Theory of profitability

The profit function and its measurement are abstract concepts. Moreover, they are not technical and operational instruments. This generally sows some doubts among outsiders or neophytes in economics and finance because they doubt the relevance of this theory. Therefore, it should be emphasized that the theory of choice in markets adequately reveals the behavior of economic agents in their consumption and investment decisions. Furthermore, the theory of profit gives a full account of the behavior of economic agents in making decisions under risk conditions.

Experimental analyses of the behavior of individuals and empirical studies of their preferences corroborate that more is always preferred to less. Moreover, as the theory proposes, the personal satisfaction of each additional unit of consumption or of a peso gained produces a decreasing satisfaction in the case of investments. This is consistent with the postulates of choice and profit theory.¹

Likewise, empirical evidence also confirms the fact that economic agents are mostly risk averse. Moreover, it is only necessary to recall that many individuals buy insurance against car theft, burglary, or damage to their homes, which reveals an attitude of risk aversion because the insurance premium is always higher than the expected loss to cover the insurer's costs and profits. In developing countries, the purchase of insurance is not very widespread, but this is not due to a passion for risk but to the very low income of a large part of the population. Nevertheless, both developed and developing country agents show an aversion to risk in experimental studies. When offered constant returns with different levels of risk, people invariably choose the lower-risk options.

Finally, even if the investor or portfolio manager does not believe that it is possible to derive profit functions, the principles of this theory offer much insight into the rational decision-making process. It clearly explains why certain investments and investment portfolios are accepted or rejected.

The presence of a paradox is worth noting: if rationality is assumed, why does the same individual buy both insurance and lottery tickets? This preference for lotteries and games of chance contradicts the theory of choice and profit postulates since its expected return is negative and reveals an attitude of passion for risk. ^{Friedman and Savage (1948) 2} suggest that the profit curve takes different forms according to the income levels of individuals. The profit curve is composed of three segments³: it tends to be concave downward for low-income levels, concave upward for middle-income levels, and again concave downward for high-income levels (decreasing, increasing, and decreasing marginal profit, respectively).

Figure 1 shows in A low-income levels where individuals present a downward concave profit curve and are risk-averse but like to participate in random games if such games can shift their wealth to PB and higher levels (where the profit curve is again downward concave). In the second category are individuals who prefer large-scale gambles. In the last segment are high-income individuals who prefer certainty relative to moderate and extreme risks. They prefer small winnings and reasonably safe games with the possibility of small losses. They are risk-averse individuals who seek a premium for taking moderate risks.⁴ This situation is depicted in Figure 2.

Source: created by the authors

Figure 1 Profit curve segmented according to income levels 

Source: created by the authors

Figure 2 Profit Curve 

The theory of profit has been enriched in the last two decades with models considering the complexity of decision making. Several factors may shape individuals' preferences. For example, individuals may be concerned about the stability of their income levels and health; if their health has declined somewhat, they may place less value on their future income because of the impossibility of enjoying it fully. Likewise, instability in income levels alters consumption and savings patterns. Hence, in the face of a possible reduction in future income, many economic agents tend to save more as a preventive measure. This is precautionary saving. Attitudes such as these have been extensively analyzed by psychologists, economists, and actuaries, often in multidisciplinary studies. Analytically, the high-order derivatives of the profit function are examined.

Methodology

Portfolio theory

The following equation expresses the investor's desired total return on their investment portfolio:

The investor's desired real return is the income they obtain from their investment, i.e., they sacrifice their present consumption for the future, investing in the purchase of shares to increase their profit.

The investor's purchasing power loss due to inflation is compensated by a surcharge on the desired real return, equal to the inflation rate (π). The desired real rate of return (r) plus the premium for inflation risk determines the "risk-free rate" (TLR). This rate has the lowest risk of bank default; in Mexico, it is commonly available from Treasury Certificates (CETES).

The investor also wants a premium to compensate for the risks acquired by investing in assets other than risk-free assets, i.e., risky assets. This requires two types of premiums: a premium for market risk and a premium for the intrinsic risk of each company.

Market risk in the investor's profits or losses is due to changes in asset prices in the financial markets. Governmental policy decisions mainly cause this risk. Intrinsic risk is composed of commercial risk and financial risk. Commercial risk is caused by possible negative impacts that affect the efficiency of each company. Financial risk is caused by possible negative impacts on the yields of financial assets.

Merton mean-variance portfolio model

Based on the abovementioned characteristics of an investment, ^{Merton (1972)} revolutionized financial thinking. He contributed with a model for the construction of optimal investment portfolios by minimizing the risk of the portfolio subject to a given level of the portfolio's expected return. The solution to this problem is known as the mean-variance portfolio. Analyzing the construction of Merton's model, it turns out that the expected return of the portfolio is the weighted average of the expected returns of the assets that make up the portfolio:

The portfolio's risk is measured by its standard deviation with respect to its expected return. ^{Luenberger (1998)} states that the variance of portfolio returns is the weighted average of the covariances of all the pairs included in the portfolio, which is expressed as:

Analogously, the variance-covariance matrix can be decomposed using

Therefore, the estimation of the variance of the portfolio can be carried out using

It is worth mentioning that this decomposition of the variance and covariance matrix is the one that will be used to incorporate the copula and GARCH models. The copula models will be incorporated through the correlation parameters to form the correlation matrix, and the GARCH models will be incorporated through the volatility parameters to form the volatility matrix.

