Question

Blair & Rosen, Inc. (B&R) is a brokerage firm that specializes in investment portfolios designed to meet the specific risk tolerances of its clients. A client who contacted B&R this past week has a maximum of $50,000 to invest. B&R's investment advisor decides to recommend a portfolio consisting of two investment funds: an Internet fund and a Blue Chip fund. The Internet fund has a projected annual return of 12%, while the Blue Chip fund has a projected annual return of 9%. The investment advisor requires that at most $35,000 of the client's funds should be invested in the Internet fund. B&R services include a risk rating for each investment alternative. The Internet fund, which is the more risky of the two investment alternatives, has a risk rating of 6 per thousand dollars invested. The Blue Chip fund has a risk rating of 4 per thousand dollars invested. For example, if $10,000 is invested in each of the two investment funds, B&R's risk rating for the portfolio would be 6(10) + 4(10) = 100. Finally, B&R developed a questionnaire to measure each client's risk tolerance. Based on the responses, each client is classified as a conservative, moderate, or aggressive investor. Suppose that the questionnaire results classified the current client as a moderate investor. B&R recommends that a client who is a moderate investor limit his or her portfolio to a maximum risk rating of 240. (a) Formulate a linear programming model to find the best investment strategy for this client. Let I = Internet fund investment in thousands B = Blue Chip fund investment in thousands If required, round your answers to two decimal places. + Max B s.t. + o X Available investment funds B X + B Maximum investment in the internet fund s B X Maximum risk for a moderate investor s I, B go > (b) Build a spreadsheet model and solve the problem using Solver. What is the recommended investment portfolio for this client? Internet Fund $ Blue Chip Fund = $ What is the annual return for the portfolio?

Answer 1

a. Let the amount invested in internet fund be "I" and blue chip fund be "B".

Nowm we have to maximize the return for the portfolio. Portfolio return = weighted average return of the two funds = 0.12I+0.09B. This is the objective function and this will have to be maximized.

Constraints:

(i) I+B<=50,000 (maximum amount avaliable for investment)

(ii) I<=35,000 (at most $35,000 of the client's funds should be invested in the Internet fund)

(iii) 6/1000*I+4/1000*B<=240 (maximum risk for a moderate investor).

Solving in excel, using the solver function, the following solution is obtained:

b.

	Internet	Blue chip
Amount invested	20000	30000
Returns	0.12	0.09
Portfolio return	5100
Constraints				Formula
50000	<=	50000		I+B<=50,000
20000	<=	35000		I<=35000
240	<=	240		6/1000*I+4/1000*B<=240

Thus I = $20,000 and B = $30,000. Annual return = $5,100. All contraints are satisfied.

c. Constraint (i) and (ii) will remain the same. The 3rd constraint will become:  6/1000*I+4/1000*B<=320

Bookl - Microsoft Excel X Clear Reapply Data Analysis ? Solver Filter Y Advanced Group Ungroup Subtotal Text to Remove Data Consolidate What if Columns Duplicates Validation Analysis Data Tools Sort & Filter Outline Analysis Home Insert Page Layout Formulas Data Review View El Connections 21 AZ Properties From From From From Other Existing Refresh 2 Sort Access Web Text Sources Connections All Go Edit Links Get External Data Connections B5 fx =(B3*B2)+(C3*C2) B С D E F 1 Internet Blue chip 2 Amount invested 20000 30000 3 Returns 0.12 0.09 G H J K L M N o P Q R 5100 5 Portfolio return 6 7 Constraints 8 50000 9 20000 10 240 <= 11 12 50000 Formula I+B<=50,000 I<=35000 6/1000*l+4/1000'B<=240 35000 240 Solver Parameters Solve MO Value of: Close 13 14 15 16 17 18 19 20 21 22 23 24 25 Set Target Cell: 5855 Equal To: © Max By Changing Cells: $B$2:$C$2 Subject to the Constraints: SA$10 <= 240 SA$8 <= 50000 SA $9 <= 35000 Guess Options Add Change Reset Al Delete Help 26 27 Sheet1 Sheet2 Sheet3 Point e 9:29 AM 7/27/2016

Solving in excel, the following solution is obtained:

	Internet	Blue chip
Amount invested	35000	15000
Returns	0.12	0.09
Portfolio return	5550
Constraints				Formula
50000	<=	50000		I+B<=50,000
35000	<=	35000		I<=35000
270	<=	320		6/1000*I+4/1000*B<=320

I = $35,000. B = $15,000 and Annual Return = $5,500

d. Constraint (i) and (ii) will remain the same. The 3rd constraint will become:  6/1000*I+4/1000*B<= 160. Solution:

	Internet	Blue chip
Amount invested	0	40000
Returns	0.12	0.09
Portfolio return	3600
Constraints				Formula
40000	<=	50000		I+B<=50,000
0	<=	35000		I<=35000
160	<=	160		6/1000*I+4/1000*B<=320

I = $0, B = $40,000 and Annual return = $3,600.

Bookl - Microsoft Excel X Clear Reapply Data Analysis ? Solver Filter Y Advanced Group Ungroup Subtotal Text to Remove Data Consolidate What if Columns Duplicates Validation Analysis Data Tools Sort & Filter Outline Analysis Home Insert Page Layout Formulas Data Review View El Connections 21 AZ Properties From From From From Other Existing Refresh 2 Sort Access Web Text Sources Connections All Go Edit Links Get External Data Connections B5 fx =(B3*B2)+(C3*C2) С D E F 1 Internet Blue chip 2 Amount invested 0 40000 3 Returns 0.12 0.09 G H J K L M N o P Q R Formula I+B<=50,000 I<=35000 6/1000*l+4/1000'B<=320 5 Portfolio return 3600 6 7 Constraints 8 40000 50000 9 OC 35000 10 160 <= 160 11 12 13 Solver Parameters 14 15 Set Target Cell: SBSS 16 Equal To: o Max Min By Changing Cells: 17 18 $B$2:$C$2 19 Subject to the Constraints: 20 $A$10 <= 160 21 SA $8 = 50000 $A$9 <= 35000 22 58$2 >= 0 23 SC$2 >=0 24 25 Solve Value of: 0 Close EN Guess Options Add Change Reset All Delete Help 26 27 HI Sheet1 Point Sheet2 Sheet3 9:36 AM 7/27/2016