The most important properties of risk measures are assumed as axioms, and the one responsible for the effect of diversification is, essentially, convexity – which makes the risk of a portfolio of securities smaller than the weighted average of the risks of its components on a stand-alone basis. Deeper results in stochastic ordering theory (see, for example, Rüschendorf (2010)) indicate that diversification opportunities are determined by the way security prices depend on one another. There exists a dependence structure, such that the risk of any portfolio equals the weighted average of the stand-alone risks for any reasonable risk measure. The worst possible dependence structure implies that security prices depend (non-linearly) on only one stochastic factor, which is a realistic hypothesis in severe market downturns. Under such conditions, diversification opportunities do not exist. From the standpoint of nonclassical models, the function of the risk measure is to translate diversification opportunities, assuming they exist, into actual allocations. Alternative frameworks have the potential to extract risk premia more efficiently than meanvariance by zooming in on the downside of the portfolio return distribution. However, as far as loss protection goes, they are just as futile.

Hedging risk Tobin (1958) showed that in the presence of a risk-free asset, efficient portfolios consist of two funds: the risk-free asset and a fund of risky assets, which is efficient in isolation and provides the highest risk-adjusted expected excess return – in other words, the risky portfolio with the highest Sharpe ratio. An investor’s risk aversion influences only the relative weight of the two funds in the portfolio. The implication for portfolio construction is that lower-risk portfolios are best obtained by increasing the allocation to the risk-free asset at the detriment of the risky portfolio with the highest Sharpe ratio, rather than by selecting a risky portfolio with lower risk but an inferior Sharpe ratio. Presence of liabilities, however, implies that the risk-free asset is not cash but a portfolio designed to hedge liability risks.

The theory of optimal asset/liability management (ALM) indicates that efficient capital allocation also involves a two-fund separation – a performance-seeking portfolio (PSP) constructed through diversification and a liability-hedging portfolio (LHP) used to deal with the variability of the stream of liabilities arising from different sources, mainly interest rates and inflation. LHPs can be designed in various ways. Cash-flow matching is a popular technique which, through investments in suitable bonds, attempts to ensure a perfect static match between the cash flows from the portfolio of assets and the commitments in the liabilities. Although the technique is simple, a big problem is finding the right bonds. The problem with bond availability exists also in the case of hedging interest rate risk through duration matching. Approaches to hedging inflation include investing in instruments such as Treasury inflation-protected securities (TIPS) or derivatives such as inflation swaps. Other asset classes may also have inflation hedging properties. Empirical studies indicate, for example, that stocks, commodities, and real estate can offer protection from inflation at lower cost.