A Socioeconomic Well-Being Index
AA Socioeconomic Well-Being Index
A. Alexandre Trindade ∗ Abootaleb Shirvani † Xiaohan Ma ‡ January 7, 2020
Abstract
An annual well-being index constructed from thirteen socioeconomic factors is proposed inorder to dynamically measure the mood of the US citizenry. Econometric models are fitted tothe log-returns of the index in order to quantify its tail risk and perform option pricing and riskbudgeting. By providing a statistically sound assessment of socioeconomic content, the index isconsistent with rational finance theory, enabling the construction and valuation of insurance-typefinancial instruments to serve as contracts written against it. Endogenously, the VXO volatilitymeasure of the stock market appears to be the greatest contributor to tail risk. Exogenously,“stress-testing” the index against the politically important factors of trade imbalance and legalimmigration, quantify the systemic risk. For probability levels in the range of 5% to 10%, valuesof trade below these thresholds are associated with larger downward movements of the indexthan for immigration at the same level. The main intent of the index is to provide early-warningfor negative changes in the mood of citizens, thus alerting policy makers and private agents topotential future market downturns.
Keywords:
Econometrics; Option Pricing; Risk Budgeting; Stress-testing. ∗ Texas Tech University, Department of Mathematics & Statistics, Lubbock TX 79409-1042, U.S.A.,[email protected]. † Texas Tech University, Department of Mathematics & Statistics, Lubbock TX 79409-1042, U.S.A.,[email protected] (Corresponding Author). ‡ Texas Tech University, Department of Economics, Lubbock TX 79409-1014, U.S.A., [email protected]. a r X i v : . [ ec on . GN ] J a n Introduction
It is an obvious statement that financial market participants dislike disruptions, especially thosethat are not based on economic fundamentals. If an economy is likely to enter a recession, fullyinformed and rational investors would store their wealth in more safe assets, or search for alternativeinvestment opportunities in emerging markets and elsewhere. Although the trough of a recessiontends to materialize slowly enough to give investors time to prepare for it, opportunist or uninformedtraders, by contrast, may decide to ride the bubble until the very end. For example, there werealready signs of crash of 2008 almost one year ago. However, the overwhelming majority of marketparticipants were content in simply riding the bubble in 2007 and early 2008. The tragedy of mostinvestors then was that they decided to upload their positions only 3 to 4 months ahead of thecrash, resulting in a ”crowding effect” where all investors ”rushed out of the door”, like in a bankrun, thus creating a market avalanche. A consequence of this is that market crowding becomes anincreasingly popular research topic. Although the reasons for it are now well known, few had anyidea about its potential severity in 2007 before the Great Recession.Nowadays, potential sharp market downturns are studied very carefully. Regulators institutedvery strict standards with the Basel III Accord, and the capital requirements for firms that are”too big to fail” are now quite severe. Even in high frequency trading, flash-crashes are becomingrare. However, financial market may still experience disruptions, due to, for example, an increasein uncertainty associated with geopolitical events or polarization of peoples opinion (e.g., the recentdiscussion concerning Californias secession from the US), the negative impact of which cannot beperfectly offset by policies. Moreover, policy makers themselves may be as uninformed as privateagents, and typically do not have perfect information on the state of the economy. Policy decisionsthus made may not be effective in stabilizing the financial market, or in serving as a useful signalfor private agents. Last but not least, the mood (sentiment) of economic agents, as exemplifiedby the crowding effect discussed earlier, may play a crucial rule in the stabilization of the financialmarket and the macroeconomy.The objective of this research is to employ state-of-the-art statistically sound financial methodsto construct a (to the best of our knowledge) reliable and dynamic aggregate index based on a varietyof macro and micro economic factors, which will provide a quantitative snapshot assessment of theUS citizenry’s level of socioeconomic content. We term our proposed index the Socioeconomic Well-Being Index (SWBI). While also ”stress-testing” against potential external factors like immigrationand trade imbalance, the index aims to determine the level of future systemic risk, thereby servingas an ”early-warning” mechanism for serious potential undercurrent issues that could precipitatefrom changes in the mood of citizens, thus leading to potential future crises that may be evenmore severe than in 2008. The SWBI assesses the tail risk (due to extreme events) and providesforward-looking distributions for economic risk factors and social well-being downturns.Specifically, the SWBI does this by being based on the log-returns of an equally-weighted linearcombination of factors. Econometric ARMA-GARCH models driven by generalized hyperbolicnoise, satisfactorily capture the serial dependence and estimate the cross-sectional distribution For example, the current yield curve is “bumpy” suggesting that markets may be uncertain about future monetarypolicy. We do not employ behavioral finance as it is not consistent with rational finance, and is thus unable to aid inthe construction of insurance-type financial instruments to serve as (financial) contracts written against the index. In contrast to many existing indices, the SWBI includes not only economic, but also diverse data related to socialwell-being.
