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#'2' A82 “"2'+*8%”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
• :.%A$*'($) A,&0$-:.+./. ':#$+' - +8-.:.0$-:.+8% ':#$+.%;
• :.%A$*'($) 4+6> +8-.:.0$-:.+8> ':#$+.+;• :.%A$*'($) A,&0$-:.+./. ':#$+' - +8-.:.0$-:.+8% ".0#1,2,%.
154 7-$18 II. 9%4##: 2":',#&".-)#'
G.4>.48, "0$%,*),%8, 42) 0,B,*$) =#$> &'4'9 + =2,:#0.**8> #'A2$('>, "0,4--#'+2,*8 + 0'A.9,% 2$-#, &789:.J".-.A $&%,0,*$) $*4$+$46'27*8> "0,4".9#,*$5 .#*.-$#,27*. 0$-:' $ 4.-
>.4*.-#$ $ *,.A>.4$%6< 42) =#./. :.*(,"($< 16*:($$ ".2,&*.-#$ %.?*. &'$%--#+.+'#7 $& .AF,5 =:.*.%$9,-:.5 #,.0$$. G0,4".2.?$%, 9#. 16*:($) ".2,&*.-#$$*4$+$4' $%,,# -2,46<F$5 +$4:
U = E(rp) " 0,5A!2p.
Y,% +8B, +,2$9$*' :.=11$($,*#' *,-:2.**.-#$ : 0$-:6A 42) $*+,-#.0', #,%+8B, “B#0'1” ".0#1,27*./. 0$-:', :.#.085 +89$#',#-) $& .?$4',%.5 4.>.4*.--#$ ".0#1,2). M' "0':#$:,, -./2'-*. $--2,4.+'*$)%, A *'>.4$#-) + "0,4,2'> .#2 4. 4, ".=#.%6 + *'B$> "0$%,0'> A64,# $-".27&.+'*. &*'9,*$, A = 3.;': :': ".-2,46<F$, +8:2'4:$ .-6F,-#+2)<#-) 42) 4+6> ':#$+.+, ",0,"$-
B,% 1.0%628 42) +89$-2,*$) 4.>.4*.-#$ $ 0$-:' ".0#1,2), "0$+,4,**8, + 0'&-4,2, 6.1, 42) 4+6> ':#$+.+.
C.>.4*.-#7 ".0#1,2): E(rp) = w1E(r1) + w2E(r2).
C$-",0-$) ".0#1,2): Var(rp) = !2p = w2
1!21 + w2
2!22 + 2w1w2cov(1,2).
R.+'0$'($) 4+6> ':#$+.+ cov(1,2) %.?,# A8#7 #':?, +80'?,*' 9,0,& :.00,-2)($<: cov(1,2) = cor(1,2)!1!2.!A.&*'9$% 4.>.4*.-#7 A,&0$-:.+./. ':#$+' :': rf . C$-",0-$) A,&0$-:.+./.
':#$+' 0'+*' *62<, $ :.+'0$'($$ - +8-.:.0$-:.+8%$ ':#$+'%$ *62,+8,. I$-:".0#1,2) 0'+,* :+'40'#*.%6 :.0*< $& 4$-",0-$$.
