# The Financial Market of Indices of Socioeconomic Well-Being

^{1}

^{2}

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^{*}

## Abstract

**:**

## 1. Introduction

## 2. World Development Socioeconomic Well-Being Indicators

#### 2.1. Constructing a Global Dollar Socioeconomic Well-Being Index

_{t}(l), by weighting it with its corresponding GDP/capita, GDP

_{t}(l), to define a US dollar-denominated index of socioeconomic well-being, the dollar socioeconomic well-being index (DWI), for country l at year t, as follows:

#### 2.2. Econometric Financial Modeling of Socioeconomic Well-Being Indices

_{0}, …, 2020, as follows:

_{t}(l)), l = 1, …, 9, is the exponentially transformed DWI for the lth country in year t, and we set l = 10 for the exponentially transformed global DWI. In each country, we model the log returns of the exponentially transformed DWI with an autoregressive AR(1) model:

_{t}= σ

_{t}ϵ

_{t}, and ϵ

_{t}are assumed to be independent and identically distributed (iid) innovations, while ϕ

_{0}and θ

_{1}are new parameters to be estimated. We model the volatility (σ

_{t}) using the best fit from among the time-varying volatility models ARCH(1), GARCH(1,1), and EGARCH(1,1). The GARCH(1,1) model is defined as

_{0}, θ

_{1}, α

_{0}, α

_{1}, and β

_{1}are parameters to be estimated (Bollerslev 1986). The sample innovations, ϵ

_{t}, are iid random variables with zero mean and unit variance (Tsay 2005).

_{t}(l) in Equation (7) using the following univariate models with standard normal iid innovations (Hamilton 2020; Tsay 2005):

- Model 1: AR(1)-ARCH(1);
- Model 2: AR(1)-GARCH(1,1);
- Model 3: AR(1)-EGARCH(1,1).

_{t}

_{31}(1;s), …, R

_{t}

_{31}(10;s)), s = 1, …, S). As a result, we obtain S scenarios for the log returns of the 10 indices for the year 2021. As the innovations are models with an NIG distribution, which captures the tail dependencies of the indices, our overall forecast of the socioeconomic market for 2021 exhibits all the “stylized facts” of a financial market (see Taylor 2011; Cont 2001).

_{2021}). For example, the estimated model for the US DWI log returns is an AR(1)-EGARCH(1,1) (with multivariate NIG sample innovations) model with the following parameters:

_{t}are iid random normal innovations.

## 3. Measuring the Tail Risk of Socioeconomic Well-Being Indices

_{α}is defined as follows:

_{X}(x) is the cumulative distribution function of the log return X.

_{α}(X) (X = −VaR

_{α}(X)) to being less than or equal to its VaR

_{α}(X) (X ≤ −VaR

_{α}(X)). Here, we refer to Y as the global DWI log returns and X as the log returns of the indices for each country in the alternative CoVaR defined in terms of the copulas in Mainik and Schaanning (2014). The CoVaR at the level of α, CoVaR

_{α}(or ξ

_{α}), is defined as

_{Y}and F

_{X}denote the cumulative distributions of Y and X, respectively, and F

_{Y}

_{|X}is the cumulative conditional distribution of Y given X. An extension of CoVaR, the conditional expected shortfall (CoES) for DWI log returns (Mainik and Schaanning 2014) at a level α, is given by

## 4. Regression and Jensen’s Alphas of the Socioeconomic Well-Being Indices with Respect to the Global Socioeconomic Well-Being Index

_{l}and Y are the log returns of the exponentially transformed DWI of country l and the global index, respectively. The terms a

_{l}, b

_{l}, and e

_{l}denote the intercept, gradient, and random error corresponding to the regression line, respectively.

## 5. Efficient Frontier of the Markets of Countries’ Well-Being Risk Measures

_{p}). The targeted return value indicates the investor’s risk tolerance: the higher the value, the more risk the investor is willing to accept. Mean-variance and mean-CVaR optimization (Uryasev and Rockafellar 2001) have the goal of minimizing the portfolio variance σ

_{p}and the portfolio CVaR, denoted by CVaR

_{p}

_{,α}, subject to a preferred expected return by using the variance and CVaR as the risk measure.

_{p},r

_{p}) results in a hyperbola curve known as the “efficient frontier” (EF), which is the region of the portfolio frontier where the projected mean returns exceed r

_{p}. Consider a portfolio consisting of n risky assets with daily return values r

_{p}, with the mean of the expected risk-adjusted returns denoted by E(r

_{p}) and the risk measure denoted by V(r

_{p}). The portfolio optimization can be summarized as follows:

_{p}

_{,0.05}, and CVaR

_{p}

_{,0.01}risk measures to contrast the effects of the central risk and tail risk on optimization and since the standard deviation is not a coherent risk measure. The DWI dynamic and historical indices referred to in Section 2.1 are used to illustrate the EFs of each country.

