**4. Preliminary data analysis**

Market-adjusted prices data were collected from Yahoo Finance and from the Investing.com databases for all assets between 2000 and 2018. Monthly data for the assets informs the computation of returns. **Figure 3** reports the fluctuations of the months returns, illustrating the synchronized behavior of the returns compared with prices (**Figure 4**). Correlation matrix and collinearity statistic were made (table to big, then only available by request) and descriptive statistics of monthly returns of the assets in **Table 4**.

The clusters are quite evident: volatility is present during the period. It is noticed also that spikes vary in time and between the assets themselves which is expected according to the propose in this study in order to create an adequate and a stable portfolio for "four stations". In general, spikes are more evident in VIX, which means this is the asset with more variation in prices (volatility). We also can see two evident clusters in this asset during the crisis of 2008 and before October of 2015 (fears about China). It is noticed also that silver had an evident cluster after April of

#### **Figure 4.**

*Accumulated returns of the 32 assets. Notes: SPX = Standard & Poor's ticker; DJI = Dow Jones Industrials ticker; STOXX = Eurostoxx 600 ticker; HSI = Hang Seng Index; EM = Emerging Markets; RE = Real Estate; B-ST = US Bonds Short Term; B-MT = US Bonds Medium Term; B-LT = US Bonds Long Term; TIP = Treasury Inflation Protected Securities; CorpB = Corporation Bonds; NGas = Natural Gas; EUR/ USD = Euro vs. USA Dollars exchange; VIX = Volatility Index (S&P 500).*

**141**

**Table 4.**

from the crisis.

2005 when other assets remained stable. What regards to equity, the most positive cluster (i.e. low spike) is present after May of 2009 when the market was recovering

*Notes: Min = Minimum; Max = Maximum; Var = Variance; SPX = Standard & Poor's ticker; DJI = Dow Jones Industrials ticker; STOXX = Eurostoxx 600 ticker; HSI = Hang Seng Index; EM = Emerging Markets; RE = Real Estate; B-ST = US Bonds Short Term; B-MT = US Bonds Medium Term; B-LT = US Bonds Long Term; TIP = Treasury Inflation Protected Securities; CorpB = Corporation Bonds; NGas = Natural Gas; EUR/USD = Euro* 

*vs. USA Dollars exchange; VIX = Volatility Index (S&P 500).*

*Descriptive statistics of monthly returns of the 32 assets.*

*Optimized Portfolios: All Seasons Strategy DOI: http://dx.doi.org/10.5772/intechopen.95122*

**Min Max Mean Standard** 

SPX −0,16,942 0,107,723 0,003389 0,041935 ,002 1208 DJI −0,14,060 0,106,046 0,004066 0,040566 ,002 1026 NASDAQ −0,22,901 0,191,947 0,004314 0,063888 ,004 1478 STOXX −0,14,134 0,134,717 0,000475 0,043185 ,002 ,981 HSI −0,19,218 0,191,929 0,003585 0,060497 ,004 ,506 EM −0,25,578 0,168,629 0,005717 0,064815 ,004 ,757 RE −0,30,435 0,325,359 0,008478 0,058845 ,003 7569 Consumer −0,16,678 0,135,928 0,006671 0,046783 ,002 1075 Health −0,14,249 0,085279 0,005354 0,035844 ,001 1247 Communications −0,19,721 0,323,605 0,000458 0,059830 ,004 4326 Energy −0,17,820 0,169,379 0,005666 0,055412 ,003 ,865 Financial −0,22,757 0,217,848 0,004584 0,054803 ,003 3504 Industrials −0,19,954 0,194,630 0,005873 0,052299 ,003 1912 Semicondoctor −0,30,345 0,238,355 0,006824 0,079641 ,006 1485 B-ST −0,01401 0,023438 0,001383 0,003911 ,000 6893 B-MT −0,05473 0,077308 0,004492 0,017663 ,000 1557 B-LT −0,13,070 0,138,855 0,004450 0,035025 ,001 3182 TIP −0,08111 0,065035 0,002477 0,015140 ,000 6606 CrorpB −0,10,723 0,133,314 0,003885 0,019498 ,000 12,713 BUND −0,03512 0,047787 0,003007 0,015447 ,000 -,203 Cocoa −0,28,082 0,345,646 0,008779 0,093491 ,009 ,874 Coffee −0,22,600 0,436,102 0,003457 0,090881 ,008 2374 Corn −0,26,536 0,221,904 0,005798 0,085273 ,007 ,433 Sugar −0,31,247 0,463,178 0,008467 0,103,947 ,011 1641 Gold −0,18,005 0,138,671 0,008176 0,048522 ,002 ,752 Copper −0,36,149 0,340,836 0,007984 0,076578 ,006 3505 Silver −0,70,670 2,047,420 0,015425 0,168,140 ,028 96,160 Crude −0,32,621 0,297,143 0,006479 0,092205 ,009 ,572 NGas −0,41,616 0,626,133 0,012380 0,157,891 ,025 1771 Commodities −0,22,325 0,137,865 0,001168 0,047982 ,002 1895 EURUSD −0,09720 0,101,047 0,001066 0,029126 ,001 1183 VIX −0,38,489 1,345,709 0,019859 0,217,513 ,047 6695

