**Abstract**

Our study explores the efficient frontier of optimal investment, taking behind the Markowitz's theory, 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. Our main goal is to design portfolios that resists to financial crisis but at the same time persists in a wealthy period. We analyze the decade where we assisted to two crashes (2000–2010) and a semi-decade where we assist to a wealthy period (2011–2018). The assets used are varied, such as Equities indexes form various countries, sector equities, bonds, commodities, EURUSD exchange and VIX. Results show that the GRG Nonlinear engine is powerful, providing excess returns in all six models.

**Keywords:** MPT, Markowitz, portfolios' formation, sharp-ratio, volatility

## **1. Introduction**

The inspiration for our work comes from the well-known investor, Ray Dalio who built a considerable personal fortune with the incredible success of the Pure Alpha strategy. In the mid-1990s he began to think about his inheritance and funds he wanted to leave behind and asked this question: "What kind of portfolio would you use if you were not already present to actively manage money?" What kind of portfolio would survive your own decision-making and would continue to support their children and their philanthropic efforts for decades? [1].

A brand-new look at asset placement. A new set of rules. And only after the portfolio has been retrospectively tested until 1925; only after having produced consistent results in a variety of economic conditions, Ray Dalio began to offer it to a narrow group. The new strategy, known as the "All Seasons" strategy, was publicly unveiled in 1996, just four years before a mass market correction put it to the test. "Passed" with distinction [1].

Conventional wisdom and the conventional management of a portfolio leave us in the hands of a model that continually shows that it cannot survive when times are tough. So, we began to explore whether we could define portfolios - asset distribution - that would perform well in any economic environment in the future, such as in the year 2008, a depression or a recession. Because no one knows what is going to happen in five years, how much more in 20 or 30 years.

According to [1], having into account this basis, we propose six different models, aiming to maximize returns but at the same time, reduce risk. Theory behind this is the Harry Markowitz's [2, 3], who is known as the father of modern portfolio theory. It explains in this way and synthesized the fundamental concept behind the

work that earned him the Nobel Prize: investments in a portfolio should not be seen individually, but as a group. There is a trade-off between risk and return, so "do not listen to just one instrument, listen to the entire orchestra". How investments behave together and how they are diversified will determine return. This advice may seem simple now, but in 1952 this thought was groundbreaking. Somehow this approach influenced virtually all portfolio managers from New York and Hong Kong.

We combine portfolios with a wide range of equity (mainly indexes from various countries and the main sectors as well), different kind of bonds (US and German treasury bonds and corporate bonds), a range of commodities (for example, different metals, agriculture commodities, energy commodities, etc.), EUR/USD exchange and VIX (volatility index of S&P 500) through a solver using the GRG (Generalized Reduced Gradient) Nonlinear engine for linear solver problems. The spam range is from 2000 to 2018 in order to cover two market crashes (2002 – technological and 2008 – subprime) and a good decade forward. Our mains investigation question is if it is possible to create a portfolio or a set of portfolios that presents robust results in a bad decade but, at the same time, in a good decade as well? Results show that definitively, is doable.

Next section, literature review, we explore the theory behind the concept of this study and empirical achievements from different authors. Section 3 presents six different models where we are going to using the solver, Section 4 preliminary analysis to the data set, in Section 5 we present the results of the models and we propose some portfolios to use as well. Finally, Section 6 concludes.

## **2. Literature review**

Many investors are naive in their financial beliefs and do not understand basic concepts such as equity or diversification [4, 5]. Benjamin Graham 1949 *apud* [6], the father of the value investing, proposed that an equilibrated portfolio should be constituted by 50% equity and 50% bonds; an intelligent investor may own 100% equity in his portfolio in certain conditions, the most important of them: only if in a crash crisis, the portfolio presented a positive return. By dividing the money by 50% for stocks and 50% for bonds (or some similar variation), many investors would think they were diversifying and reducing their risk. But later, when [2] presented his work about the efficient portfolio, concluded that what investors are doing is taking more risks than they think. Because, according to Ray Dalio *apud* [1–3] shares can be three times riskier (i.e., volatile) than bonds. In fact, by having a 50/50 portfolio, we have something more like a 95% risk distribution in stocks. Below, **Figure 1** represents a chart with a 50/50 portfolio. The left side shows the money divided by shares and bonds, in percentage. The right side shows the same portfolio, but divided in terms of risk, between stocks and bonds.