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G,0+') $& *'B$> .AF$> &'4'9 +8A.0' ".0#1,2) +:2<9',# ".4A.0 :.%A$*'($$.4*./. A,&0$-:.+./. ':#$+' (asset0) $ .4*./. +8-.:.0$-:.+./. (asset1), 4.->.4*.-#$ :.#.08> -9$#'<#-) *,:.00,2$0.+'**8%$. C'**8, "0$+,4,*8 *'0$-. 6.10: A,&0$-:.+') -#'+:' 4.>.4*.-#$ 0'+*' 1%, ' ':#$+ asset1 $%,,# 0$-:$ 4.>.4*.-#7, ".>.?$, *' ".:'&'#,2$ .A2$/'($5 $& 0'--%.#0,**./. 0'*,, ".0#-1,2). L&%,*)) 4.2< $*+,-#$0.+'*$) + asset1 ()9,5:' H14) + 4$'"'&.*, .# 0 4.100%, %.?*. ".269$#7 "0)%6< 2$*$<, -.,4$*)<F6< #.9:6 “0$-:–4.>.4*.-#7”42) A,&0$-:.+./. ':#$+' asset0 $ -..#+,#-#+6<F6< #.9:6 42) asset1. H#'2$*$) ".:'&'*' *' 0$-. 6.11.L*4$+$46'27*8, $*+,-#.08 A646# +8A$0'#7 :.*:0,#*6< #.9:6 *' 4'**.5 "0)-
%.5 + &'+$-$%.-#$ .# &*'9,*$) $> :.=11$($,*#' *,-:2.**.-#$ : 0$-:6 ()9,5-:' R6). M' 0$-. 6.11 ".:'&'*' :0$+') ".2,&*.-#$ (+ +$4, 4$-:0,#*8> #.9,:, &*'-
+,-.- 6. /01#2#3-4#* #).'$1#4#"))"5" 0"&16',* 155
6,). 6.10. 389$-2,*$) + =2,:#0.**8> #'A2$('> 42) &'4'9$ 1, 2$-# &789:
6,). 6.11. @'+$-$%.-#7 ".2,&*.-#$ $ 4.>.4' .# 0$-:' 42) &'4'9$ 1
9,*$) :.#.08> .#.A0'?,*8 *' "0'+.5 .-$) 42) ".0#1,2,5 - 0'&2$9*8%$ "0.".0-($)%$ asset1 "0$ :.=11$($,*#, *,-:2.**.-#$ : 0$-:6 3 (+ +$4, .#4,27*8>#.9,:, &*'9,*$) :.#.08> .#.A0'?,*8 *' "0'+.5 +,0#$:'27*.5 .-$ /0'1$:'). G.-2,&*.-#7 *'9$*',#-) -. &*'9,*$) 0,01 (:./4' +-, 100% $*+,-#$06<#-) + asset0)$ ".-#,",**. 6+,2$9$+',#-) ". %,0, 6+,2$9,*$) 4.2$ asset1 4. 4.-#$?,*$)."#$%'27*.5 #.9:$, ".-2, 9,/. *'9$*',# 6A8+'#7 4. &*'9,*$) 0,006 (:./4' 100%$*+,-#$06<#-) + asset1).
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!"#$%'27*') #.9:' *' "0)%.5 -..#+,#-#+6,# #.9:, %':-$%'27*.5 ".2,&*.-#$,:.#.06< %.?*. ".269$#7, *'>.4) 4.2< asset1 ()9,5:' *14) $& -2,46<F,/.-..#*.B,*$):
w"1 =
E(r1) " rf
A!21
.
C2) 4'**./. "0$%,0' + ."#$%'27*.% ".0#1,2, 35,9% -0,4-#+ $*+,-#$06,#--) + +8-.:.0$-:.+85 ':#$+, 9#. 4',# &*'9,*$, ".2,&*.-#$ 0,012 (#.,. =:+$+'2,*#/'0'*#$0.+'**.5 4.>.4*.-#$ + 1,2%). C2) +89$-2,*$) &*'9,*$) 4.2$ +8-.:.0$--:.+./. ':#$+' ()9,5:' M15) A82' 0'&0'A.#'*' 16*:($) Prob1OptimalRisky-Weight (0'&4,2 6.11).!A0'#$#, #':?, +*$%'*$,, 9#. ."#$%'27*') 4.2) -0,4-#+, +2.?,**8> + +8-.-
:.0$-:.+85 ':#$+ (' *, + ':#$+ asset0), &'+$-$# .# -#,",*$ *,-:2.**.-#$ : 0$-:6:.*:0,#*./. $*+,-#.0'. L*+,-#.0 - A.2,, +8-.:$% :.=11$($,*#.% *,-:2.**.-#$: 0$-:6 (*'"0$%,0, 4) 6%,*7B$# 4.2< +8-.:.0$-:.+8> ':#$+.+ + -+.,% ".0#1,2,4. 27%.
