Empirical evidences on the interconnectedness between sampling and asset returns' distributions

Orlando, Guiseppe; Bufalo, Michele Article Empirical evidences on the interconnectedness between sampling and asset returns' distributions Risks Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Orlando, Guiseppe; Bufalo, Michele (2021) : Empirical evidences on the interconnectedness between sampling and asset returns' distributions, Risks, ISSN 2227-9091, MDPI, Basel, Vol. 9, Iss. 5, pp. 1-35, https://doi.org/10.3390/risks9050088 This Version is available at: https://hdl.handle.net/10419/258176 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ risks Article Empirical Evidences on the Interconnectedness between Sampling and Asset Returns’ Distributions Giuseppe Orlando 1,* and Michele Bufalo 2   Citation: Orlando, Giuseppe, and Michele Bufalo. 2021. Empirical Evidences on the Interconnectedness between Sampling and Asset Returns’ Distributions. Risks 9: 88. https:// doi.org/10.3390/risks9050088 Academic Editor: M. Martin Boyer Received: 9 March 2021 Accepted: 23 April 2021 Published: 8 May 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1Department of Economics and Finance, Università degli Studi di Bari Aldo Moro, Via C. Rosalba 53, 70124 Bari, Italy 2Department of Methods and Models for Economics, Università degli Studi di Roma “La Sapienza”, Territory and Finance, Via del Castro Laurenziano 9, 00185 Roma, Italy; [email protected] *Correspondence: [email protected]; Tel.: +39-080-504-9218 Abstract: The aim of this work was to test how returns are distributed across multiple asset classes, markets and sampling frequency. We examine returns of swaps, equity and bond indices as well as the rescaling by their volatilities over different horizons (since inception to Q2-2020). Contrarily to some literature, we find that the realized distributions of logarithmic returns, scaled or not by the standard deviations, are skewed and that they may be better fitted by t-skew distributions. Our finding holds true across asset classes, maturity and developed and developing markets. This may explain why models based on dynamic conditional score (DCS) have superior performance when the underlying distribution belongs to the t-skew family. Finally, we show how sampling and distribution of returns are strictly connected. This is of great importance as, for example, extrapolating yearly scenarios from daily performances may prove not to be correct. Keywords: return distributions; t-skew; market volatility; correlation; equity markets; bond markets; FX JEL Classification: G10; C10; C20; C16 1. Introduction The aim of this article was to investigate the interconnectedness between sampling and asset returns’ distributions. To this end, we empirically perform a number of analyses across asset classes, markets and for several sampling frequencies. The topic is quite relevant as both vendors and financial institutions may rely on scenarios generated under the assumption that financial series returns are normally distributed. There are some works which claim that standardized daily returns “are approximately unconditionally normally distributed” Andersen et al. (2001) or that “are IID Gaussian, with variance equal to 1” Rogers (2018). A more realistic work hypothesis is that time series follow a t-skew distribution. The t-skew distribution can be seen as a mixture of skew-normal distributions Kim (2001) which generalize the normal distribution thanks to an extra parameter regulating the skewness. By construction, then, they can model heavy tails and skews that are common in financial markets. Thus, their adoption in finance is gaining momentum for modeling distributions Harvey (2013) and risk Gao and Zhou (2016). Further, t-skew has the power to link-up with observation-driven models such as the dynamic conditional score (DCS) Creal et al. (2013) or based on data partitioning Orlando et al. (2019,2020). This paper tries to help in gaining insights on returns’ distributions and on the most suitable way of fitting them. In particular, according to the tests carried out on our dataset, the distributions of log-returns do not seem to be normally distributed. The same applies on the returns standardized by the standard deviation. In a different context, Tiwari and Gupta (2019) found that the Jarque–Bera test strongly rejects the hypothesis of Gaussian distribution for all considered time series concerning G7 stock markets. Therefore, through the paper, Risks 2021,9, 88. https://doi.org/10.3390/risks9050088 https://www.mdpi.com/journal/risks Risks 2021,9, 88 2 of 35 we report a number of tests to decide the better distribution between Gaussian, t-skew, generalized hyperbolic, generalized Pareto and exponential Pareto. In terms of applications, being able to correctly identify the distribution is important for risk management as the tail conditional expectation provides information about the mean of the tail of the loss distribution, “while the tail variance (TV) measures the deviation of the loss from this mean along the tail of the distribution” Eini and Khaloozadeh (2020). Another application is in option