I'm looking for the solution to a practical maths problem that is way beyond my level of competence

Graeme_Sm
Graeme_Sm used Ask the Experts™
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I'm not sure if this is the appropriate site to pose this, and if not I'm open to suggestions of sites where I might more appropriately directed it (if that is not against the EE terms-of-use).

The problem below I thought at first this would be a fairly simple, but it turns out to be a lot more complicated than I first anticipated.  I was good at maths at school, but that was 20 years ago and after attempting this I've realized it is beyond my scope..

I have 3 known inputs, LongStdDev, ShortStdev and MarketStdev.  And I have two targeted inputs, PortfolioBeta and PortfolioStdev.

I only have two variables, LongWeight and ShortWeight.

PortfolioBeta = LongWeight - ShortWeight
PortfolioStdev = Sqrt((LongWeight*LongStdev)^2 + (ShortWeight*ShortStdev)^2 + (abs(LongWeight-ShortWeight)*MarketStdev)^2)

From these two formulas and five inputs is it possible to calculate the LongWeight (and by Inference the ShortWeight)?  Trying to use what I remember from high school maths I hit a wall when I try to substitute {PortfolioBeta-LongWeight} for {ShortWeight} and then square it.
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Dave BaldwinFixer of Problems
Most Valuable Expert 2014

Commented:
Since your first equation has two unknowns in it, the only way to solve the system of equations is by substitution.  A spreadsheet could make the calculations go quickly.  I'm not sure that there is only a single answer though.

Author

Commented:
It is easy enough to get it down to one unknown since LongWeight = ShortWeight + PortfolioBeta.  However I'm not sure how to move that one variable to the left hand side of the equation because it ends up in a quadratic formula (a + bX + cX^2)
Dave BaldwinFixer of Problems
Most Valuable Expert 2014

Commented:
No, LongWeight = ShortWeight + PortfolioBeta is still 2 unknowns.  Actually, it's 3 because you don't have a value for any of them.  There is simply not enough info for a numerical solution.
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Author

Commented:
PortfolioBeta is a known, the five inputs are all known.  So once either the Long or Short Weight is calculated the other is known.  

The formula can be substituted to :

PortfolioStdev^2 = (LongWeight*LongStdev)^2 + ((LongWeight-PortfolioBeta)*ShortStdev)^2 + (abs(PortfolioBeta)*MarketStdev)^2

so that LongWeight is the only unknown variable.  However this expands into a quadratic equation so I don't know how to move LongWeight to the left hand side.
Dave BaldwinFixer of Problems
Most Valuable Expert 2014

Commented:
I can't help you then.  Click on "Request Attention" above to get others to look at your question.  Since it is Friday evening, it may take a while.
A quadratic

aX²  + bX + c = 0

has solutions

x = ( -b ± sqrt( b² - 4ac) )/2a

This has 2 solutions. My guess for your problem one of them won't make physical sense and as such can be discarded.

Author

Commented:
Thanks Gwyn,

That was exactly what I was looking for.

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