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6,). 6.12. 389$-2,*$) + =2,:#0.**8> #'A2$('> 42) &'4'9$ 2, 2$-# &789:
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: 6%,*7B,*$< 0$-:', ".:' *, A64,# 4.-#$/*6#' #.9:' ".0#1,2) - %$*$%'27*.54$-",0-$,5 (/4, 87,7% -0,4-#+ +2.?,*. + asset1), -#'*4'0#*., .#:2.*,*$, :.-#.0./. 0'+*. 9,77%. G.:'&'#,2$ 42) ".0#1,2) - %$*$%'27*.5 4$-",0-$,5 *,&'+$-)# .# :.=11$($,*#' *,-:2.**.-#$ : 0$-:6 $ %./6# 0'--%'#0$+'#7-) :': +,-'".0#1,2) 42) $*+,-#.0' - ".2*.5 *,-:2.**.-#7< : 0$-:6.@*'9,*$, 4.2$ ",0+./. ':#$+' (w1) + ".0#1,2, - %$*$%'27*.5 4$-",0-$,5
0'+*.!2
2 " "12!1!2
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2 " 2"12!1!2.
C2) 64.A-#+' .A.&*'9$% 4.2< ",0+./. ':#$+' + ".0#1,2, - %$*$%'27*.5 4$--",0-$,5 9,0,& wmv
1 (87,7%, :': ".:'&'*. + )9,5:, H24), #./4' 4.2) +#.0./. ':-#$+' 0'+*' (1 " wmv
1 ). ;':$% .A0'&.%, ,-2$ 87,7% ".0#1,2) +2.?,*. + asset1,#. 4'**') :.%A$*'($) >'0':#,0$&6,#-) %$*$%'27*8% 0$-:.% (9,77%). !A0'#$#,+*$%'*$,, 9#. =#. %,*7B,, 9,% 0$-: :'?4./. $& ':#$+.+ + .#4,27*.-#$.L*+,-#.08, :.#.08, -#0,%)#-) : A.27B,%6 0$-:6, A646# 6%,*7B'#7 4.2<,
+2.?,**6< + asset1, $ 6+,2$9$+'#7 4.2< ':#$+' asset2. M' 0$-. 6.13 42)$*+,-#.0' - :.=11$($,*#.% *,-:2.**.-#$ : 0$-:6, 0'+*8% 3, ".:'&'*' 16*:($)".2,&*.-#$ (+ +$4, 4$-:0,#*8> #.9,: -. &*'9,*$)%$, .#.A0'?,**8%$ *' "0'+.5.-$), :.#.0') +.&0'-#',# 4. 4.-#$?,*$) 0$-:' "0$A2$&$#,27*. 15%. L*+,-#.0A64,# /.#.+ +&)#7 *' -,A) A.27B$5 0$-:, ".:' 4.2) ",0+./. ':#$+' *, 6%,*7B$#-)4. 35,2% (-#'*4'0#*., .#:2.*,*$, ".0#1,2) + =#.5 #.9:, 0'+*. 14,7%).D.?*. +8+,-#$ &*'9,*$, ."#$%'27*.5 4.2$ ",0+./. ':#$+' + +8-.:.0$-:.+.%
".0#1,2,:
wopt1 = wmv
1 +E(r1) " E(r2)
A(!21 + !2
2 " 2"12!1!2).
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
0,& ."0,4,2,**6< ".27&.+'#,2,% 16*:($<. H#$ +89$-2,*$) 0,'2$&.+'*8 + +$4,16*:($$ Prob2OptimalRiskyWeight(), :.#.0') ."$-'*' + 0'&4,2, 6.11.