2f the data, as predicted by contemporary best-practices financial theory (Massing, 2019). Thisframework also allows for the generation of Monte Carlo based future price scenarios, leading tooption pricing and risk budgeting for the SWBI. This enables the construction and valuation ofinsurance-type financial instruments. Among the component series of the SWBI, the VXO volatilitymeasure of the stock market is the greatest contributor to tail risk. Completing the rational finance-based valuation, stress-testing of the SWBI against external factors like trade imbalance and amountof legal immigration, quantifies the level of systemic risk. In this regard we find that the level oftrade imbalance tends to be associated with a larger impact on negative well-being than doesimmigration.There have been several recent noteworthy attempts by academics and non-academics aliketo quantify the well-being of the nation, and/or to explore its implications on economic perfor-mance. Most studies approach this issue via surveys, thereby producing subjective indicators ofcontentment. For example, the Gallup-Sharecare Well-Being Index is constructed from monthlytelephone interviews on how people perceive and experience their daily lives through five perspec-tives: purpose, social, financial, community, and physical. The Gallup-Healthways Well-BeingIndex interviews 1,000 US adults daily to provide real-time measurement of health and well-being.The World Happiness Report combines various global household surveys to construct three mainhappiness measures: life evaluations, positive effect, and negative effect. The National Accountsof Time Use and Well-being is used by Krueger (2009), among others, to construct the subjectivewell-being of nations.The index we propose here differs from the above measures by being based on time series ofimportant macroeconomic aggregates, which would arguably be more comprehensive, as well asimmune from possible response bias associated with subjective surveys . A strand of the literaturehas already focused on such socioeconomic factors affecting people’s well-being. Blanchflower andOswald (2004) documents happiness trends in the US and Great Britain, and quantitatively esti-mates the dollar values of events like unemployment and divorce as sources influencing happiness.Ferrer-i Carbonell and Frijters (2004) develops a conditional estimator for the fixed-effect orderedlogit model to re-evaluate the micro-level determinants of happiness. Di Tella et al. (2003) showsthat macroeconomic movements, such as gross domestic product and unemployment benefits, havesignificant impact on national well-being.The main contribution of our paper, however, is in constructing a historical national well-beingindex based on financial econometric modeling and dynamic asset pricing theory. It can thereforecan be used for macroeconomic forecasting and the issuing of marketable financial contracts, suchas options and futures.The rest of the paper is structured as follows. Section 2 describes the set of socioeconomicfactors to be used to quantify the index, the construction of which is detailed in Section 3. This isfollowed by econometric time series modeling of the index and its marginal density estimation inSection 4, this being a prerequisite step for the option pricing and risk budgeting steps in Sections5 and 6, respectively. The paper ends with a stress-testing analysis in Section 7, where the effectof external adverse socioeconomic factors on the tail risk is examined. A discussion rounds out thepaper. See McLean (2014) for a survey of national and international indices of well-being. Data Description
The variables we select are those that have been shown in the literature to affect the well-being ofeconomic agents, such as in Krueger (2009), Blanchflower and Oswald (2004), Di Tella et al. (2003),Ferrer-i Carbonell and Frijters (2004), among others. Abbreviated names for our list of 13 factorsas well as their precise description is as follows. (We append the prefix
Neg to some of these if wewish to also consider the reversed-sign version.)
Confidence . The Consumer Confidence Index provides an indication of future developments ofhousehold consumption and savings, calculated based upon answers regarding their expectedfinancial situation, their sentiment about the general economic situation, unemployment, andcapability of savings.
