**Benefits of diversification**

From the Figure 01, we can see that the average risk of 1-stock portfolio is the largest (10.54%) and the risk of equally weighted 30-stock portfolio is the lowest (6.10%). These results reflect the concept of portfolio diversification: the risk of more diversified portfolio tends to be lower than the risks of less diversified portfolio. In a large diversified portfolio, the risk may reach to the average risk of a portfolio with a very large stock. As we can see from the shape of the curve, the risk of portfolio tends to decrease if the number of stock in portfolio increases. In other words, the more diversified portfolio has systematic risk (covariance) and has less unsystematic risk from individual stocks. However, when the number of stocks in portfolio is large enough, adding a more stock does not have a significant effect in reducing portfolio risk.

**Portfolio Performance**

The Figure 02 is a plot of risks and returns of equally weighted portfolios created in Part A and 8 30-stock portfolios of the weights WMVP1U, WMVP1C, WMVP2U WMVP2C, WOpt1U, WOpt1C, WOpt2U and WOpt2C in the 2^{nd} half. From the graph that we can draw a line through the numbers of points plotted to present the relationship between risk and average return. The points above the line represent portfolio which have higher return than the points below the line given a risk level.

**Choosing portfolios according to the objective**

In this part, we will examine the choice of portfolio weights done by Solver on 30-stock portfolio for the 1^{st} half and 2^{nd} half data. The discussion will be based on the table of 30-stock portfolio.

For the purpose to minimize the risk of portfolio, we use Solver function to find the weights of a portfolio that has minimal risk or minimal variance. In the first half, in case of unconstraint, Solver tends to pick stocks of lowest average risks in a large portions to hold (e.g 10.61% of J:AJ@N(RI) with the lowest std. dev. of 6.89%, 25.39% of J:ASMR(RI) with the second lowest std. dev. 7.17%) and stocks with highest risks for short sales in a large portions (e.g -15.17% of J:IH@N(RI) of std. dev. 14.38%). In the 2^{nd} half, in case of unconstraint, Solver suggests the largest weight of 38.63% of J:AJ@N(RI) with the lowest std. dev. (4.92%) and the largest weight for short sales at -21.45% of J:NK@N(RI) with the highest std. dev. 17.07%. In case of constraint, in both 1^{st} and 2^{nd} half, Solver also picks stocks with the lowest risks in large portions but is does not pick any stocks with negative weights for short sales due to the constraint that all the weights must be positive. The negative weights in case of unconstraint become zero in case of constraint. The largest positive weights in case on unconstraint remain the largest weights in case of constraint because those weights have minimal risks. This suggests that largest weights chosen by Solver are consistent with the objective of minimizing portfolio risks and some stocks are weighted more heavily because they have lower risks among available stocks in the portfolio. However, Solver does not pick stocks by choosing from the lowest risk stock to the higher risk stocks. For light weights, Solver tends to choose stocks with higher returns.

For the purpose to maximize the sharpe ratio, we use Solver function to find the weights of a portfolio that has maximal sharpe ratio or the maximal return on portfolio. In the first half, in case of unconstraint, Solver tends to pick stocks of highest average return in the heaviest weights (e.g 103.06% of J:SG@N(RI) with the highest return 3.36%). In the 2^{nd} half, in case of unconstraint, Solver suggests the largest weight of 237.13% of J:HE@N(RI) with the highest return (2.69%). In case of constraint, in both 1^{st} and 2^{nd} half, Solver also picks the highest return stocks in large portions but is does not pick any stocks with negative weights for short sales due to the constraint that all the weights must be positive. The negative weights in case of unconstraint become zero in case of constraint. The largest positive weights in case on unconstraint remain the largest weights in case of constraint because those weights have minimal risks. This suggests that the largest weights chosen by Solver are consistent with the objective of maximizing portfolio returns and some stocks are weighted more heavily because they have higher returns among available stocks in the portfolio. However, Solver does not pick stocks by choosing from the highest return stock to the lower return stocks. For light weights, Solver tends to choose stocks with lower risks.

In case of unconstraint, portfolios are quite diversified and include all stocks to achieve the objectives. Solver tends to find the optimal solutions with a very large diversification to achieve the objective. Portfolios with minimal risks are more diversified than portfolios of maximal returns. In case of constraint, portfolios are also diversified but do not include all stocks. Portfolios with minimal risks are less diversified than portfolios of maximal returns.

**Performance and portfolio strategy**

Now we will discuss about the performance of 30-stock portfolio in the 2^{nd} half. The feasible portfolios are portfolios of weights WMVP1U, WMVP1C, WOpt1U, WOpt1C and the perfect foresight portfolios are those of weights WMVP2U WMVP2C, WOpt2U and WOpt2C.

According to the calculation results, the feasible portfolios have much poorer performances than the perfect foresight portfolios. In general, the feasible portfolios have much lower returns but higher risks than the perfect foresight portfolios. For example, average return of feasible portfolios is -0.41% vs. 3.26%; average std. dev. of feasible portfolios is 11.01% vs. 5.45%; average sharpe ratio of feasible portfolios is 2.39% vs. 42.88%. The largest sharpe ratio of the feasible portfolio WMVP1C in the 2^{nd} half is 11.53% which is smaller than all sharpe ratios of the foresight portfolios. This portfolio comes closest to the foresight portfolios because it has the highest sharpe ratio (or highest return) and the lowest risk among 4 feasible portfolios. In case of unconstraint, the best feasible portfolio is a feasible portfolio that comes very close to the WMVP2U foresight portfolio in terms of risk and return. However, for risk-averse investors, the best feasible portfolio can come near to WOpt2U portfolio because the return is highest among perfect foresight portfolios. In case of constraint, the best feasible portfolio is a feasible portfolio that comes very close to the WOpt2C perfect foresight portfolio.

Even though the data are from the last 10 years but we can see totally different effects of the 1^{st} half data and the 2^{nd} half data on the portfolios in the 2^{nd} half. If we use the weights of optimal portfolios in the 1^{st} half to evaluate the portfolio in the 2^{nd} half, we do not get the same results. Moreover, the results obtained seem to be very far away from the results obtained by using the last 5 years data. This may be due to the fact that the 1^{st} 5-year data, even if they are not so out-dated, do not reflect well the present market situation. When using such historical data we ignore recent changes in the market and therefore we might lose some opportunities. The performance of optimal portfolios in the part may be very far away from the perfect foresight portfolios. In fact, we can construct feasible portfolios by using past data but it is difficult to construct an optimal feasible portfolio where we can maximize returns and minimize risks. Even we can find such a portfolio, its performance is still far away from the perfect foresight portfolio

This suggests that we should not rely on diversification to match the performance of some market index to evaluate the performance of portfolios. We also should not assume that marketplace can reflect all available information in the market. Rather than relying on the simple diversification, we should take an active portfolio strategy, using available information and forecasting techniques to seek better performance and take advantages of the market such as finding mispriced securities.

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