/��!�GH&%*-,<&4� � t�� ³ � ;�&/24��P/!���F,.��7/,.�/F���,-6(7"!G+��&(2¿!�7/�4� �"!�,.���$���D,<;��I����*.#5!�,-1%����>��a¯P/� �/;��%K3���I#5*-��&0��� ��&/��!^8@� #5*-!�7�#5�#SG+��#5;�!�,-&(�c&5G@!�7�#5!#5;�7/,-��1%��F�B"J�!�7/� � t�� ³��¨F�� �/&"!���F»¶�� � t�� ³�·c,.��!�7/�I!�#%B/*-� � KcLM7/�4~��D��!���&58M�|&5GM!�7/�I!�#%B/*-���D� ��&(��!|!�7/�1%&%*.#5!>,<*-,-!�J�&5G�!�7/�^;�&(�/��!�,-!�P/� �"!M��!>&(;�T%�:#%�/F�!�7/�;�&/������*.#5!�,-&(��BC��!E8@��� ��!�7/��,.�M����!�P����/�}K

#%�/FA;�&(������*.#5!>,<&/��,.�"GH&(��24#5!>,<&/�W,.�=Ly#%B/*-�W� � �¾!�7/�$8@� #5*-!�7'#5;e7/,-��1%��FpB"JW!�7/��P��/,<1"� ����#5*@��&(��!DG�&%*-,-&%�4#%�/FA!�7/�f �I� � � )EP��CFQ#5!���#5�^8@��*-*�#5�|!�7/�4B�����!3;�&(�/��!?#%�%!D)E��� B�#5*.#%�/;���FU�C&(��!�GH&%*-,-&RB�#%����*-J�&/P/!���� �sGH&(��2 !?7/�4,.�/F�,<1/,-FQP�#5*��!�&/;�T%� K

Ld7/�N�P/*xG��Â�:³º#%�/FUn/;�7/*hP�2 !�7������?)+��!�&/;�TR�C&(��!�GH&%*-,-&R�/*-&%!�!>��FU,.�'À�,<6/P���� !���g�7/,.B/,-!��3��&(23�4,.�"!�� ������!>,h�/6B�� 7�#}1/,-&(��K3n/;e7/*.P�2 ��T"J¯��&/;�T5��!��^B���!�8@��� �WFQ#}J(�+"([%[%[�#"�/F " �%[%[�� � ��#%B/*-,.�/6R,-!^!�&�&/P/!���� �sGH&(��2 !?7/�iY"Y})+J%� #%�� t�� ³b#5!M!?7�#5!���&",h�"! KMn/;�7/*.P�2 *h#k!�� �MF���;�*-,.�/���,.�c1O#k*hP/�%��#%�/FWB�&%!�7�!�7/��Y"Y})+J%� #%� � t�� ³o#"�/F0!�7/�3f �4� � � )P���FQ#k!��&(P/!?��� �sGH&(��2¸n/;�7/*hP�2¿&51%� �@!�7/�^� �"!�,.����C� ��,-&(F�K@�¨Gz!�7/�^��g���� �D,h2i� �%!:� �/F���FR#%�D&(P��/F�FQ#}J "��"[%[��/!�7/� �!�7/�fO�$� � � )EP���FQ#5!��|8@&(P/*-F�J(,-��*-F0*-�����:8@� #5*-!�7I!�7�#%�S��,.24�/*-J�,.�%1%���>!�,.�/6$,.�0n/;�7/*.P�2�Kz�s�/F�����F��Q!�7/� � t�� ³=GH&(�"��%["[M!��>#5F�,.�/6FQ#}J(��,-��,h�"1%����!>��F48�7/&%*-*-J3,h�In/;�7/*hP�2��%8�7/,-*-��!?7/��� ��t ³cG�&/��!�7/�M� �"!�,.����Y%Y})+J"� #%�y�C� ��,-&(FI,.�%1"����!"'"��Ã,.�6�P/*xG�� ! "��Ã,.�0�:³p#%�/FWY%Y��Ã,.�0n/;�7/*.P�2�K

À &(�^!?7/���s��&(a�P/&%,-���%V:,.�A{^��T��C&(��!�GH&%*-,-&Q��À�,-6(P���� "W��7/&58M�!�7/����� �D;�� �%!�#k6%�$&kG�!�7/�I8�� #k*<!?7�,.�"1%����!���FA,.��+�D&(a�P/&%,-�3GH&(�BC&%!�7'!�7/��fO�$� � � )EP���FQ#5!>�0#%�/FU!?7/��P��/,-1%� ���?#5*M��&(��!DG�&%*-,-&�#5*-6%&(��,-!�7�2�K0{:*-!�7/&(P/6(7=BC&%!�7A#5*-6%&5)��,-!�7�23�^!>� �/F���FR!>&�T5��� �c��,.23,-*.#%�|��&(��!DG�&%*-,-&%� ��!?7/��P��/,-1%� ����#k*z��&(��!DG�&%*-,-&0#5*-6%&(�D,<!?7�2�;�7�#%�/6%��FW,-!��^,.�%1"����!�23� �"!����&(�C&(��!�,-&(�/�:23&(���#56"6(��������,-1%��*-JI8�7/� �0V:,.��{`��TI��&%�>�#%��&(P��/F�FQ#}J/� !%[%[ � �%[%[�K

� Y

Page 14: Mathematical Finance, 8(4):325-347, 1998.web.cs.iastate.edu/~honavar/portfolio-selection.pdf · Mathematical Finance, 8(4):325-347, 1998. ! #" %$'&)( * +% -,. */0 1 #+324,. " 5%687

0

50

100

150

200

0 1000 2000 3000 4000 5000

BCRPEG

UniversalBest stock

À�,-6(P����0��� t &(24��#"��,-��&(�c&5G�8@� #5*-!�7/�3#5;�7/,<��1"��F�B"J0!�7/�IB�����!|;�&(�/��!�#%�"!�)E��� B�#k*h#"�/;���F��C&(��!�GH&%*-,-&Q��!�7/�If �4� � � )P���FQ#k!��%�5!�7/�MP��/,-1%� ����#5*Q�C&(��!�GH&%*-,-&`#k*<6"&(��,-!�7�2Á#%�/F3!�7/�@B�����!���!>&(;�TdG�&/�y#:��&(�D!�G�&"*<,-&^;�&(�/�>,<�>!�,.�/6^&5G t &(242i� ��;�,.#5*½W��!�#5*-��#%�/F0V:,.�c{^��TQK����� � PQ%��?��� ��#&PQ� +ÀQ&"*<*-&58M,.�/6 t &O1"� �zXZw�\E�58@�M#5*-��&^!����>!���F3!�7/�M; #5���@8�7/� �D��!?7/���C&(��!�GH&%*-,-&; #%�3B�P/J3��!�&/;�T%��&(�324#%��6",h��K�LM7/,-��; #5���; #%��BC��23&/F���*-��FRB"J�#5F�F�,.�/6$#"��#5F�F�,-!�,-&(��#5*M¶�2S#%��6%,.�I;�&(24�C&(�/� �"!�·3GH&(��� #5;e7I�>!�&(;�T3!�&3!�7/�^1"��;�!�&(��&5Gz����,-;������*.#5!�,-1%���}K�����#5�>��P�23��F4!�7�#k!z#5*-*�24#%�D6%,.�4��P���;�7�#5�����z8@� ���:24#5F�� �%[ �ÁF�&58��3#%�/FI8M,-!�74# �"[��º*-&(#%��K�LM7/P/� �!�7/��2S#%��6%,.�I����,-;��`����*.#5!>,<1"�MGH&(�z#:�>!�&(;�T3Õ�&/�3FQ#}J�2�,-�MY%Ò 4Ô � � � � 8�7/� ��� � ,-��!?7/��FQ#5,-*-J4,.�"!�� ������!@��#5!���������; #5*-*!�7�#5!@Ò 4Ô ,-�M!?7/������,-;������*.#5!�,-1%�:&5Gy��!>&(;�T3Õ � K¾����!>����!���FI!�7/,-��; #k���M8M,-!�7 � � [��x[%[%[%Y'! !M8�7/,-;�7I;�&(�������?��&(�/F��@!�&#%��#%���/P�#5*�,.�"!�� ������!^��#5!>�^&5GH_ ��KzLM7/�3������P/*-!��^#%���^6%,-1%� ��,h�cLy#"B/*<�3Y�K@�H!M,-�:;�*<� #"�@G+��&(2¸!?7/�^!�#%B/*-�!�7�#k!M!�7/�GH&(P��s)+,.�%1"����!�23� �"!���&/��!�GH&%*-,<&S;�&(�%!?#5,.�/,h�/6I!�7/�^��#"23�^!�8�&3��!>&(;�T%�M�/*.P/�I¶�B�P/J(,.�/6I&(�I!�7/�24#"��6%,.�/·$������P/*-!��d,h�c#6(��� #k!�� ��8�� #k*<!?7�KzLM7/�� �I;�,<� �/;�J�&kGÂ&(P��dP���FQ#5!��3��P/*-�� ��#%B/*-���^P/��!>&�!>����!M&(P��MP��CFQ#5!����M&/��23&(���^!�7�#"�I!E8@&��!�&/;�T%� K@½W&(�D��&O1"� ����#5���?7/&O8:�0B"J�!�7/��#%��#k*<J/��,-� ��!�7/�|8@� #5*-!�7=¶�*-&%�>!�·$B"J0&(P��M#k*<6"&(��,-!�7�23�:;�&(24��#%�D��F0!�&I!�7/�B����>!M;�&(�/��!�#"�%!�)E�D� B�#5*.#%�/;���F��C&(��!�GH&%*-,-&4��; #5*-���M*-,.T5�"!�� 6 Í � �(8:7/� ��� #5��G�&/�z!?7/�B�&(P��/F��M&/� t &51%� ��u���P��/,-1%� ���?#5*��&/��!�GH&%*-,<&I#5*-6%&(��,-!�7�23�}�(!?7/��*-&%���d,h�I8@� #5*-!�7S,-�M*-,.�/� #%�@,.��!�7/�^�/P�2�BC� �@&5G�,.�%1"����!�23� �"!M&(�/!�,-&(�/�MÍAK�LM7/P/� �/&(P��#5*-6%&(�D,<!?7�2¸,-�`23&/���^*-,hTk��*<Jc!�&3!�&%*-� ��#5!>�#5F�F�,-!�,-&(��#5*�,h�"1%����!?23� �%!d&(�/!�,-&(�/� ���?P/;e7�#5�:B�P/J(,.�/6I&(��24#%��6",h��K

