Risk Return Relationship in the Portfolio Selection Models

In this paper, we calculate four different kinds of means—AM, GM, HM, and GDM—to investigate the risk-return contour using Markowitz risk minimization and Sharpe’s angle maximization models. For a given k value (target portfolio return), the rank order of risk or variance-covariance (υ) can change. In the vertical segment of an efficient frontier curve, we observed v(GDM) > v(HM) > v(GM) > v(AM). At higher k values, the rank changes to v(GDM) > v(HM) > v(AM) > v(GM). That is to say, ranking a portfolio using different kinds of means may well give different rankings depending on what k value one is evaluating. It is also shown the harmonic mean should not be used in the case of a small negative growth rate in stock prices.


Introduction and Literature Reviews
The foundation of modern investment theory is laid upon the quadratic program portfolio selection model developed more than half century ago by Harry Markowitz [1] [2] [3]. The optimization (risk-minimization) process over meanvariance-covariance space can trace out the efficient frontier curve, which provides the solution space for investors. However, an exact solution cannot be found without the knowledge of a risk free rate on a government bond and an investor's attitude toward risk. To this end, Sharpe [4] formulated and solved the angle-maximization model in which the risk (standard deviation) adjusted port-How to cite this paper: Hung, K., Yang folio return (net of risk free rate) was maximized. The Sharpe model provides a convex combination of risk free government bonds and a portfolio of stocks selected based on the criterion of risk minimization. Attitude toward risk such as 20% on bond and 80% on stocks will give investor an exact solution without the knowledge of the indifference (isoutility) curve. Soto and Su [5] proposed a "sparse" estimator of the inverse covariance matrix that achieves significant outof-sample risk reduction and improves certainty equivalent returns after transaction costs. Yang et al. [6] proved that Markowitz risk minimization and Sharpe angle-maximization models are mathematically equivalent given some required portfolio returns and risk free bond rate. Gil-Bazo [7] found the return dynamics is related to the investor's portfolio choice for different investment horizons and that return predictability under stationarity which may induce both positive and negative horizon effects in the optimal allocation to the risky asset. Best and Grauer [8] investigated and revealed when only a budget constraint is imposed on the investment problem, the analytical results indicate that an MV-efficient portfolio's weights, mean, and variance can be extremely sensitive to changes in asset means.
Does the choice of mean returns of stocks in the portfolio matter in the selection process? If so, how different are the optimum solution sets? In this note, we first apply the well-known means: arithmetic, geometric and harmonic means to five companies stocks. In addition, we add a golden mean to the simulation for comparison. The organization of the paper is as follows. Next section introduces the Markowitz risk minimization and Sharpe angle maximization models. Section III describes data and four different means. Section IV performs computer simulations via LINGO [9] to trace out corresponding efficient frontier curves.
Section V gives a conclusion.

Portfolio Selection Models with Different Means
Given a set of n selectable stocks, the purpose of the Markowitz portfolio model is to minimize the weighted risk in terms of variance and covariance of n stock returns, or where i x = the weight or proportion of investment in stock i;  , , , n x x x  ) for * 0 i x ≥ are the framework under which weighted risk υ can be calculated. Along with a set of k values, we have geometric means calculated according to the growth rate formula (next section) and a set of risk-return values on which an efficient frontier curve can be traced.
However, the exact location cannot be determined with a risk-free bond rate F R . To expand to risk minimization model, Sharpe [4] proposed an anglemaximizing model with a highest straight line from F R that is tangent to the efficient frontier derived from the Markowitz model.
As is shown by Yang et al. [6], a given F R corresponds to a k value in equation (2) of the Markowitz model. Furthermore, the denomination of Equation

Description of Data and Characteristics of Different Mean Returns
Monthly Note that when

A Comparison of Simulation Results
An efficient frontier generally consists of two parts: a vertical section and a concave part. The vertical part indicates Equation (2) holds with strict inequality (> k). That is the portfolio return at optimality exceeds the minimum required rate k. For k = 2% (annualized), which is obviously too low. The optimum portfolio return far exceeds k = 2% in the case of AM. If we arbitrarily impose Equation (2) with an equality sign (k = 2%), there will exist no feasible solution, for the lowest annual average rate of return is 7.317% (IBM), and as such the vertical sections of the efficient frontier curve starts to bend at k ≈ 20%, k = 19%, k = 18%, and k = 31% in the cases of AM, GM, HM and GDM respectively ( Table  3). A perusal of Table 3 indicates that for range of k < 18%, the portfolio risks in   Table 3). spectively. HM has shown smallest risk due to negative covariance. However, it must be pointed out that small negative returns are biased and misleading in the sense that the double reciprocity formula assigns too much weight to small negative return (slight price decrease in stock). The risk associated with AM for 34% k ≥ is slightly less than that of GM since GM is less sensitive to outliers than is AM, and as such AM has a slightly larger choice set (smaller risk) at high k value. The four efficient frontiers are shown in Figure 1.
For a given risk free rate F R , the greatest tangent angle attainable between the ray from F R and the efficient frontier curve can be calculated from Equation (5). By verifying F R we find the optimal tangent angle via LINGO and report the portfolio returns portfolio risk (υ) and tangent angles (tan θ) in Table   4. An inspection of

Concluding Remarks
In this paper, we calculate four different kinds of means-AM, GM, HM, and GDM-to investigate the risk-return contour using Markowitz risk minimization and Sharpe's angle-maximization models. For a given k value (target portfolio return), the rank order of risk or variance-covariance (υ) can change. In the vertical segment of an efficient frontier curve, we observed v(GDM) > v(HM) > v(GM) > v(AM). At higher k values, the rank changes to v (GDM) > v(HM) > v(AM) > v(GM). That is to say, ranking a portfolio using different kinds of means may well give different rankings depending on what k value one is evaluating. When a risk free-rate is added to the Markowitz model, we arrive at the angle-maximization solution. It seems that risk-return combinations under GDM dominate those under HM for the former has high returns and less risk. The same can be said of AM and GM. The combination using AM seems to have greater portfolio returns but less risk. When does these exists a trade-off, i.e., higher risk coupled with higher return, Sharpe's index (tanθ) favors both GDM and AM over HM and GM respectively for greater angle translates into high portfolio return (net of f R ) per risk. Care must be exercised though; the results from this paper are limited to the stocks that we take in the sample. However, comparative evaluations are needed for a comprehensive analysis on a portfolio performance. In sum, GM is less sensitive to outliers and hence is more suitable for conservative strategy. GDM is determined by the size of its range and tends to offer an optimistic forecast on mean return especially when there exist a few unusually large positive returns. Finally, HM is not appropriate if some of the returns (%) are small and negative. In that case, HM ought to be removed from the analysis.