CPI . Inflation as measured by consumer price index (CPI), is the growth rate of CPI for all urbanconsumers, which is a measure of the average change rate in the price for goods and servicespaid by urban consumers between any two time periods. (Reversed-sign version:
NegCPI .) CrimeRate . Crime rate represents the estimated amounts of violent crimes per 100,000 people.(Reversed-sign version:
NegCrimeRate .) DispIncome . Disposable income represents real disposable personal income, which is inflation ad-justed personal income after payment of taxes.
GDP . Real GDP (Gross Domestic Product) is the inflation adjusted value of the final goods andservices produced by labor and property located in the United States.
GenderParity . Gender parity is an index measuring the relative access to primary and secondaryeducation for males and females, calculated as the quotient of the number of females to thenumber of males enrolled in the given stage of education. (Reversed-sign version:
NegGenderParity .) GovTrans . Government transfer is the amount of government social benefits provided to the un-employed, also known as unemployment insurance.
Inequality . Inequality is the Gini index of income inequality, measuring household income dis-persion. (Reversed-sign version:
NegInequality .) LifeExpect . Life expectancy indicates the number of years a newborn infant would live if prevailingpatterns of mortality at the time of its birth were to stay the same throughout its life.
Sentiment . The index of Consumer Sentiment is an economic indicator that measures how op-timistic consumers perceive about their financial conditions and the state of the economy,constructed based on survey questions in the Survey of Consumers.
Uncertainty . Uncertainty indicates the US policy uncertainty index based on newspaper coveragefrequency, the increase of which is found to foreshadow declines in investment, output, andemployment in the United States. (Reversed-sign version:
NegUncertainty .) Unemploy . Unemployment represents the number of unemployed as a percentage of the labor force(people 16 years of age and older, who currently do not reside in institutions, and who arenot on active duty in the Armed Forces). (Reversed-sign version:
NegUnemploy .)4 XO . The VXO index is the CBOE S&P 500 Volatility Index, calculated by the Chicago Board Op-tions Exchange (CBOE), and measuring the overall short-term volatility in the stock market.(Reversed-sign version:
NegVXO .)Of these, real GDP, inflation, unemployment, government transfer, life expectancy, disposableincome, and VXO, are obtained from the data set of the Federal Reserve Bank of St. Louis.Inequality is obtained from the US Census Bureau. Crime rate is calculated by the FBI. Theuncertainty index is developed by Baker et al. (2016). Consumer confidence is provided by OECD(Organization for Economic Co-operation and Development) in its publication of Main EconomicIndicators: Business tendency and consumer opinion surveys. Gender parity index is obtained fromUNESCO (The United Nations Educational, Scientific and Cultural Organization).The common period of these 13 yearly factors is 1986-2016. Figures 1a and 1b display time seriesof their actual values. They seem to be of two distinct types:
GDP , CPI , Inequality , LifeExpect , CrimeRate , and
DispIncome display trending behavior, while the remaining 7 appear to be sta-tionary. (However, at the 5% level of significance the Augmented Dickey-Fuller test does not rejectthe null hypothesis of unit-root nonstationarity for any of the series.) G D P C P I . . . . . . I nequa li t y Time L i f e E x pe c t C r i m e R a t e D i s p I n c o m e Time (a) Trending. U ne m p l o y G o v T r an s f e r U n c e r t a i n t y S en t i m en t Time C on f i den c e . . . . . G ende r P a r i t y VX O Time (b) Stationary.
