My Math Forum Op. Res: What next?

 Economics Economics Forum - Financial Mathematics, Econometrics, Operations Research, Mathematical Finance, Computational Finance

 January 25th, 2011, 05:59 PM #1 Senior Member   Joined: Sep 2009 Posts: 251 Thanks: 0 Op. Res: What next? Operations Research Questions: You have $5000 to invest for five years. At the beginning of each year, you can invest in one- or two-year bond at 4% and 9% total interest, respectively. At the start of the second year, you can invest in three-year 15% total interest bond. Find a linear program to maximize total cash after five years. My answer (so far): $y_i$: amount available to invest end of year i ($); i=0,1,2,3,4,5 $x_{i,j}$: amount invested at end of year i in j-length bonds (\$); i=0,1,2,3,4; j=1,2,3 ex: x_{0,1} is the amount invested at the end of year 0 (beginning of year 1) in 1-year bonds Objective: Maximize $y_5$. Constraints: $y_0=5000\\ x_{i,j} \ge 0\\ x_{i,j}=0, \forall i \lt 0 x_{0,3}=0$ The amount of money available at end of year i equals the amount of money invested in 1-, 2-, and 3- year bonds at the end of year i: $y_0=5000=x_{0,1}+x_{0,2} \hspace{10mm} (+x_{0,3})\\ y_1=x_{1,1}+x_{1,2}+x_{1,3}\\ y_2=x_{2,1}+x_{2,2}+x_{2,3}\\ y_3=x_{3,1}+x_{3,2}+x_{3,3}\\ y_4=x_{4,1}+x_{4,2}+x_{4,3}\\$ The amount of money available at end of year i equals the amount of money rolling off investments from 1, 2, and 3 years earlier: $y_1=1.04x_{0,1}\\ y_2=1.04x_{1,1}+1.09x_{0,2}\\ y_3=1.04x_{2,1}+1.09x_{1,2}+1.15x_{0,3}\\ y_4=1.04x_{3,1}+1.09x_{2,2}+1.15x_{1,3}\\ y_5=1.04x_{4,1}+1.09x_{3,2}+1.15x_{2,3}$ Equate the $y_i$s (and rearrange a little bit): $y_5=1.04x_{4,1}+1.09x_{3,2}+1.15x_{2,3}\\ \\ 5000=x_{0,1}+x_{0,2}\\ x_{1,1}+x_{1,2}+x_{1,3}-1.04x_{0,1}=0 x_{2,1}+x_{2,2}+x_{2,3}-1.04x_{1,1}-1.09x_{0,2}=0\\ x_{3,1}+x_{3,2}+x_{3,3}-1.04x_{2,1}-1.09x_{1,2}-1.15x_{0,3}=0\\ x_{4,1}+x_{4,2}+x_{4,3}-1.04x_{3,1}-1.09x_{2,2}-1.15x_{1,3}=0\\$ So what's next? I have 5 equations and 15 unknowns. I've tried thinking of ways to substitute to reduce variables, but can't find any. My classmate says it's five equations and five unkowns. TIA.

 Tags res

 Thread Tools Display Modes Linear Mode

 Contact - Home - Forums - Cryptocurrency Forum - Top