Markowitz with a Risk-Free Asset

Abstract

In this chapter, a risk-free asset is added to the set of investable securities and the optimal portfolios are now derived in this augmented economy. One of the results obtained is that, in the risk/return analysis, the parameters (σ, m) of any investment portfolio lay either on or inside a certain cone $\mathcal {C}(\sigma , m)$. The upper side of the cone represents the efficient investment portfolios and is called the Capital Market Line . We show that the portfolios on the Capital Market Line can be built as an allocation between the risk-free asset and a particular investment portfolio, made of risky assets only, called the Tangent Portfolio . The Tangent Portfolio appears to define the tangent point between the cone $\mathcal {C}(\sigma , m)$ and the hyperbola $\mathcal {F}(\sigma , m)$ delimiting the (σ, m) of all investment portfolios made of risky assets only. Based on some economic reasoning, the Tangent Portfolio is sometimes assimilated to the Market Portfolio, for which the investment in each asset is proportional to its relative market capitalisation. We also show in this chapter that the problem of optimal allocation can be segmented into two steps. First, the investor decides on the risk exposure he is ready to take, secondly, he calculates the allocation to the Tangent Portfolio which gives him this level of risk (the rest of the money being invested in the risk-free asset). This paradigm for investing is known as the Separation Theorem of James Tobin (Nobel Prize in Economics in 1981, see Tobin) and shows that whatever the risk appetite is, there is only one way to take risk exposure efficiently.