pricing. For instance, Yeap et al. (2018) propose a t-skew model with “a fat-tailed, skewed distribution and infinite-activity (pure jump) stock dynamics, which is achieved through modeling the length of time intervals as stochastic”. Having described the framework of our investigation, now we are in position to perform some tests on swaps, equities (for both developed and emerging markets) and corporate bonds (for both developed and emerging markets) sampled on weekly, monthly and yearly basis. Section 2contains a literature review, Section 3describes the dataset and the methods we intend to adopt for our analysis, Section 4reports the results we obtained on the original data as well as on the time series of the rescaled returns, the last Section 5 draws the conclusions. 2. Literature Review Distribution of returns is important because econometric models depend upon specific distributional assumptions, and in the case of implied volatilities, on further assumptions concerning the distributional and dynamic properties of stock market volatility. Wrong assumptions call into question the robustness of findings based on those models. One may always opt for an alternative approaches such as those based on squared returns over a given horizon, that provides model-free unbiased estimates of the ex-post realized volatility. Unfortunately, however, “squared returns are also a very noisy volatility indicator and hence do not allow for reliable inference regarding the true underlying latent volatility” Andersen et al. (2001). To overcome such limitations, Andersen et al. (2001) suggested a model free volatility estimate by summing squares and cross-products of intraday highfrequency returns. That approach, however, relies on a reliable high-frequency return observation which, often is not guaranteed. Moreover, it is not necessarily true that the characteristics of a time series are independent on the time horizon and the sampling frequency so that, for example, one may extrapolate seeminglessly from daily data monthly or yearly distributions. Furthermore, time horizon and sampling frequency not only may influence the moments of a given distribution of returns but, also, the way in which data are hierarchically and spatially organized Tumminello et al. (2007). According to McNeil et al. (2015), a bivariate t-Student distribution can describe a pair of daily stock returns. In fact, multivariate stock returns could be modeled by the means of a t-Student copula Aas et al. (2009) and Nikoloulopoulos et al. (2012). However, t-Student imposes symmetric dependence on the joint upper and lower tails which contrasts with empirical studies, e.g., Ang and Chen (2002); Longin and Solnik (2001); Patton (2006). Among alternatives, parametric approaches skew normal distributions, as introduced by Azzalini (1985) and Henze (1986), have became quite popular because they suit well in modeling skewed data defined as follows f(z;θ) = 2ϕ(z)Φ(zθ)z∈R, (1) where θ is the parameter controlling the skewness and ϕ and Φ denote the N( 0, 1 ) are the standard normal density and the normal cumulative distribution, respectively. By further enhancing those distributions, Kim (2001) proposes a family of t-skew distributions in terms of a scale mixture of skew-normal distributions. The random variable X is said to be t-skew distributed with parameter θ and ν if its probability density function is f(x;θ,ν) = 2E[z1/2 ϕ(z1/2x)Φ(z1/2θx)|z]x∈R, (2) Risks 2021,9, 88 3 of 35 where z∼Γ(ν/ 2, 2 /ν) , ϕ and Φ are the standard normal density and the normal cumulative distribution, respectively. The salient features of the family of t-skew distributions are their mathematical tractability, the inclusion of the normal law and the shape parameter regulating the skewness, apart from the ability of fitting heavy-tailed and skewed data, scale mixtures of skew-normal densities come from a family of t-skew distributions. Such an extension allows for a continuous variation from normality to non-normality and it has found a number of applications on fitting heavy-tailed and skewed data. Other applications of such distributions are related to copula modeling. Yoshiba (2018) found that the AC t-skew copula describes well the dependence structures of both “unfiltered daily returns and GARCH or EGARCH filtered daily returns of the three stock indices: the Nikkei225, S&P500 and DAX”. This is because financial time series are characterized by asymmetry. For instance, Patton (2006) reported evidence that “the mark–dollar and yen–dollar exchange rates are more correlated when they are depreciating against the dollar than when they are appreciating”. A drawback of t-skew models is that it is more difficult to handle compared to Gaussian distributions and that there is no closed form analytic formula for computing the elements of the expected information matrix. However, numerical methods are available, e.g., Martin et al. (2020). Moreover, the multiple questions related to parametric models and model-free have led to further development of observation-driven models such as the so-called dynamic conditional score (DCS) Creal et al. (2013) where the updating of the score function is a mechanism that acts as a kind of partitioning of the dataset Lavielle and Teyssiere (2006); Orlando et al. (2020). DCS models, often based on t-skew distributions, have been conceived for describing the distribution of returns Harvey (2013) and they find a number applications in finance from forecasting Value at Risk (VaR) and expected shortfall (ES) Gao and Zhou (2016) to FX Ayala and Blazsek (2019), from commercial and residential mortgage defaults Babii et al. (2019) to hedging for crude oil future Gong et al. (2019). For a review, see Blazsek and Licht (2020). 