Figure 1: Time series of actual values for the factors with trending and with stationary behavior.At this point we examined the existence of cointegration relationships among the series ineach of the two groups. For the trending group, the strong trends coupled with small sample sizedoes not allow for estimation of an underlying VAR model necessary for carrying out Johansen’scointegration test (Johansen, 1988). For the stationary group, we find 2 cointegrating relationshipsat the 5% level of significance based on a VAR(2). (Higher order VAR models could not be fitteddue to the similar issue of collinearity and small sample size.) Figure 2 displays cross-sectionalscatterplots of actual values for all 13 factors. We note that the large degree of collinearity is easilyspotted from the correlations on the upper triangular portion, which are displayed with font sizeproportional to the magnitude. There is therefore the question of whether all series are necessaryfor the construction of the SWBI; an issue to be explored in the next section.5 DP llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . lllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllll
120 200
CPI lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Unemploy llll l llllllllll lllllll l llllllll llll l llllllllll lllllll l llllllll llll l llllllllll lllllll l llllllll llll l llllllllll lllllll l llllllll llll l llllllllll lllllll l llllllll llll l llllllllll lllllll l llllllll llll l llllllllll lllllll l llllllll llll l llllllllll lllllll l llllllll llll l llllllllll lllllll l llllllll llll l llllllllll lllllll l llllllll
20 80 140
GovTransfer lllllllllllllll lllllll l lllllllllllllllllllllll lllllll l lllllllllllllllllllllll lllllll l lllllllllllllllllllllll lllllll l lllllllllllllllllllllll lllllll l lllllllllllllllllllllll lllllll l lllllllllllllllllllllll lllllll l lllllllllllllllllllllll lllllll l lllllllllllllllllllllll lllllll l llllllll
Inequality lllllll lllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllll llllllllllllllllllllllll lllllll llllllllllllllllllllllll
75 77 79
LifeExpect lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll −0.92−0.89 −0.11 −0.51 −0.84 −0.91
CrimeRate llll lllllllllllllllllllllllllll llll lllllllllllllllllllllllllll llll lllllllllllllllllllllllllll llll lllllllllllllllllllllllllll llll lllllllllllllllllllllllllll
400 600 llll lllllllllllllllllllllllllll
60 120 180 −0.25
Uncertainty l lll lllllll lll lllllll l ll l lllllll lll lllllll lll lllllll l ll l lllllll lll lllllll lll lllllll l ll l lllllll lll lllllll lll lllllll l ll l lllllll lll lllllll lll lllllll l ll l llllll −0.069 −0.19 −0.74 −0.61 −0.21 −0.73
Sentiment llllll ll lllllllllllllll llll l lllllllll ll lllllllllllllll llll l lllllllll ll lllllllllllllll llll l lll
60 80 100 llllll ll lllllllllllllll llll l lll
97 100 −0.10 −0.22 −0.75 −0.62 −0.24 −0.70
Confidence ll llll ll ll l llllll llllll lll ll lllll llll ll ll l llllll llllll lll ll lllll llll ll ll l llllll llllll lll ll lll −0.92 −0.15 −0.18
DispIncome lllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllll −0.23 −0.32 −0.19−0.18 −0.33 −0.25 −0.12 −0.26
GenderParity llllll lll ll l lll llllllll lllll lll −0.12−0.16 . −0.18 −0.11 −0.24−0.20 −0.14
15 30
VXO
Figure 2: Cross-sectional scatterplots of actual values for the 13 factors.
In this section we detail the methodology for constructing the SWBI. It employs the 13 factorsdiscussed in the previous section. As is typical of macroeconomic variables, the Augmented Dickey-Fuller Test gives large p-values for all series, confirming that they are all I (1), or integrated oforder one. Thus, and in order to arrive at plausibly stationary series, we transformed the actualvalues to log-returns, i.e., if P ( i, t ) denotes the value of the i -th factor at time t , then r ( i, t ) =log P ( i, t ) − log P ( i, t −
1) denotes its log-return.In order to have positive values equate to greater well-being, we reversed the sign of thelog-return value associated with the following factors:
CPI , Unemploy , Inequality , CrimeRate , Uncertainty , GenderParity , and
VXO . On their original scale, it can be argued that large valuesof these factors would tend to be associated with decreased well-being. We further set the 1986return value to zero for all factors, so that our starting point is the panel of log-returns: { r ( i, t ) } , i = 1 , . . . , N = 13 , t = 1 , . . . , T = 30 . (1)Time series plots of r ( i, t ) for each of i (each factor) are displayed in Figure 3a. For comparison,the group whose sign was reversed is shown in a different color and line type. All series appear tobe stationary now; at least with respect to trends and cycles.The formation of the SWBI value at time t , r t , is now obtained as an equally weighted linearcombination of the r ( i, t ) values for each of the 13 factors at time t . More specifically, we implement6he following algorithm. Algorithm 1 (Formation of SWBI)