At first glance, with 50% of the money in shares, it seems relatively balanced. But, as it turns out here it would have been about 95% risk, given the volatility of its composition in stocks. So, if shares sink, the whole portfolio sinks. And the balance is lost. How does this concept work into real life? From 1973 to 2013, the S&P 500 lost money nine times and accumulated losses totaled 134%. In the same period, bonds (represented by the Barclays Aggregate Bond index) lost money only three times and accumulated losses were 6%. Therefore, having a 50/50 portfolio, the S&P 500 would have accrued 95% of the losses.

Placing assets is the only key that can differentiate us from all investors [2, 3]. Nobel Prize winner and father of modern portfolio theory (MPT) said that "diversification is the only free lunch." Why? Because spreading the money for different investments lowers the risk and increases the possibility of gains over time and costs nothing.

**133**

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

*Allocation versus risk. Source: adapted from [1].*

When we look at most portfolios, they usually hold up well in good periods, but they fall in bad periods. And then, the strategy is simply to wait for the stock to go up. This conventional approach to diversifying investments is not at all diversified (Dalio *apud* [1]). According to [7] "Financial crises occur in all market economies, although sometimes there are long periods of quiet. Crises occur in developed countries, not just emerging markets. Crises occur in economies with and without a

**GROWTH INFLATION** RISE ↑ Higher than expected economic growth Higher than expected inflation FALL ↓ Lower than expected economic growth Lower than expected inflation

Competitive pressures and market efficiency turn difficult to financial forecast particularly to predict asset returns - is very difficult compared to standard forecasting problems in macroeconomics, in which the presence of a sizeable persistent

Dalio *apud* [1] revealed the simplest and most important distinction of all. There

In this way, it all boils down to four possible environments, or economic "seasons," which will ultimately determine whether investments (asset prices) rise or fall - except that, unlike the seasons, there is no predetermined order in the

central bank and with and without deposit insurance".

*The four things that drive asset prices. Source: adapted from [1].*

are only four things that drive asset prices (**Table 1**):

component makes forecasting easier [8].

1.Inflation

**Figure 1.**

**Table 1.**

2.Deflation

succession. They are:

3.High economic growth

4.Declining economic growth

#### **Figure 1.**

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

as well? Results show that definitively, is doable.

**2. Literature review**

some portfolios to use as well. Finally, Section 6 concludes.

portfolio, but divided in terms of risk, between stocks and bonds.

S&P 500 would have accrued 95% of the losses.

work that earned him the Nobel Prize: investments in a portfolio should not be seen individually, but as a group. There is a trade-off between risk and return, so "do not listen to just one instrument, listen to the entire orchestra". How investments behave together and how they are diversified will determine return. This advice may seem simple now, but in 1952 this thought was groundbreaking. Somehow this approach

We combine portfolios with a wide range of equity (mainly indexes from various countries and the main sectors as well), different kind of bonds (US and German treasury bonds and corporate bonds), a range of commodities (for example, different metals, agriculture commodities, energy commodities, etc.), EUR/USD exchange and VIX (volatility index of S&P 500) through a solver using the GRG (Generalized Reduced Gradient) Nonlinear engine for linear solver problems. The spam range is from 2000 to 2018 in order to cover two market crashes (2002 – technological and 2008 – subprime) and a good decade forward. Our mains investigation question is if it is possible to create a portfolio or a set of portfolios that presents robust results in a bad decade but, at the same time, in a good decade

Next section, literature review, we explore the theory behind the concept of this study and empirical achievements from different authors. Section 3 presents six different models where we are going to using the solver, Section 4 preliminary analysis to the data set, in Section 5 we present the results of the models and we propose