_{p}

_{,0.05}and CVaR

_{p}

_{,0.01}. There are overall qualitative similarities in the behaviors of the CVaR EFs compared to the mean-variance EFs. While the EFs change “smoothly” and are convex, the variation in the behavior of the EFs is more pronounced under the CVaR risk measures compared to the mean-variance EF.

#### Efficient Frontier and Risk Measures of the Market for Countries with High GDPs

_{p}

_{,0.05}and CVaR

_{p}

_{,0.01}. The behaviors of the CVaR’s EFs of the DWI for countries with high GDPs and those for all countries are comparable. Again, it can be seen that the risk of investing in the DWI for all countries is less than the risk for countries with high GDPs at the same expected return. The variation in the behavior of the EFs is more pronounced under the CVaR

_{p}

_{,0.01}risk measure.

## 6. Pricing Options on Socioeconomic Well-Being Indices

_{t}for a given F

_{t}

_{−1}, as distributed on a real-world probability space (P), as follows:

_{t}is transformed into the risk-neutral probability density using the Esscher transform.

_{t}for a given F

_{t}

_{−1}is distributed on the risk-neutral probability (Q) using the Esscher transformation as follows:

_{t}is the solution to MGF (1 + θ

_{t}) = MGF (θ

_{t}) e

^{rt}

^{′}, and MGF is the conditional moment-generating function of R

_{t}

_{+1}given F

_{t}.

- Fit GARCH(1,1) with NIG innovations to R
_{t}and forecast ${a}_{1}^{2}$ by setting t = 1; - Repeat steps (a)–(d) for t = 3, 4, …, T, where T is the time to maturity of the DWI call option from t = 2:
- Estimate the model parameter ${\theta}_{t}$ using $MGF(1+{\theta}_{t})=MGF\left({\theta}_{t}\right)\text{}{e}^{{r}_{t\prime}}$, where MGF is the conditional moment-generating function of R
_{t}_{+1}given F_{t}on P; - Find an equivalent distribution function for ϵ
_{t}on Q; - Generate the value of ϵ
_{t}_{+1}under the assumption ${\u03f5}_{t}~\text{}NIG(\lambda ,\text{}\alpha ,\text{}\beta ,\text{}\sqrt{{a}_{t}\text{}}{\theta}_{t}+\delta ,\mu )$ on Q; - Compute the values of R
_{t}_{+1}and a_{t}_{+1}using a GARCH(1,1) model with NIG innovations.

- Generate future values of R
_{t}for t = 1, …, T on Q, where T is the time to maturity. Recursively, future values of the DWI are obtained as follows:DWI_{t}= exp(R_{t}) · DWI_{t}_{−1}. - Repeat steps 2 and 3 10,000 (N) times to simulate N paths to compute future values of the DWI.

## 7. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Notes

1 | The study, “Report by the Commission on the Measurement of Economic Performance and Social Progress”, was led by Joseph Stiglitz, Amartya Sen, and Jean-Paul Fitoussi (see Stiglitz et al. 2009). |

2 | The Basel II Accord requires 10,000 scenarios in the generation of future portfolio returns to properly assess the tail risk portfolio of returns (Orgeldinger 2006; Jacobson et al. 2005). |

3 | For the Ljung-Box test, please refer to Ljung and Box (1978). |

4 | Regarding the CVaR as a coherent risk measure, please refer to Acerbi and Tasche (2002). |

5 | The historical regression assumes iid dependent variables. However, our econometric analysis shows that the dependent variables form a time series with characteristics quite different from white noise. The log returns of the socioeconomic well-being indices display heavy-tailed marginal distributions and volatility clustering. Thus, it is essential to employ a time series forecast, as demonstrated above, and conduct OLS and RR regressions on a sample of S = 10,000 Monte Carlo iid scenarios. |

6 | In real financial markets, Jensen’s alpha is generally nonzero, as they often operate in a pre-equilibrium state with price fluctuations (Soros 2015). |

7 | For more information about discrete stochastic volatility-based models, refer to, for example, Duan (1995), Barone-Adesi et al. (2008), and Chorro et al. (2012). |

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**Figure 1.**(

**a**) US development indicators from World Bank reports (IBRD 2022) for 1990−2020 and (

**b**) the US dollar socioeconomic well-being index (DWI) constructed using Equation (3) and the log returns of the exponentially transformed DWI with constraints from Equation (5).

