Converting a Portfolio Beta Constraint into Linear Inequalities

# optimisation problem with linear constraint


# optimisation problem with linear constraint












I have an optimisation problem.

I wish to maximise a function subject to a constraint. It is the constraint that is causing me problems. I am using an addin in Matlab which does the optimisation however the constraints that I have used before have been in the format of the line below.

```
 b_l <= Ax <= b_u
```

The constraint is,

```
 Sum(x .* stock)*BetaBM - 0.1 <= Sum(x .* stock.*BetaSK) <= Sum(x .*stock)*BetaBM + 0.1
```

where,

```
 x is 2000 by 1 vector
 stock is 2000 by 1 vector
 BetaBM is a scalar
 BetaSK is 2000 by 1 vector

 x - is the weight of each stock in the fund. It cannot be more than 100% but can be less.
 stock - I am looking at M&A deals. The stock variable is a number between 0 and 1 which represents how much of the deal is being paid for in the acquires stock. 0 would mean the deal is purely cash. If there is part of the deal being paid in stock I will hedge the beta exposure against the S&P Index.

 BetaBM - is the S&P beta.
 BetaSK - contains all the individual beta for all the stocks in the fund
```

I need to get the constraint in the format b_l <= Ax <= b_u if at all possible?

## Answer by Richi Wa (score 2, accepted)

https://quant.stackexchange.com/a/14891

If you have a vector of weights $w=(w_1,\ldots,w_n)^T$ then $(1,\ldots,1)* w = \sum_{i=1}^n w_i$ thus a sum condtion can be formulated by multiplication with a row of ones. A $\le$ can be put into an $\ge$ by multiplying with $(-1)$ and if you have to put all your constraints into on $A$ then you usually stack all the row vectors together. In your case the matrix $A$ will consist of rows of ones. As long as you don't explain your constraint I can not add more.

I will try: If you want to put a constraint on your porfolio beta then you should use a constraint: $$(\beta_1,\ldots,\beta_n) * w \le \beta_{BM}+0.1$$ and $$(\beta_1,\ldots,\beta_n)*w \ge \beta_{BM}-0.1$$ which the same as $$ -(\beta_1,\ldots,\beta_n)*w \le 0.1-\beta_{BM}$$. So you could use a matrix $$A = (\beta_1,\ldots,\beta_n;-\beta_1,\ldots,-\beta_n)$$ and right hand side $b = (\beta_{BM}+0.1;0.1-\beta_{BM})$ then your condition is: $$ A w \le b. $$

EDIT after a remark by PO: If you have $$ (\sum_{i=1}^n w_i stock_i) \beta_{BM} -0.1 \le \sum_{i=1}^n w_i stock_i \beta_i $$ then this is equivalent to $$ \sum_{i=1}^n w_i stock_i (\beta_{BM} - \beta_i) \le 0.1 $$ then you define scalars $k_i = stock_i (\beta_{BM} - \beta_i)$ and the same as above holds for $\sum_{i=1}^n w_i k_i \le 0.1$. For the right hand side you get $$ \sum_{i=1}^n w_i (-k_i) \le 0.1. $$ Your matrix $A$ has two rows, one $(k_1,\ldots,k_n)$ and one $(-k_1,\ldots,-k_n)$ the rhs is $(0.1, 0.1)$.