158 7-$18 II. 9%4##: 2":',#&".-)#'
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G.-2,4*)) .AF') &'4'9' 0'--%'#0$+',# :.%A$*'($< +8-.:.0$-:.+./. ".0#-1,2) - A,&0$-:.+8% ':#$+.%. H#' &'4'9' 0,B',#-) + 4+' =#'"'. M' ",0+.% =#'",*,.A>.4$%. *'5#$ ."#$%'27*6< :.%A$*'($< +8-.:.0$-:.+8> ':#$+.+ (9'-#*85-269'5 &'4'9$ 2), *' +#.0.% — -..#*.B,*$, %,?46 A,&0$-:.+8% ':#$+.% $ +8-.-:.0$-:.+8% ".0#1,2,% (9#., ". -6F,-#+6, "0,4-#'+2),# -.A.5 &'4'96 1). C'**8,42) +89$-2,*$5 "0$+,4,*8 *' 0$-. 6.14.J*'9'2', $/*.0$06) *'2$9$, A,&0$-:.+./. ':#$+', *,.A>.4$%. *'5#$ ."#$-
%'27*., -..#*.B,*$, %,?46 4+6%) +8-.:.0$-:.+8%$ ':#$+'%$ + ".0#1,2,. 3 "0,-4846F,% 0'&4,2, A82. *'54,*. "0$A2$&$#,27*., 0,B,*$, 35%:65%. G0,4".2.-?$%, 9#. ."#$%'27*85 ".0#1,27 - 0$-:.% R $%,,# .?$4',%6< 4.>.4*.-#7 E(rR)$ -#'*4'0#*., .#:2.*,*$, !R, ' 4.2) ",0+./. ':#$+' + ".0#1,2, 0'+*' F1. M,.A>.-4$%. +8A0'#7 &*'9,*$, F1, %':-$%$&$06<F,, .#*.B,*$, “4.>.4*.-#7–0$-:” 42)4'**./. ".0#1,2) R - 69,#.% +:2<9,*$) A,&0$-:.+./. ':#$+' *' +#.0.% =#'",&'4'9$, #.,. %':-$%$&$0.+'#7 &*'9,*$,
E(rR) " rf
!R.
G0$%,*$+ *,:.#.08, '2/,A0'$9,-:$, +8:2'4:$ ($2$ /0'1$9,-:$5 %,#.4, :':+ [1]), %.?*. ".:'&'#7, 9#. 6/.2 *':2.*' "0)%.5, -.,4$*)<F,5 ".0#1,2$ *'
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F opt1 =
!22E(R1) + cov(1,2)E(R2)
!22E(R1) + !2
1E(R2) " cov(1,2)(E(R1) + E(R2)),
/4, E(R1) $ E(R2) — .?$4',%8, 4.>.4*.-#$ -+,0> *.0%8 rf .!A0'#$#, +*$%'*$,, 9#. +,-' ."#$%'27*./. ".0#1,2) *, &'+$-)# .# :.=11$-
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:.0$-:.+./. ".0#1,2) +2.?,*. + ':#$+ asset1, ' 89,5% — + asset2.M' +#.0.% =#'", 0,B,*$) &'4'9$ ."#$%'27*85 ".0#1,27 $& 4+6> +8-.:.0$-:.-
+8> ':#$+.+, ".269,**85 *' ",0+.% =#'",, :.%A$*$06,#-) - A,&0$-:.+8% ':#$+.%42) 4.-#$?,*$) %':-$%'27*.5 ".2,&*.-#$. J2,4.+'#,27*., 4.2) +8-.:.0$-:.+./.".0#1,2) +89$-2),#-) :':
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A!2R
,
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160 7-$18 II. 9%4##: 2":',#&".-)#'
3 0,&627#'#, ".269,* ."#$%'27*85 ".0#1,27, /4, 68,0% "0$*'42,?$# +8--.:.0$-:.+.%6 ".0#1,2< $ 32% +2.?,*. + A,&0$-:.+85 ':#$+. G.2,&*.-#7 "0$=#.% 6+,2$9$2'-7 - 0,0292 *' ",0+.% =#'", 4. 0,0347. (!"#$%'27*., -..#*.-B,*$, +8-.:.0$-:.+8> ':#$+.+ %.?*. #':?, ".269$#7 - ".%.F7< 16*:($$Prob1OptimalRiskyWeight(), :': + )9,5:, H50.)I'-"0,4,2)) 68% $*+,-#$($5 + ",0+.*'9'27*8, ':#$+8, %8 ".269',% 4.2$