**Figure 2.**(

**a**) Dollar socioeconomic well-being indices constructed using Equation (3) and (

**b**) the global DWI proposed in Equation (4).

**Figure 3.**Robust regression for historical and dynamic log returns in the US. Regression lines for both historical and dynamic data result in upward forecasts.

**Figure 6.**Conditional value-at-risk portfolio optimization: (

**a**) CVaR

_{p}

_{,0.05}and (

**b**) CVaR

_{p}

_{,0.01}EFs.

**Figure 7.**Variation of the weight composition of the Markowitz optimal portfolios along each efficient frontier (as a function of standard deviation): (

**a**) historical portfolio and (

**b**) dynamic portfolio.

**Figure 8.**Variation of the weight composition of the CVaR

_{p}

_{,α}optimal portfolios along each efficient frontier (as a function of α): (

**a**) historical portfolio and (

**b**) dynamic portfolio.

**Figure 9.**Markowitz efficient frontier for all countries and countries with high: (

**a**) historical indices and (

**b**) dynamic indices.

**Figure 10.**Comparing conditional value-at-risk portfolio optimization for dynamic indices: (

**a**) CVaR

_{p}

_{,0.05}and (

**b**) CVaR

_{p}

_{,0.01}efficient frontiers.

**Figure 11.**Option prices for the US DWI at time t with varying strike prices K using a GARCH(1,1) model with NIG innovations: (

**a**) call prices and (

**b**) put prices.

**Figure 12.**Implied volatilities of the US DWI based on the time to maturity (T) and moneyness (M = S/K) using a GARCH(1,1) model with NIG innovations.

**Table 1.**Estimated dynamic model comparison for the log returns of DWIs in Equation (6) based on the AIC and BIC (Model 1: AR(1)-ARCH(1), Model 2: AR(1)-GARCH(1,1), Model 3: AR(1)-EGARCH(1,1)).

Country | Model 1 | AIC Model 2 | Model 3 | Model 1 | BIC Model 2 | Model 3 |
---|---|---|---|---|---|---|

US | 2.4627 | 2.5209 | 2.5960 | 2.6495 | 2.7544 | 2.8498 |

Australia | 1.5981 | 0.8630 | 0.5255 | 1.7849 | 1.0965 | 0.8058 |

Brazil | 1.6695 | 1.7362 | 1.4492 | 1.8563 | 1.9697 | 1.7294 |

China | 1.8168 | 1.4721 | 1.4353 | 1.0037 | 1.7056 | 1.7156 |

Germany | 2.2366 | 2.3033 | 2.1957 | 2.4235 | 2.5368 | 2.4760 |

India | 0.9228 | 0.9895 | 0.5342 | 1.1097 | 1.2230 | 0.8144 |

Japan | 2.7870 | 2.8536 | 2.4123 | 2.9738 | 3.0872 | 2.6926 |

SA | 2.1488 | 2.2155 | 2.1473 | 2.3357 | 2.4490 | 2.4276 |

UK | 2.5711 | 2.6378 | 2.4302 | 2.7580 | 2.8713 | 2.7105 |

Global DWI | 2.4346 | 2.5013 | 2.4220 | 2.6214 | 2.7348 | 2.7022 |

**Table 2.**Comparison of Pearson’s R and left-tail systemic risk measures (VaR, CoVaR, CoES, and CoETL) of the joint densities of the global DWI and DWI of each country at different stress levels for dynamic and historical log returns.

Dynamic Log Return Left-Tail Risk Measures | |||||||||
---|---|---|---|---|---|---|---|---|---|

Country | Pearson’s R | VaR %95 | VaR %99 | CVaR %95 | CVaR %99 | CoES %95 | CoES %99 | CoETL %95 | CoETL %99 |

