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Math Help - Diversification benefit

  1. #1
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    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!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by sharpe View Post
    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.

    Alternatively Google for "Lagrange multipliers"

    CB
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  3. #3
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    Thanks for the reply - that is helpful

    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?
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  4. #4
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    yes, lagrange multipliers work when the constraint is "equal to"
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by sharpe View Post
    Thanks for the reply - that is helpful

    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
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  6. #6
    Grand Panjandrum
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    Quote Originally Posted by sharpe View Post
    Thanks for the reply - that is helpful

    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?
    Have you tried the other suggestion of minimising with one or more of the variables set to zero and comparing the resulting minima?

    CB
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  7. #7
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    Thank you - I will try this. I think it is likely the solution, as the diversified benefit formula is a quadratic for each x. Will let you know when I have something. Thanks again
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