1) Formulation of the Problem as a Linear Programming Model

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1 1) Formulation of the Problem as a Linear Programming Model Let xi = the amount of money invested in each of the potential investments in, where (i=1,2, ) x1 = the amount of money invested in Savings Account x2 = the amount of money invested in Belfort Stock x3 = the amount of money invested in Bayside Stock x4 = the amount of money invested in Goodies Stock x5 = the amount of money invested in Basic Insurance program x = the amount of money invested in Government bonds x7 = the amount of money invested in Intra CO Stock x = the amount of money invested in GR Stock Min Z Z= 0x1 + 0x2 + 0,25x3 + 0,3x4 + 0,2x5 + 0,15x +0,5x7 + 0,40x The above investment strategy minimizes the total risk value. Constraints: x1 + x2 + x3 + x4 + x5 + x +x7 + x = ,04x1 +0,052x2 +0,071x3 + 0,1x4 + 0,02x5 + 0,05x +0,2x7 + 0,125x x1 + x2 + x5 +x x1 + x3 + x4 +x7 + x x1 + x x1, x2, x3, x4, x5, x,x7, x 0 Katrakaza Sevina Σελίδα 1

2 2) Optimal Solution of the Problem Using Excel s Solver tool Answer Report (my Excel was in Greek, so unfortunately the answer report of Solver is also in Greek) Target Cell (Min) ί προορισμού (Ελάχιστο) Κε λί Όνομ α Αρχική , 7 Adjustable Cells Ρυθμιζόμενα κελιά ί Όνομα Αρχική $B$3 x1 0 17,33 $C$3 x2 0 12, 7 $D$ 3 x3 0 0 $E$3 x4 0 22, 7 $F$3 x5 0 47,33 $G$ 3 x 0 0 $H$3 x7 0 0 $I$3 x 0 0 Constraints Τιμή Κατάστασ Απόκλισ ί Όνομα κελιού Τύπος η η Constrai nts =$L$ υποχρεωτικ 7>=$L $7 77,33 >=$L $ υποχρεωτικ ός 27, >=$L 0 Katrakaza Sevina Σελίδα 2

3 9 $9 ός 10<=$ L$10 Sensitivity Report Adjustable Cells Ρυθμιζόμενα κελιά Μειωμέν ο Αντικειμε νικός Κε λί Όνο μα κόστος συντελεσ τής αύξηση μείωση $B $3 x1 17, , ,005 $C $3 x2 12, ,005 0, $D $3 x3 0 0,04 7 0,25 1E+30 0,04 7 $E$ 3 x4 22, 7 0 0,3 0, , $F$ 3 x5 47, ,2 0, ,005 $G $3 x 0 0,00 7 0,15 1E+30 0,00 7 $H $3 x7 0 0,01 7 0,5 1E+30 0,01 7 $I$ 3 x 0 0,01 7 0,4 1E+30 0,01 7 Constraints Σκιώδης ί Όνομα - Constra 0,073 ints Περιορι σμός R.H. Side αύξηση μείωση 434,14 341, , Katrakaza Sevina Σελίδα 3

4 7 77, , , ,33 1E ,11 2, ,33, 3) The Optimal Investment Strategy that the consultant should suggest & The Total Risk Value with this Strategy The optimal investment Strategy that the consultant should suggest can be found on the Answer Report. The solution is given in the Final Values (Τελικές Τιμές), as we can see underneath with red color. Target Cell (Min) ί προορισμού (Ελάχιστο) Κε λί Όνομ α Αρχική , 7 Adjustable Cells Ρυθμιζόμενα κελιά ί Όνομα Αρχική Katrakaza Sevina Σελίδα 4