**Deviation**

**Var Kurtosis**

#### *Optimized Portfolios: All Seasons Strategy DOI: http://dx.doi.org/10.5772/intechopen.95122*

*Quality Control - Intelligent Manufacturing, Robust Design and Charts*

Market-adjusted prices data were collected from Yahoo Finance and from the Investing.com databases for all assets between 2000 and 2018. Monthly data for the assets informs the computation of returns. **Figure 3** reports the fluctuations of the months returns, illustrating the synchronized behavior of the returns compared with prices (**Figure 4**). Correlation matrix and collinearity statistic were made (table to big, then only available by request) and descriptive statistics of monthly

The clusters are quite evident: volatility is present during the period. It is noticed also that spikes vary in time and between the assets themselves which is expected according to the propose in this study in order to create an adequate and a stable portfolio for "four stations". In general, spikes are more evident in VIX, which means this is the asset with more variation in prices (volatility). We also can see two evident clusters in this asset during the crisis of 2008 and before October of 2015 (fears about China). It is noticed also that silver had an evident cluster after April of

*Accumulated returns of the 32 assets. Notes: SPX = Standard & Poor's ticker; DJI = Dow Jones Industrials ticker; STOXX = Eurostoxx 600 ticker; HSI = Hang Seng Index; EM = Emerging Markets; RE = Real Estate; B-ST = US Bonds Short Term; B-MT = US Bonds Medium Term; B-LT = US Bonds Long Term; TIP = Treasury Inflation Protected Securities; CorpB = Corporation Bonds; NGas = Natural Gas; EUR/*

*USD = Euro vs. USA Dollars exchange; VIX = Volatility Index (S&P 500).*

**4. Preliminary data analysis**

returns of the assets in **Table 4**.

**140**

**Figure 4.**


*Notes: Min = Minimum; Max = Maximum; Var = Variance; SPX = Standard & Poor's ticker; DJI = Dow Jones Industrials ticker; STOXX = Eurostoxx 600 ticker; HSI = Hang Seng Index; EM = Emerging Markets; RE = Real Estate; B-ST = US Bonds Short Term; B-MT = US Bonds Medium Term; B-LT = US Bonds Long Term; TIP = Treasury Inflation Protected Securities; CorpB = Corporation Bonds; NGas = Natural Gas; EUR/USD = Euro vs. USA Dollars exchange; VIX = Volatility Index (S&P 500).*

#### **Table 4.**

*Descriptive statistics of monthly returns of the 32 assets.*

2005 when other assets remained stable. What regards to equity, the most positive cluster (i.e. low spike) is present after May of 2009 when the market was recovering from the crisis.

If compared to the next figure (**Figure 4**), it is illustrated the synchronized behavior of the returns compared with prices. The spikes are much more evident. It also offers a clear picture of the volatility clusters.

What refers to correlation matrix and collinearity statistic (table available by request), US markets are very correlated and collineated with European market and all equity-sectors, although there is no correlation with the Chinese market, bonds, commodities and EUR/USD exchange. Also, it is shown that VIX is inverse correlated to the equity market in general. What regards to bonds, there is a high correlation and collinearity between themselves (except for the short-term bonds which are only a little correlated to medium-term bonds) but there is not (in general) with commodities and VIX. Commodities, in general, are not correlated with themselves (except crude oil that is correlated to all commodities index) and not correlated either to VIX or EUR/USD exchange. It is interesting to note that agriculture commodities are not correlated at all to themselves (cocoa, coffee, corn or sugar) but neither metals, for example, are not correlated between themselves (gold, copper or silver).

As the table above shows, standard deviation presents higher values rather the mean which means that volatility is present for all types of assets. Also, kurtosis presents value higher than 3 for Real State (equity), communications, financial services, short-term bonds, long-term bonds, TIPs, corporate bonds, copper, VIX and an exceptional high value (higher than 96) for silver. This may mean that the monthly returns distribution is non-normal for this kind of assets.

### **5. Results**

This study uses GRG Nonlinear engine for linear solver problems. **Table 5** reports the returns from each portfolio (Model 1 – Model 6) and **Table 6**, the constitution of assets for each portfolio.