':#$+.+ + )9,5:'> H52:H54. !"0,4,2,**') ".27&.+'#,2,% 16*:($) Prob3Opti-malWeightsVec(), +89$-2)<F') +,:#.0 +,-.+ 42) .:.*9'#,27*./. ."#$%'27-*./. ".0#1,2), ."$-'*' + 0'&4,2, 6.11 ()9,5:$ I52:I54).M' /0'1$:,, "0$+,4,**.% *' 0$-. 6.15, :0$+') =11,:#$+*./. %*.?,-#+' "0,4-
-#'+2),# +-, +.&%.?*8, :.%A$*'($$ ':#$+.+ asset1 $ asset2. T9$#8+') *'-2$9$, A,&0$-:.+./. ':#$+', %.?*. .#%,#$#7, 9#. -6F,-#+6,# ,4$*-#+,**85 ."-#$%'27*85 ".0#1,27: ,%6 -..#+,#-#+6,# #.9:', /4, "0)%'), -.,4$*)<F') A,&0$--:.+85 ':#$+ - =11,:#$+*.5 /0'*$(,5 42) +8-.:.0$-:.+./. ".0#1,2), )+2),#-):'-'#,27*.5 : :0$+.5 =11,:#$+*.5 /0'*$(8. (;':6< #.9:6 :'-'*$) *' =11,:-#$+*.5 /0'*$(, 4',# “-:.00,:#$0.+'**')” &'4'9' 2.) R'-'#,27*') : =11,:#$+*.5/0'*$(, *'9$*',#-) -. 100% $*+,-#$($5 + A,&0$-:.+85 ':#$+, &'#,% 4.2) "'4',#4. 0% (#.9:' :'-'*$)) $ &':'*9$+',#-) 4.2,5 150% + +8-.:.0$-:.+.% ".0#1,2,$ .#0$('#,27*.5 4.2,5 –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
0'A.#:, %.4,2$ CAPM, "0,4-#'+2,**.5 + -2,46<F,5 /2'+,, ' $%,**. ",0,+,-#$"0,4".9#,*$) $*4$+$46'27*8> $*+,-#.0.+ *' 60.+,*7 +-,/. 08*:', ".269$+ #'-:$% .A0'&.% #,.0$< .(,*:$ -#.$%.-#$ ':#$+.+.
6.10 H-"#$#/#%%?# -'/+:';1*#/#3 .4%8A,,S$-# Module1 -.4,0?$# :.4 42) ."0,4,2,**8> ".27&.+'#,2,% 16*:($5, $--
".27&6,%8> + 0'A.9,5 :*$/, Equity1.xls. H#$ 16*:($$ $22<-#0$06<# .A0'-A.#:6 %'--$+.+ + ."0,4,2,**8> ".27&.+'#,2,% 16*:($)> — '0/6%,*#8 %'--$+.+,16*:($$ %'--$+', +89$-2,*$) -. -:'2)0*8%$ +,2$9$*'%$ $ %'--$+'%$, ' #':?,".269,*$, 0,&627#'#.+ + +$4, %'--$+'. M'"0$%,0, 16*:($) PortfolioReturn()&'4'*' #'::Function PortfolioReturn(retvec, wtsvec)If Application.Count(retvec) = Application.Count(wtsvec) Then
+,-.- 6. /01#2#3-4#* #).'$1#4#"))"5" 0"&16',* 161
6,). 6.15. @'+$-$%.-#7 ".2,&*.-#$ $ 4.>.4*.-#$ .# 0$-:' 42) &'4'9$ 3
If retvec.Columns.Count <> wtsvec.Columns.Count Thenwtsvec = Application.Transpose(wtsvec)
End IfPortfolioReturn = Application.SumProduct(retvec, wtsvec)
ElsePortfolioReturn = -1
End IfEnd Function