��!

Page 15: Mathematical Finance, 8(4):325-347, 1998.web.cs.iastate.edu/~honavar/portfolio-selection.pdf · Mathematical Finance, 8(4):325-347, 1998. ! #" %$'&)( * +% -,. */0 1 #+324,. " 5%687

2

4

6

8

10

12

14

0 1000 2000 3000 4000 5000

BCRPEG

Best stock

À�,-6(P����$Y � t &(24��#"��,-��&(�c&5G�8@� #5*-!�7/�3#5;�7/,<��1"��F�B"J0!�7/�IB�����!|;�&(�/��!�#%�"!�)E��� B�#k*h#"�/;���F��C&(��!�GH&%*-,-&Q��!�7/�If �4� � � )P���FQ#k!���#%�/F�!?7/�B�����!:��!�&/;eTiGH&(�@#4��&(�D!�G�&"*<,-&I;�&(�/��,-��!�,.�/6I&5GÂ� �z½ #%�/F t &/; # t &%*.#�K

���B� � �XPQ%T�O��� � %TP.'T�����GH&(P��/F!�7�#k!y*-� #%���/,.�/63��#5!�����#%��&(P��/F �3� [��Z['�d#%���z6"&(&/F3;e7/&%,-;�����GH&(�¾!�7/�Mf �4� � � )EP���FQ#k!��%K��¨G�!?7/���>!�#%��!�,.�/6��&/��!�GH&%*-,<&�,-�:G+#%��G���&/2�!�7/�3B�����!^;�&/�/��!�#%�"!�)E��� B�#5*.#%�/;���F��C&(��!�GH&%*-,-&0#%�/F�f �¸P/�����^#I1%� ��J�*-&58»*-� #%���/,.�/6W��#5!��%�!�7/� �I,-!:8M,-*<*���� �sGH&(��2��C&(&/��*-J0B���; #%P/���^,-!:F�&(���:�/&%!�23&51%�^#}8�#}J|G+��&(2Ã!�7/�&/��,-6%,.��#5*y��&/��!�GH&%*-,<&iG�#5�>!�� �/&/P/6(7�K]^�S!�7/�M&%!�7/� ��7�#%�/F���#7/,-6(7S*-� #%���/,.�/6$��#5!>��; #"�$#5*-��&3B��^B�#5F$#5��,-!�; #"P/������fO�j!�&3B��^23,-��*-��F0B%J3FQ#}J")+!�&5)+FQ#}J��P/;�!�P�#5!>,<&/�/� K3ÀQ&(�:!?7/�iYO)+��!�&/;eT���g¯#%2S�/*<�S6%,-1%� �W,.�W!�7/�4,.�"!���&/FQP/;�!�,-&(����!�7/�4f �I� � � )EP��CFQ#5!��$#5;�!?P�#5*-*<J'*<&"�����23&(�/��JI8�7/� ��!�7/�^*-� #%���/,.�/60��#5!��:,-�`#"B�&51%�^&(�/�%K

�b,-!�7/,.�=�>&(23�023,-F�F�*-�U��#%�/6"�%�z,-!I!�P����/�4&(P/!S!�7�#5!4!�7/�W��� �sGH&(��2S#%�/;���&5Gf �_,-���/&%!I&O1"� ��*-JU��� �/�>,<!>,<1"�!�&�!?7/�0��#%��!�,-; P/*.#%�S;e7/&%,-;��c&5G`*-� #%�>�/,h�/6=��#5!>� � K�¹�� #%���/,.�/6��>#5!����3G+��&(29[�KZ[C�$!>&U[�K.���W#5*-*�#k;e7/,-��1%��F�6(��� #5!8@� #5*-!�7���6/��� #5!�� �:!�7�#"�0!?7/�48@� #5*-!�7�#5;�7/,-��1%��FUB"JR!?7/�$P��/,-1%� ����#5*@��&(�D!�G�&"*<,-&W#5*-6%&(��,-!�7�2_#%�/F',h�A24#%�"J0; #k�����;�&(24��#"��#%B/*-�3!�&4!�7/�38@� #5*-!�70#k;e7/,-��1%��F�B"J�!�7/�3;�&(�/��!�#%�"!�)E��� B�#k*h#"�/;���FR��&/��!�GH&%*-,<& K`Ld7/�8@� #5*-!�70#k;e7/,-��1%��FWGH&(�F�,���� ��� �%!:*-� #%���/,.�/60��#5!>���MGH&(�@!�7/��G�&/P��s)+,.�%1%���>!�23� �"!���&(�D!�G�&"*<,-&�F�,-��; P/���>��Fµ#"B�&51%����!E8@&3��!�&/;�T%���/*.P/�24#"��6%,.� �#%���^6",<1"� ��,.�0L�#%B/*-� !�K

� "

Page 16: Mathematical Finance, 8(4):325-347, 1998.web.cs.iastate.edu/~honavar/portfolio-selection.pdf · Mathematical Finance, 8(4):325-347, 1998. ! #" %$'&)( * +% -,. */0 1 #+324,. " 5%687

0

10

20

30

40

50

60

70

80

0 1000 2000 3000 4000 5000

BCRPEG

Best stock

À�,-6(P���� ! � t &(24��#"��,-��&(�c&5G�8@� #5*-!�7/�3#5;�7/,<��1"��F�B"J0!�7/�IB�����!|;�&(�/��!�#%�"!�)E��� B�#k*h#"�/;���F��C&(��!�GH&%*-,-&Q��!�7/�If �4� � � )P���FQ#k!��%�Q#%�/Fc!�7/�B�����!:��!�&/;�T|G�&/�M#4��&/��!�GH&%*-,<&S;�&(�/��,-��!�,.�/6I&5GH�|P/*xGD�Q�:³p#"�/FRn/;�7/*hP�2�K

��� %���W � ��� )�#&%'& P.'T��#&�����#5*-��&W!�����!���F'!�7/���C� �sGH&(��24#%�/;��I&5G�&/P����&/��!�GH&%*-,<&�P��CFQ#5!��0#5*-6%&(�D,<!?7�2_8�7/� �A��,-F��I,.�"G�&(�>24#5!�,-&(�',-�$�����?)��� �"!���F�KWÀQ&/�!?7/�����$��g���� �D,h2i� �%!��}�z!�7/�I��,-F���,.�"G�&/��24#5!�,-&(�'8M#5�4P/!�,-*-,ED���FbP/��,.�/6�!�7/�023��!?7/&(FAF�����; ��,.B���F�,.�n/��;�!�,-&(�WY�K "3&5G���#"��!�,-!�,-&(�/,.�/6$!?7/�����a�P/� �/;��3,h�"!�&I��P�B/����a�P/� �/;����B�#5����Fc&(��!�7/�^15#5*.P/�3&5GÂ!?7/���,-F��3,.�"G�&(�>24#})!�,-&(��K�LM7/� ���:#%����24#"�%J3��&%���>,hB/*-�dG�&/��23��&5G���,-F���,.�"GH&(��24#5!>,<&/�3&(�|8�7/,-;e7S!�7/�����:#5*-6%&(��,-!�7�23�@23,-6(7"!zB��M!����>!���F�K�+� &(P��3��g���� ��,.23� �"!�� �@8@��;�7/&%����!�&'F��?~��/�0!�7/�c��,-F��0,.�"G�&/��24#5!�,-&(�A15#5*.P/�W!�&�B��0!?7/�0,.�/F���g=&5G^!�7/����!�&/;eT8M,-!�7W!?7/�iBC����!6/��&58M!�7�&kG�8@� #5*-!�7W&(�c!�7/�S*h#k��!4� [%[S!���#5F�,.�/6cFQ# J/�

