Identify and formulate an investment strategy for Charles

Solution Preview

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See the attached files. I have attached the MS files also.

a) Let X1, X2 and X3 is the amount of bonds Charles sell in 5th, 6th and 7th year respectively.
Also, T1, T2 and T3 is the amount of tax paid by Charles in 5th, 6th and 7th year respectively.

Objective function:
Maximize X1*1.5001+X2*1.6351+X3*1.7823 - T1 - T2 - T3

Subject to
X1+X2+X3 = 30000
T1>=(X1*0.5001-6100)*0.30 or 0.15003X1-T1<=1830
T2>=(X2*0.6351-6100)*0.30 or 0.19053X2-T2<=1830
T3>=(X3*0.7823-6100)*0.30 or 0.23469X3-T3<=1830
X1,X2,X3,T1,T2,T3>=0

Use Management Scientist to solve the problem. The optimal solution is
X1=0
X2=$9604.79
X3=$20395.21

Taxes paid will be $0 in 5th year, $0 in 6th year and $2956.55 in 7th year.
The amount of money Charles will have at the end of 7th year will be $49098.62 (principal plus interest after paying taxes).

Output:
LINEAR PROGRAMMING PROBLEM

MAX 1.5001X1+1.6351X2+1.7823X3-1T1-1T2-1T3

S.T.

1) 1X1+1X2+1X3=30000
2) 0.15003X1-1T1<1830
3) 0.19053X2-1T2<1830
4) 0.23469X3-1T3<1830

OPTIMAL SOLUTION

Objective Function Value = 49098.62278

Variable Value Reduced Costs
-------------- --------------- ------------------
X1 0.00000 0.04751
X2 9604.78665 0.00000
X3 20395.21335 0.00000
T1 0.00000 1.00000
T2 0.00000 0.54081
T3 2956.55262 0.00000

Constraint Slack/Surplus Dual Prices
-------------- --------------- ------------------
1 0.00000 1.54761
2 1830.00000 0.00000
3 0.00000 0.45919
4 0.00000 1.00000

OBJECTIVE COEFFICIENT RANGES

Variable Lower Limit Current Value Upper Limit
------------ --------------- --------------- ---------------
X1 No Lower Limit 1.50010 1.54761
X2 1.54761 1.63510 1.73814
X3 1.73479 1.78230 1.86979
T1 No Lower Limit -1.00000 0.00000
T2 No Lower Limit -1.00000 -0.45919
T3 -1.20244 -1.00000 -0.62721

RIGHT HAND SIDE RANGES

Constraint Lower Limit Current Value Upper Limit
------------ --------------- --------------- ---------------
1 17402.30678 30000.00000 No Upper Limit
2 0.00000 1830.00000 No Upper Limit
3 0.00000 1830.00000 4230.23849
4 No Lower Limit 1830.00000 4786.55262

Part b) The investment advisor is considering only the interest to be earned in a single year. But when we are paying taxes on selling the bonds, the tax is applicable on interest earned for all the years. So if we are withdrawing the investment in 7th year, we are paying tax on the interest earned for all the seven years. So in 7th year we are paying 30% tax on 0.7823 for every $1 invested which is about $0.23469 and effectively we are ending with $1.54761, where if we can withdraw this amount in 6th year and save taxes we will get $1.6351. Thus, it is better to withdraw and save taxes and keep it under the carpet rather than invest and pay more tax.

Part c)
We add two more variable. C1 and C2 is the amount invested in CDs at the end of 5th and 6th year respectively

Objective function:
Maximize X1*1.5001+X2*1.6351+X3*1.7823 - T1 - T2 - T3 + C1*0.04+C2*0.04

Subject to
X1+X2+X3 = 30000
C1-X1*1.5001+T1<=0 ...