H#' "0./0'%%' +8".2*),# "0.+,0:6 0'&%,0*.-#$ %'--$+.+, &'4,5-#+.+'**8> + +8-9$-2,*$)>. Z6*:($) SumProduct() +.&+0'F',# -:'2)0*6< +,2$9$*6, *. #0,A6-,#, 9#.A8 4+' '0/6%,*#'-%'--$+' $%,2$ .4$*':.+6< 0'&%,0*.-#7. H#. 4.-#$/',#-)"6#,% "0.+,0:$ #./., $%,<# 2$ %'--$+8 retvec $ wtsvec .4$*':.+., :.2$9,--#+. =2,%,*#.+. @'#,%, ,-2$ :.2$9,-#+. -#.2A(.+ + 4+6> %'--$+'> 0'&*.,, %'--$+wtsvec #0'*-".*$06,#-). K-2$ %'--$+8 $%,<# 0'&*., :.2$9,-#+. =2,%,*#.+,16*:($) +.&+0'F',# &*'9,*$, .B$A:$ –1. M,.A>.4$%.-#7 "0.+,0)#7 0'&%,0*.-#7+8&+'*' #':?, #,%, 9#. -",($'27*') 16*:($) %.?,# ".269'#7 -+.$ '0/6%,*#8 :':+ +$4, -#.2A(', #': $ + +$4, -#0.:$.Z6*:($) HLPortfolioWeights() (:.4 :.#.0.5 #':?, "0$+,4,* + 2$-#,
Module1) "0$%,9'#,27*' .-.A,**.-#)%$ 0'A.#8 - 16*:($)%$ %'--$+.+. G0,?4,+-,/. *,.A>.4$%. -2,4$#7 &' 0'&%,0*.-#7< %'--$+.+ +*6#0$ 16*:($$. 3 VBA.4*.%,0*8, %'--$+8 >0'*)#-) + +$4, -#0.:, ".=#.%6 $ .4*.%,0*8, %'--$+8, :.-#.08, +.&+0'F'<# -",($'27*8, 16*:($$, #':?, A646# +,:#.0'%$--#0.:'%$. G.--2,46,% =#.%6 "0'+$26 $ .A,-",9$%, 9#.A8 ."0,4,2,**8, ".27&.+'#,2,% 16*:($$
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+.&+0'F'2$ + Excel .4*.%,0*8, %'--$+8 VBA + 1.0%, +,:#.0.+--#0.:. ;,% *,%,*,, +*6#0$ ."0,4,2,**8> ".27&.+'#,2,% 16*:($5 269B, $-".27&.+'#7 406/.,"0'+$2., -./2'-*. :.#.0.%6 +,:#.08 &'"$-'*8 + +$4, +,:#.0.+--#.2A(.+. M'"0$-%,0, "0$ *,.A>.4$%.-#$ ",0,%,**8, uvec $ retvec #0'*-".*$06<#-) 42) "0,-.A0'&.+'*$) + +,:#.08--#.2A(8.M,.A>.4$%. #':?, +*$%'#,27*. -2,4$#7 &' "0.%,?6#.9*8%$ ",0,%,**8%$,
".269,**8%$ $& 16*:($5 %'--$+.+.Dim i As Variantl = Application.MMult(vinvmat, retvec)m = Application.MMult(vinvmat, uvec)
G.-:.27:6 ",0,%,**8, l $ m -'%$ )+2)<#-) %'--$+'%$, ".269,**8%$ $&16*:($$ %'--$+', $> *6?*. .AE)+$#7 :': #$" Variant, *., #': :': .*$ A646#$-".27&.+'*8 + :'9,-#+, '0/6%,*#.+ 42) 16*:($5 %'--$+', A,& &'4'**.5 0'&%,0-*.-#$. 3 16*:($$ Mmult() '0/6%,*# vinvmat — =#. %'#0$(' [n, n], ' retvec —+,:#.0 [n, 1]. J2,4.+'#,27*., %'--$+, ".269,**85 :': $> "0.$&+,4,*$,, A64,# +,:-#.0.%--#.2A(.% [n, 1].J2,46<F$5 %.%,*#, :.#.085 *,.A>.4$%. 69,-#7, :'-',#-) +89$-2,*$) 9,#8-
0,> -:'2)0*8> +,2$9$*: a, b, c $ d. 3,2$9$*' d +89$-2),#-) $> #0,> 406/$>,:'?4') $& :.#.08> ".269,*' $& 16*:($$ %'--$+'. M'"0$%,0, a +89$-2),#-) $&%'--$+' ,4$*$9*./. +,:#.0' uvec $ %'--$+' 1.a = Application.Sum(Application.MMult(Application.Trans-pose(uvec), 1))