5 $B$3 x1 0 17,3 3 $C$3 x2 0 12, 7 $D$ 3 x3 0 0 $E$3 x4 0 22, 7 $F$3 x5 0 47,3 3 $G$ 3 x 0 0 $H$3 x7 0 0 $I$3 x 0 0 The optimal investment strategy is 17.,33 invested in Savings Account, 12.,7 invested in Belfort Stock, 22.,7 invested in Goodies Stock, and 47.,33 invested in Basic Insurance program. The total risk value with this Strategy is 1.2,7. 4) Suggestion in order for the consultant to invest in Bayside or in Intra CO s stocks If we go to the Answer report, we can easily see that the investment programs x3 and x7 are not being suggested (Bayside and Intra CO s). If we go to the Sensitivity Report, we can see that in order to propose to invest to Bayside stock, the risk factor should be decreased by 0,047, so the factor should be less than 0,2033, or 20,33%. Similarly, in order to propose to invest to Intra Co s stock, the risk factor should be decreased by 0,017, so the factor should be less than 0,, or 3,33%. Katrakaza Sevina Σελίδα 5

6 Ρυθμιζόμενα κελιά Μειωμέν ο Αντικειμε νικός Κε λί Όνο μα κόστος συντελεσ τής αύξηση μείωση $B $3 x1 17, , ,005 $C $3 x2 12, ,005 0, $D $3 x3 0 0,04 7 0,25 1E+30 0,04 7 $E$ 3 x4 22, 7 0 0,3 0, , $F$ 3 x5 47, ,2 0, ,005 $G $3 x 0 0,00 7 0,15 1E+30 0,00 7 $H $3 x7 0 0,01 7 0,5 1E+30 0,01 7 $I$ 3 x 0 0,01 7 0,4 1E+30 0,01 7 5) The Change in the total risk value and in the final investment strategy, if the consultant wishes to invest at least 30% of the available budget in immediately liquid investments. Katrakaza Sevina Σελίδα

7 The 10% reduction in the available budget in immediately liquid investment, should change our constraint of to , so we would have a change of We go to the Sensitivity report, and we can see that we are in the range of feasibility (allowable decrease 2..9 ), so the shadow price is valid, and the total risk value will be reduced by *0,04=400 to an new Z= 15.,7. Σκιώδης ί Όνομα Περιορι σμός R.H. Side αύξηση μείωση Constra ints , , , , , ,33 1E , , , , ,33, However, as we can see the constraint is binding, so the final investment strategy changes. We have to run solver again to find the new investment strategy. ί Όνομα Τιμή κελιού Constrai nts , Τύπος =$L$ 7>=$L $7 >=$L $ 9>=$L $9 10<=$ L$10 Κατάστασ Απόκλισ η η υποχρεωτικ υποχρεωτικ 27,33 ός Υποχρεωτι κ Katrakaza Sevina Σελίδα 7

8 ) The consultant wants to know if he could follow the same investment strategy by increasing the % investment of the available budget in one of the A-rated or immediate liquid, or zero-risk investments. Σκιώδης ί Όνομα Τιμή Περιορι σμός R.H. Side αύξηση μείωση Constra ints , , , , , ,33 1E , , , , ,33, The consultant wants to know if he can increase the amount of money available to invest in one of the investment groups, the A-rated, immediate liquid or zero risk. As we can see from the sensitivity report, we have with red color the allowable increase of the 3 constraints. If the increase is in the range of feasibility: For the A-rated investments the allowable increase is 27.,33. For the Immediately liquid investement, the allowable increase is ,11. For the Zero-risk investment the allowable increase is 17.,33. If they are in the feasibility range, then the shadow price is valid, and we can calculate the total risk value by adding the result of Increase*Shadow price. Τιμή Κατάστασ Απόκλισ ί Όνομα κελιού Τύπος η η Constrai nts =$L$ υποχρεωτικ 7>=$L $7 77,33 >=$L 27,33 Katrakaza Sevina Σελίδα

