       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 -> 0.246238, x -> 0.0909659,
x -> 0.214026, x -> 0., x -> 0.44877}}

```

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