The portfolio optimization by the

Markowitz model was performed by the Solver function of the Excel software, based on the long-term interest rate of the following countries: the United States, Japan, Germany, France, Australia, Canada, and Great Britain.

A third reason we have favored small portfolios is that some of the modifications to the original

Markowitz model create more problems than they seem to solve, for instance, the presence of a risk-free rate that in reality is stochastic or customary inclusion of textbook (naive) short sales that violate Federal Reserve regulations and make little economic sense.

The

Markowitz model assumes that investors would like to maximize return under a certain risk level or minimize the risk with a certain return level [6] and this model makes use of the mean and variance of normalized historical asset prices to compute the expected portfolio return and risk [24], respectively.

There are some major limitations for the application of

Markowitz model to the real world.

Determining the optimized portfolio based on

Markowitz model has a lot of complications including the volume of the calculation and the number of variables so that in a market with N investors, [n.sup.2] + 3n + 2/2 variable should be calculated.

However, the impact of adding CVaR constraint to the

Markowitz model on the portfolios with low benefit payment ratio is not as significant as that on the relatively high benefit payment ratio portfolios.

In an analogy with the

Markowitz model, where investors minimize portfolio variance for a given level of expected return, I consider a regional economic activity characterized by some return (its overall value added growth rate), uncertainty (sector growth rate variability), and portfolio structure (sector composition of economic activity).

Markowitz (1952) proposed the mean-variance (MV) or

Markowitz model by using variance as the measure of risk while mean return as the expected return.

Sharpe (1964), Lintner (1965), Mossin (1966) and others have utilized the choice-theoretic structure of the

Markowitz model as the basis for a positive theory of equilibrium capital asset pricing under conditions of uncertainty.

The

Markowitz model is a relatively simple nonlinear programming model designed to maximize wealth over a single time horizon.

The Sharpe Model for an individual security can be employed to construct a model alternative to the

Markowitz Model of portfolio.

The

Markowitz model assumes that for any given rate of growth there is a minimum level of volatility, and for any given level of volatility, there is a maximum rate of growth.