,.�"GH&(��24#5!>,<&/�0!?7�#5!^8@&(P/*-F�;�� ��!�#5,.�/*-JB��4#}1O#k,<*.#%B/*-�I!�&�#I,h�"1%����!>&(�M,.��#I��� #5*�!���#kF�,h�/6��>,<!?P�#5!�,-&(��KLM7/P/� ��!�7/�4�C&%����,.B/*-�4����!^&5Gz15#5*.P/���^GH&(�d!�7/�3��,-F��,.�"G�&/��24#5!�,-&(�I,-� :(�%�}� � ���-, < 8�7/� �D� , � ÍAK

Ld7/�������P/*-!���#%�D�^��P�2424#%�D,ED���F�,.�cLy#%B/*-� "QKz�H!d,<�d��1(,-F�� �"!MG+��&(2Ã!�7/�^��g�#%24�/*-���:6%,-1%� �I,.��!�7/�^!�#"B/*<�|!�7�#5!P/��,.�/63!�7/�M��,-F��:,.�"G�&(�>24#5!�,-&(�W�¨,HK��%K.�/T5��� �/,.�/6iÍ ��&/��!�GH&%*-,<&31%��;�!>&(��� � �����?P/*<!>�z,.�$#���,-6(�/,x~Q; #%�"!�,.24���D&O1"� 23� �%!�,.�!�7/�:8@� #5*-!�7$#k;e7/,-��1%��F��/��1%� �I8�7/� �IP/��,.�/64��P/;�7S��,.24�/*-�#%�/F��D� #5F�,-*<J�#}1O#5,-*.#%B/*-��>,<F��^,.�"GH&(��24#k!�,-&(��K��M&58@��1%� ���!�7/�:6(#%�IB���!�8@��� �4!?7/�`BC����!@��,-F��^,h�"GH&(��2S#5!�,-&(�SF�� ��� �/F�� �"!�;�&(�/�>!�#%�"!�)E��� B�#5*.#%�/;���F���&(�D!�G�&"*<,-&$#"�/F$!�7/�:8@� #5*-!�7#5;�7/,-��1%��FpB%J'!�7/��fO�$� � � )EP���FQ#5!>��8M,-!�7��>,<F���,.�"GH&(��24#k!�,-&(�A,-�i�/&58�23P/;e7'*.#%��6%� ��KR]^�/��&5G:!�7/���D� #5��&(�/�|,<�!�7�#5!d8��3P/����F0!?7/���#%2i�^*<� #"���/,.�/60��#5!������6/#%��F�*-�����:&5Gz!�7/��>,<F��|,.�"G�&(�>24#5!�,-&(��1O#5*.P/�%K�¹¾#%��6%�|*<� #"���/,.�/60��#5!����; #%P/���^!?7/�P���FQ#5!��#k*<6"&(��,-!�7�23�M!>&�a�P/,-;eT"*-J0#%������&(#k;e7S!�7/�B����>!M;�&(�/��!�#"�%!�)E�D� B�#5*.#%�/;���F0��&(�D!�G�&"*<,-&Q� B�P/!�2S#%T5�,-!4F�,��$; P/*-!IG�&/�3!�7/�0#5*-6%&(�D,<!?7�29!�&U��� #k;e7A!�7/,-�$�C&(��!�GH&%*-,-&U��g�#5;�!�*-J%Kp]^� !�7/��&%!�7/� �S7�#%�/F��@��24#5*-*M*-� #%�>�/,h�/6

���

Page 17: Mathematical Finance, 8(4):325-347, 1998.web.cs.iastate.edu/~honavar/portfolio-selection.pdf · Mathematical Finance, 8(4):325-347, 1998. ! #" %$'&)( * +% -,. */0 1 #+324,. " 5%687

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0 1000 2000 3000 4000 5000

Fra

ctio

n of

"Ir

oquo

is"

BCRPEG

Universal

À�,-6(P���� " �ALM7/�cG���#k;�!�,-&(��&5G!�7/�W8@� #5*-!�7�,h�"1%����!>��Fo,.�j�s��&/a¯P/&%,-��&51%� �S!�,.23��B%J�!�7/��P��/,-1%� ����#5*�C&(��!�GH&%*-,-&#5*-6%&(�D,<!?7�2�#%�/F0!?7/�f �4� � � )EP���FQ#k!��^GH&(�M#3��&/��!�GH&%*-,<&I;�&(�/��,-��!>,h�/6I&5Gz�+��&/a�P/&%,-�`#%�/F0V:,.��{`��T K

��#5!>����#5,-FA;�&(�%1"� ��6%� �/;��4!�&c!�7/��B�����!|;�&(�/��!�#%�"!�)E��� B�#k*h#"�/;���F��C&(��!�GH&%*-,-&Q�zB�P/!�2S# JW; #%P/�>�$!�7/�I#5*-6%&(��,-!�7�2 !�&���C� �/F�#:*-&(�/63!�,.23�@G+#%��#}8M#}J:G���&/2º!�7/,-��15#5*.P/�%KzLM7/� ���?GH&(���%�"8�7/� �3!�7/�d��,-F���,.�"GH&(��24#5!>,<&/�3���/*-,-!��@!�7/���/P�2�BC� �&5G@!���#5F�,.�/6�FQ#}J(�^P��/��1%� �/*-J%��F�,���� �D� �%!^*-� #%���/,.�/6W��#5!����dG�&/�:!�7/�3F�, ��� ��� �"!^��,-F��4,.�"GH&(��24#5!>,<&/�015#5*.P/����24#}J0BC�����a�P/,.����F�K���$ � #&%('*)�#&����#&+ ����'�� & #&% �6'�� PQ� ' �N# +*'1#�/��+LM7/�@!�7/��&(�D��!�,-; #5*¯BC&(P��/F4&(��!�7/�@8@� #5*-!�7#5;�7/,-��1%��F$B"J^!�7/�Mf �I� � � )EP��CFQ#5!��M��P/6"6%����!���!�7�#5!�!?7/��fO�$� � � )EP���FQ#5!>�#5*-6%&(�D,<!?7�2 8@&(P/*-F���; #k*<�BC��!�!�� ��!�7�#%� t &O1"� ��u��zP��/,-1%� ����#k*y��&(�D!�G�&"*<,-&4#5*-6%&(�D,<!?7�2�#5�@!�7/�^�(P�23B�� ��&5Gy��!�&/;�T%��&(�#5���>��!���,.�c!�7/�3��&(��!DG�&%*-,-&I6(��&58M� K@À�P���!�7/� ��23&/���%��!�7/�|!�,.23�4����a¯P/,.����F�B"J�!�7/�4f �4� � � )EP��CFQ#5!��3!�&$23&/F�,xG�Jc!�7/���&/��!�GH&%*-,<&�#}GH!�� �:� #5;�7W!��>#5F�,.�/6RFQ#}Jc,<�3;�&(�/�>!�#%�"!���� ����!�&/;eTc8�7/,-*-��!�7/�4!>,h2i�$;�&(2S�/*<��g/,-!�J�&5GM!�7/�$P��/,-1%� ���?#5*��&/��!�GH&%*-,<&I#5*-6%&(��,-!�7�2¸6/��&58M�z��g���&(�/� �"!�,.#5*-*-J�G+#5��!�8M,-!�7I!�7/��/P�2�BC� �@&5Gy�>!�&(;�T"��,.��!�7/�`�C&(��!�GH&%*-,-&QKzLM7/���>�^!E8@&T5��J3G+#5;�!�&/���@��P/6%6%����!�!�7�#5!@!�7/�fO�4� � � )EP���FQ#5!��2i,<6/7%!MB��#|1/,.#%B/*-��#5*-!�� ����#k!�,-1%�#5�M#3����&51O#%B/*-JI;�&(24����!>,<!>,<1"���&/��!�GH&%*-,<&S����*-��;�!�,-&(�W#5*-6%&(��,-!�7�2�K

L¾&W� 24�/,.��,-; #5*-*-Jp1%� ��,xGHJ�!�7�#5!3!�7/�0#"B�&51%��#5F�1O#%�"!�#56"���i7/&"*<F=,.�=����#k;�!�,-;��%�z8@����� �sGH&(��23��F=��g���� �D,h2i� �%!��8M,-!�7A��&(�D!�G�&"*<,-&%�3&5GM��,FD��0Í � Y�!�&WÍ � Y "�G+��&(2_!?7/��ld��8¸m�&(��TWn/!�&/;eT�fqg(;�7�#%�/6%�%K����$;�&/24��#%����FA!�7/�8@� #5*-!�7�#%�/Fc!�7/�^!�&%!�#k*Q!�,.23�3����a¯P/,.����Fc!�&$P��CFQ#5!��!?7/���&(��!DG�&%*-,-&%�M!?7���&(P/6(7I!�7/�^� �"!�,.���3Y%Y})+J%� #%�@�C� ��,-&(F�K

�[_

Page 18: Mathematical Finance, 8(4):325-347, 1998.web.cs.iastate.edu/~honavar/portfolio-selection.pdf · Mathematical Finance, 8(4):325-347, 1998. ! #" %$'&)( * +% -,. */0 1 #+324,. " 5%687

�o,-!�7/&/P/!M*-&(#%�/� �b,-!�702S#%��6%,.�I*-&(#%�/�t &(2423� �D;�,h#k*�½W��!�#k*<� �%Y��Z[ � w�� CV:,h��{`�>T " �.� [��Z[� t � ³ � "'"Q�Z[ YT_%Y�� " ��/,-1%� ���?#5*y��&/��!�GH&%*-,<& C ��� � w ��� "f �4� � � �"� [��ZY � Y��%�Zwf�g�#5;�!�f �4� � � �"� [��.� � Y%Y��Z[

L�#%B/*-�0Y � t &(24��#%��,-��&/��&5GM!�7/����&/��!�GH&%*-,<&W����*-��;�!�,-&(�=#k*<6"&(��,-!�7�23�|8�7/� �=24#%��6",h�µ*<&/#%�/�|G�&(��� #5;�7���!>&(;�TW#%���#}1O#5,-*.#%B/*-�%K

f �� t�� ³ :�/,<1"� ����#5* �3� [C�Z[�� �� [C�Z[%Y �3� [��Z['� �3� [��.� [ �3� [�� ��� �3� [��xY%[YT_%Y�K " w �CK " �%� w�Kxw � YC�%K " � YC�%KZw �%� !�K ! � ['!�K.� wC�%K !

