**3. Model framework**

Six portfolio models are proposed: first, it is used a solver, where the selected solving method is the Generalized Reduced Gradient (GRG) Nonlinear engine for linear solver problems. The form is:

$$\max f(\boldsymbol{\omega}) : h(\boldsymbol{\omega}) = 0, L \le \boldsymbol{\omega} \le U,\tag{1}$$

Where *h* has dimension *m*. The method supposes can be partition *x = (v,w)* such that:


As in the linear case, for any *w* there is a unique value, *v(w)*, such that *h(v(w),w)* = 0 (c.f., Implicit Function Theorem), which implies that


*Notes: S&P = Standard & Poor's; TIPS = Treasury Inflation Protected Securities; EUR/USD = Euro vs. USA Dollars exchange; VIX = Volatility Index (S&P 500).*

#### **Table 3.**

*Assets list.*

( ( )) ( ) <sup>−</sup> =∇ ∇<sup>1</sup> / *dv dw h x h x v w* . The idea is to choose the direction of the independent variables to be the reduced gradient: ∇ − ( ( ) ( )) *<sup>T</sup> <sup>w</sup> f x yhx* , where ( ( )) ( ) <sup>−</sup> = =∇ ∇<sup>1</sup> / *v w y dv dw h x h x* . Then, the step size is chosen, and a correction procedure applied to return to the surface, *h(x)* = 0.

The main steps (except the correction procedure) are the same as the reduced gradient method, changing the working set as appropriate.

The constitution of the portfolios is set from the Solver and varies. The six portfolios present different risks and returns, depending of the profile of each investor. There are conservative portfolio and aggressive portfolios. The solver configuration of each portfolio is showed above.

With regards to variable cells, the percentage of weighing of the assets type are the changeable ones. It is used a wide asset as equity indexes, bonds, commodities and other. The detail of each family of assets used in the model are listed table above. In total, it is used 32 assets:

With regards to subject to the constraints, the sum of the percentage of each asset is equal to 1, i.e., 100%:

$$
\Sigma \propto \text{Assets}(a) = 1 \tag{2}
$$

**137**

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

where occurred two crashes:

is a recover from the last decade.

prepared for crashes or deflationary periods.

= monthly returns; *Pt*

deviation, computed as given by the following steps:

Where *S* = Accumulated value; *P* = Principal.

But is need the rate for the numerator:

**3.1 Model 1: maximize sharp ratio**

returns, computed as given by:

Where *Rt*

tives of each one:

That is:

(years).

respectively.

a.Between 2001 and 2002 the technological crash;

b.Between 2008 and 2009 the subprime crisis.

1.The first decade (2000 to 2010) was very turbulent for financial markets,

2.The second semi-decade not completed yet, between 2011 to 2018 where there

Then is important to study some robust portfolios that can provide some return to investors and at the same time with lower risk, mean, volatility, in order to be

The model uses past returns (monthly returns) for each asset and the portfolios are re-equilibrated monthly according to the optimal weighting of each one. The benchmark, to compare results is the S&P 500 index. It is relied on monthly

> − = − 1 1 *<sup>t</sup>*

Finally, it is presented above, for each model, the own specifications and objec-

The set objective of this model is to maximize the sharp ratio for all the period (2000–2018). It is relied by the division by the average year return and the standard

= + (1 )

= + (1 )

Where *FV* = Future Value; *PV* = Present Value; *i* = rate; *n* = number of periods

<sup>=</sup> *<sup>n</sup> FV <sup>i</sup> PV* *n*

*n*

*SP i* (4)

*FV PV i* (5)

(3)

(6)

are the assets prices at moments *t* and *t-1*

*t <sup>P</sup> <sup>R</sup> P*

*t*

and *Pt*<sup>−</sup><sup>1</sup>

Where x = coefficient; a = each type of asset as showed in **Table 3** – Assets list. Note: it is forced to make unconstrained variables non-negative.

The period is set between 2000 to 2018, but in some analysis the two decades are separated (2000 to 2010; and 2011–2018). The reason of the period spam used is important because:

NASDAQ Treasury 20+ years Corn EuroStoxx 600 TIPS Sugar Hang Seng Corporate bonds Gold Emerging Markets Bunds (Germany) Copper Real Estate Silver Consumer Crude Healthcare Natural Gas

Communications General commodities index

**Equities index Bonds Commodities Other** S&P 500 Treasury 1–3 years Cocoa EUR/USD Dow Jones 30 Treasury 7–10 years Coffee VIX

( ( )) ( ) <sup>−</sup> =∇ ∇<sup>1</sup>

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

of each portfolio is showed above.

above. In total, it is used 32 assets:

asset is equal to 1, i.e., 100%:

= =∇ ∇<sup>1</sup>

Energy Financials Industrials Semiconductors

**Table 3.** *Assets list.*

( ( )) ( ) <sup>−</sup>

variables to be the reduced gradient: ∇ − ( ( ) ( )) *<sup>T</sup>*

gradient method, changing the working set as appropriate.

procedure applied to return to the surface, *h(x)* = 0.