3. Data and Methods 3.1. Data In order to have a representative dataset, we diversified the investigation across asset classes (equity, bonds, swaps), maturity (from 1 month to 10 years), issuer (government, corporate), market (developed, emerging). In Table 1, we report the data as retrieved from Ice Data Indices and Bloomberg. Risks 2021,9, 88 4 of 35 Table 1. Dataset. Index Code Description Asset Class Market Time Frame a USBAAC USD Basis Swap 1Mv3M Swap Developed 12 February 2007–30 March 2020 b SPX S&P 500 Equity Developed 30 December 1927–29 May 2020 c IBOV Bovespa Equity Emerging 5 January 1927–29 May 2020 d BAMLCC0A2AATRIV AA US Corp.TR Bond Corporate Developed 23 December 1988–29 May 2020 e BAMLEM1BRRAAA2ACRPITRIV AAA-A Em. Mkt Corp TR Bond Corporate Emerging 8 January 1999–29 May 2020 f DGS3MO 3-M Treasury Const. Mty Bond Government Developed 11 January 1982–25 May 2020 g DGS10 10-Y Treasury Const. Mty Bond Government Developed 8 January 1962–25 May 2020 a: USD Basis Swap 1Mv3M (Bloomberg ticker USBAAC) returns which is a swapping 1 month (reference index US0001M) versus 3 months (reference index US0003M), taken from 12 February 2007 to 30 March 2020; b: S&P 500 index returns, taken from 30 December 1927 to 29 May 2020; c: Bovespa index returns, taken from 5 January 1990 to 29 May 2020; d: ICE BofA AA US Corporate Index Total Return Index Value [BAMLCC0A2AATRIV], taken from 23 December 1988 to 29 May 2020; e: ICE BofA AAA-A Emerging Markets Corporate Plus Index Total Return Index Value [BAMLEM1BRRAAA2ACRPITRIV], taken from 8 January 1999 to 29 May 2020; f: 3-Month Treasury Constant Maturity Rate [DGS3MO], taken from 11 January 1982 to 25 May 2020; g: 10-Year Treasury Constant Maturity Rate [DGS10], taken from 8 January 1962 to 25 May 2020. Risks 2021,9, 88 5 of 35 3.2. Volatility Rescaled Returns Following Rogers (2018), we test if the rescaled log returns series of our dataset are or not normally distributed. In order to rescale the (standardized) returns Xi, we let e Xi+1=Xi+1 bσi+1 (3) where bσ2 i+1=βY2+ (1−β)bσi with Y=max(−Kbσi,min(Kbσi,Xi)), and bσ2 0=1 n n ∑ i=1 X2 i. Firstly, we set the parameters (K , β) equal to ( 4, 0.025 ) , as suggested by the author. In a second moment, we compute (K∗,β∗)by solving the following optimization problem (K∗,β∗) = arg min (K,β)∑ iFi−b Fi(K,β)2, (4) where b F is the empirical CDF of the series defined in (3) and F is the (standard) normal CDF (evaluated at the same points of b F). 3.3. Methods In this section, we describe the statistical procedure that will be carried out in previous Section 3.1 on different return time series. For each series, we analyze weekly, monthly and yearly. As yearly data may display high levels of autocorrelation that can alter model’s forecasts, we randomly shuffle those returns and we check their properties as well. Among the analysis we perform, we mention the moments, the histograms and the socalled quantile-quantile (Q-Q) plot Wilk and Gnanadesikan (1968) where we consider the normal distribution versus the t-skew distribution, etc. 3.3.1. Analysis on the Normality of Returns Kolmogorov–Smirnov Normality Test To confirm evidence on the graphical analysis resulting from the (Q-Q) plot, we use the Kolmogorov–Smirnov normality (K-S) test Kolmogorov (1933); Stephens (1992). It is a nonparametric test of the equality of probability distributions that can be used to compare a sample with one reference probability distribution (one-sample K–S test). The Kolmogorov–Smirnov statistic is D=sup x∈R|b F(x)−F(x)|, where b F(x) , F(x) are the empirical distribution function and the theoretical distribution chosen as benchmark, respectively. For large n (being n the sample size), the null hypothesis H0(i.e., the sample is drawn from the reference distribution) is rejected at level αif D>r−ln(α/2) n. Notice that in our tests we use α=0.01, as usual. Risks 2021,9, 88 6 of 35 Dvoretzky–Kiefer–Wolfowitz Bounds A second comparison between the empirical and the normal CDF for a given time series, is based on the Dvoretzky–Kiefer–Wolfowitz (DKW) inequality Dvoretzky et al. (1956) . To assess how close the above-mentioned CDFs are, given ε> 0, one has to solve the following one-sided estimate Psup x∈Rb F(x)−F(x)>ε≤e−2nε2∀ε≥rln 2 2n, which also implies a two-sided estimate Psup x∈R|b F(x)−F(x)|>ε≤2e−2nε2∀ε>0. This strengthens the Glivenko–Cantelli Theorem Tucker (1959) by quantifying the rate of convergence as n goes to infinity; it also estimates the tail probability of the Kolmogorov–Smirnov statistic D. The interval that contains the true CDF F(x) , with probability 1 −α is often specified as [b F(x)−ε,b F(x) + ε], where ε=rln(2/α) 2n. Finally we introduce the variable, named "DKW exceeds", which enumerates the percentage of points of Fthat exceed the DKW upper and lower bounds. 