L�#%B/*-� ! � t &/24��#%��,-��&(�µ&5G�!�7/�S8�� #k*<!?7�#5;�7/,<��1"��FUB%JW!�7/�If �4� � � )EP���FQ#k!��$#%�/FWGH&(�:15#%��,-&(P/�3*-� #%���/,.�/6���#5!����#%�/F�!�7/�3P��/,-1%� ����#5*���&(��!DG�&%*-,-&$#5*-6%&/��,-!�7�2��/GH&(�@!�7/�^�>!�&(;�T"�M;�&(�/��,-F�� ����Fc,h�cLy#"B/*<�3Y�K

L¾&�;�&(2S��P/!��0P���FQ#5!>���4GH&(�|!�7/�RP��/,-1%� �D��#5*��C&(��!�GH&%*-,-&�#5*-6%&/��,-!�7�2���8��0P/�>��Fo#W��#"24�/*-,h�/6A!���;�7��/,-a¯P/�W#5�F�����; ��,.BC��FoB"J t &O1"� �4XZw}\�K�ÀQ&(�|��&(�D!�G�&"*<,-&%�3&5GM��,FD��0Í �¸w���8@�$#%������&5g/,h2S#5!���F=f�a�P�#5!�,-&(�b��� � B"JU#�~��/,<!>���P�2´!�#%T5� �c&51%� �:#5*-*���&"����,.B/*-���C&(��!�GH&%*-,-&01%��;�!�&/��� � &5G@!�7/�|G�&/��2Ê�HÕ � �-�5�?Õ � �-�5�}� �����?ÕHÓ �-� � 8�7/� ���3� #5;�7�Õ � ,<�#I�/&(���/��6(#5!�,-1%�S,h�"!���6%� �:#%�/F�Õ � � * [ ��ÕHÓ � �(K|�+�c&(P��d��g¯�C� ��,.23� �%!>� ��8@�4P/����F � � � [�K^LM7/,-�^��#"24�/*-,h�/6!���;�7��/,-a¯P/�^a�P/,-;eT"*-J�BC��;�&(23���@,.�"G�� #5�>,hB/*-����GH&(�z#|��&(�D!�G�&"*<,-&3;�&(�/��,-��!>,h�/6S&5Gyw^��!�&/;�T%� �",-!�!?#%T5����#"B�&(P/!@w�� �7/&/P����!�&|; #5*-; P/*h#k!��!�7/�^P��/,-1%� ����#k*��C&(��!�GH&%*-,-&4P���FQ#5!����@&51%� ��!�7/�^Y%Y})+J%� #"��!���#kF�,h�/64�C� ��,-&(F�KzLd7/���#%�/,-F$,.�"GH� #5��,.B/,-*-,.!�J&5G�!�7/��P��/,-1%� �D��#5*y�C&(��!�GH&%*-,-&�#5*-6%&/��,-!�7�2�; #"��#5*-��&�BC������ ��,.��À�,-6(P���� �38�7/,-;e7c��7/&58M�d!�7/�^!�,.23���D��a¯P/,.����FW!�&;�&(24��P/!>�^!�7/��P��/,-1%� ���?#5*y#5*-6%&(�D,<!?7�2¸;�&(24��#%�D��F�!�&$f �4� � � KL¾&�7�#%�/F�*-�M!?7/,<��F�,��$; P/*-!�J%�5GH&(����&/��!�GH&%*-,<&"�y;�&(�/�>,<�>!�,.�/6^&5G�23&(����!?7�#%�wd��!�&/;eT"� �}8@�z,.�/��!�� #5F3#%������&5g/,h2S#5!���Ff�a�P�#5!�,-&(����� � P/��,.�/6�#:*.#%��6%�M�/P�23B�� �M��� [ � � &5G���#%�/F�&(2i*<J|;e7/&"��� �31%��;�!�&/��� � �5� #5;�7|����*-��;�!���F$,.�/F�� �C� �/F�� �%!�*-J#%�/F3P��/,ZGH&(��2i*<J:G+��&(2�!�7/������#5;���&5GC#k*<*/����&(B�#%B/,-*-,-!EJ|1%��;�!�&(�D� K�����; #%P/���z!?7/,<��,-�y&/�/*<J#"�^#%������&5g(,.24#k!��z23��!�7/&/F��!�7/�@J/,<��*-F��@&5G�!�7/��P��/,-1%� ����#k*Q��&(��!DG�&%*-,-&�2i,<6/7%!zBC��P��/F�� ������!�,.24#k!���F4FQP/�M!�&^�?��#%�����@��#%2S�/*<,.�/64#%��!>,ZG+#5;�!���8�7/,-;�7#%���P���#}1%&",<FQ#"B/*<�|6%,-1%� ��!?7/�^!�,.23�;�&(2S�/*<��g/,-!�J�&5Gy!�7/�3#5*-6%&(��,-!�7�2�K

À�,-6(P��D� �c#5*-��&�6%,-1%���|;�&(24��P/!�#5!>,<&/�0!>,h2i���^GH&(�:!�7/�$fqg¯#5;�!MfO�$� � � )EP���FQ#5!��%K t *-� #%��*-J%��#5��/&%!���FW� #%�D*<,-� ���!�7/,-�^23��!�7/&/F�,<�:��,-6(�/,x~Q; #%�"!�*-J0��*-&58@� ��!�7�#%��f �I� � � 8�7/,-*-��#5;�7/,-��1(,.�/60#5*.23&%�>!M,-F�� �%!�,-; #5*�����!�P��>�/� K

L¾&0!>����!��C� �sG�&/��24#%�/;��I&5GM!�7/����&(��!DG�&%*-,-&0�>��*<��;�!>,<&/�=#5*-6%&(��,-!�7�2 &(�'*.#%��6%� �|��&(��!DG�&%*-,-&%� ��8@�S~�����!; �D� #5!���F~Q1%�4��&/��!�GH&%*-,<&"�^;�&(�/��,-��!�,.�/6�&5G^� Y4&/��Y "I��!�&/;�T%�:� #5;�708:7/,<;�7W8@� ���|;e7/&%�>� �R#k;�;�&(��F�,.�/6�!�&I1%&%*.#5!�,-*-,-!EJ%K3L¾&�BC����C��;�,Z~Q;%��8@���>#%��T5��FW!�7/� !T_I��!�&/;�T%�^B�#5����FW&(�c!�7/��,.�:1%&%*.#5!>,<*-,-!�J�#5�23� #5��P��D��FµB"J�!�7/�3��!�#"�/FQ#%��FWF���1(,.#5!>,<&/�&5G�!�7/�*-&%6/#%��,-!�7�2¸&5G�!�7/�3����,-;�������*.#5!�,-1%���4�¨�����%�CGH&(�@,.�/��!�#%�/;��%���:P/*<*@X.� C \ � K����^!�7/� ��F�,<1/,-F���F�!�7/� !T_3��!�&/;�T%�,.�%!>&!�7������^��&/��!�GH&%*-,<&"� ��!�7/�@~�����!�;�&(�/��,-��!���FI&5G�!�7/��}Y^��!�&/;�T%��8M,-!�74*-&58@����!�1%&%*.#5!>,<*-,-!�J�ÄC!�7/���/��g/!z;�&(�/��,-��!>��F$&5G!�7/�3� Y|��!�&/;eT"�@8M,-!�7I7/,<6/7/����!M1%&%*.#5!>,<*-,-!�J�Ä #%�/F�!�7/�:!�7/,.��F�;�&(�/��,-��!���F�&kGy!�7/�^��� 24#5,.�/,.�/6R�}Y^��!�&/;�T%�@&5GÂ2i��F�,hP�21%&%*.#5!>,<*-,-!�J%K����^#5*-��&!>����!���F4&(�i!E8@&^*.#%��6"� �z��&/��!�GH&%*-,<&"��;�&/�/��,-��!�,.�/63&5G�!�7/��Y "|��!�&/;eT"��&5G�*-&O8@����!�1%&%*.#5!�,-*-,-!�J$#%�/F!�7/�Y "S��!�&/;eT"�@&5Gq7/,-6(7/���>!�1%&"*h#k!�,-*<,-!�J%K

