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Re: Using a Correlation Matrix to reduce risk
*To*: mathgroup at smc.vnet.net
*Subject*: [mg114397] Re: Using a Correlation Matrix to reduce risk
*From*: Ray Koopman <koopman at sfu.ca>
*Date*: Fri, 3 Dec 2010 05:20:30 -0500 (EST)
*References*: <id7t4n$l8c$1@smc.vnet.net>
On Dec 2, 2:41 am, Garapata <warsaw95... at mypacks.net> wrote:
> I have a problem for school and hoped someone could assist me.
>
> I have price data on 5 stocks and calculate a correlation matrix:
>
> cMatrix = Correlation[data]
>
> {{1.,0.635562,0.698852,0.404792,-0.32746},
> {0.635562,1.,0.410075,0.314375,-0.0636438},
> {0.698852,0.410075,1.,0.374416,-0.260137},
> {0.404792,0.314375,0.374416,1.,0.293135},
> {-0.32746,-0.0636438,-0.260137,0.293135,1.}}
>
> cMatrix //TableForm
>
> 1.000000 0.635562 0.698852 0.404792 -0.32746
> 0.635562 1.000000 0.410075 0.314375 -0.0636438
> 0.698852 0.410075 1.000000 0.374416 -0.260137
> 0.404792 0.314375 0.374416 1.000000 0.293135
> -0.32746 -0.0636438 -0.260137 0.293135 1.000000
>
> Now I want to construct a portfolio of the 5 stocks
> that minimizes its correlation or concentration risk.
>
> It's easy to understand this if for instance I had just 3 stocks,
> with 2 of them having correlations of 1 (100%) and the third at 0,
> I think it's matrix would look like this:
>
> {{1,1,0},{1,1,0},{0,0,1}}
>
> Than it would make sense to put 25% in each of the 2 correlated stocks
> and 50% in the uncorrelated one. This offsets the risk of
> concentrating in correlated instruments.
>
> But I can not think of how to use the correlation matrix to do this
> (especially for the 5 instruments). I keep searching for a solution
> on the internet and at the library but can not find a specific
> discussion on this.
>
> I hope someone can help or point me in the right direction.
>
> Thank you.
> G
You want to minimize the variance:
Clear[x]; p = Array[x, Length@cMatrix];
Minimize[{p.cMatrix.p, Tr@p == 1 && And@@Thread[p >= 0]}, p]
{0.306671, {x[1] -> 0.246238, x[2] -> 0.0909659,
x[3] -> 0.214026, x[4] -> 0., x[5] -> 0.44877}}
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