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The last 2 calls must be changed from library(portfolio.r) library(portfolio_noshorts.r) to source(portfolio.r) source(portfolio_noshorts.r) The correct files must be availbale at http://faculty.washington.edu/ezivot/econ424/

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In: MODERN PORTFOLIO THEORY AND INVESTMENT ANALYSIS - ELTON, GRUBER, BROWN, GOETZMANN , chapter 5, you can find a good explanation of your issues.

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Your question is very important! In formal way to demonstrate it is very interesting ... but a bit complicated ... and boring for non mathematicians. We may move around this demonstration to explain most of portfolio theory. However, to give the idea, if we have N risky assets we obtain, as efficient frontier, a semi-parabola and the weights of the countless ...

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You can do 2 things: incremental risk: Calculate the volatility with the asset and with the asset replaced by cash. The difference gives you the (non-linear) incremental risk contribution of the asset. They don't sum up to $\sigma$. contributions to volatility (Euler allocation) As $\sigma = \sigma^2/\sigma$ you can define risk contributions by  ...

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