Ld7/�������P/*-!��@GH&(��!�7/�����:~Q1%�3��&(�D!�G�&"*<,-&%�:#%���:6%,-1%� ��,.��L�#%B/*-� ��KzLd7/,<�d!�#%B/*-��#5*-��&3��7/&58M�d!�7/�^1%&%*.#5!�,-*-,-!�J$&5G� #5;�74&kGy!�7/�^��&(��!DG�&%*-,-&4����*-��;�!�,-&(�c#5*-6%&(��,-!�7�2i� K�LM7/�����������P/*-!��@,.�/F�,-; #5!��|!�7�#5!z!?7/�����23��!�7/&/F��M����&(FQP/;��2i&(���8@� #5*-!�7c8�7/� ��23&/���31%&%*.#5!�,-*-�4��!�&/;�T%�#%�D�iP/�>��F�K3{M!^!�7/�3��#%23�3!�,.23�%��f �4� � � #%�/F�!?7/�iP��/,-1%� �D��#5*��C&(��!�GH&%*-,-&

� C

Page 19: Mathematical Finance, 8(4):325-347, 1998.web.cs.iastate.edu/~honavar/portfolio-selection.pdf · Mathematical Finance, 8(4):325-347, 1998. ! #" %$'&)( * +% -,. */0 1 #+324,. " 5%687

�b,-!�7/&(P/!�n/,-F��4�+�"GH&(��24#k!�,-&(� �b,-!�70n/,-F��3�+�"GH&(��24#5!>,<&/�n/!�&/;eT"� � t�� ³ f �4� � � ��/,-1�K � t�� ³ fO�$� � � ��/,-1�K�+�D&(a�K � V:,.��{^��T C !�K C C [�Kxw "([�KZ[ !%[ C KZw w"w�K " �T_�K _t &(2�K � V:,.�c{^��T � "'" KZ[ �%� [�KxY �%[�K � "��C�%K ! Y'� C KZY �%����K Ct &(2�K � ½W��,-;�;�& � ['!CKZ[ w C Kxw C "QK.� "�! _�KZY ���T_�K.� �%� [�Kxw� ��½ � t &(Tk� ����K.� � "QKxw � "QKZY �%� ��K � �"w�KZw Y��%K �

L�#%B/*-� " � t &(24��#"��,-��&(�3&5G�!�7/�M8@� #5*-!�74#k;e7/,-��1%��F�B%J3!�7/�MBC����!@;�&(�/��!�#%�"!�)E��� B�#k*h#"�/;���F$��&(�D!�G�&"*<,-&�� � t�� ³ � #%�/F!�7/�fO�4� � � )EP���FQ#5!��^8:7/� ���/&��>,<F��|,.�"G�&(�>24#5!�,-&(�I,-������&51/,-F���F0#%�/F�8�7/� �I��,-F��^,.�"G�&/��24#5!�,-&(��#"B�&(P/!@!�7/�BC����!��!�&/;�T3,.�I!�7/��*.#5�>!`�}[%[|!���#5F�,.�/63FQ#}J(�@,-�M��������� �"!���F�K����7�#}1%��P/�>��F�!�7/�^��#%23�d*<� #"���/,.�/6���#k!���� �3� [��x['� � GH&(�B�&"!�7I; #5����� K

1

10

10^2

10^3

10^4

10^5

2 4 6 8 10 12

Run

Tim

e (s

econ

ds)

Portfolio Size

UniversalEG

‘‘Exact’’ EG

À�,-6(P������ �4LM7/����P����/,.�/6�!�,.23���3&5GM!�7/��P��/,-1%� ����#5*H�@f �4� � � ��#%�/F=f�g�#5;�!�fO�4� � � �C&(��!�GH&%*-,-&R�>��*<��;�!>,<&/�=#5*-6%&5)��,-!�7�23�^#5�^#3G+P��/;�!�,-&(�c&5Gz!�7/�3�(P�23BC� �:&5Gz��!�&/;eT"�M,.�c!�7/�4��&(�D!�G�&"*<,-&QKd{�*-*���g���� ��,.23� �"!��^8@� ���|;�&(�/FQP/;�!���FW&(�#%��nV�|�z½��s³zn � � ["[%[%[3��P����/,.�/60#5!^� w'�|½��ZD%K#5*-6%&(�D,<!?7�23��#%���^*-�����:1%&%*.#5!>,<*-�|!�7�#%��#%�"J$&kGÂ!�7/�^;�&/�/��!�,-!�P/� �"!M��!�&/;eT"� K

ld��g/! ��8@�!�����!>��FR!?7/�i�C� �sGH&(��24#%�/;��3&5G@!�7/�4#5*-6%&(��,-!�7�2i�#5�`#SG+P��/;�!�,-&(�W&kGz!�7/�4�(P�23B�� ��&5Gz��!�&/;�T%�:,.��#��&/��!�GH&%*-,<&3;�&(�/�>,<�>!�,.�/6�&kG¾�>#%�/F�&(23*-J3����*-��;�!���F���!�&/;�T%� K¾ÀQ&(�q� #5;�74��&(�D!�G�&"*<,-&3��,FD��^ÍU�"8@����#%�/F�&(2i*<JI�/,<;�T5��FW� [%[F�,���� ��� �%!3��P�B/�>��!����¨&/P/!&5GM!�7/� � �Ó �4�C&%����,.B/*-����P�B/����!�� � #"�/FU��#%�'!�7/�$P��/,-1%� ���?#5*M��&(��!DG�&%*-,-&R#k*<6"&(��,-!�7�29#%�/F!�7/�4f �I� � � P��CFQ#5!��%K3���3!�7/� �c; #5*-; P/*.#5!���FA!�7/�36%��&(23��!?��,-;i2i� #%�W&5Gz!�7/�38@� #5*-!�7/�|&(B/!�#5,.�/��FW&51%� �:!�7/��� [%[F�,���� ��� �%!^�?P�B/����!��&kG���,FD��$ÍÊ#"�/FU�/*-&%!�!���FW!�7/�$�D����P/*-!��^,.�WÀ�,-6(P����N_I,.��#%���/P�#5*z�C� ��;�� �%!^J/,-��*-F�� K4{:*-��&Q��GH&(�� #5;�7���P�B/�>��!M&5Gz��!�&/;�T%� �/8@�; #k*<; P/*.#5!>��FR!?7/�^8�� #k*<!?70#5;�7/,-��1%��FRB"J�!�7/���C&(��!�GH&%*-,-&$�>��*<��;�!>,<&/�R#k*<6"&(��,-!�7�23��#k��#G+��#5;�!�,-&(�c&5Gz!�7/�S8�� #k*<!?7R#k!�!�#5,.�/��F�B%Jc!�7/�4B�����!^;�&/�/��!�#%�"!`�D� B�#5*.#%�/;���FU��&/��!�GH&%*-,<& K�LM7/�36%��&/23��!���,-;4#}1%� ��#56%�&5G�!�7/�����^8@� #5*-!�7���#k!�,-&%�M,-�`�/*-&%!>!���F�,.��À�,-6(P���� C K

��!@,<�d;�*<� #"��G��D&(2Á!�7/�:~Q6/P������@!�7�#5!@!�7/�:8�� #k*<!?7�#5;e7/,-��1%��F0B"J4!�7/�fO�4� � � )EP���FQ#5!>�^,<�d;�&(�/��,-��!�� �"!�*-J�7/,-6(7/� �!�7�#%�|!�7�#k!�&5G�!�7/�:P��/,<1"� ����#5*���&(��!DG�&%*-,-&#5*-6%&(�D,<!?7�2��(#%�/FS!�7/�MF�,-��; ��� ��#%�/;�J36(�D&O8d�z#5��!�7/�@��,FD���&kG�!�7/���C&(��!�GH&%*-,-&,.�/; ��� #5�����}KR�d&58���1"� ����#5�4�/&%!���F�� #"��*-,<� ����!�7/�4��,-6(�/,x~Q; #%�"!4F���6(�>#5FQ#5!�,-&(�W,.�=�C� �sG�&/��24#%�/;��4&kG�!?7/��P��/,-1%� ���?#5*��&/��!�GH&%*-,<&�#5*-6%&(�D,<!?7�2�8�7/� �cÍ 6%��!��M*.#%��6"� ��23,-6(7"!�BC�,.����P/� �/;���FUB"J�!�7/���#"24�/*-,h�/6k)EB�#5����F�#%������&5g/,h2S#5!�,-&(�!���;�7��/,-a¯P/�48:7/,<;�7W8M#5��P/�>��FWG�&/��Í � w�Ä ,.�cG�#k;�! �Q!?7/,<�^�-����;�!:����� 23��F��>&$6(�D� #5!MGH&(�:Í � � Y�� !�7�#5!M8@�3F�,-F