Z6*:($) Mmult() 6%*.?',# +,:#.0--#0.:6 [1, n] *' +,:#.0--#.2A,( [n, 1]$ +.&+0'F',# %'--$+ [1, 1] (-:'2)0*6< +,2$9$*6). 3 Excel =#. *, +8&.+,# *$:':$>#064*.-#,5, *. + VBA *,.A>.4$%. $-".27&.+'#7 16*:($< Sum() 42) "0,.A0'&.-+'*$) %'--$+' + -:'2)0*6< +,2$9$*6.M':.*,(, *,.A>.4$%. 0,B$#7 +."0.- -%,B'**.5 .A0'A.#:$ +89$-2,*$5, /4,
&'4,5-#+.+'*8 :': -:'2)0*8, +,2$9$*8, #': $ =2,%,*#8 %'--$+.+. 3 Excel =#.#':?, +8".2*),#-) A,& "0.A2,%, :': ".:'&'*. + )9,5:'> I24:I26 42) +,:#.0' g. M.+ VBA #':$, +89$-2,*$) 269B, +-,/. +8".2*$#7 + ($:2,.Dim wtsvec() As Variantn = Application.Count(retvec)ReDim wtsvec(n)For i = 1 To ngi = b * m(i, 1)-a * l(i, 1)hi = c * l(i, 1)-a * m(i, 1)wtsvec(i) = (gi + hi * expert)/d
Next i
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+,-.- 6. /01#2#3-4#* #).'$1#4#"))"5" 0"&16',* 163
.",0'#.0' ReDim. 3 ($:2, %.?*. 6%*.?$#7 -:'2)0*8, +,2$9$*8 *' .#4,27*8,=2,%,*#8 %'--$+'.
6.11 54%8A,, $/& *"#( 'DI,( :1$12-'"*.#/+%'@' ,%;#)*,"';1%,&
Z6*:($) Prob1OptimalRiskyWeight() - 9,#807%) '0/6%,*#'%$ )+2),#--) "0)%.5 0,'2$&'($,5 + "0./0'%%, VBA 1.0%628, "0,4-#'+2,**.5 + 0'&4,2, 6.7.O0/6%,*#'%$ 16*:($$ )+2)<#-) 4.>.4*.-#7 $ 0$-: (r1 $ sig1) +8-.:.0$-:.+./.':#$+', A,&0$-:.+') -#'+:' 4.>.4*.-#$ (rf) $ :.=11$($,*# *,-:2.**.-#$ : 0$-:6(rraval).Function Prob1OptimalRiskyWeight(r1, rf, sig1, rraval)Prob1OptimalRiskyWeight = (r1-rf)/(rraval * sig1^2)
End Function
M$?, "0$+,4,* :.4 16*:($$ Prob2OptimalRiskyWeight(),:.#.06< %.?-*. $-".27&.+'#7 42) 0,B,*$) &'4'9 2 $ 3. !*' $%,,# +-,/. -,%7 '0/6%,*#.+: r1,r2, sig1 $ sig2 — 4.>.4*.-#$ $ -#'*4'0#*8, .#:2.*,*$) 42) 4+6> +8-.:.0$--:.+8> ':#$+.+; corr12 — :.00,2)($) %,?46 4+6%) +8-.:.0$-:.+8%$ ':#$+'%$;rraval — :.=11$($,*# *,-:2.**.-#$ : 0$-:6 42) :.*:0,#*./. $*+,-#.0'; rf —A,&0$-:.+') -#'+:' 4.>.4*.-#$. C2) &'4'9$ 2 A,&0$-:.+') -#'+:' 4.>.4*.-#$ &'-4'*' 0'+*.5 *62<.Function Prob2OptimalRiskyWeight(r1, r2, sig1, sig2, corr12,rraval)Dim cov12, vqr1, var2,minvarw, w, xr1, xr2cov12 = corr12 * sig1 * sig2var1 = sig1^2var2 = sig2^2If rf <= 0 Then
minvarw = (var2--cov12)/(var1 + var2-2*cov12)w = minvarw + (r1--r2)/(rraval*(var1 + var2-2*cov12))
Elsexr1 = r1--rfxr2 = r2--rfw = xr1*var2-xr2*cov12w = w /(xr1*var2 + xr2*var1-(xr1 + xr2)*cov12)
End IfProb2OptimalRiskyWeight = w
End Function
Z.0%62' 42) ."#$%'27*.5 4.2$ ':#$+' &'"$-'*' 9,0,& :.+'0$'($< (.A.&*'9,*-*6< :': cov12).G,0,%,**') minvarw— =#. ."#$%'27*') 4.2) ':#$+' + ".0#1,2,
