Portfolios and linear programming 2

How To Make The Most Out Of Your Divorce

Alice’s husband cheats on her with Betty. Now, Alice wants a divorce. She has two initial options: she can take a cash settlement of 55K, or take a fixed percentage of his assets each year. Husband’s assets are his salary (100K), his investments (20K), and his savings (130K). She may only take a maximum of 30% from his salary, 20% from his investments and 20% from his savings, and the total percentage she takes from all his assets must be less than or equal to 50%. The risk associated with the salary is 10%, his investment is 30%, and no risk on his savings. Husband must pay 12K every year in child support on top of this. How can Alice maximize the percentage she takes of Husband’s assets? Should she just take the cash settlement and child support for Husband Jr.?

The Toy Model

Assets	Value	Risk (%)	Percentage
Salary	100K	10	x1
Investments	20K	30	x2
Savings	130K	0	x3
Child Support	12K

${\text{maximize}}\;z=100x_{1}+20x_{2}+130x_{3}$
${\begin{aligned}{\text{s.t.}}\;&x_{1}\leq 0.3\\&x_{2}\leq 0.2\\&x_{3}\leq 0.2\\&x_{1}+x_{2}+x_{3}\leq 0.5\\&-0.1x_{1}\leq 0\\&x_{1},x_{2},x_{3}\geq 0\\\end{aligned}}$

The objective function displays each asset and our decision variables are the percentage that Alice may take from each asset. Our constraints are the maximum percentage that Alice may take from each asset. We represent the risk of Husband’s assets as a combined constraint:

${\dfrac {0.1x_{1}+0.2x_{2}}{x_{1}+x_{2}}}<0.2x_{1}+x_{2}$

$-0.1x_{1}\leq 0$

Results

This problem has a maximum value of 56K at x₁ = 0.3, x₂ = 0, x₃ = 0.2. If we include the 12K Husband must pay in child support, Alice would get 68K by taking 30% of his salary and 20% of his savings.

The Dual Problem

The dual problem represents the risk associated with each asset. The shadow prices reflect a change in the optimal value (we would have to increase by partial units as 1=100% in our model).

${\text{minimize}}\;w=0.3y_{1}+0.2y_{2}+0.2y_{3}+0.5y_{4}$
${\begin{aligned}{\text{s.t.}}\;&y_{1}+y_{4}-0.1y_{5}\geq 100\\&y_{2}+y_{4}\geq 20\\&y_{3}+y_{4}\geq 130\\&y_{1},y_{2},y_{3},y_{4},y_{5}\geq 0\\\end{aligned}}$

Results

This problem has a minimum value of 56K at y₁ = 80, y₂ = 0, y₃ = 110, y₄ = 20, and y₅ = 0.

Actual Portfolio

The actual portfolio minimizes risk while maximizing the settlement of assets. This portfolio has a 10% risk associated with Husband's income and 0% risk associated with Husband's savings. Therefore, including risk, we see a minimal value of: <br\> $(30\times 0.90)+(20\times 1)+12=65K$ <br\> and a maximal value of: <br\> $(30\times 1.10)+(20\times 1)+12=71K$.<br\> <br\> The cash settlement has a total value of 67K.

Results

So we see that the most Alice can receive from Husband’s assets from the second option is 71K total. But if we associate the risk, then the least amount of money she can get is 65K. If Alice wants to play it safe, she can opt for the cash settlement of 67K total.

In our model, we see that if Alice were to take a division of assets, she would have ⅔ probability that it would be greater than the lump sum, assuming that the risk has uniform distribution. As the risk increases, there probability decreases (approaching 50%) that she would obtain a payout greater than the lump sum but this payout would be much larger. This also applies that, as the risk increases, the probability increases (approaching 50%) that she will obtain a payout lesser than the lump sum with the payout being much smaller.

When this problem is modelled linearly, we find it easy to form and work out even as it is not an integer valued problem. The simplex method provides an optimal basic solution. But when it is modelled quadratically, we receive a complementary solution. If we had more assets and more complicated values, a quadratic solution would be ideal. Thus, it is better to view Alice’s divorce settlement as a Linear Programming problem in this specific model.