���

Page 20: Mathematical Finance, 8(4):325-347, 1998.web.cs.iastate.edu/~honavar/portfolio-selection.pdf · Mathematical Finance, 8(4):325-347, 1998. ! #" %$'&)( * +% -,. */0 1 #+324,. " 5%687

n/!�&(;�TS1%&%*.#5!�,-*-,-!EJ � t�� ³ f �$� � � :�/,<1"� ����#5*³�&(��!�GH&%*-,-& 2i,h� #}1/6 24#5g 8�� #k*<!?7 8�� #k*<!?7 1%&%*HK 8@� #5*-!�7 1%&"*�K¹�&58 � Y [CKZ[��%��� [�KZ[�� !%Y [�Kx[�� "Q� �[_�K _ ����K _ [�KZ[%[ "/w � Y�K " [�KZ["[ �'!½R��F��}Y [CKZ[�� "(Y [�KZ[�� �'! [�Kx[�� C [ � "QK.� !��%K _ [�KZ[%[T_C� �[_�KZw [�KZ["[ � ��M,-6(7'� Y [CKZ[�� C � [�KZ[%Y _%[ [�Kx[ "(w � � C "QK ! ���%[�KZY [�KZ[�� [ ! ���%KZY [�KZ[C�%� C¹�&58»Y " [CKZ[��%��� [�KZ[�� "�! [�Kx[�� C [ � "QK.� � C K.� [�KZ[%['�C��M,-6(7cY " [CKZ[�� "(Y [�KZ[%Y"[T_ [�Kx[ "(w � Y'�%[�K _ ���T_�K.� [�KZ[%[%w"[

L�#%B/*-� � � t &(24��#"��,-��&(�|&5G�!?7/�M8@� #5*-!�74#5;�7/,-��1%��F$B"J�!?7/��B�����!�;�&(�/��!�#"�%!�)E�D� B�#5*.#%�/;���F$��&/��!�GH&%*-,<&�� � t�� ³ � �5!�7/�P��/,-1%� ����#k*��C&(��!�GH&%*-,-&4#%�/F�!�7/�^f �4� � � )EP���FQ#k!��:G�&/�z��&/��!�GH&%*-,<&"��&kGy1O#%�DJ(,.�/631%&%*.#5!�,-*-,-!�,-����;�&/�/��,-��!�,.�/64&5G��}Y^&(�@Y "��!�&/;�T%��� #5;e7�K�½W,.�/,.2�P�2��¯#}1%� �>#56%��#%�/FI24#5g/,h23P�2 1%&%*.#5!�,-*-,-!�,-���`#%�D�M6%,-1%� �|GH&(��!?7/����!>&(;�T%��,.�I� #5;�74�C&(��!�GH&%*-,-&Q�#5�M8@��*-*y#5�d!�7/�^1%&%*.#5!�,-*-,-!�,-���^&5GzP��/,<1"� ����#5*y#"�/F0f �$� � � �C&(��!�GH&%*-,-&%� K

16

18

20

22

24

26

28

5 10 15 20

Ave

rage

Ann

ual R

etur

n (%

)

Number of Stocks

BCRPEG

UniversalUniversal (sampling)

À�,-6(P���� _ �4LM7/��#}1%� ��#56%�I#%���(P�#k*��C� ��;�� �"!�J/,-��*-Fp&5G�� t�� ³���!�7/�0P��/,-1%� ����#k*��C&(��!�GH&%*-,-&�#5*-6%&(��,-!�7�29#%�/FA!�7/�f �I� � � )EP��CFQ#5!���G�&(�d��#%�/F�&(2Ã��P�B/����!>��&5G���!�&/;�T%��G+��&(2¸��,FD��3Y3!�&4Y "QK�/&%!^�/*-&%!#%�"J0������P/*-!��^,.�c!�7/,-�7/,<6/7/� ���#%�/6"�%K�]^�W!�7/,-�^FQ#5!?#���,-!^����� 23�^;�*-� #%�:!�7�#k!�!�7/�|6%&/&(F���� �EG�&(�>24#%�/;��#%�/F�1%� �DJ$*-&58»;�&/24��P/!�#5!�,-&(�I!>,h2i�����a¯P/,.����F�B"J$!?7/�f �$� � � )EP���FQ#k!�������&51/,-F����`#31/,.#%B/*-�4#5*-!�� ����#5!�,-1%�^!>&4!�7/�P��/,-1%� ����#k*y��&(�D!�G�&"*<,-&�#5*-6%&(��,-!�7�2N8M,-!�70����&515#%B/*-�^;�&(24����!>,<!>,<1"� �/�����B�&(P��/F��}K

{:�@!�7/��&(����!>,<; #k*�#%��#5*-J/��,-�M,.24�/*-,-��� �Q!?7/��6(#"�$BC��!E8@��� �S!�7/�^# 1"� ��#56%�MJ/,-��*-F�&5Gy!�7/�^f �4� � � )EP��CFQ#5!��#%�/FI!�7/�� t�� ³b,.�/; ��� #k�����^8M,-!�7c!�7/�3��,FD��3&5Gz!�7/���C&(��!�GH&%*-,-&QK^�M&58@��1%� ��� ,-!^,-�^,h�"!�� �����>!�,.�/6�!�&��/&"!��!�7�#k!�!�7/�3#}1%� ��#56%�#%���/P�#5*y����!?P����I&5G�!�7/�f �I� � � )EP��CFQ#5!���#k*<�>&4,.�/; ��� #5�����^#5�M!�7/��C&(��!�GH&%*-,-&$,.�/;�*.P/F����23&(�D�^��!�&/;eT"� K

� ����Ì� ���Ì�Ì������ Ë:�� �5������¾Ç_��Ç@Ì�Ç@ËM�¾ ��{:*<!?7/&(P/6(7'!�7/�4��g���� ��,.23� �"!�#5*@������P/*-!��4��������� �"!���FA,h�'!�7/,-����#"��� �3#%���4� �/;�&/P���#56%,.�/6Q��8@�47�# 1"�4,-6(�/&(����FA&(�/�,.24��&/��!�#%�"!�#5�?����;�!&kGd#c��� #5*z24#"��T5��! �^!?��#5F�,.�/60;�&%�>!�� K$L�J��/,-; #5*-*<J%��!�7/� ���I#%���4!�8@&$!�J������3&5G@;�&(2423,-����,-&(�/�,.24��&"����F4,.�$#:��� #5*Q2S#%��T5��! K��+�3!?7/�z~��D��!�; #5���%�"!�7/�@,h�"1%����!>&(���/����F��z!�&��#}J3#^��� ��;�� �"!�#56%�@&5G�!�7/�@!���#%�/�?#5;�!�,-&(�!�&$#SB���&(T5� ��K@�+��!�7/,-��; #k���%��8@�; #%����!�,-*-*y8:��,-!��^F�&58��0#3;�*-&%�>��F�GH&(��2¸��g�����������,-&(��GH&(�@!�7/�^8@� #5*-!�7c#5;�7/,<��1"��F

� w

Page 21: Mathematical Finance, 8(4):325-347, 1998.web.cs.iastate.edu/~honavar/portfolio-selection.pdf · Mathematical Finance, 8(4):325-347, 1998. ! #" %$'&)( * +% -,. */0 1 #+324,. " 5%687

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

2 4 6 8 10 12 14 16 18 20 22 24

Fra

ctio

n of

BC

RP

Number of Stocks

EGUniversal

Universal (sampling)

À�,-6(P���� C � t &(24��#"��,-��&(�c&5G�!?7/�$#}1%� ��#56%�38@� #5*-!�7c&5GM!�7/�4P��/,-1%� ����#k*���&/��!�GH&%*-,<&W#5*-6%&(��,-!�7�2 #%�/F�!�7/�If �4� � � )P���FQ#k!��4GH&(�|��#%�/F�&(2 ��P�B/�>��!��3&5GM��!�&/;eT"�:G��D&(2_��,FD���Y�!�&WY "QKILM7/���D����P/*-!���#%�D�46%,-1%� �A#5�^G+��#5;�!�,-&(�/�|&5GM!�7/�8@� #5*-!�7I&5G�!�7/� � t�� ³�K#5!|� #5;e7'!�,.23�4�>!�� ��8:7/,<*-��!�#"T%,.�/60!?7/�$!��>#5F�,.�/6R;�&"��!��,.�"!�&R#k;�;�&(P��%!}K0�M&58@��1%� ���¾!�7/�48@� #5*-!�7µG�P��/;�!>,<&/��8@�#%���:!���J/,.�/64!�&324#5g/,h2i,ED��3B���;�&(2i����7/,-6(7/*-J0�/&(�")+*-,h�/� #"����#"�/F$,-!M,-��7�#%�DF$!�&3F�� ��,-1%�3#%�IP���FQ#5!���>P/*<�%K t &51%� ��u��P��/,-1%� ����#k*`�C&(��!�GH&%*-,-&�#5*-6%&/��,-!�7�2Ê8M#5�4����;�� �"!�*-Jp��g/!�� �/F���Fb#%�/Fo#%��#k*<J0D���Fb,.�p!?7/,<��24#%���/� �SB%J ��*.P�2r#%�/FV^#5*.#5,zX C \EK

