30 March 2013

I do not pretend that I understand all aspects of risk. But being a stock investor who learn some good stuff at SA, I feel that I should share my thoughts about this important subject.

We know that stock investors face many flavors of risk and there are endless debates about this term in literature.

There are many aspects of risk which can be mostly reduced to the absolute risk /probability to loose money (invested capital)/ and the relative risk /probability to loose purchasing power of money/.

"A good starting point [in the measurement of investment risk] is the preservation and enhancement of your purchasing power in real terms." - David Dreman

IMO each stock investor should try to decide about coherency of his/her absolute risk.

A coherent risk measure R = R(NYSE:X) is satisﬁed 4 postulated conditions such as cash equivariance, monotonicity, positive homogeneity, and subadditivity [4]. The last one **R(X + Y) does not exceed R(X) + R(NYSE:Y)** (or using a bit uncommon symbols R(X+Y)</= R(X)+ R(Y), where R is risk, X and Y are positions) directly related with diversification. If your principal risk is subadditive you should finish up with infinitely large portfolio (assuming no economical limitations related to portfolio expansion). If your principal risk is additive (i.e., **R(X + Y) =** **R(X) + R(Y)** by definition) you can do a job with one or very few well-selected positions (each to compensate different risk factors).

Of course MPT proponents like the coherent risk measure which axiomatic seems invented to satisfy mainstream of so-called finance science. One caveat of the coherent risk is a stock liquidity. Due to the positive homohenity /i.e. R(A*X) = A*R(X) for any A > 0/ and the subadditivity of the coherent risk this measure becomes a non-sense for any position X with limited liquidity. For example doubling of weight of obscure company's stock R(X) ===> R(X+X)=R(2X) in portfolio reduced it risk because R (X+X) </= 2*R(X) by the subadditivity definition.

Of course stock pickers like non-coherent risk measure such as volatility or VAR. One caveat of such measure is extreme losses at negative event with stock issuer (i.e. company) and "too often" black swans appearance [2] because of position invariance **R(X) + R(Y)** = **R(Y) + R(NASDAQ:Z)**, **X-Z < S** where S is a small number (i.e. positions X and Z are similar but not equal).

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Simple equal-weighed diversification helps to spread the absolute risk to some extend, and proper handling of coherent risk reduce the absolute risk. Compounding (at firm level or at investor level) helps to deal with the relative risk.

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DGI is a form of the relative risk reduction, stocks (actually firms) selection is a form of the absolute risk reduction. Fortunately many DG firms are suitable for diversified portfolio which copes with both risks

<to be continued>

References:

[1] Artzner, P., Delbaen, F., Eber, J.-M., Heath, D., 1999. Coherent measures of risk. Mathematical Finance, Vol. 9, p. 203-228.

[2] Taleb, N. N., 2010, Common Errors in the Interpretation of the Ideas of The Black Swan and Associated Papers,Critical Review, Vol. 21, No 4