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

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

also offers a clear picture of the volatility clusters.

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

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).

monthly returns distribution is non-normal for this kind of assets.

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

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

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%)

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**5. Results**

constitution of assets for each portfolio.


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

#### **Table 5.** *Results from the 6 models.*


*Notes: SR = Sharp Ratio; 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 6.**

*Constitution of the 6 models.*

because metals came across with a big drop which is a big part of the constitution of this portfolio (38% commodities, which 26% silver). The remain constitution: 33% equity and 29% VIX. It is seen 3 negative years (2001, 2013 and 2017) which is

**145**

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

> **S&P 500**

Portfolio2 −0,23 0,30 1

Portfolio1 0,11 1

*Correlation matrix of portfolios vs. benchmark.*

S&P 500 1

**Table 7.**

**Portfolio 1**

Portfolio3 −0,09 0,41 0,81 1

Portfolio4 −0,18 0,40 0,97 0,89 1

Portfolio5 0,04 0,35 0,94 0,86 0,95 1

**Portfolio 2**

**Portfolio 3**

**Portfolio 4**

**Portfolio 5**

**Portfolio 6**

perfectly acceptable when comparing to the benchmark (7 negative years in 19 years total). Portfolio 5 is very similar to Portfolio 4, in results and in constitution. Here, the difference is to assure a sharp ratio equal or superior to 1 for the first decade and for the second decade as well. There was a little improvement comparing to the last one, the model will "steal" 1% of the returns from the first decade and return it to the second decade, i.e., instead of 18,28% vs. 10,25% (Portfolio 4), we get 17,12% vs. 11,42%. Also, instead of 3 negative years, there is 2 negative years (2001 and 2013) and the worst year instead of −10,14% (Portfolio 4), −8,10%. Finally, Portfolio 6 we maximize the minimum return (yearly). We may say that this portfolio is an upgrade from the first one (Portfolio 1), because 1. there are no negative years, 2. the worse year presents a positive return of 2,58% and 3. it maximizes more returns to the investor. The overall return is 12,9% yearly (vs 4,49% - Portfolio 1) and sharp ratio is superior to 1 for both decades. What regards to its constitution: 38% equities,

Portfolio6 −0,06 0,51 0,69 0,89 0,81 0,77 1

As can be seen, all portfolios come across to the benchmark, portfolio 1 with less spikes, although, S&P500 is almost touching the line of the portfolio. Portfolio 2 seems to be the most volatile. **Table 7** shows the correlation matrix of portfolios and

As can be seen, there is no correlation between S&P 500 and any portfolio, meaning that our proposed portfolios behave quite independently from the stock market. Portfolio 1, where we maximize the sharp ratio has no correlation with others 5 portfolios at all. Portfolio 2 to 5 are highly correlated between themselves and

Our study shows that is possible to create robust portfolios where the risk is minimized, and the return is maximized. Theory behind is [2] which study focus on 'efficient frontier of optimal investment', while advocating a diversified portfolio to reduce risk. To perform it, six portfolio models are proposed, and its formation are made by a solver, where the selected solving method is the GRG Nonlinear engine for linear solver problems. Then we compare results with the benchmark (S&P 500), a linear regression model (available for request) and other "popular" portfolios (already known by many investors – also, only available by request) as well. Results show that the GRG Nonlinear engine is powerful, providing excess returns to all six models. We design models for three types of investors: conservative, moderate and aggressive. For a conservative investor, portfolio 1 fits the best

35% commodities, 21% VIX and 5% treasury bonds.

Portfolio 6 (max min) is highly correlated to Portfolios 3 to 5.

benchmark between themselves.

**6. Conclusions**

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


#### **Table 7.**

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

**Model 2 Max return**

**Model 3 Equal return**

RE 18% 11% 31% 16% 19% Consumer 9% 23% 3% 3%

Semiconductors 9% 2% 16% 16%

B 7-10y 1% B 20 + y 1% TIPS 2%

BUND 33% 1% Cocoa 3% 2% 2% 17%

GOLD 3%

Silver 2% 35% 19% 26% 27% 11%

Natural Gas 16% 4% 9% 11% 3%

VIX 3% 31% 32% 29% 28% 21% *Notes: SR = Sharp Ratio; 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).*

**Model 4 Max return/ SR**

**Model 5 Model 4 (ver2)**

**Model 6 Max min**

**Model 1 Max SR**

SP 500

NASDAQ STOX 600 HANG SENG Emerging Mkts

DJ 30 6%

Healthcare 1%

B 1-3y 32%

Sugar 1%

EUR/USD 9%

*Constitution of the 6 models.*

Communications

Energy Financials Industrials

Corporate B

Coffee Corn

Cooper

Crude

Commodities

because metals came across with a big drop which is a big part of the constitution of this portfolio (38% commodities, which 26% silver). The remain constitution: 33% equity and 29% VIX. It is seen 3 negative years (2001, 2013 and 2017) which is

**144**

**Table 6.**

*Correlation matrix of portfolios vs. benchmark.*

perfectly acceptable when comparing to the benchmark (7 negative years in 19 years total). Portfolio 5 is very similar to Portfolio 4, in results and in constitution. Here, the difference is to assure a sharp ratio equal or superior to 1 for the first decade and for the second decade as well. There was a little improvement comparing to the last one, the model will "steal" 1% of the returns from the first decade and return it to the second decade, i.e., instead of 18,28% vs. 10,25% (Portfolio 4), we get 17,12% vs. 11,42%. Also, instead of 3 negative years, there is 2 negative years (2001 and 2013) and the worst year instead of −10,14% (Portfolio 4), −8,10%. Finally, Portfolio 6 we maximize the minimum return (yearly). We may say that this portfolio is an upgrade from the first one (Portfolio 1), because 1. there are no negative years, 2. the worse year presents a positive return of 2,58% and 3. it maximizes more returns to the investor. The overall return is 12,9% yearly (vs 4,49% - Portfolio 1) and sharp ratio is superior to 1 for both decades. What regards to its constitution: 38% equities, 35% commodities, 21% VIX and 5% treasury bonds.

As can be seen, all portfolios come across to the benchmark, portfolio 1 with less spikes, although, S&P500 is almost touching the line of the portfolio. Portfolio 2 seems to be the most volatile. **Table 7** shows the correlation matrix of portfolios and benchmark between themselves.

As can be seen, there is no correlation between S&P 500 and any portfolio, meaning that our proposed portfolios behave quite independently from the stock market. Portfolio 1, where we maximize the sharp ratio has no correlation with others 5 portfolios at all. Portfolio 2 to 5 are highly correlated between themselves and Portfolio 6 (max min) is highly correlated to Portfolios 3 to 5.