US | 0.55 | 2.88 | 2.55 | 3.56 | 3.66 | 0.43 | 0.61 | 1.54 | 2.65 |

Australia | −0.25 | 0.52 | 0.39 | 0.52 | 0.66 | 0.47 | 0.65 | 0.31 | 0.53 |

Brazil | −0.63 | 0.96 | 0.52 | 0.70 | 1.24 | 0.47 | 0.65 | 0.59 | 0.96 |

China | −0.76 | 0.76 | 0.40 | 0.62 | 1.08 | 0.47 | 0.65 | 0.36 | 0.76 |

Germany | 0.00 | 1.31 | 0.39 | 0.58 | 1.70 | 0.46 | 0.63 | 0.55 | 1.22 |

India | −0.41 | 2.11 | 1.10 | 1.39 | 2.89 | 0.47 | 0.65 | 1.06 | 2.12 |

Japan | 0.40 | 0.60 | 0.26 | 0.47 | 0.75 | 0.45 | 0.63 | 0.28 | 0.58 |

SA | −0.58 | 1.53 | 0.84 | 1.11 | 2.00 | 0.47 | 0.65 | 0.89 | 1.54 |

UK | 0.24 | 0.67 | 0.37 | 0.47 | 0.80 | 0.45 | 0.64 | 0.43 | 0.66 |

Historical log return left-tail risk measures | |||||||||

US | 0.79 | 2.74 | 4.87 | 4.30 | 5.66 | 1.85 | 3.83 | 1.16 | 4.70 |

Australia | −0.65 | 1.81 | 3.14 | 2.91 | 3.46 | 2.44 | 3.97 | 1.93 | 3.15 |

Brazil | −0.79 | 1.81 | 3.47 | 3.19 | 3.86 | 2.44 | 3.97 | 1.97 | 3.49 |

China | −0.82 | 2.09 | 3.73 | 3.36 | 4.23 | 2.44 | 3.97 | 2.18 | 3.75 |

Germany | 0.04 | 2.35 | 2.80 | 2.67 | 2.97 | 2.44 | 3.97 | 2.35 | 2.80 |

India | −0.72 | 1.42 | 2.64 | 2.34 | 3.05 | 2.44 | 3.97 | 1.47 | 2.65 |

Japan | 0.57 | 2.77 | 2.85 | 2.83 | 2.87 | 2.44 | 3.97 | 2.54 | 2.85 |

SA | −0.77 | 2.28 | 3.69 | 3.42 | 4.06 | 2.44 | 3.97 | 2.39 | 3.70 |

UK | −0.06 | 3.13 | 4.57 | 4.24 | 5.02 | 2.09 | 3.97 | 2.70 | 4.37 |

Regression Type | |||||
---|---|---|---|---|---|

Data Type | RR | ||||

Coefficient | p-Value | Standard Error | RMSE | ||

Historical | Intercept (a) | −0.026 | 0.705 | 0.070 | 0.381 |

Gradient (b) | 0.673 | 0.000 | 0.093 | ||

Dynamic | Intercept (a) | 0.275 | 0.000 | 0.007 | 0.489 |

Gradient (b) | 1.313 | 0.000 | 0.017 | ||

OLS | |||||

Historical | Intercept (a) | −0.008 | 0.937 | 0.093 | 0.512 |

Gradient (b) | 0.862 | 0.000 | 0.125 | ||

Dynamic | Intercept (a) | 0.378 | 0.000 | 0.008 | 0.603 |

Gradient (b) | 1.360 | 0.000 | 0.022 |

**Table 4.**Robust regression for historical and dynamic log returns in the US. Regression lines for both historical and dynamic data result in upward forecasts.

Country | Estimated Gradient | Standard Error |
---|---|---|

US | 1.36 | 0.0174 |

Japan | 0.11 | 0.0025 |

UK | 0.07 | 0.0031 |

Germany | 0.01 | 0.0125 |

Australia | −0.22 | 0.0092 |

Brazil | −0.77 | 0.0095 |

China | −1.04 | 0.0087 |

India | −1.11 | 0.0025 |

SA | −1.14 | 0.0157 |

Country | Historical Index | Dynamic Index |
---|---|---|

Japan | −0.048 | −0.5247 |

Germany | 0.0087 | −0.2615 |

US | −0.0075 | −0.2264 |

UK | 0.0232 | 0.1487 |

Australia | 0.0242 | 0.189 |

China | 0.0558 | 0.1951 |

Brazil | 0.0198 | 0.4666 |

India | 0.0894 | 0.5306 |

South Africa | 0.0404 | 0.5359 |

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## Share and Cite

**MDPI and ACS Style**

Mahanama, T.V.; Shirvani, A.; Rachev, S.; Fabozzi, F.J.
The Financial Market of Indices of Socioeconomic Well-Being. *J. Risk Financial Manag.* **2024**, *17*, 35.
https://doi.org/10.3390/jrfm17010035

**AMA Style**

Mahanama TV, Shirvani A, Rachev S, Fabozzi FJ.
The Financial Market of Indices of Socioeconomic Well-Being. *Journal of Risk and Financial Management*. 2024; 17(1):35.
https://doi.org/10.3390/jrfm17010035

**Chicago/Turabian Style**

Mahanama, Thilini V., Abootaleb Shirvani, Svetlozar Rachev, and Frank J. Fabozzi.
2024. "The Financial Market of Indices of Socioeconomic Well-Being" *Journal of Risk and Financial Management* 17, no. 1: 35.
https://doi.org/10.3390/jrfm17010035