A Reinterpretation of the Optimal Demand for Risky Assets in Fund Separation Theorems - EDHEC-Risk Institute research article in Management Science

Written on 12 November 2018.


We are pleased to enclose an EDHEC-Risk Institute research article published in the September 2018 issue of Management Science entitled "A Reinterpretation of the Optimal Demand for Risky Assets in Fund Separation Theorems". In this article, authors Romain Deguest, Lionel Martellini and Vincent Milhau introduce new expressions for the optimal portfolio in a multi-period model, and they propose a criterion to assess whether the introduction of a new asset in the investment universe adds to an investor's welfare.

Dynamic portfolio optimisation models have been developed to find optimal investment strategies for investors who have the possibility to revise their asset allocation over time. They provide optimal investment rules that explain how the percentage allocation to each asset should depend on market conditions at each future date, these conditions being summarised in the level of interest rates, the risk premia of constituents (e.g. the equity and the bond risk premia), as well as their volatilities and correlations. In a series of pioneering articles published between 1969 and 1973, the economist Robert Merton, who was awarded the Nobel Memorial Prize in Economic Sciences in 1997 with Myron Scholes, showed that at each point in time, the optimal portfolio is an overlay of different portfolios that have well-defined roles. The first is the “myopic portfolio”, which maximises the Sharpe ratio at a short horizon instead of the actual, possibly remote, investment horizon. Then come a series of “hedging portfolios”, which are specially designed to have good performance when market conditions deteriorate (and conversely display low returns when conditions improve). This result is known as the “fund separation theorem”.

The intuition for the presence of the hedging demands is as follows: bad market conditions mean poor risk and return perspectives for investors, so they have negative impact on their welfare, but this effect can be offset if the portfolio displays higher returns. It is the purpose of the hedging portfolios to deliver these higher returns in the “bad scenarios”. For instance, lower interest rates mean lower expected returns on portfolios, but by purchasing long-term bonds, the prices of which increase when interest rates decrease, an investor can hedge against this event. The hedging portfolio associated with the interest rate level will thus be mostly invested in long-term bonds. More generally, there is one hedging portfolio for each risk factor behind market conditions.

An asset, whether it is stocks, bonds, a sub-class of these or an alternative class, can be part of all building blocks, so the total demand for this asset is the sum of a myopic demand and a series of hedging demands. The first contribution of our paper is to provide a new way to compute the weight of an asset in each elementary portfolio, which turns out to clarify the economic intuition behind the composition of the portfolios: an asset is included in a portfolio if, and only if, this improves the criterion optimised by the portfolio. Thus, it is part of the myopic portfolio if it leads to a higher Sharpe ratio, and it enters a hedging portfolio if it contributes to a better correlation with the risk factor to hedge. These rules can be expressed mathematically with the help of regression parameters, which convey easier economic interpretation than the matrix formulas traditionally used to calculate portfolio weights. The second contribution of our paper is to find “irrelevance conditions” for a new asset, that is conditions under which the introduction of this asset in the investment universe does not improve welfare. This question has been asked in the literature to study the benefits of diversifying investment equities across countries, or those of adding alternative asset classes to a stock-bond mix, but it has always been studied with “mean-variance spanning tests”, which test whether or not the new asset is useful in the myopic portfolio. In a dynamic context, an asset can be irrelevant in the myopic portfolio, but still be useful in the hedging portfolios. Thus, an asset does not contribute to welfare if it does not enter any of the building blocks of the fund separation theorem.

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