/ *dv dw h x h x v w* . The idea is to choose the direction of the independent

*Notes: S&P = Standard & Poor's; TIPS = Treasury Inflation Protected Securities; EUR/USD = Euro vs. USA Dollars* 

The main steps (except the correction procedure) are the same as the reduced

The constitution of the portfolios is set from the Solver and varies. The six portfolios present different risks and returns, depending of the profile of each investor. There are conservative portfolio and aggressive portfolios. The solver configuration

With regards to variable cells, the percentage of weighing of the assets type are the changeable ones. It is used a wide asset as equity indexes, bonds, commodities and other. The detail of each family of assets used in the model are listed table

With regards to subject to the constraints, the sum of the percentage of each

Where x = coefficient; a = each type of asset as showed in **Table 3** – Assets list.

The period is set between 2000 to 2018, but in some analysis the two decades are separated (2000 to 2010; and 2011–2018). The reason of the period spam used is

Note: it is forced to make unconstrained variables non-negative.

/ *v w y dv dw h x h x* . Then, the step size is chosen, and a correction

*<sup>w</sup> f x yhx* , where

∑ = *x Assets a*( ) 1 (2)

**136**

important because:

	- a.Between 2001 and 2002 the technological crash;
	- b.Between 2008 and 2009 the subprime crisis.

Then is important to study some robust portfolios that can provide some return to investors and at the same time with lower risk, mean, volatility, in order to be prepared for crashes or deflationary periods.

The model uses past returns (monthly returns) for each asset and the portfolios are re-equilibrated monthly according to the optimal weighting of each one. The benchmark, to compare results is the S&P 500 index. It is relied on monthly returns, computed as given by:

$$R\_t = \frac{P\_t}{P\_{t-1}} - \mathbf{1} \tag{3}$$

Where *Rt* = monthly returns; *Pt* and *Pt*<sup>−</sup><sup>1</sup> are the assets prices at moments *t* and *t-1* respectively.

Finally, it is presented above, for each model, the own specifications and objectives of each one:

#### **3.1 Model 1: maximize sharp ratio**

The set objective of this model is to maximize the sharp ratio for all the period (2000–2018). It is relied by the division by the average year return and the standard deviation, computed as given by the following steps:

$$S = P\left(\mathbf{1} + \mathbf{i}\right)^{\mathbf{v}} \tag{4}$$

Where *S* = Accumulated value; *P* = Principal. That is:

$$FV = PV\left(\mathbf{1} + i\right)^{n} \tag{5}$$

Where *FV* = Future Value; *PV* = Present Value; *i* = rate; *n* = number of periods (years).

But is need the rate for the numerator:

$$\dot{a} = \sqrt[n]{\frac{FV}{PV}}\tag{6}$$

The, the standard deviation (σ) – the denominator – is a measure of how widely values are dispersed from the average value (the mean), using the "n" method. It is used the following formula:

$$
\sigma = \sqrt{\frac{\sum \left(x - \overline{x}\right)^2}{n}} \tag{7}
$$

Finally, the Sharp Ratio (SR) formula:

$$SR = \frac{\dot{i}}{\sigma} \tag{8}$$

This is a measure of stability, if SR > 1 it means that returns overcome the standard deviation (volatility).

#### **3.2 Model 2: maximize rate return**

In this model, the concern is to bring the maximum return to the investor, ignoring the volatility, then we can argue that model 2 presents the higher risk comparing to others:

**Set objective:** Global rate return **To:** Maximum

#### **3.3 Model 3: two decades, equals returns**

The model equals the return of the two decades (*i i* 2000 2010 2011 2018 − − = ). Comparing to other models, it may generate a more distributed income to investors. Then, in a decade of crises the investor may generate the same return as in a period of expansion. Also, model 3 "guarantees" a minimum return of half of percent each year:

**Set objective:** Global rate return **To:** Maximum **Additional subject to the constrains:** *i i* 2000 2010 2011 2018 − − = ; *ni* > 0,5%

#### **3.4 Model 4: maximize rate and sharp-ratio**

In this model, the concern is to bring some extra return to the investor. It may generate more income than the model 1 but still with the concern of a stability, lowering a little bit the volatility of the portfolio:

**Set objective:** Global rate return **To:** Maximum **Additional subject to the constrains:** Sharp Ratio > 1 (all model)

#### **3.5 Model 5: maximize rate and sharp-ratio (version 2)**

In this model, in a similar way of the previous model (model 4), the concern is to bring some extra return to the investor but still with the concern of a stability, lowering a little bit the volatility of the portfolio:

**139**

**Figure 3.**

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

**To:** Maximum

(period 2011–2018)

**To:** Maximum

**Set objective:** Global rate return

decade but in the second decade as well.

**3.6 Model 6: maximize the minimal year return**

positive returns each year. Basically, it maximizes the minimum: **Set objective:** Minimum return of each year (2000–2018)

**Additional subject to the constrains:** SR > 1 (period 2000–2010); SR > 1

Comparing to the previous model, the SR >1 appears twice in the constrains and not in the whole model (2000–2018). This measure will provide stability in the first

The model "guarantees" a minimal return year by year. Then, it may generate

**Additional subject to the constrains:** Year return > Minimum return of each year

*Monthly 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).*

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

**Set objective:** Global rate return **To:** Maximum **Additional subject to the constrains:** SR > 1 (period 2000–2010); SR > 1 (period 2011–2018)

Comparing to the previous model, the SR >1 appears twice in the constrains and not in the whole model (2000–2018). This measure will provide stability in the first decade but in the second decade as well.