9 $ 9>=$L $9 10<=$ L$10 υποχρεωτι κός Υποχρεωτι κ Υποχρεωτι κ However, as we can see the A-rated investment is not binding, and its shadow price is 0, so for A-rated investments the investment strategy will not change if the increase is within 27.,33. In order to have the final investment strategy for the other 2 investments, we have to run again the solver. 7) The change in the total risk factor and in the final investment strategy, if the consultant decides to invest in zero risk investments. Σκιώδης ί Όνομα Περιορι σμός R.H. Side αύξηση Μείωση Constra ints , , , , , ,33 1E , , , , ,33, If we go again to the sensitivity report, we can see that the constraint referring to the zero-risk investments, and we can also see that it is binding. If we decrease the zero risk investment from to , then we must check if the within the feasibility range of the allowable decrease (.,33 ). As we can see they are within the range. As a consequence the total risk value will increase to 5.000*0,1=500, so Z=1.7,7. Again, since the constraint is binding, we should run again the solver in order to find which will be the change to the final investment strategy. Katrakaza Sevina Σελίδα 9

10 ) The changes in the total risk value and in the investment strategy, if the risk factor for Goodie s stock becomes 2%. Ρυθμιζόμενα κελιά Μειωμέν ο Αντικειμε νικός Κε λί Όνο μα κόστος συντελεσ τής αύξηση μείωση $B $3 x1 17, , ,005 $C $3 x2 12, ,005 0, $D $3 x3 0 0,04 7 0,25 1E+30 0,04 7 $E$ 3 x4 22, 7 0 0,3 0, , $F$ 3 x5 47, ,2 0, ,005 $G $3 x 0 0,00 7 0,15 1E+30 0,00 7 $H $3 x7 0 0,01 7 0,5 1E+30 0,01 7 $I$ 3 x 0 0,01 7 0,4 1E+30 0,01 7 The Goodies stock is the x4. If instead of 0,2, the new risk factor would be 0,2, we must check for a 0,04 decrease, which as we can see is not in the feasibility range. As a consequence we do not know either the change of the total risk value, or the final investment strategy, and we have to run the solver again. 9) The change in the total risk factor and in the investment strategy, if the amount invested in A-rated investments is increased by 10%. ί Όνομα Τιμή κελιού Constrai nts , Τύπος =$L$ 7>=$L $7 >=$L $ 9>=$L $9 Κατάστασ Απόκλισ η η υποχρεωτικ υποχρεωτι 27,33 κός Katrakaza Sevina Σελίδα 10

11 <=$ L$10 As we can see the A-rated investment is not binding. We are discussing of an increase of 10%, so the at least budget should have an increase of As we can see we are within feasibility range, and concerning the binding term, and the allowable increase as we can see underneath from the sensitivity report. Σκιώδης ί Όνομα Τιμή Περιορι σμός R.H. Side Αύξηση μείωση Constra ints , , , , , ,33 1E , , , , ,33, As a consequence, there will be no change either in the total risk value, or the final investment strategy. We can also see in order to check if we are correct, that the optimal solution, is proposing to invest 77. to A rated investments, so there is no change in the solution. Katrakaza Sevina Σελίδα 11

12 10) Assume that the consultant is thinking to add in the portfolio an extra potential investment with a 9% annual expected return, A-quality rating and immediate liquidity. What is the risk factor that would be recommended in order to be possible for this new investment to be chosen? We shall name the new investment x9.we go to the sensitivity report of the affected constraints: Σκιώδης ί Όνομα Τιμή Περιορι σμός R.H. Side αύξηση Μείωση Constra ints , , , , , ,33 1E , , , , ,33, - The new investment would affect the 1 st constraint, since it will be added to the total budget of The decrease of the RHS is shadow price * x9 = -0,073x9 - It also affects the 2 nd constraint, since it should have annual return if it was chosen. Its annual return is 9%, so it would be 0,09 decrease * shadow price = 0,09*3,33=0,2997x9 - It is an A-rated investment so the 3 rd constraint is affected, but the shadow price is 0, so the total risk factor will not change. - It is also an immediate liquidity investment so the 4 th constraint changes as well, and the RHS constraint will decrease by shadow price* x9 = 0,04x9 Since the change in the total risk value can be calculated using the shadow prices for each constraint, we can find the risk factor for the new investment: -0,073x9 + 0,2997x ,04x9 = 0,27. So the risk factor of the new investment that is recommended, in order for it to be chosen in the portfolio is 2,7%. Katrakaza Sevina Σελίδα 12