Many investors are naive in their financial beliefs and do not understand basic concepts such as equity or diversification [4, 5]. Benjamin Graham 1949 *apud* [6], the father of the value investing, proposed that an equilibrated portfolio should be constituted by 50% equity and 50% bonds; an intelligent investor may own 100% equity in his portfolio in certain conditions, the most important of them: only if in a crash crisis, the portfolio presented a positive return. By dividing the money by 50% for stocks and 50% for bonds (or some similar variation), many investors would think they were diversifying and reducing their risk. But later, when [2] presented his work about the efficient portfolio, concluded that what investors are doing is taking more risks than they think. Because, according to Ray Dalio *apud* [1–3] shares can be three times riskier (i.e., volatile) than bonds. In fact, by having a 50/50 portfolio, we have something more like a 95% risk distribution in stocks. Below, **Figure 1** represents a chart with a 50/50 portfolio. The left side shows the money divided by shares and bonds, in percentage. The right side shows the same

At first glance, with 50% of the money in shares, it seems relatively balanced. But, as it turns out here it would have been about 95% risk, given the volatility of its composition in stocks. So, if shares sink, the whole portfolio sinks. And the balance is lost. How does this concept work into real life? From 1973 to 2013, the S&P 500 lost money nine times and accumulated losses totaled 134%. In the same period, bonds (represented by the Barclays Aggregate Bond index) lost money only three times and accumulated losses were 6%. Therefore, having a 50/50 portfolio, the

Placing assets is the only key that can differentiate us from all investors [2, 3]. Nobel Prize winner and father of modern portfolio theory (MPT) said that "diversification is the only free lunch." Why? Because spreading the money for different investments lowers the risk and increases the possibility of gains over time and costs nothing.

influenced virtually all portfolio managers from New York and Hong Kong.

**132**

*Allocation versus risk. Source: adapted from [1].*


#### **Table 1.**

*The four things that drive asset prices. Source: adapted from [1].*

When we look at most portfolios, they usually hold up well in good periods, but they fall in bad periods. And then, the strategy is simply to wait for the stock to go up. This conventional approach to diversifying investments is not at all diversified (Dalio *apud* [1]). According to [7] "Financial crises occur in all market economies, although sometimes there are long periods of quiet. Crises occur in developed countries, not just emerging markets. Crises occur in economies with and without a central bank and with and without deposit insurance".

Competitive pressures and market efficiency turn difficult to financial forecast particularly to predict asset returns - is very difficult compared to standard forecasting problems in macroeconomics, in which the presence of a sizeable persistent component makes forecasting easier [8].

Dalio *apud* [1] revealed the simplest and most important distinction of all. There are only four things that drive asset prices (**Table 1**):

#### 1.Inflation


In this way, it all boils down to four possible environments, or economic "seasons," which will ultimately determine whether investments (asset prices) rise or fall - except that, unlike the seasons, there is no predetermined order in the succession. They are:


The price of a stock (or a bond) already incorporates what we (the market) "expect" about the future. Many authors [9–11], claim that there is literally a picture of the future when looking at prices today. In other words, the stock price of a company today already incorporates the expectations of investors, who believe that the company will continue to grow at a certain pace [12–14] – this phenomenon also known as efficient market hypothesis (EMH). This is why is sometimes heard that the stock price will fall when companies announce that their future growth (their profits) will be lower than they had originally forecast – see also the post-earnings announcement drift (PEAD) phenomenon [15–17].

It is the surprises that will ultimately determine which asset class will behave well. If the news announces that there will be sustained growth, this will be very good for stocks and not so good for bonds. If we watch a surprise fall in inflation this will be good for the bonds [18, 19]. If there are only four potential economic environments, or seasons, one should therefore have 25% of the risk in each of the categories. That is why this approach is called "All Seasons" because there are four possible seasons in the financial world, and no one really knows which season will come next – EMH/Random walk [12, 20–22]. With this approach each season, each quadrant is always covered, so the portfolio is always protected. Let us imagine, then, four portfolios, each with an equivalent amount of risk. This means that we will not have exposure to any particular environment. We are not trying to predict the future, because no one knows what the future will bring [12, 22–24]. What is known is that there are only four potential seasons that we will have to face. Using this investment strategy, we can know that we are protected - not just hopeful - and that the investments are safe and will perform well in any season that comes.

"All Seasons": today we can structure a portfolio that will behave well in 2029, even if we have no chance of knowing what the world will look like in 2029. Below is a table that shows the four potential seasons and the type of investment that will perform best in each of these environments, categorizing each of them in each of the seasons (**Table 2**).