Rate means the yearly return of the portfolio, and as can be seen the best result of 15,06% belongs to portfolio 2 which is expected because we are maximizing this metric (rate), although in a less consistent way since sharp ratio presents the lowest value compared to other portfolios. This means that portfolio 2 is the most volatile, i.e., in terms of sharp ratio almost equals to the benchmark (S&P 500). Still it has fewer negative years when comparing to the benchmark (3 versus 7). Although, it loses power in the good semi-decade (2011–2018), showing a return of 8,55% (annually), when in the bad decade (2000–2010) the average return was 20,03%. Portfolio 1 presents the highest sharp ratio with no negative years; the worst year presented a positive return of 2,18%. The average yearly return is 4,49% in the overall and its maximum presented a value of 6,52% (much lower comparing to 29,60% - the benchmark). It means that this portfolio is adequate to a very conservative investor. The rate is only a little higher than a deposit rate, which is expected according to its constitution (see **Table 6**) because 65% is constituted by treasury bonds, then only 16% equity, 7% commodities and 12% others (EUR/USD and VIX). In Portfolio 3 we try to create a portfolio that, in general, the rate of a bad decade is almost equal to a good decade. It is expected a yearly rate of 14,15% overall and equal for both decades. Sharp ratio still present positive values (superior to 1) and the investor should not present any negative years with Portfolio 3 (there was this condition as well in this model). To accomplish that, portfolio constitution is curious: 42% equity, 32% VIX, 26% commodities and no bonds (see **Table 6**). In Portfolio 4 the overall rate is maximized but with the constrain of the overall sharp ratio equal or superior to 1 and the solver obtained the result successfully. The overall rate is 14,83% by year which is an excellent result, but it loses "power" in the good decade (18,28% 2000–2010 vs. 10,25% 2011–2018). The worse year was in 2013 (−10,14%)

**143**

**Table 5.**

*Results from the 6 models.*

*Optimized Portfolios: All Seasons Strategy DOI: http://dx.doi.org/10.5772/intechopen.95122*

> **SP500 Benchmark**

**Model 1 Max SR** **Model 2 Max return**

 −5,32% 4,96% 29,62% 9,18% 19,89% 20,50% 8,75% −13,04% 4,02% −13,37% 0,50% −4,00% −8,10% 11,23% −23,37% 4,79% 27,08% 6,96% 20,85% 12,26% 12,82% 26,38% 5,31% 10,43% 12,23% 11,57% 19,94% 13,20% 8,99% 4,54% 4,65% 0,50% 6,69% 1,79% 2,59% 3,00% 4,21% 73,67% 37,70% 54,60% 59,50% 27,36% 13,62% 3,96% 17,76% 16,80% 21,49% 16,33% 15,64% 3,53% 6,08% 39,86% 32,98% 31,60% 32,30% 26,42% −38,49% 3,61% 13,02% 10,51% 8,95% 2,49% 7,33% 23,45% 5,20% 9,57% 10,67% 12,29% 17,95% 18,50% 12,78% 6,17% 27,57% 23,86% 26,61% 26,33% 17,49% Rate −0,93% 4,80% 20,03% 14,15% 18,28% 17,12% 14,44% ER 0 5,74% 20,97% 15,09% 19,22% 18,06% 15,37% SR −0,05 5,97 0,94 1,22 1,24 1,00 2,00

 −2,22% 4,76% 11,76% 14,59% 12,16% 9,32% 2,58% 13,41% 4,87% 7,65% 8,29% 8,33% 8,03% 7,78% 29,60% 3,64% −13,94% 0,50% −10,14% −5,21% 2,58% 11,39% 6,52% 12,74% 25,68% 20,72% 19,27% 24,30% −0,73% 2,70% 8,75% 17,01% 12,14% 11,00% 14,63% 9,54% 2,18% 13,28% 7,31% 9,07% 16,05% 3,34% 19,42% 3,09% −5,01% 2,24% −2,37% 1,93% 2,98% −6,24% 4,81% 41,10% 43,30% 38,71% 35,52% 32,36% Rate 6,26% 4,06% 8,55% 14,15% 10,25% 11,42% 10,83% ER 0 −2,19% 2,29% 7,90% 4,00% 5,17% 4,57% SR 0,52 3,04 0,57 1,07 0,75 1,00 1,01

Rate 3,14% 4,49% 15,06% 14,15% 14,83% 14,69% 12,90% ER 0 1,35% 11,92% 11,02% 11,70% 11,55% 9,77% SR 0,75 4,00 0,76 1,15 1,00 0,96 1,43 AVG 4,51% 4,50% 16,64% 14,78% 15,75% 15,64% 13,26% MED 8,99% 4,76% 12,74% 10,67% 12,16% 16,05% 12,82% MIN −38,49% 2,18% −13,94% 0,50% −10,14% −8,10% 2,58% MAX 29,60% 6,52% 73,67% 43,30% 54,60% 59,50% 32,36% (+) 12 19 16 19 16 17 19 (−) 7 0 3 0 3 2 0 *Notes: SR = Sharp Ratio; ER = Excess return (comparing to the benchmark); AVG = Average annually returns; MED = Median annually returns; MIN = Minimum annually returns; MAX = Maximum annually returns; (+)* 

**Model 3 Equal return**

**Model 4 Max return/SR** **Model 5 Model 4 (ver2)**

**Model 6 Max min**

**Panel A: Decade 2000–2010**

**Panel B: Semi-decade 2011–2018**

**Panel C: All period 2000–2018**

*count of positive years; (*−*) count of negative years.*
