Question

An investment broker has received $250,000 to invest in a 12-month commitment. The money can be placed in Treasury notes (with a return of 8% and a risk score of 2) or in municipal bonds (with a return of 9% and a risk score of 3). The broker’s client wants diversification to the extent that between 50% and 70% of the total investment must be placed in Treasury notes. Also, because of fear of default, the client requests that the average risk score of the total investment should be no more than 2.42. How much should the broker invest in each security so as to maximize return on investment?

Answer 1

Investment made in treasury notes = 175,000

Investment made in municipal bonds = 75,000

Maximum return = 20750

Dole Poge Let hunted Let het investment made in treasury notes = x Let investment made in municipal bonds We must maximize the returns Objective function: Max. Z=0.08>k+ 0.0 Constraints Given that between 5% & 70%. of in total investment must be made treasury ey notes. z 0.7&ty) x 20.56+ y 2 Total investment constraint :- £250,000 Risk score investment constraint :- 2x+3y L2.42 (x+y) Non-zero constraints : O xy Scanned by TapScanner 2L ä ty

There fore Max. Z=0.08xt +0.094 Subject to : xty £250,000 x 20.7Cty) 2x + 3y - 2.42 Caty) sys Find points of equations : Х 230.5.Cxty 4 20 * EQUATION xty = 250000 Y 1250000 0 | 250000 30.7Gety x = 0. 5(x +y) 2x+3y = 2.426xty) 350000 105000 O o 250000 250000 O 138095-2100000 pe Find paints of intersection of Octy = 250000 * = 0.1(x+y) - Scanned by TapScanner

Date Dage 26 (itty - 777250,000 - dok D in @ = 0,7(250,000 -y+y) y = 75,000 in Put value of a 250000-y from Put value of y at 75,000 = 250,000 2:175,000 Point of intersection (175 000, 75000) # Find g points. intersection - xty = 250000 m a 2x +3y = 2.42(x+y) Put value of a from 2 (250000-y) + 3y = 2.42 x(250000-y ty y = 105,000 0 끼 in Put value of y in (I Scanned by TapScanner

Dole Page 105.000 + x = 3250000 2- 145000 Point of intersection = (145000, 105000) Hook book csak ery Hook (25олоо x=0.5% + 150k A Hook Bxz6.76sty 150* 0 300k 350k 50k 100k 200K 150K OAB represents the feasible region. Scanned by TapScanner

Data Page Value of 2 Conner Points 0 (0,0) IA (145 000, 105000) B C175000, 75000) 6:08 xo 40.0480 -0. 0.08x145000 +0.09x 10 Sood. => 2 1050 0.08 x 175000+ 0.09% 75000 => 20750 Maximum value occurs at point B. Therefore, Investment made in treasury notes = 175.000 Investment made in municipal o bonds = 75.000 Maximum Return => 20750 Scanned by TapScanner