Starting from the panel of time series r ( i, t ) in (1) , proceedas follows:(i) For each i = 1 , . . . , N , standardize r ( i, t ) according to its factor-level mean and standarddeviation: R ( i, t ) = r ( i, t ) − m ( i ) s ( i ) , m ( i ) = 1 T T (cid:88) t =1 r ( i, t ) , s ( i ) = 1 T − T (cid:88) t =1 [ r ( i, t ) − m ( i )] . (ii) Form the standardized index return series by weighting equally across all factors: R ( t ) = 1 √ N N (cid:88) i =1 R ( i, t ) . (iii) Form the annual index return series by undoing the standardization in (i) according to theaverage of the factor means and standard deviations: r t = m + s R ( t ) , m = 1 N N (cid:88) i =1 m ( i ) , s = 1 N N (cid:88) i =1 s ( i ) . For ease of reference in subsequent analyses, we call the resulting r t series simply as the Index orSWBI. A time series plot is displayed in the top left panel of Figure 4.At this point, and with plausibly stationary factor log-returns as gleaned from Figure 3a, onecan properly investigate the issue of whether all 13 factors that make up r t are needed. A principalcomponents analysis (PCA) on these series suggests we need almost all factors (correlation matrixbased). Although the 1st PCA explains 32% of the variability, there is a very gradual contributionfrom each additional component, so that it is not until the 9th PCA that we obtain an explanatorycapability from these factors exceeding 95%. Moreover, the weights in each PCA are spread overalmost all factors in each of these first 9 PCAs. In summary, there is not an overwhelmingly clearindication that we should reduce the dimensionality of the vector of 13 factors. In this section we fit time series models to r t . This will allow for evaluation of risk measures andthe performing of option pricing for the Index . Two models were entertained here. Searching forthe best-fitting ARIMA model via AIC and BIC points to an ARIMA(0,0,0) with zero mean, i.e.,white noise. However, since the option pricing relies on an ARMA-GARCH configuration, we fitalso an ARMA(1,1)-GARCH(1,1) with zero mean and normal innovations. The innovations fromthis fit (the raw residuals divided by the GARCH conditional standard deviation estimate) aredisplayed in the top right panel of Figure 4.The generation of multiple scenarios from the (stationary distribution of the) fitted model for r t in order to calculate risk measures, a step we loosely call “forecasting”, can be performed once anappropriate model for the innovations has been determined. The standard parametric family in this7 ear1990 1995 2000 2005 2010 2015 − . . . sign not reversedsign reversed (a) Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8 Comp.9 Comp.11 Comp.13 V a r i an c e s (b) Figure 3: Analysis for the panel of log-returns, r ( i, t ), defined in (1). (a) Time series plots withcolor and line type coded according to whether the sign was reversed (dotted) or not (solid). (b)Screeplot from a PCA giving the proportion of contribution from each component toward explaningthe 13-dimensional correlation matrix.context is the Generalized Hyperbolic (GH) distribution, originally introduced by Barndorff-Nielsen(1977). Jorgensen (1982) is the classical reference on its properties, while an accessible overviewis given by Paolella (2007). Briefly, the GH features both heavy tails and skewness, includes thefamiliar elliptical family as a special case (normal, t, etc.), exhibits tail-dependence (Schmidt, 2007),and is infinitely divisible, a necessary and sufficient condition to build Levy processes (ubiquitousin financial time-series due to their continuity and ability to model jumps). For these and otherreasons it has become the default distribution in modeling the returns of equity indices at varioustemporal scales (Massing, 2019).Notationally, we designate by X ∼ GH ( λ, α, β, δ, µ ) a random variable following a GH withparameters λ ∈ R (tail heaviness), α > β ∈ R (skewness) such that α − β > δ > µ ∈ R (location). (There are at least two alternative parametrizations in commonusage). The density function is derived by mixing normals according to a Generalized InverseGaussian distribution (a special case of GH), and does not therefore in general have a closed-formrepresentation. Software implementation of the GH is provided through the R library ghyp (Luethiand Breymann, 2016) and its accompanying vignette. Two special cases of the GH we focus onin this study are the Variance Gamma (VG), which is a GH with λ > δ = 0, and theNegative Inverse Gaussian (NIG), a GH with λ = − /