The expression of Equation (3) in its matrix form is equal to

Therefore, the standard deviation of the portfolio (risk) is

It is necessary to find the optimal weights (w₁, w₂, …,w_n) that minimize the risk for each expected return to find the efficient frontier of the investment portfolios. Thus, the optimization problem posed by Merton is expressed as follows:

subject to:

The restriction is that the weights at least obtain a value of 0 to avoid allowing short sales in the model.

Copula theory

A copula function is a multivariate distribution function defined on the unit cube [0,1]ⁿ × [0,1], with uniformly distributed marginals. The mathematical basis of copula functions is established by ^{Hoeffding (1940)} and is specified by ^{Sklar's Theorem (1959)}.

Sklar's theorem

Let it be an n-dimensional distribution function F with continuous marginal distributions F₁,…,F_n; there exists a unique n-copula C: [0,1]ⁿ→[0,1], such that:

Then, the copula combines the marginals to form a multivariate distribution. This theorem provides a parameterization of the multivariate distribution and a copula construction diagram.

This theorem allows the choice of different marginals and a given dependence structure to be used in the construction of a multivariate distribution⁵. It differs from the traditional way of constructing multivariate distributions, where one has the restriction that the marginals are usually of the same type.

The copula functions to be used in this work are bivariate copulas—copulas composed of only two marginal distribution functions. It is important to note that the marginal distribution functions that will be used to construct the bivariate copula functions will be marginal distribution functions corresponding to the distributional structure that will be proposed through the copulas.

Although there are many copula functions, this paper will only focus on the copulas of the Elliptic family: the Gaussian copula and the t-Student copula.

The main characteristic of these copulas is that they are associated with random variables with a symmetrical multivariate distribution function. Therefore, the contour lines generated by this type of copulas have an elliptical shape.

Gaussian copula

The Gaussian copula is a generalization of a multivariate Gaussian distribution. For the bivariate case Φ denotes the (cumulative) Gaussian distribution, and Φ_ρ,2 denotes the standard 2-dimensional Gaussian distribution with correlation matrix ρ.

The bivariate Gaussian copula with correlation matrix ρ is

As noted above, the marginal distribution functions used in this paper for the estimation of bivariate normal copulas correspond to the distributional structure of the copula, in this case Gaussian marginal.

T-Student copula

The t-Student copula is a generalization of a multivariate t-Student distribution. For the bivariate case, a 2-dimensional t-Student distribution T_2,p,ν with ν degrees of freedom and a correlation matrix ρ is defined as

And its respective bivariate t-Student copula with correlation matrix ρ and ν degrees of freedom is

where T_ν is the univariate t-Student distribution with ν degrees of freedom.

As indicated, the marginal distribution functions used for estimating the bivariate t-Student copulas correspond to the copula's distributional structure, in this case, t-Student marginals.

Estimation of copula parameters

The estimation of the parameters of the Elliptic copulas proposed in this work will be through the maximization of their log-likelihood function.

The log-likelihood function is defined as

where θ is the set of parameters of both the marginals and the copula.

The log-likelihood function is maximized, and the maximum likelihood estimator is obtained:

The estimated copula parameter (dependence parameter) will serve as the correlation parameter in the Merton portfolio. Thus, two correlation matrices will be generated, one with the parameters estimated via the Gaussian copula and the other with the parameters estimated via the t-Student copula.

GARCH model

^{Bollerslev (1986)} extends the ARCH model, proposing the GARCH model⁶. The generalized autoregressive conditional heteroscedasticity (GARCH) model estimates that the conditional variance depends on two factors: the innovations of the squared residuals ε² _t-1, ε² _t-2, ε² _t-p, known as the ARCH effect, with their respective autoregressive coefficients; and the prior conditional variances σ² _t-1, σ² _t-2, σ² _t-p, known as the GARCH effect, with their respective autoregressive coefficients; an intercept term, γ₀; and an error term, u_t..

The GARCH model is expressed as

The GARCH model exemplifies a GARCH (p,q) model, where p and q determine the number of lags. It is necessary to mention that the sum of the autoregressive coefficients of the ARCH and GARCH effects must be less than or equal to 1, i.e., α₀ + α₁ +∙∙∙+ α_p + β₀ + β₁ + ∙∙∙ + β_q ≤ 1, so that the GARCH process is ensured to be stationary and ergodic.

This study uses the GARCH model with errors distributed in two different ways, specifically via the two distributional proposals proposed in the copula models: the Gaussian distribution and the t-Student distribution. Once the conditional variance of both models is obtained, its square root is extracted to obtain the adjusted standard deviation (volatility) that will be used in the configuration of the variance and covariance matrices that will be used in the configuration of the adjusted portfolios.