3.3.2. Comparison with Other Distributions In order to enforce our thesis, we compare the t-skew distribution with the following distributions. • Generalized hyperbolic (GH) distribution, f(x;λ,α,β,δ,µ) = γλ √2π δλ+1γKλ·αλ+1/2pδ2+ (x−µ)2Kλ−1/2 (δ2+ (x−µ)2)1/4−λ/2 (x∈R), (5) where β∈R is the asymmetry parameter, δ∈R is the scale parameter, µ∈R is the location, γ=pα2−β2 , α∈R , (α2>β2) and Kλ(λ∈R) denotes the modified Bessel function of the second kind. • Generalized Pareto (GP) distribution, (1+ξx−µ σ)−(1/ξ+1) σ(x>µ), (6) where ξ∈R is the shape parameter, σ∈R+ is the scale parameter and µ∈R is the location. • Exponential distribution, obtained by the GP distribution (6) when ξ=µ=0. 3.3.3. Analysis on the Autocorrelation Ljung-Box Q-Test Generally, one can assess the presence of autocorrelation at a given lag by the sample autocorrelation function (ACF) and by the partial autocorrelation function (PACF). Among the more qualitative tests used to detect the autocorrelation, we adopt the Ljung- Box Q-test Ljung and Box (1978) and the ARCH test Engle (1982). The null hypothesis of the Ljung-Box Q-test is that the first m autocorrelations are jointly zero, i.e., H0:ρ1=ρ2=... =ρm=0. Risks 2021,9, 88 7 of 35 Hassani and Silva (2015); Hassani and Yeganegi (2019) warn on the sensitivity of the test to large values of m . In our case we avoid the problem by setting m=ln(n) , where n is the sample size. The Ljung-Box test statistics are given by Q(m) = n(n+2) m ∑ h=1 ρ2 h n−h; (7) it follows a χ2 mdistribution. ARCH Test An uncorrelated time series can still be seriously dependent because of the dynamic conditional variance process. A time series whose squared residuals exhibit conditional heteroscedasticity or autocorrelation is said to have autoregressive conditional heteroscedastic (ARCH) effect. The ARCH test is a Lagrange multiplier test to assess the significance of ARCH effects. Under the assumption that the squared residuals e2 t follow an AR(m) process, i.e., e2 t=a0+ n ∑ h=1 ahe2 t−h+εt, being εta white noise, the ARCH test null hypothesis becomes H0:a0=a1=... =am=0. One way to choose m is to compare log-likelihood values for different choices of m , e.g., the likelihood ratio test or AIC-BIC information criteria. 3.3.4. Analysis on the Stationarity KPSS Test In order to understand whether returns follow a t-skew distribution, we investigate about the presence of unitary roots, i.e., about the absence of stationarity, and consequently the persistence of fat tail, that is in contrast with the normal distribution. Among all, we choose the Kwiatkowski, Phillips, Schmidt and Shin (KPSS) test, which assesses the null hypothesis that a (univariate) time series xt is trend stationary against the alternative that it is a nonstationary unit root process. This test uses the stochastic model (xt=yt+θt+ε1,t yt=yt−1+ε2,t, where θ is the trend coefficient, ε1,t is a stationary process and ε2,t is an independent and identically distributed process with mean 0 and variance σ2. The null hypothesis is σ2= 0, meaning that yt is a constant random walk. The alternative hypothesis is σ2> 0 that introduces the unit root in the random walk. The test statistic is ∑n t=1S2 t (τn)2, where St is the partial sum of the (absolute) errors coming from the regression on xt , n is the sample size and τ2is the Newey–West estimate of the long-run variance. Hassani Test The Hassani’s-1/2 Theorem in Hassani (2009); Hassani and Yeganegi (2019) states that the sum of sample autocorrelation function for any stationary time series with arbitrary length n≥2 and lag h≥1 is SACF = n−1 ∑ h=1 ρh=−1 2. (8) Risks 2021,9, 88 8 of 35 Note that ρ(h) for large h tends to one. In fact, if h=n− 1, there is only one sample. Therefore, a rule of thumb is not to evaluate ρ(h)for h>n/3. 4. Empirical Results In this section, we report the statistical properties of the considered times series and then we perform some additional analysis ad detailed in Sec. on the original, averaged and volatility rescaled returns on the USBAAC, respectively. Further analyses for the remaining time series are reported in the Appendix A. 4.1. Statistical Properties and Analysis on Original Log Returns With regard to statistical properties of the considered time series, Table 2summarizes the moments. As expected, log returns are not normally distributed and they are heavily skewed. Moreover, Table 3shows that in most cases there is statistical evidence to reject the null hypothesis of trend stationarity. Finally, for all time series considered SACF =0.5. Table 2. Statistics on returns. Statistical Characteristics of Weekly Returns Mom.\Des. USD Swap 1Mv3M S&P 500 Bovespa AA US Corp.TR Em Mk 3m Tbill 10Y Tbond St. Dev. 0.0480 0.0250 0.0616 0.0054 0.0051 0.0001 0.0262 Mean −7.0055 ×10−50.0011 0.0092 0.0012 0.0012 0.0001 −0.0006 Kurtosis 27.4840 9.6489 19.7571 19.4851 23.2641 2.7099 26.0274 Skew 0.3850 −0.6135 1.5259 −1.3492 −2.2882 0.5302 −1.3782 Statistical Characteristics of Monthly Returns Mom.