2; both of these possessing exponential tails.The latter in particular is pointed out by Barndorff-Nielsen (1997) and Barndorff-Nielsen (2007)as being especially appropriate for stochastic volatility modeling based on a homogeneous Levyprocess, and the option pricing models that can consequently be constructed.Fitting VG and NIG marginal models to the sample of ARMA(1,1)-GARCH(1,1) innovationsseen as the rug and histogram plot in the lower left panel of Figure 4, leads to the VG densitydisplayed as the solid line in that figure (the NIG being an inferior fit). The appropriateness of thefit is confirmed by the Kolmogorov-Smirnov and Anderson-Darling tests with p-values exceeding0.9. Simulating 10,000 scenarios from the overall fitted ARMA(1,1)-GARCH(1,1) model with VG8 he Index r(t)
Year1990 1995 2000 2005 2010 2015 − . − . − . . . . Innovations from ARMA−GARCH fit to r(t)
Time1990 1995 2000 2005 2010 2015 − − Estimated density of innovations −3 −2 −1 0 1 . . . . . kernel densityvariance−gamma Simulated density of r(t) with VG innovations −0.8 −0.6 −0.4 −0.2 0.0 0.2 . . . . . . . . Figure 4: Time series of the
Index r t and its innovations from an ARMA-GARCH fit (top panels).The bottom right panel displays 10,000 scenarios from the marginal of r t based on a VG distributionfitted to the ARMA-GARCH innovations (bottom left panel).innovations, then leads to the marginal of r t seen as the rug and histogram in the lower right panelof Figure 4; the solid line being a kernel density smoother. Summary statistics and left tail riskmeasures for these scenarios are shown on Table 1. Here, the Value-at-Risk (VaR) and expectedshortfall (ES) risk measures are reported (Pflug and Werner, 2007, Ch. 2).Table 1: Summary statistics and left tail risk measures for the 10,000 scenarios generated from theARMA(1,1)-GARCH(1,1) model with VG innovations fitted to the Index .Summary StatisticsMinimum/Maximum Mean/Median Skewness/Kurtosis (excess)-0.8492/0.2894 0.0064/0.0351 -1.181/2.094Left Tail Risk Measures1% VaR/ES 5% VaR/ES 10% VaR/ES-0.4411/-0.5458 -0.2655/-0.3720 -0.1826/-0.29609
Option Pricing
Options can be used for hedging, speculating, and calculating the risk of investments. The mostcommon option pricing models are Black-Scholes, Binomial, Trinomial tree, Monte-Carlo simula-tion, and finite difference. Recently, the discrete stochastic volatility based model has receivedconsiderable attention, particularly with regard to explaining some well-known mispricing phenom-ena. The pricing model in a discrete-time conditional heteroskedasticity setting was first consideredby Duan (1995). He used a GARCH driven by normal innovations for asset returns in order to priceoptions. We follow his strategy by considering a standard GARCH model with GH innovations tocalculate the fair value of an option of our
Index (SWBI). Specifically, we assume the log-returns r t follow the process: r t = log I t I t − r (cid:48) t + λ (cid:112) h t − h t + (cid:112) h t (cid:15) t , (2)where h t = V ( r t | F t − ) is the conditional variance at time t , with F t − denoting the informationset consisting of all linear functions of past returns available up to time t − r (cid:48) t is the risk-less rateof return at time t , and λ is the risk premium for the SWBI. We use a GARCH(1,1) with GHinnovations to model the conditional variance as follows: h t = m + a h t − + b (cid:15) t − , { (cid:15) t } ∼ iid GH ( λ, α, β, δ, µ ) . (3)Blaesild (1981) proved that under this model, the conditional distribution of r t given F t − on the real-world probability space P is distributed as r t ∼ GH (cid:18) λ, α √ h t , β √ h t , δ (cid:112) h t , r (cid:48) t + m t + µ (cid:112) h t (cid:19) , with m t = λ (cid:112) h t − h t . (4)Options are priced on the risk-neutral probability (probability space of future outcomes adjustedfor risk). In an incomplete market to price options, the crucial issue is to identify an equivalent mar-tingale measure to obtain a consistent price for contingent claim. Using the Gerber and Shiu (1994)Esscher transformation, Chorro (2012) proved that, under model (3), the conditional distributionof r t given F t − on the risk-neutral probability space Q is distributed as r t ∼ GH (cid:18) λ, α √ h t , β √ h