## 4 Numerical Experiments
### Numerical Analysis  
In the above experiment, we conduct in-sample test and out-of-sample test of MCVSK and MVSK portfolio optimization models by GWO algorithm, PSO algorithm and GA respectively, then we compare them with FTSE100 index and the control group \(y\). In this section, we mainly consider four indicators: skewness, kurtosis, VaR or CVaR and \(E[g(x,\xi)]\). From the results of backtesting, it's shown that the skewness of portfolio derived from MCVSK and MVSK model are higher than FTSE100 index and control group \(y\). Simultaneously, its value of kurtosis is rather lower than FTSE100 index and control group \(y\). On the indicator of risk measure, MCVSK model take CVaR as risk measure while MVSK model's risk measure is VaR. It can be seen from the experimental results that the portfolio optimized by GWO algorithm are generally much more profitable than FTSE100 index and control group \(y\) while having lower risks.  
Specifically speaking, when the confidence coefficient \(\alpha\) is \(1\%\), \(5\%\) or \(10\%\), the CVaR of portfolio obtained by MCVSK model is \(-2.7743\), \(-2.1648\) and \(-1.7413\) respectively, which is lower than \(-3.2503\), \(-2.3205\) and \(-1.8086\). At the same time, the return of portfolio obtained by MCVSK model is \(0.1430\), \(0.1478\) and \(0.1512\) respectively, which is greatly higher than \(0.1017\). Compare the MCVSK model with the MVSK model, it can be found that portfolio obtained by MCVSK model generally has a better performance than MVSK model on the rate of return indicator. Specifically speaking, when the confidence coefficient \(\alpha\) is \(1\%\), the return of portfolio obtained by MCVSK model is \(0.1430\) while the portfolio obtained by MVSK model is \(0.1402\). When the confidence coefficient \(\alpha\) is \(5\%\) and \(10\%\), there is also the same phenomenon. Therefore, in a sense, CVaR has its special advantage as a type of risk measure.  
Then we compare the GWO algorithm with PSO algorithm and GA. From the experimental results, it can be seen that all three algorithms can obtain excess returns when all the constraints on the portfolio optimization model are met. However, as a main evaluation index, the return to portfolio optimized by GWO algorithm is much higher than that of PSO algorithm and GA. Specifically speaking, when the confidence level is \(1\%\), the GWO algorithm optimizes the MCVSK model to obtain a return on portfolio of \(0.1430\). Under the same conditions, the PSO algorithm optimizes the return on portfolio to \(0.1107\) and GA to \(0.1215\). This phenomenon also exists when the confidence level is \(5\%\) and \(10\%\). While having a higher rate of return, the risk of portfolio optimized by GWO algorithm is similar to the portfolios optimized by PSO algorithm or GA. This shows that the GWO algorithm has a higher optimization capability in optimizing the MCVSK and MVSK portfolio optimization model.  
From the results of out-of-sample tests, the MCVSK model and MVSK model optimized by GWO algorithm also have a wonderful performance. Generally speaking, the experimental results prove that  
Figure 14: The return of optimal portfolio by model (33) in each period with model (34) by GWO algorithm and the FTSE100 Index portfolio in out-of-sample test with \(\alpha=0.10\).  
the GWO algorithm has great potential in solving single-objective portfolio optimization problems. The GWO algorithm not only has a faster optimization speed, but also maintains a high optimization quality. It's believed that GWO algorithm will have a much wider application.