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

be the amount invested in bond

.

Let

be the expected return of Bond

.

Let

be the worst case return of Bond

.

Let

be the duration of bond

We have n = 4 bonds.

Also, I have

as constraint (a)

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: