# Math Help - Diversification benefit

1. ## Diversification benefit

Hello,

I am working in the insurance industry. One of the problems we have is how to work out how various risk diversify with one another. One approach used is to hold a certain amount of capital with respect to each risk the company faces. Then work out a correlation factor between each risk.

For example, an insurance company might have three risks; say, Equity risk, interest rate risk, mortality. The company might think it needs 100, 50, 20 for each risk individually.

We take a matrix A = (x1, x2, x3) = (100, 50, 20) in this case.

The correlation matrix between each risk might look like:

1 0.5 0
0.5 1 0
0 0 1

We call this matrix B. We can calculate the amount of capital the insurer holder allowing for diversification by doing:

Diversified capital = Sqrt(Transpose(A) * B * A )

One question I had is there must be an optimal level for each x that minimises the diversified capital relative to the sum of the x's (sum of x's in example = 100+50+20). [In practice this would give the optimal balance of risks the insurer should take to get the most diversification benefit]. There is a numerical answer using solver in excel, but I wanted a closed form solution.

I had a go at this and tried differentiating the diversified capital / sum of x's and set this equal to zero (as a minimum). This works and gives a closed form answer. However, we have a constraint that is that the minimum for any of the x's is zero. Again this can be solved using solver in Excel. However is there a closed solution allowing for the constraint that the x's must be bigger than zero?

I was thinking the likely way to look at this is quadratic programming; in which case using solver in excel is actually the best way to do it.

Any help would be most appreciated!

2. Originally Posted by sharpe
Hello,

I am working in the insurance industry. One of the problems we have is how to work out how various risk diversify with one another. One approach used is to hold a certain amount of capital with respect to each risk the company faces. Then work out a correlation factor between each risk.

For example, an insurance company might have three risks; say, Equity risk, interest rate risk, mortality. The company might think it needs 100, 50, 20 for each risk individually.

We take a matrix A = (x1, x2, x3) = (100, 50, 20) in this case.

The correlation matrix between each risk might look like:

1 0.5 0
0.5 1 0
0 0 1

We call this matrix B. We can calculate the amount of capital the insurer holder allowing for diversification by doing:

Diversified capital = Sqrt(Transpose(A) * B * A )

One question I had is there must be an optimal level for each x that minimises the diversified capital relative to the sum of the x's (sum of x's in example = 100+50+20). [In practice this would give the optimal balance of risks the insurer should take to get the most diversification benefit]. There is a numerical answer using solver in excel, but I wanted a closed form solution.

I had a go at this and tried differentiating the diversified capital / sum of x's and set this equal to zero (as a minimum). This works and gives a closed form answer. However, we have a constraint that is that the minimum for any of the x's is zero. Again this can be solved using solver in Excel. However is there a closed solution allowing for the constraint that the x's must be bigger than zero?

I was thinking the likely way to look at this is quadratic programming; in which case using solver in excel is actually the best way to do it.

Any help would be most appreciated!
A function either has a calculus like minimum within the feasible region (which from what you say yours does not), or it has a minimum on the boundary. So try the minimisation under the assumption that $x_1=0$, $x_2=0$, $x_3=0$, ( $x_1=0$ and $x_2=0$), ( $x_1=0$ and $x_3=0$), .... or ( $x_1=0$ and $x_2=0$ and $x_3=0$) then keep the minimum.

CB

With no constraints there is a calculus minimum.

If you add in a constraint x's >= 0 is it possible there is a solution with all the x's > 0?

I did try lagrange multipliers - are they useful when the constraint is equal to; rather than greater than or equal to?

4. yes, lagrange multipliers work when the constraint is "equal to"

5. Originally Posted by sharpe

With no constraints there is a calculus minimum.

If you add in a constraint x's >= 0 is it possible there is a solution with all the x's > 0?

I did try lagrange multipliers - are they useful when the constraint is equal to; rather than greater than or equal to?
For inequality constraints you convert them to equality constraints by introducing slack variables.

CB

6. Originally Posted by sharpe