\Des. USD Swap 1Mv3M S&P 500 Bovespa AA US Corp.TR Em Mk 3m Tbill 10Y Tbond St. Dev. 0.0737 0.0540 0.1302 0.0125 0.0144 3.4106 ×10−40.0591 Mean 3.3382 ×10−40.0042 0.0355 0.0049 0.0047 4.2310 ×10−4−0.0023 Kurtosis 8.3043 12.4736 8.7592 5.7354 39.0341 2.6788 25.9171 Skew −0.6069 −0.4424 1.0511 −0.3696 −3.7233 0.52482 −2.1663 Statistical Characteristics of Yearly Returns Mom.\Des. USD Swap 1Mv3M S&P 500 Bovespa AA US Corp.TR Em Mk 3m Tbill 10Y Tbond St. Dev. 0.2386 0.1954 0.9815 0.0504 0.0505 0.0040 0.2029 Mean −0.0046 0.0526 0.4455 0.0602 0.0588 0.0052 −0.0170 Kurtosis 6.3841 6.9968 8.3539 2.5708 5.2510 2.0359 7.7485 Skew 0.8520 −1.1589 2.4618 −0.0347 0.1058 0.3025 −1.0258 Risks 2021,9, 88 15 of 35 Table 10. K-S test to detect the normality of the rescaled returns with (K∗,β∗)(see Equation (4)). Normal Index Sampling resp. p-Value DKW Exceeds K∗β∗ USBAAC Weekly 1 1.5841 ×10−533.76% 2.0166 0.3653 Monthly 1 0.0028 6.62% 12.9821 0.3374 Yearly 1 0.0063 1.22% 6.7090 0.1032 Yearly Shuffled 1 2.1026 ×10−10 40.24% 7.6792 0.0053 S&P 500 Weekly 1 1.0061 ×10−744.27% 8.3221 0.1905 Monthly 1 2.0385 ×10−532.34% 9.5246 0.1070 Yearly 1 1.2624 ×10−24 57.09% 5.5153 0.0050 Yearly Shuffled 1 1.0678 ×10−27 73.56% 8.5980 0.0393 Bovespa Weekly 1 0.0029 8.89% 6.7990 0.1540 Monthly 0 0.0260 0% 1.1396 0.1014 Yearly 0 0.5326 0% 5.2612 0.0116 Yearly Shuffled 1 1.3433 ×10−98 80.01% 3.9341 0.0293 US Corp. Weekly 0 0.0626 0% 9.6300 0.1515 Monthly 0 0.2102 0% 7.8913 0.0015 Yearly 0 0.4253 0% 5.5781 0.0040 Yearly Shuffled 0 0.0886 0% 2.3842 0.0948 EmMkt Corp. Weekly 1 0.0016 5.95% 11.7189 0.2936 Monthly 0 0.0109 0% 4.6068 0.3794 Yearly 1 1.3835 ×10−422.16% 5.9900 0.0044 Yearly Shuffled 0 0.0234 0% 4.5304 0.0279 DGS3M Weekly 1 2.3686 ×10−10 49.58% 4.2862 0.0244 Monthly 1 1.1081 ×10−29 64.83% 5.0489 0.0161 Yearly 1 1.4193 ×10−67 70.98% 33.7053 0.0185 Yearly Shuffled 1 2.9770 ×10−48 73.56% 62.4370 0.1358 DGS10Y Weekly 0 0.2669 0% 7.4724 0.1314 Monthly 1 0.0060 0.26% 7.6785 0.1516 Yearly 1 0.0024 2.54% 5.3205 0.0047 Yearly Shuffled 1 4.2387 ×10−740.33% 10.0382 0.0009 5. Conclusions According to the tests carried out on our dataset, the distributions of log-returns do not seem to be normally distributed. The same applies on the returns standardized by the standard deviation. In a different context, Tiwari and Gupta (2019) found that the Jarque–Bera test strongly rejects the hypothesis of Gaussian distribution for all considered time series concerning G7 stock markets. A more realistic work hypothesis is that time series follow a t-skew distribution. The tskew distribution can be seen as a mixture of skew-normal distributions Kim (2001) which generalize the normal distribution thanks to an extra parameter regulating the skewness. By construction, then, they can model heavy tails and skews that are common in financial markets. Thus, their adoption in finance is gaining momentum for modeling distributions Harvey (2013) and risk Gao and Zhou (2016). Further, t-skew has the power to link-up with observation-driven models such as the dynamic conditional score (DCS) Creal et al. (2013) or based on data partitioning Orlando et al. (2019,2020). This paper tries to help in gaining insights on returns’ distributions and on the most suitable way of fitting them. According to the empirical results we reported, the distributions that fit better the data are the t-skew and the hyperbolic Pareto. As the latter is more difficult to handle, this research suggests that the t-skew represents a suitable alternative. That is relevant in terms of policy implications because risk management or option pricing Mininni et al. (2020) should rely on models able to describe fat tails, skewed distributions and jumps in assets’ dynamics Orlando et al. (2018) rather than on Gaussian distributions that may underestimate the extremes (and leave the investors exposed to unexpected losses). For those reasons, regulators and financial institutions should pay particular attention to model risk (i.e., risk resulting from using insufficiently accurate models) when they choose a particular distribution. Risks 2021,9, 88 16 of 35 Last but not least, t-skew models could be used as linkages between financial markets. To this end, Yoshiba (2018) provides a solution for computing the MLE and keeping the correlation matrix positive semi-definite during the optimization process. Author Contributions: Conceptualization, G.O.; methodology, G.O.; software, G.O. and M.B.; validation, G.O. and M.B., formal analysis, G.O. and M.B.; investigation, G.O. and M.B.; resources, G.O. and M.B.; data curation, G.O. and M.B.; writing—original draft preparation, G.O.; writing—review and editing, G.O. and M.B.; visualization, G.O. and M.B.; supervision, G.O.; project administration, G.O. