EXAMPLE 14.4 Innls Investments Is a small. famlly-owned buslness that manages personal flnanclal portfollos. The company manages slx mutual funds and has a cllent that has as qulred $500,000 from an lnherftance. lBharactel'fstlcs of the funds are glven In Table 14.5. Innls Investments uses a proprfetary algonthm to establlsh a measure of rfsk for Its funds based on the hlstoncal volatlllty of the Investments. The hlgher the volatlllty, the greater the rfsk. The company recom- mends that no more than $200.000 be Invested In any In- dlvldual fund, that at least 5550.000 be Invested In each of the multlnatlonal and balanced funds. and that the total amount Invested In Income equlty and balanced funds be at least 40% of the total Investment. or $200,000. The cllenf would Ilke to have an average return of at least 5% but would llke to mlnlmlze rlle IIlll'hat portfollo would achleve thls'? Let X1 through 1., represent the dollar amount In- vested In funds 1 through E, respectlvely. The total rlslt would be measured by the welghted rfsk of the port- follo, where the welghts are the proportlon of the total Investment In any fund (Jg-f500.000}. Thus. the objective function ls Innis Investments mlnlmlze total nsk = 10.5371 +1B.22Xa + 14.02Xa + 2.3034 + 0.00% + 16135 500.000 The rst constralnt ensures that $500,000 ls Invested: X1 +XZ+XH+Xq+XE+XE=5DDIDm The next constralnt ensures that the welghted return Is at least 5%: max. + scant, + 156.37.. + 3.6212, + 7.75m. + 4.4M, 500,000 The next constralnt ensures that at least 40% be Invested In the Income equlty and balanced funds: 2500 x... + x, a easement}: The followl ng constralnts speclfy that at least 050.000 be Invested In each of the multlnatlonal and balanced funds: x, a some it... a some Flnally, we restrfct each Investment to a maxlmum of $200,000 and Include nonnegatlvlty: 1952mm: for}: 1,...,s )(I-ED forj=1....,0