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Thread: Linear Programming

  1. #1
    Member Maccaman's Avatar
    Joined
    Sep 2008
    Posts
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    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 $\displaystyle X_i$ be the amount invested in bond $\displaystyle i$.
    Let $\displaystyle E_i$ be the expected return of Bond $\displaystyle i$.
    Let $\displaystyle W_i$ be the worst case return of Bond $\displaystyle i$.
    Let $\displaystyle D_i$ be the duration of bond $\displaystyle i$

    We have n = 4 bonds.

    Also, I have

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

    $\displaystyle \sum_i^n (W_i - 8) x_i \geq 0$ 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]
    Last edited by Maccaman; Mar 30th 2010 at 02:32 AM.
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  2. #2
    Member Maccaman's Avatar
    Joined
    Sep 2008
    Posts
    85
    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?
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by Maccaman View Post
    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 $\displaystyle X_i$ be the amount invested in bond $\displaystyle i$.
    Let $\displaystyle E_i$ be the expected return of Bond $\displaystyle i$.
    Let $\displaystyle W_i$ be the worst case return of Bond $\displaystyle i$.
    Let $\displaystyle D_i$ be the duration of bond $\displaystyle i$

    We have n = 4 bonds.

    Also, I have

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

    $\displaystyle \sum_i^n (W_i - 8) x_i \geq 0$ 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:

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

    The objective is:

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

    or as a percentage:

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

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

    CB
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