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Restrictions apply to the availability of these data. Data was obtained from Ice Data Indices and Bloomberg are available from the authors with the permission of Ice Data Indices and Bloomberg. Conflicts of Interest: The authors declare no conflict of interest. Appendix A In the following, we report the analysis we have performed on the indices from (b) through (g) of Table 1. Figures A1,A6,A11,A16,A21 and A26 display log-returns histograms. Figures A2,A7,A12,A17,A22 and A27 show monthly and yearly log-returns Q-Q plots. Figures A3,A8,A13,A18,A23 and A28 exhibit yearly windowed and yearly windowed shuffled log-returns Q-Q plots. Figures A4,A9,A14,A19,A24 and A29 show the empirical CDF versus the standard normal CDF. Figures A5,A10,A15,A20,A25 and A30 display log-returns autocorrelations. Tables A1,A3,A5,A7,A9 and A11 report the K-S test to detect the original distribution. Finally, Tables A2,A4,A6,A8,A10 and A12 exhibit the Ljung-Box Q-test and ARCH test to detect autocorrelation. Appendix A.1. Analysis on S&P 500 Index Figure A1. S&P 500 log-returns histograms. Risks 2021,9, 88 17 of 35 -4 -3 -2 -1 0 1 2 3 4 Standard Normal Quantiles -0.2 -0.1 0 0.1 Quantiles of Input Sample QQ Plot of Sample Data versus Standard Normal - Weekly Log Returns -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Quantiles of tlocationscale Distribution -0.4 -0.2 0 0.2 0.4 Quantiles of Input Sample QQ Plot of Sample Data versus T-Skew - Weekly Log Returns -4 -3 -2 -1 0 1 2 3 4 Standard Normal Quantiles -0.4 -0.2 0 0.2 0.4 0.6 Quantiles of Input Sample QQ Plot of Sample Data versus Standard Normal - Monthly Log Returns -0.5 0 0.5 Quantiles of tlocationscale Distribution -0.5 0 0.5 Quantiles of Input Sample QQ Plot of Sample Data versus T-Skew - Monthly Log Returns Figure A2. S&P 500 monthly and yearly log-returns Q-Q plots. -4 -3 -2 -1 0 1 2 3 4 Standard Normal Quantiles -1.5 -1 -0.5 0 0.5 1 Quantiles of Input Sample QQ Plot of Sample Data versus Standard Normal - Yearly Log Returns -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Quantiles of tlocationscale Distribution -2 -1 0 1 2 Quantiles of Input Sample QQ Plot of Sample Data versus T-Skew - Yearly Log Returns -4 -3 -2 -1 0 1 2 3 4 Standard Normal Quantiles -1.5 -1 -0.5 0 0.5 1 Quantiles of Input Sample QQ Plot of Sample Data versus St. Normal - Yearly Shuffled Log Ret. -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Quantiles of tlocationscale Distribution -2 -1 0 1 2 Quantiles of Input Sample QQ Plot of Sample Data versus T-Skew - Yearly Shuffled Log Ret. Figure A3. S&P 500 yearly windowed and yearly windowed shuffled log-returns Q-Q plots. Risks 2021,9, 88 18 of 35 -10 -8 -6 -4 -2 0 2 4 6 0 0.2 0.4 0.6 0.8 1Weekly Log Ret. SP500 Empirical CDF Standard Normal CDF Lower bounds Upper bounds -8 -6 -4 -2 0 2 4 6 8 0 0.2 0.4 0.6 0.8 1Monthly Log Ret. SP500 Empirical CDF Standard Normal CDF Lower bounds Upper bounds -8 -6 -4 -2 0 2 4 6 0 0.2 0.4 0.6 0.8 1Yearly Log Ret. SP500 Empirical CDF Standard Normal CDF Lower bounds Upper bounds -8 -6 -4 -2 0 2 4 6 0 0.2 0.4 0.6 0.8 1Yearly Shuffled Log Ret. SP500 Empirical CDF Standard Normal CDF Lower bounds Upper bounds Figure A4. Empirical CDF versus standard normal CDF for S&P 500 returns. The dotted black lines represent the DKW upper and lower bounds. 0 0.2 0.4 0.6 0.8 1 Sample Autocorrelation Aut. of Weekly Log Ret. SP500 0 5 10 15 20 Lag 0 0.2 0.4 0.6 0.8 1 Sample Autocorrelation Aut. of Monthly Log Ret. SP500 0 5 10 15 20 Lag 0 0.2 0.4 0.6 0.8 1 Sample Autocorrelation Aut. of Yearly Log Ret. SP500 0 5 10 15 20 Lag 0 0.2 0.4 0.6 0.8 1 Sample Autocorrelation Aut. of Yearly Shuffled Log Ret. SP500 0 5 10 15 20 Lag Figure A5. S&P 500 log-returns autocorrelations. Risks 2021,9, 88 19 of 35 Table A1. K-S test to detect the original distribution. The response is a boolean where 0 indicates that there is no evidence to reject the null hypothesis, and the value 1 is the opposite case. Normal t-skew Gen. Hyperbolic Gen. Pareto Exp. Pareto resp. p-Value DKW Exceeds resp. p-Value resp. p-Value resp. p-Value resp. p-Value Weekly 1 1.6502 ×10−19 74.92% 0 0.0482 0 0.7256 1 0 1 0 Monthly 1 8.3615 ×10−958.32% 0 0.1985 0 0.9714 1 0 1 0 Yearly 1 2.7038 ×10−30 68.55% 1 8.6098 ×10−71 0 1 0 1 0 Yearly Shuffled 1 2.7038 ×10−30 68.55% 1 8.6098 ×10−71 0 1 0 1 0 Table A2. Ljung-Box Q-test and ARCH test to detect autocorrelation. The response is a boolean where 0 indicates that there is no evidence to reject the null hypothesis, and the value 1 is the opposite case . Ljung-Box Q-Test ARCH Test m=ln(n)m= (n−1) resp. p-Value resp. p-Value resp. p-Value Weekly 1 1.9720 ×10−51 0 1 0 Monthly 1 0.0031 1 0 1 0 Yearly 0 0.7234 1 0 0 0.2308 Yearly Shuffled 1 0 0 0.9306 1 0 Appendix A.2. Analysis on Bovespa Index Figure A6. Bovespa log-returns histograms. Risks 2021,9, 88 20 of 35 -4 -3 -2 -1 0 1 2 3 4 Standard Normal Quantiles -0.4 -0.2 0 0.2 0.4 0.6 Quantiles of Input Sample QQ Plot of Sample Data versus Standard Normal - Weekly Log Returns -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Quantiles of tlocationscale Distribution -1 -0.5 0 0.5 1 Quantiles of Input Sample QQ Plot of Sample Data versus T-Skew - Weekly Log Returns -4 -3 -2 -1 0 1 2 3 4 Standard Normal Quantiles -0.4 -0.2 0 0.2 0.4 0.6 Quantiles of Input Sample QQ Plot of Sample Data versus Standard Normal - Monthly Log Returns -1.5 -1 -0.5 0 0.5 1 1.5 Quantiles of tlocationscale Distribution -1 -0.5 0 0.5 1 Quantiles of Input Sample QQ Plot of