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- %$*$%'27*.5 4$-",0-$,5 wmv1 , ' ",0,%,**') w — &*'9,*$, wopt
1 , %':-$%$&$06-<F,, ".2,&*.-#7 ".0#1,2).C2) &'4'9$ 3 16*:($) $-".27&6,# 16*:($$, 0,B'<F$, &'4'9$ 1 $ 2. Z6*:-
($) Prob3OptimalWeightsVec() +.&+0'F',# #0$ ."#$%'27*8, 4.2$ ':#$+.++ +$4, +,:#.0'. G.2*85 :.4 16*:($$ "0$+,4,* *$?,.Function Prob3OptimalWeightsVec(r1, r2, sig1, sig2, corr12,rraval)Dim w0, w1, w2, rr, sigrw1 = Prob2OptimalRiskyWeight(r1, r2, rf, sig1, sig2, corr12, _rrava)w2 = 1-w1rr = w1*r1 + w2*r2sigr = Sqr((w1*sig1)^2 + (w2*sig2)^2 + 2*w1*w2*corr12*sig1* _sig2)w0 = 1-Prob1OptimalRiskyWeight(rr, rf, sigr, rraval)w1 = (1-w0)*w1w2 = (1-w0)*w2Prob3OptimalWeightsVec = Array(w0, w1, w2)
End Function
X,& 69,#' A,&0$-:.+./. ':#$+' 4'**') &'4'9' -+.4$#-) : &'4'9, 2 $ +89$-2),#-..#*.B,*$, %,?46 ':#$+'%$ asset1 $ asset2:w1 = Prob2OptimalRiskyWeight(r1, r2, rf, sig1, sig2, corr12, _rrava)w2 = 1-w1
L& =#$> +,-.+ %.?*. +89$-2$#7 4.>.4*.-#7 $ 0$-: +8-.:.0$-:.+./. ".0#1,2):rr = w1*r1 + w2*r2sigr = Sqr((w1*sig1)^2 + (w2*sig2)^2 + 2*w1*w2*corr12*sig1*sig2)
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6.12 J'-'/%,*#/+%?# 318"')?D':0.-8, $-".27&6,%8, 42) ".269,*$) =11,:#$+*.5 /0'*$(8 - ".%.F7<
*'4-#0.5:$ !"#$% &'(')#* (Solver), -.4,0?'#-) + 2$-#, %.462) ModuleM. M'4--#0.5:6 !"#$% &'(')#* *,.A>.4$%. 6-#'*.+$#7 4.".2*$#,27*., ,-2$ .4*.$%,*-*') :.%'*4' *, .#.A0'?',#-) + %,*< ;'&.#$ (Tools). R0.%, #./., 9#.A8 $-".27&.-+'#7 16*:($$ *'4-#0.5:$ !"#$% &'(')#* + VBA, + 2$-#, %.462) *6?*. 6:'&'#7--82:6 *' 1'52 *'4-#0.5:$ SOLVER.xla.Sub EffFrontier1()SolverReset
+,-.- 6. /01#2#3-4#* #).'$1#4#"))"5" 0"&16',* 165
Call SolverAdd(Range("portret1"), 2, Range("target1"))Call SolverOk(Range("portsd1"), 2, 0, Range("change1"))Call SolverSolve(True)SolverFinish
End Sub
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Loop
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SolverResetCall SolverAdd(Range("portwts2"), 3,Range("portmin2"))Call SolverAdd(Range("portwts2"), 1,Range("portmax2"))Call SolverOk(Range("portret2"), 2, 0,Range("change2"))Call SolverSolve(True)SolverFinishprmin = Range("portret2").ValueCall SolverOk(Range("portret2"), 1, 0, Range("change2"))Call SolverSolve(True)SolverFinish
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