Ld7/�^����;�&(�/F0!�J����^&5G�;�&(242i,<�>��,-&(��,-�:!�&$��#}J�#�~Qg(��FW#%23&(P��"!M��� �d!���#%�/��#k;�!�,-&(����!�7�#5!@,-� ����� �:��P���;�7�#5���&(����#k*<�:&5G�#^��!�&/;�TQK�LM7/� ���?GH&(���%�%!?7/� ����23,-6(7"!�BC�MFQ#}J(�qG�&(��8:7/,<;�7S!�7/�M8@� #5*-!�738M,-*-*�BC��*.#%�D6%� ��,xGÂ�/&!��>#5F�,.�/6^,<���� �EG�&(�>23��F��5��������;�,.#5*-*-J4,xG�!?7/����&/��!�GH&%*-,<&:1%��;�!�&/�y#}GH!�� ��!?7/���/��8A!���#kF�,h�/6^FQ#}J^,-��;�*<&"���@!�&�!?7/�zF�����,.����F$�C&(��!�GH&%*-,-&1%��;�!�&/��K3���4; #%�cF��?~��/��#���� 2i,Z)+;�&/�/��!�#%�"!�)E��� B�#5*.#%�/;���F=��&/��!�GH&%*-,<&c8�7/,-;e7µ,<�3��� B�#5*.#%�/;���FA&(�/*-J�&(�W#��?P�B/����!&5Gz!�7/�4�C&%����,.B/*-�4!���#kF�,h�/6�FQ#}J/� K�ld&58� ,h�'#5F�F�,-!�,-&(�c!�&$!?7/�iBC����!^;�&(�/��!?#%�%!D)E��� B�#5*.#%�/;���F���&(�D!�G�&"*<,-&Q��8@�3�/����F#5*-��&�!�&S~��/F�!�7/�4BC����!^��P�B/����!^&5G@!�7/�3����a�P/� �/;��4!�7�#5!^������P/*-!��|,h�c!�7/�424#kg(,.24#5*�8@� #5*-!�7�K����3��P/������;�!|!�7�#5!~��/F�,.�/6$!?7/�B�����!M�?P�B/����!M,-�M;�&(24��P/!?#5!�,-&(��#5*-*-J�7�#"��F�Kzn/!�,-*-*H�Q,-!M,-���/&%!d;�*<� #"�@8�7/��!�7/� ��~��/F�,.�/6�#|;�&(24����!>,<!>,<1"�#%������&5g/,.24#5!�,-&(�I,-�^7�#%��F0#5�M8@��*-*HK

Ld7/,<�W��#%��� �c#%�/F»23&"��!�&%!�7/� �I8@&(��T &(�b,.�"1%����!�23� �"!���!��>#5!���6%,-���0� 2S�/*<&5Jj#A!�#5;�,-!W#5����P�24�/!>,<&/�b!�7�#5!!�7/�024#"��T5��!4,-�I��!�#5!�,-&(��#"��JU#%�/F=����� T=#µ��!���#5!>��6%J�!�7�#5!S��P/;�;�������G+P/*-*<J�;�&(24�C��!�����#k6(#5,.�/��!3!�7/�RBC����!�¤�±Ð£"²(¬-ª;�&(�/��!?#%�%!D)E��� B�#5*.#%�/;���F���&(�D!�G�&"*<,-&QKM�d&58@��1%� ����!�7/,-�#5�>��P�24�/!�,-&(��,<�:G+#%��G+��&(2 B���,.�/60��� #k*<,-��!>,<;%K^LM7/,-�^��P/6"6%����!��#%���/*-J/,h�/6c!���;�7��/,<a�P/���F���1"��*<&/����F'G�&(�d!���#k;eT",.�/60#IFQ��,xG�!�,.�/6�;�&(�/;�� �/!IX !/�y�*_}\Â!>&�&(�")+*-,h�/�I��&(��!DG�&%*-,-&0�>��*<��;�!>,<&/�,.�0#3;�7�#%�/6%,.�/6�24#%��Tk��! KyLM7/,-�^#%������&(#k;e7I8M#5�M����;�� �"!�*-J���g¯�/*-&(�D��FRB"J0n/,h�/6"� �XZY"Y�\�K

Ld7/� ���$,-�$#k*<�>&µ2i&(���4!�7/��&/����!�,-; #5*@8@&(��Tc!�&�B���F�&(�/�I,.�A&(��F�� �|!�&�P��/F�� ����!?#%�/FU8:7%JUfO�$� � � �>��� 23�4!�&��� �EG�&(�>2�B���!�!>� �@!�7�#%� t &51%� ��u��M#5*-6%&(��,-!�7�2¸F����?�/,<!>�!�7/�|;�*<� #"�@!�7/��&(����!>,<; #k*���P���� �D,<&/��,-!EJ�&kGq7/,-�^#5*-6%&(��,-!�7�2�K

�¸ � ��� )¸È�Ç@ �^É9ÇM�:��ÌLM7�#%��T"�M!�&4L¾&(2 t &51%� �@#%�/F�fz��,.TI]^��F�� �%!>*<,-;�7�G�&(�@����&51/,-F�,h�/6IP/�M8M,-!�7I!�7/�^��!>&(;�T424#%�>T5��!@FQ#5!�#��¨&(��,-6%,.��#5*-*-J6%� �/� ��#k!���F�B"J���#k*�n/!�� ��� � P/����FA,h�A&/P��^��g���� ��,.23� �"!�� Kc���$#%�D��#5*-��&c6(��#5!>�?G�P/*�!�&Wfz��,.T�]^�DF�� �%!�*-,-;�7'G�&(�|#; #%���?G+P/*y��� #kF�,h�/6�#%�/F07/��*.�"G+P/*y;�&(2S23� �%!>�M&(��#%�I� #%�D*<,-� �dFQ��#}GH! K

Y%[

Page 22: Mathematical Finance, 8(4):325-347, 1998.web.cs.iastate.edu/~honavar/portfolio-selection.pdf · Mathematical Finance, 8(4):325-347, 1998. ! #" %$'&)( * +% -,. */0 1 #+324,. " 5%687

� Ç � Ç@��Çd�� zÇ@ÌX.�e\�³�K��|K�{�*-6%&/��! K :�/,-1%� ����#5*��>;e7/� 23����G�&/��������F�,-;�!�,-&(����6(#"2�B/*-,.�/6Q��#%�/FA��&(��!DG�&%*-,-&�����*-��;�!�,-&(��K��^£ £Q¦%¬x¤¢�°�@©E¢/«�¦(«�±Ð¬-±¨¥ �5�QY%[ �Zw"[�� � w " �%�(�}w%w%Y�K

XZYO\�³�K(�|K5{:*<6"&(��!@#%�/F$L^K"½�K t &O1"� ��K�{:��J¯2S�/!�&%!�,-;@&(�/!�,.24#5*-,-!�J$#%�/F$#k��J¯2S�/!�&%!�,-;@��a¯P/,.��#%�D!�,-!�,-&(�$���D&(��� ��!>,<���&5G�*-&%65)+&(�/!>,h23P�2¿,.�%1%���>!�23� �"! K��^£Q£ ¦%¬x¤`¢�°��@©�¢(«�¦/«�±¨¬<±¨¥ �5���[_Q�HY � � � C _ � �%w �/��� w � �CK

X !O\�³�K�{^P/� �^#%�/FW½�K�V3K �U#%��23P/!�7�K^L���#5;�T%,.�/6�!�7/�3B�����!^F�,-� (�P��/;�!�,-&(��K3�+����5¥! �^£ £ �/¦%¬���� 'M®�¢5¤�± ��' ¢"£%�¢'�/£Q­%¦"¥H±Ð¢%£C¤¢�°� z¢-'d® �/¥Hª ©���¡}±Ðª}£Q¡?ª ��� w%w'�CKX "k\`{3K �@#%����&(�3#%�/F4L^K}½=K t &51%� ��K"{oB�&/P��/F4&(�|!�7/��~���#%�/;�,.#5*Q15#5*.P/�M&5G�,h�"GH&(��2S#5!�,-&(��K��) ) ) � ©�¦%£C¤e¦"¡ ¥H±¨¢%£�¤

¢%£���£�°e¢"©1'3¦%¥H±¨¢%£ � (ª?¢%©-�O� ! " �.� ["w C � �%�}[%[��/� w � ��KX �O\ � K ����*-*z#%�/FRL^K ½=K t &51%� ��K t &(2S����!�,-!�,-1%�3&(�/!�,.24#k*<,-!�J�&5Gz*-&%6(#"��,-!�7�23,-;3,h�"1%����!?23� �%!}K��R¦%¥! (ª�'3¦%¥H±Ð¡�¤�¢�°� ®�ª ©�¦%¥H±¨¢%£�¤��@ª�¤�ª�¦%©�¡3 (� � �.�[_�� � �[_T_����}w �%[�K

