Are independent risks substitutes according to the Generalized Sharpe Ratio?

Independent risks are substitutes if the opportunity to invest in one risk cuts down the demand in the others. Intuition seems to sustain this idea, but if the problem is tackled in a normative framework, no consensus in the literature is found. In this paper we investigate what does happen if the decision rule is based on the Generalized Sharpe Ratio. The answer depends on the risk measure under consideration. For the standard deviation, the standard semi-deviation and the Conditional Value-at-Risk (CVaR), the just mentioned intuitive idea is confirmed, but in general, this is no longer true in case of the Value-at-Risk (VaR). A related question is whether a sequence of independent risks which are individually unacceptable according to the Generalized Sharpe Ratio, can become acceptable when it is only long enough. In the context of expected utility theory this problem is known as “fallacy of large numbers”. The answer depends again on the underlying risk measure. We show for some common risk measures that any sequence of independent, identically distributed risks with positive expectations will eventually become acceptable if the the risk measure is non-degenerate and continuous in a certain sense.