### Picking an investor’s preferred portfolio from the points on the efficient frontier

One’s preferred level of risk affects how one invests. Different levels of your risk coefficient will create different ideal portfolios as illustrated in the figure below. This post describes the math behind the different allocations for different risk preference levels.

The classic Markowitz optimization creates a set of portfolios called the efficient frontier which returns the maximum expected return for a given volatility. Determining the best portfolio from this set for a specific investor requires additional steps. The first step is an assessment of the investor’s aversion to risk.

An investor’s level of risk aversion can be thought of as an adjustment to the expected return of an investment. Risk neutral investors evaluate each investment by the expected return of the investment. More risk averse investors evaluate the investment by subtracting a penalty for more volatile investments.

Using the Arrow-Pratt model of constant absolute risk aversion (CARA), each investor’s risk aversion is summarized by a single coefficient, A, which maps any investment with an uncertain return (represented by the volatility, σ and the expected return of μ) to a zero-volatility investment returning

μ– ½ * A σ^{2}

Elsewhere, (Meziani & Noma, submitted for publication) we talk about how to estimate an investor’s risk aversion.

Once the investor’s risk aversion has been assessed, one can determine the investor’s preferred portfolio. If the investor chooses not to employ leverage and to be fully invested, then the points on the efficient frontier make up the candidate portfolios. In that case, the investor’s utility is calculated (using the investor’s risk coefficient, A) for each point on the efficient frontier and the portfolio with the highest utility is the investor’s preferred investment portfolio.

The following is the efficient frontier created (using daily returns from 2008 to 2017) by investing in 20 listed equities that are frequently held by institutional investors: GOOG, MSFT, AAPL, JNJ, JPM, RDS.A, CSCO, WFC, GILD, PFE, VZ, CVS, V, FB, RHHBY, CMCSA, HD, PEP, MRK, XOM. The efficient frontier is shown in figure 1 along with portfolios preferred for various levels of risk aversion.

If the investor can employ leverage, one creates a capital market line based on levering or delevering the portfolio (on the efficient frontier) with the highest Sharpe ratio. The capital market line contains the risk-free rate where the volatility is zero and the highest Sharpe portfolio on the efficient frontier. On this line, the point of maximum utility is the point that maximizes the CARA utility function (with risk aversion, A). This is the solution to

Max_{(σ, μ)} μ– ½ * A σ^{2} such that R = β * σ + α

Using a Lagrange multiplier, the investor’s preferred portfolio has a volatility of σ = β / A.

If the investor is risk neutral (A = 0) then the investor should seek as much leverage as is possible. By contrast, a totally risk aversion investor, would investment only in the risk-free instrument. Note that the risk-free rate of return (α) does not appear in the calculation of the volatility of the most-preferred portfolio, but is used in determining beta when calculating the Sharpe ratio for each point on the efficient frontier.

Figure 2 shows the same set of the investments, with the investor’s preferred portfolio along the capital market line.

The leverage ratio is calculated by dividing the volatility of the preferred portfolio by the volatility of the maximum beta portfolio on the efficient frontier. Figure 3 shows relationship between the investor’s A coefficient and the amount of leverage in the preferred investment. A leverage of one indicates that the entire amount is invested in the maximum beta portfolio on the efficient frontier. In this case, this occurs when A = 11.84.

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