The original "All Seasons" is composed by equity, bonds and commodities which became a popular asset over the past decade [25]. [26] argues that MPT is the formula of diversification, which selects a collection of assets that has collectively lower risk than individually. In sum, for a given amount of risk, MPT describes how to select a portfolio with the highest possible expected return [27, 28]. Below it is presented, in **Figure 2**, the efficient frontier.


**135**

such that:

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

line is the efficient frontier.

*Efficient frontier. Source: [29].*

**Figure 2.**

**3. Model framework**

linear solver problems. The form is:

eracy assumption);

• ∇*vh x*( ) is nonsingular at *x = (v,w)*.

• *v* has dimension *m* (and *w* has dimension *n-m*);

The hyperbola is sometimes referred to as the "Markowitz Bullet" and is the efficient frontier if no risk-free asset is available. With a risk-free asset, the straight

The Capital Asset Pricing Model (CAPM), for example, was the next step, it approached the risk of an individual asset through the diversification theory [30]. Based on this theory background and MPT, we present in the next chapter six different portfolios aiming to a certain risk, produce the maximum return to the investor.

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

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

• the values of *v* are strictly within their bounds: *L vU v v* < < (this is a nondegen-

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

max : 0, , *f x hx L x U* ( ) ( ) = ≤≤ (1)

#### **Table 2.**

 *List of assets for each "season".*

The price of a stock (or a bond) already incorporates what we (the market) "expect" about the future. Many authors [9–11], claim that there is literally a picture of the future when looking at prices today. In other words, the stock price of a company today already incorporates the expectations of investors, who believe that the company will continue to grow at a certain pace [12–14] – this phenomenon also known as efficient market hypothesis (EMH). This is why is sometimes heard that the stock price will fall when companies announce that their future growth (their profits) will be lower than they had originally forecast – see also the post-earnings

It is the surprises that will ultimately determine which asset class will behave well. If the news announces that there will be sustained growth, this will be very good for stocks and not so good for bonds. If we watch a surprise fall in inflation this will be good for the bonds [18, 19]. If there are only four potential economic environments, or seasons, one should therefore have 25% of the risk in each of the categories. That is why this approach is called "All Seasons" because there are four possible seasons in the financial world, and no one really knows which season will come next – EMH/Random walk [12, 20–22]. With this approach each season, each quadrant is always covered, so the portfolio is always protected. Let us imagine, then, four portfolios, each with an equivalent amount of risk. This means that we will not have exposure to any particular environment. We are not trying to predict the future, because no one knows what the future will bring [12, 22–24]. What is known is that there are only four potential seasons that we will have to face. Using this investment strategy, we can know that we are protected - not just hopeful - and

that the investments are safe and will perform well in any season that comes.

"All Seasons": today we can structure a portfolio that will behave well in 2029, even if we have no chance of knowing what the world will look like in 2029. Below is a table that shows the four potential seasons and the type of investment that will perform best in each of these environments, categorizing each of them in each of

The original "All Seasons" is composed by equity, bonds and commodities which became a popular asset over the past decade [25]. [26] argues that MPT is the formula of diversification, which selects a collection of assets that has collectively lower risk than individually. In sum, for a given amount of risk, MPT describes how to select a portfolio with the highest possible expected return [27, 28]. Below it is

**GROWTH INFLATION**

Commodities/gold

Treasury bonds Equities

TIPS

1.Inflation higher than expected (rising prices)

2.Inflation lower than expected (deflation)

3.Economic growth higher than expected

4.Economic growth lower than expected

announcement drift (PEAD) phenomenon [15–17].

**134**

**Table 2.**

the seasons (**Table 2**).

RISE ↑ Equities

 *List of assets for each "season".*

FALL ↓ Treasury bonds

presented, in **Figure 2**, the efficient frontier.

Treasury Inflation-Protected Securities (TIPS)

Corporate bonds Commodities/gold

**Figure 2.** *Efficient frontier. Source: [29].*

The hyperbola is sometimes referred to as the "Markowitz Bullet" and is the efficient frontier if no risk-free asset is available. With a risk-free asset, the straight line is the efficient frontier.

The Capital Asset Pricing Model (CAPM), for example, was the next step, it approached the risk of an individual asset through the diversification theory [30].

Based on this theory background and MPT, we present in the next chapter six different portfolios aiming to a certain risk, produce the maximum return to the investor.
