A Reinterpretation of the Optimal Demand for Risky Assets in Fund Separation Theorems

In a continuous-time portfolio selection model with N risky assets and K state variables driving their risk and return parameters, we derive simple expressions for the allocation to each asset in the K + 1 risky funds of the (K + 2)-fund separation theorem. We show that the allocation to any given risky asset in each fund can be written in terms of the parameters of a regression of the excess returns of this asset on those of the N - 1 remaining assets. We also use these parameters to provide quantitative measures of the increase in Sharpe ratio of the speculative demand, or in the maximum correlation of each hedging demand with respect to the corresponding risk factor, associated with the introduction of a new asset in the investment universe. Finally, we show that in a multiperiod setting, an asset is "spanned" by others if and only if it improves neither the maximum Sharpe ratio of the speculative demand nor the maximum correlations of the hedging demands with the risk factors.