A second version of the Markowitz portfolio model maximizes expected return subject to a constraint that the variance of the portfolio must be less than or equal to some specified amount. Consider the Hauck Financial Service data.

Click on the datafile logo to reference the data.

Annual Return $(\%)$

Mutual Fund Year $1$ Year $2$ Year $3$ Year $4$ Year $5$

Foreign Stock $10\mathrm{.}0613\mathrm{.}1213\mathrm{.}4745\mathrm{.}42-21\mathrm{.}93$

Intermediate$-$Term Bond $17\mathrm{.}643\mathrm{.}257\mathrm{.}51-1\mathrm{.}337\mathrm{.}36$

Large$-$Cap Growth $32\mathrm{.}4118\mathrm{.}7133\mathrm{.}2841\mathrm{.}46-23\mathrm{.}26$

Large$-$Cap Value $32\mathrm{.}3620\mathrm{.}6112\mathrm{.}937\mathrm{.}06-5\mathrm{.}37$

Small$-$Cap Growth $33\mathrm{.}4419\mathrm{.}403\mathrm{.}8558\mathrm{.}68-9\mathrm{.}02$

Small$-$Cap Value $24\mathrm{.}5625\mathrm{.}32-6\mathrm{.}705\mathrm{.}4317\mathrm{.}31$

$($a$)$ Construct this version of the Markowitz model for a maximum variance of $38.$

Let:

FS $=$ proportion of portfolio invested in the foreign stock mutual fund

IB $=$ proportion of portfolio invested in the intermediate$-$term bond fund

LG $=$ proportion of portfolio invested in the large$-$cap growth fund

LV $=$ proportion of portfolio invested in the large$-$cap value fund

SG $=$ proportion of portfolio invested in the small$-$cap growth fund

SV $=$ proportion of portfolio invested in the small$-$cap value fund

$=$ the expected return of the portfolio

Rs $=$ the return of the portfolio in years

If required, round your answers to two decimal places. For subtractive or negative numbers use a minus sign even if there is a $+$ sign before the blank $($Example: $-300).$ If the constant is $"1"$ it must be entered in the box. If your answer is zero enter $0.$

Max

s$.$t$.$

$10\mathrm{.}06$

FS $+$

$17\mathrm{.}64$

IB $+$

$32\mathrm{.}41$

LG $+$

$32\mathrm{.}36$

LV $+$

$33\mathrm{.}44$

SG $+$

$24\mathrm{.}56$

SV

$=$

R $1$

$13\mathrm{.}12$

FS $+$

$3\mathrm{.}25$

IB $+$

$18\mathrm{.}71$

LG $+$

$20\mathrm{.}61$

LV $+$

$19\mathrm{.}40$

SG $+$

$25\mathrm{.}32$

SV

$=$

R $2$

$13\mathrm{.}47$

FS $+$

$7\mathrm{.}51$

IB $+$

$33\mathrm{.}28$

LG $+$

$12\mathrm{.}93$

LV $+$

$3\mathrm{.}85$

SG $+$

$-6\mathrm{.}70$

SV

$=$

R $3$

$45\mathrm{.}42$

FS $+$

$-1\mathrm{.}33$

IB $+$

$41\mathrm{.}46$

LG $+$

$7\mathrm{.}06$

LV $+$

$58\mathrm{.}68$

SG $+$

$5\mathrm{.}43$

SV

$=$

R $4$

$-21\mathrm{.}93$

FS $+$

$7\mathrm{.}36$

IB $+$

$-23\mathrm{.}26$

LG $+$

$-5\mathrm{.}37$

LV $+$

$-9\mathrm{.}02$

SG $+$

$17\mathrm{.}31$

SV

$=$

R $5$

$1$

FS $+$

$1$

IB $+$

$1$

LG $+$

$1$

LV $+$

$1$

SG $+$

$1$

SV

$=$

$1$

$0\mathrm{.}20$

$=$

$0\mathrm{.}20$

$<=$

$38$

FS$,$ IB$,$ LG$,$ LV$,$ SG$,$ SV

$>=$

$0$

$($b$)$ Solve the model developed in part $($a$).$

If required, round your answers to two decimal places. If your answer is zero, enter $0.$

FS

$\%$

IB

$\%$

LG

$\%$

LV

$0$

$\%$

SG

$0$

$\%$

SV

$\%$

Portfolio Expected Return $=$

$\%$