X _O\ � K ����*-*�#%�/F�L^K5½=K t &O1"� ��KX�#"23�?)+!�7/��&(����!>,<;d&(�/!�,.24#5*���&(��!DG�&%*-,-&%� K��W¦%£Q¦�²('3ª £ ¥���¡ ±¨ª £Q¡?ª � ! " � C Y " � C !'!��� w� ��KX C \`{:1¯�D,h2 �z*.P�29#%�/FA{�FQ#"2 V^#k*h#k,�K :�/,<1"� ����#5*M��&/��!�GH&%*-,<&"��8d,<!?7U#%�/FA8M,-!�7/&(P/!|!���#%�/�?#5;�!�,-&(�';�&%��!�� K��s��@©E¢}¡�ª?ª?­%±Ð£"²%¤¢�°4¥! (ª � ª £ ¥! ���£Q£ �/¦%¬� z¢%£�°eª}©Eª}£Q¡?ª¢%£� z¢-'d® �/¥H¦%¥H±¨¢%£Q¦"¬ � ª?¦%©?£Q±¨£%² � (ª?¢%©-�5����#56%����!%[%w �!�� !�K/{ t ½Ê³z�����>� ��� w%w C K

X �O\�L^K5½=K t &51%� ��K"{`�3#5*-6%&(�D,<!?7�2bGH&(��24#5g/,.23,FD�,.�/6��g�����;�!���F4*-&%6:,.�%1"����!�23� �"!z����!�P�����K��) ) ) � ©�¦%£C¤e¦"¡ ¥H±¨¢%£�¤¢%£���£�°e¢"©1'3¦%¥H±¨¢%£ � (ª?¢%©-�O� !%[ � !T_"w � ! C !C��� w � "QK

XZwO\�L^KC½�K t &51%� ��K :�/,-1%� ����#k*y��&(�D!�G�&"*<,-&%�}K��R¦%¥! /ª/'3¦%¥H±¨¡�¦%¬ %z±¨£Q¦%£ ¡�ª}�Q�(��� � �.� � Y"w���� w%w��"KX.� [O\�L^K�½=K t &51%� �#"�/F 4K ��*.P/��� KUf�24�/,.��,-; #5*��@#}J%���3��!�&/;eT'24#%��T5��!3��&(�D!�G�&"*<,-&%�}K��:­%Ï ¦"£Q¡?ª�¤$±¨£��@®"®�¬-±Ðª?­

�W¦%¥! (ª/'3¦%¥H±¨¡�¤�� C �¾� w �T_�KX.�%�e\�L^Kv½�K t &51%� ��#%�/F�fMK/]^��F�� �%!>*<,-;�7�K� ��/,-1%� ���?#5*��C&(��!�GH&%*-,-&%�@8M,-!�74�>,<F��:,.�"G�&/��24#5!�,-&(��K��4) ) ) � ©�¦%£C¤e¦"¡ ¥H±¨¢%£�¤

¢%£���£�°e¢"©1'3¦%¥H±¨¢%£ � (ª?¢%©-�O� "(YQ�HY � �Q� w%w _�KX.� YO\ � K À�*-��!�;�7/� ��K���©�¦%¡}¥H±Ð¡?¦%¬�'3ª ¥! (¢}­5¤�¢D°¢?®�¥H±!'�±���¦%¥H±Ð¢"£QK��b,-*-��J%�y� w� C KX.��!O\ 4K��:#%P/����*-� ���@lK�¹�,-!�!>*<���>!�&(�/�%�M#%�/FA½=KÂV3Kq�U#%��23P/!�7�K�³�����F�,-;�!�,.�/6 :O[C� � < )HG+P��/;�!�,-&(�/�|&(�=��#%�/F�&(2i*<J

FQ��#}8����C&%,.�%!>� K���£�°e¢%©�'3¦%¥H±¨¢%£W¦%£Q­ z¢ 'M®��(¥H¦%¥H±¨¢%£ ���%���Q��Y � �ZY � " � Y%w !���� w"w "QKX.� "k\ 4K/³yK��d��*.2�BC&%*-F�� �QK/V:,<1/,.�/� ����#"�/F$½=K%V3K/�U#%��23P/!�7�K ��&(�D��!�)+; #5���d*<&"���MB�&(P��/F���GH&(����,-6(2i&%,-F���F�*<,.�/� #%�

�/� P���&/�/� K��s�!�:­%Ï ¦%£ ¡�ª�¤`±¨£�"`ª �(©�¦%¬#��£�°�¢%©�'�¦"¥H±Ð¢%£��@©E¢}¡�ª�¤�¤�±¨£%²$� �}¤�¥Hª/'|¤�%%��� w%w _�KX.���O\ 4K�³�K��d��*h23BC&%*-F�� � KyfMK�n/;�7�#%�/,.���%�¾m�K�n/,h�/6"� ���y#%�/F'½�K�V3K��A#%��23P/!�7�K^{¸;�&(24��#"��,-��&(�c&5G��/��8¸#%�/F

&%*-FW#5*-6%&(��,-!�7�23��G�&(�d#423,-g(!?P����^����!�,.24#5!>,<&/�0����&(B/*-� 2�K&�R¦%¡3 (±Ð£ ª � ª?¦%©?£Q±¨£%²/�(Y C �D� � �Zw C � �%� w��/� w%w C KX.�[_O\`½=Ky�d� ��B/��!>� ��#"�/F�½=KyV3K��U#%��23P/!�7�K�L���#5;�T%,.�/6W!�7/�$BC����!3��g¯�C� ��! K0�s�'�@©�¢ ¡?ª�ª?­%±¨£%²%¤3¢�°�¥! (ª � � ª ¬ °�¥!

��£ ¥Hª ©?£Q¦%¥H±¨¢%£ ¦%¬� z¢%£�°�ª ©�ª}£Q¡?ª¢%£��R¦"¡2 (±¨£Qª � ª?¦%©?£Q±¨£%²/�(��#56%���:Y �T_ � Y"w "Q�Q�}w%w'��KX.� C \ �QK t K��:P/*<*HK � ®�¥H±¨¢%£C¤1�/° �/¥ �/©Eª�¤1�@¦%£Q­I¢%¥! (ª}©­%ª}©�±¨Ï ¦"¥H±ÐÏ}ª�¤�K³���� �"!�,-;����:#5*-*���� w%w C KX.���O\ �SK �(P�24#%�D,<�%K(��ª}¬-¦%¥H±ÐÏ}ª±¨£�°�¢%©�'�¦"¥H±Ð¢%£ Kzn�����,.�/6%� �s) ��� �D*h#k6Q�y�}w%w%[�K

Y��

Page 23: Mathematical Finance, 8(4):325-347, 1998.web.cs.iastate.edu/~honavar/portfolio-selection.pdf · Mathematical Finance, 8(4):325-347, 1998. ! #" %$'&)( * +% -,. */0 1 #+324,. " 5%687

X.� wO\ �QK�¹�K/V:��*-*<J%K�{»�/��8b,.�"!�� ��������!�#k!�,-&(�3&5G�,.�"G�&/��24#5!�,-&(�I��#5!>�%K��Mª ¬¨¬ ���}¤e¥Hª�'¤ � ª�¡3 (£Q±¨¡�¦"¬��¯¢'�/©?£Q¦%¬-��! � �Zw�� C �w%Y _���� w'�T_�KXZY%[O\ �%J���T",yV:,-1(,.�/� �W#%�/F�½�#%�"G+����F�V3K��U#%��23P/!�7�K¾{�F�F�,-!�,-1%�|1%� ����P/�d��g¯�C&(�/� �%!>,h#k!���F06(�>#5F�,-� �%!MP���FQ#k!����MGH&(�

*-,.�/� #%�^������F�,-;�!�,-&(��K(��£�°�¢%©�'3¦%¥H±Ð¢"£R¦"£Q­� z¢-'d® �/¥H¦%¥H±Ð¢"£Q����!%YQ��� � �.� � _ "Q���/#%�(P�#"��JR�}w%w C KXZY��e\�l|K�¹�,<!>!�*-����!�&(�/�%K � ��FQP��/FQ#%�"!0�/&%,-��Jp#5!�!?��,.B�P/!���� �@#5!�!���,.B�P/!���� �>��&(��� �@#%�/F�*<,.�/� #%�I!?7�������7/&%*-Fp*-� #%�>�/,h�/6

P/��,.�/6��b,.���/&58KW�+����©�¢}¡�ª?ª�­"±Ð£"²%¤�¢D°�¥! (ª %�¢'�(©?¥! �^£ £ �/¦%¬���¢%©�� ¤� (¢?®j¢%£ �¢-'d® �(¥H¦"¥H±Ð¢%£ ¦%¬ � ª�¦%©?£Q±¨£%²� /ª�¢%©-�5�Q��#56%���|� " C � � �T_��Q�}w%w��%K

XZY%YO\`m�&(�>#%2�n/,.�/6%� ��K@n/8M,-!�;�7/,h�/6���&/��!�GH&%*-,<&"� K���£ ¥Hª ©?£Q¦%¥H±¨¢%£Q¦"¬���¢'�/©�£ ¦%¬�¢�° "`ª �(©�¦%¬#� �}¤e¥Hª�'|¤e�/!�&4#%���C� #%��K

Y%Y