# Math Help - Linear Programming

1. ## Linear Programming

Hello,
I have a solution to the following problem, but I know its not correct.
You work for a company who has 1 million dollars and they need your advice about which of four bonds should they invest the money in. The company wants to maximize the expected return from the bond investments. The information for the four bonds is as follows:

Bond 1
Expected Return: 13%
Worst Case Return: 6%
Duration: 3

Bond 2
Expected Return: 8%
Worst Case Return: 8%
Duration: 4

Bond 3
Expected Return: 12%
Worst Case Return: 10%
Duration: 7

Bond 4
Expected Return: 14%
Worst Case Return: 9%
Duration: 9

(duration of a bond is its sensitivity to interest rates).

The following constraints must be observed:
(a) The worst-case return return of the portfolio must be at least 8%
(b) The weighted average duration of the portfolio must be at most 6.
(c) $400,000 is the maximum investment in any single bond. How much should they invest in each bond to maximize the expected return subject to the constraints? Here is all I have so far: Let $X_i$ be the amount invested in bond $i$. Let $E_i$ be the expected return of Bond $i$. Let $W_i$ be the worst case return of Bond $i$. Let $D_i$ be the duration of bond $i$ We have n = 4 bonds. Also, I have $\sum_i^n (D_i - 6) x_i \leq 0$ as constraint (a) $\sum_i^n (W_i - 8) x_i \geq 0$ as constraint (b) I'm having trouble defining mathematically constraint (c), and finding the objective function. Can anyone please help? I'm guessing the objective function to be: [Math]Maximize \sum_i^n(1000000 - (x_i E_i))[/tex] 2. I've had an attempt and I'm not confident with the solution: Bond 1 -> 230769.23 Bond 2 -> 230769.23 Bond 3 -> 230769.23 Bond 4 -> 307692.30 Is this wrong? 3. Originally Posted by Maccaman Hello, I have a solution to the following problem, but I know its not correct. You work for a company who has 1 million dollars and they need your advice about which of four bonds should they invest the money in. The company wants to maximize the expected return from the bond investments. The information for the four bonds is as follows: Bond 1 Expected Return: 13% Worst Case Return: 6% Duration: 3 Bond 2 Expected Return: 8% Worst Case Return: 8% Duration: 4 Bond 3 Expected Return: 12% Worst Case Return: 10% Duration: 7 Bond 4 Expected Return: 14% Worst Case Return: 9% Duration: 9 (duration of a bond is its sensitivity to interest rates). The following constraints must be observed: (a) The worst-case return return of the portfolio must be at least 8% (b) The weighted average duration of the portfolio must be at most 6. (c)$400,000 is the maximum investment in any single bond.

How much should they invest in each bond to maximize the expected return subject to the constraints?

Here is all I have so far:

Let $X_i$ be the amount invested in bond $i$.
Let $E_i$ be the expected return of Bond $i$.
Let $W_i$ be the worst case return of Bond $i$.
Let $D_i$ be the duration of bond $i$

We have n = 4 bonds.

Also, I have

$\sum_i^n (D_i - 6) x_i \leq 0$ as constraint (a)

$\sum_i^n (W_i - 8) x_i \geq 0$ as constraint (b)

I'm guessing the objective function to be:

$Maximize \sum_i^n(1000000 - (x_i E_i))$
c) $X_i\le 400000,\ i=1,2,3,4$

The objective is:

$Ob=\sum_{i=1}^4 \frac{E_i}{100} X_i$

or as a percentage:

$Ob^*=100 \times \frac{\sum_{i=1}^4 \frac{E_i}{100} X_i}{\sum_{i=1}^4 X_i}$

I expect the first of these is what is really required.

CB