Sample Data versus T-Skew - Monthly Log Returns Figure A7. Bovespa monthly and yearly log-returns Q-Q plots. -4 -3 -2 -1 0 1 2 3 4 Standard Normal Quantiles -2 0 2 4 Quantiles of Input Sample QQ Plot of Sample Data versus Standard Normal - Yearly Log Returns -60 -40 -20 0 20 40 60 Quantiles of tlocationscale Distribution -50 0 50 Quantiles of Input Sample QQ Plot of Sample Data versus T-Skew - Yearly Log Returns -4 -3 -2 -1 0 1 2 3 4 Standard Normal Quantiles -2 0 2 4 Quantiles of Input Sample QQ Plot of Sample Data versus St. Normal - Yearly Shuffled Log Ret. -60 -40 -20 0 20 40 60 Quantiles of tlocationscale Distribution -50 0 50 Quantiles of Input Sample QQ Plot of Sample Data versus T-Skew - Yearly Shuffled Log Ret. Figure A8. Bovespa yearly windowed and yearly windowed shuffled log-returns Q-Q plots. Risks 2021,9, 88 21 of 35 -10 -5 0 5 10 15 0 0.2 0.4 0.6 0.8 1Weekly Log Ret. Bovespa Empirical CDF Standard Normal CDF Lower bounds Upper bounds -4 -2 0 2 4 6 0 0.2 0.4 0.6 0.8 1Monthly Log Ret. Bovespa Empirical CDF Standard Normal CDF Lower bounds Upper bounds -2 -1 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1Yearly Log Ret. Bovespa Empirical CDF Standard Normal CDF Lower bounds Upper bounds -2 -1 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1Yearly Shuffled Log Ret. Bovespa Empirical CDF Standard Normal CDF Lower bounds Upper bounds Figure A9. Empirical CDF versus standard normal CDF for Bovespa returns. The dotted black lines represent the DKW upper and lower bounds. 0 0.2 0.4 0.6 0.8 1 Sample Autocorrelation Aut. of Weekly Log Ret. Bovespa 0 5 10 15 20 Lag 0 0.2 0.4 0.6 0.8 1 Sample Autocorrelation Aut. of Monthly Log Ret. Bovespa 0 5 10 15 20 Lag 0 0.2 0.4 0.6 0.8 1 Sample Autocorrelation Aut. of Yearly Log Ret. Bovespa 0 5 10 15 20 Lag 0 0.2 0.4 0.6 0.8 1 Sample Autocorrelation Aut. of Yearly Shuffled Log Ret. Bovespa 0 5 10 15 20 Lag Figure A10. Bovespa log-returns autocorrelations. Risks 2021,9, 88 22 of 35 Table A3. K-S test to detect the original distribution. The response is a boolean where 0 indicates that there is no evidence to reject the null hypothesis, and the value 1 is the opposite case. Normal t-skew Gen. Hyperbolic Gen. Pareto Exp. Pareto resp. p-Value DKW Exceeds resp. p-Value resp. p-Value resp. p-Value resp. p-Value Weekly 1 4.0876 ×10−19 69.62% 0 0.7870 0 0.9870 1 0 1 0 Monthly 1 8.4152 ×10−731.55% 0 0.6679 0 0.9977 1 0 1 0 Yearly 1 6.7479 ×10−103 81.43% 1 3.7439 ×10−12 1 0 1 0 1 0 Yearly Shuffled 1 6.7479 ×10−103 81.43% 1 3.7439 ×10−12 1 0 1 0 1 0 Table A4. Ljung-Box Q-test and ARCH test to detect autocorrelation. The response is a boolean where 0 indicates that there is no evidence to reject the null hypothesis, and the value 1 is the opposite case . Ljung-Box Q-test ARCH test m=ln(n)m= (n−1) resp. p-Value resp. p-Value resp. p-Value Weekly 1 0 1 0 1 0 Monthly 1 0 1 0 1 1.3679 ×10−4 Yearly 1 0 1 0 1 0 Yearly Shuffled 0 0.8933 0 0.5147 0 0.7809 Appendix A.3. Analysis on US Corporate Index Figure A11. US Corp. log-returns histograms. Risks 2021,9, 88 23 of 35 -4 -3 -2 -1 0 1 2 3 4 Standard Normal Quantiles -0.06 -0.04 -0.02 0 0.02 0.04 Quantiles of Input Sample QQ Plot of Sample Data versus Standard Normal - Weekly Log Returns -0.03 -0.02 -0.01 0 0.01 0.02 0.03 Quantiles of tlocationscale Distribution -0.06 -0.04 -0.02 0 0.02 0.04 Quantiles of Input Sample QQ Plot of Sample Data versus T-Skew - Weekly Log Returns -4 -3 -2 -1 0 1 2 3 4 Standard Normal Quantiles -0.06 -0.04 -0.02 0 0.02 0.04 Quantiles of Input Sample QQ Plot of Sample Data versus Standard Normal - Monthly Log Returns -0.06 -0.04 -0.02 0 0.02 0.04 0.06 Quantiles of tlocationscale Distribution -0.06 -0.04 -0.02 0 0.02 0.04 Quantiles of Input Sample QQ Plot of Sample Data versus T-Skew - Monthly Log Returns Figure A12. US Corp. monthly and yearly log-returns Q-Q plots. -4 -3 -2 -1 0 1 2 3 4 Standard Normal Quantiles -0.2 -0.1 0 0.1 0.2 0.3 Quantiles of Input Sample QQ Plot of Sample Data versus Standard Normal - Yearly Log Returns -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 Quantiles of tlocationscale Distribution -0.2 -0.1 0 0.1 0.2 0.3 Quantiles of Input Sample QQ Plot of Sample Data versus T-Skew - Yearly Log Returns -4 -3 -2 -1 0 1 2 3 4 Standard Normal Quantiles -0.2 -0.1 0 0.1 0.2 0.3 Quantiles of Input Sample QQ Plot of Sample Data versus St. Normal - Yearly Shuffled Log Ret. -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 Quantiles of tlocationscale Distribution -0.2 -0.1 0 0.1 0.2 0.3 Quantiles of Input Sample QQ Plot of Sample Data versus T-Skew - Yearly Shuffled Log Ret. Figure A13. US Corp. yearly windowed and yearly windowed shuffled log-returns Q-Q plots. Risks 2021,9, 88 24 of 35 -15 -10 -5 0 5 10 0 0.2 0.4 0.6 0.8 1Weekly Log Ret. US Corp. Empirical CDF Standard Normal CDF Lower bounds Upper bounds -6 -4 -2 0 2 4 0 0.2 0.4 0.6 0.8 1Monthly Log Ret. US Corp. Empirical CDF Standard Normal CDF Lower bounds Upper bounds -3 -2 -1 0 1 2 3 0 0.2 0.4 0.6 0.8 1Yearly Log Ret. US Corp. Empirical CDF Standard Normal CDF Lower bounds Upper bounds -3 -2 -1 0 1 2 3 0 0.2 0.4 0.6 0.8 1Yearly Shuffled Log Ret. US Corp. Empirical CDF Standard Normal CDF Lower bounds Upper bounds Figure A14. Empirical CDF versus standard normal CDF for US Corp. returns. The dotted black lines represent the DKW upper and lower bounds. 0 0.2 0.4 0.6 0.8 1 Sample Autocorrelation Aut. of Weekly Log Ret. US Corp. 0 5 10 15 20 Lag 0 0.2 0.4 0.6 0.8 1 Sample Autocorrelation Aut. of Monthly Log Ret. US Corp. 0 5 10 15 20 Lag 0 0.2 0.4 0.6 0.8 1 Sample Autocorrelation Aut. of Yearly Log Ret. US Corp. 0 5 10 15 20 Lag 0 0.2 0.4 0.6 0.8 1 Sample Autocorrelation Aut. of Yearly Shuffled Log Ret. US Corp. 0 5 10 15 20 Lag Figure A15. US Corp. log-returns autocorrelations. Risks 2021,9, 88 31 of 35 Table A9. K-S test to detect the original distribution. The response is a boolean where 0 indicates that there is no evidence to reject the null hypothesis, and the value 1 is the opposite case. Normal t-skew Gen. Hyperbolic Gen. Pareto Exp. Pareto resp. p-Value DKW Exceeds resp. p-Value resp. p-Value resp. p-Value resp. p-Value Weekly 1 1.6567 ×10−20 47.48% 1 1.7558 ×10−20 11.46 ×10−20 1 0 1 0 Monthly 1 1.0664 ×10−71 75.69% 1 3.6783 ×10−71 0 1 0 1 0 Yearly 1 2.0644 ×10−286 79.83% 1 2.2533 ×10−26 1 0 1 0 1 0 Yearly Shuffled 1 3.0634 ×10−297 80.24% 1 1.3943 ×10−28 1 0 1 0 1 0 Table A10. Ljung-Box Q-test and ARCH test to detect autocorrelation. The response is a boolean where 0 indicates that there is no evidence to reject the null hypothesis, and the value 1 is the opposite case. Ljung-Box Q-Test ARCH Test m=ln(n)m= (n−1) resp. p-Value resp. p-Value resp. p-Value Weekly 1 0 1 0 1 0 Monthly 1 2.4962 ×10−41 0 1 0.0283 Yearly 1 0 1 0 1 0 Yearly Shuffled 1 3.0097 ×10−50 0.6966 1 0 Appendix A.6. Analysis on 10-Year Treasury Constant Maturity Rate Figure A26. DGS10Y log-returns histograms. Risks 2021,9, 88 32 of 35 -4 -3 -2 -1 0 1 2 3 4 Standard Normal Quantiles -0.4 -0.2 0 0.2 Quantiles of Input Sample QQ Plot of Sample Data versus Standard Normal - Weekly Log Returns -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 Quantiles of tlocationscale Distribution -0.4 -0.2 0 0.2 0.4 Quantiles of Input Sample QQ Plot of Sample Data versus T-Skew - Weekly Log Returns -4 -3 -2 -1 0 1 2 3 4 Standard Normal Quantiles -0.6 -0.4 -0.2 0 0.2 Quantiles of Input Sample QQ Plot of Sample Data versus Standard Normal - Monthly Log Returns -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 Quantiles of tlocationscale Distribution -0.6 -0.4 -0.2 0 0.2 0.4 Quantiles of Input Sample QQ Plot of Sample Data versus T-Skew - Monthly Log Returns Figure A27. DGS10Y monthly and yearly log-returns Q-Q plots. -4 -3 -2 -1 0 1 2 3 4 Standard Normal Quantiles -1.5 -1 -0.5 0 0.5 1 Quantiles of Input Sample QQ Plot of Sample Data versus Standard Normal - Yearly Log Returns -1.5 -1 -0.5 0 0.5 1 1.5 Quantiles of tlocationscale Distribution -2 -1 0 1 Quantiles of Input Sample QQ Plot of Sample Data versus T-Skew - Yearly Log Returns -4 -3 -2 -1 0 1 2 3 4 Standard Normal Quantiles -1.5 -1 -0.5 0 0.5 1 Quantiles of Input Sample QQ Plot of Sample Data versus St. Normal - Yearly Shuffled Log Ret. -1.5 -1 -0.5 0 0.5 1 1.5 Quantiles of tlocationscale Distribution -2 -1 0 1 Quantiles of Input Sample QQ Plot of Sample Data versus T-Skew - Yearly Shuffled Log Ret. Figure A28. DGS10Y yearly windowed and yearly windowed shuffled log-returns Q-Q plots. Risks 2021,9, 88 33 of 35 -15 -10 -5 0 5 10 0 0.2 0.4 0.6 0.8 1Weekly Log Ret. DGS10Y Empirical CDF Standard Normal CDF Lower bounds Upper bounds -15 -10 -5 0 5 0 0.2 0.4 0.6 0.8 1Monthly Log Ret. DGS10Y Empirical CDF Standard Normal CDF Lower bounds Upper bounds -8 -6 -4 -2 0 2 4 0 0.2 0.4 0.6 0.8 1Yearly Log Ret. DGS10Y Empirical CDF Standard Normal CDF Lower bounds Upper bounds -8 -6 -4 -2 0 2 4 0 0.2 0.4 0.6 0.8 1Yearly Shuffled Log Ret. DGS10Y Empirical CDF Standard Normal CDF Lower bounds Upper bounds Figure A29. Empirical CDF versus standard normal CDF for DGS10Y returns. The dotted black lines represent the DKW upper and lower bounds. 0 0.2 0.4 0.6 0.8 1 Sample Autocorrelation Aut. of Weekly Log Ret. DGS10Y 0 5 10 15 20 Lag 0 0.2 0.4 0.6 0.8 1 Sample Autocorrelation Aut. of Monthly Log Ret. DGS10Y 0 5 10 15 20 Lag 0 0.2 0.4 0.6 0.8 1 Sample Autocorrelation Aut. of Yearly Log Ret. DGS10Y 0 5 10 15 20 Lag 0 0.2 0.4 0.6 0.8 1 Sample Autocorrelation Aut. of Yearly Shuffled Log Ret. DGS10Y 0 5 10 15 20 Lag Figure A30. DGS10Y log-returns autocorrelations. Table A11. K-S test to detect the original distribution. The response is a boolean where 0 indicates that there is no evidence to reject the null hypothesis, and the value 1 is the opposite case. Normal t-skew Gen. Hyperbolic Gen. Pareto Exp. Pareto resp. p-Value DKW Exceeds resp. p-Value resp. p-Value resp. p-Value resp. p-Value Weekly 1 1.2613 ×10−19 74.57% 0 0.7075 0 0.9404 1 0 1 0 Monthly 1 2.3517 ×10−639.47% 0 0.93367 0 0.9775 1 0 1 0 Yearly 1 3.2079 ×10−741.77% 1 0.0020 1 0 1 0 1 0 Yearly Shuffled 1 3.2079 ×10−741.77% 1 0.0020 1 0 1 0 1 0 Risks 2021,9, 88 34 of 35 Table A12. Ljung-Box Q-test and ARCH test to detect autocorrelation. The response is a boolean where 0 indicates that there is no evidence to reject the null hypothesis, and the value 1 is the opposite case. 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