t + θ t , δ (cid:112) h t , r (cid:48) t + m t + µ (cid:112) h t (cid:19) , (5)where, and with M ( · ) denoting the conditional moment generating function of r t +1 given F t on P , θ t solves the equation M (1 + θ t ) = M ( θ t ) exp { r (cid:48) t } . (6)(See Duffie (2001) for details on the precise definitions of the probability spaces P and Q .)To price the SWBI call option up to a given time to maturity T , we use Monte Carlo simulationto generate future values of SWBI via the scheme implemented by Chorro (2012), as follows:1. Fit the GARCH(1,1) model (3) to the available historical log-returns of SWBI.2. Set t = 0 and forecast h , the conditional variance at time t = 1.3. Starting from t = 1, repeat steps (a)–(c) below under Q for t = 1 , . . . , T :10a) solve (6) to find θ t ;(b) generate (cid:15) t +1 from the stationary distribution of (cid:15) t ∼ GH ( λ, α, β + √ h t θ t , δ, µ );(c) compute r t +1 and h t +1 .4. This scheme yields { r , . . . , r T } under Q . Thus the future price of the SWBI at time T isgiven by I T = I exp (cid:40) T (cid:88) t =1 r t (cid:41) . (7)5. Repeat step (3)–(4) in order to simulate N = 10 ,
000 future price values of the SWBI: { I (1) T , . . . , I ( N ) T } .The approximate call ( ˆ C ) and put ( ˆ P ) option prices at time t ≤ T are then computed, for a given strike price K , as the Monte Carlo averages:ˆ C ( t, T, K ) = e − r (cid:48) t ( T − t ) N N (cid:88) i =1 (cid:16) I ( i ) T − K (cid:17) + , ˆ P ( t, T, K ) = e − r (cid:48) t ( T − t ) N N (cid:88) i =1 (cid:16) K − I ( i ) T (cid:17) + . Figures 5-7 show call option prices, put option prices, and call-put option prices, respectively,plotted against both maturity ( T ) and strike ( K ). The graphs reveal the relationships among timeto maturity, strike, and option prices. As expected, for a fixed K we see a decline in price as T increases. Figure 8 plots the implied volatility surface (the market’s view of the future value ofvolatility) against time to maturity and Moneyness (defined as
S/K , where S is the price of thestock). Again as expected, we observe that as time to maturity decreases, the volatility surfaceincreases, reflecting the fact that a higher level of uncertainty exists regarding whether or not theoption will be exercised. Risk budgets are often used to allocate the risk of a portfolio by decomposing the total portfolio riskinto the risk contribution of each component position. Portfolio standard deviation (Std), Value-at-Risk (VaR), and expected tail loss (ETL) budgets are the most popular strategies used to betterunderstand the center-risk and tail-risk contributions. Chow and Kritzman (2001), Litterman(1996), Maillard et al. (2010), and Peterson and Boudt (2008) studied the use of portfolio Std andVaR in risk budgeting. Boudt et al. (2013) reviewed the ETL budgets. Here, we use the Std andETL risk budgets as an investment strategy under the condition of an equal-weights portfolio. Theequal-weights portfolio is widespread in practice because it does not require information on the riskand return, and supposedly provides a diversified portfolio.Thus, we have the vector of portfolio weights w = ( w , . . . , w N ) at each time point t , where N = 13 and t = 1 , . . . ,
31. We first define the marginal risk and risk contribution of the i th asset inthe portfolio, and then calculate the respective Std and ETL risk contributions. Let R ( w ) : R n → R w , then the marginal risk contribution of the12igure 7: Call and Put-Option prices against time to maturity and strike price.Figure 8: Implied volatility against time to maturity and Moneyness. i -th asset denoted by RC i is RC i ( w ) = w i ∂R ( w ) ∂w i . (8)13he marginal risk contribution of the k -th subset is RC M k ( w ) = (cid:88) i ∈ M k RC i ( w ) , (9)where M k ⊆ { , . . . , N } for each k = 1 , . . . , s , denotes s subsets of portfolio assets.Table 2 reports the estimated risk allocation of the equal-weights portfolio. It seems NegVXO has a relatively higher risk than the other factors. Meanwhile,
GovTrans factor has the lowesttail risk contribution in both cases, and
NegUnemploy has the lowest center risk contributions.Thus, the tail risk diversifiers are
GovTrans , DispIncome , GDP , NegInequality , NegCrimeRate , NegGenderParity , and
LifeExpect . The tail risk contributors are the remaining factors. Notethat
NegVXO is the main risk contributor among all factors.Table 2: Standard deviation and ETL Risk BudgetFactors MCTRETL (95) PCTRETL (95) MCTRETL (99) PCTRETL(99) MCTR(Std) PCTR(Std)GovTrans -0.70% -7.25% -1.17% -9.38% 3.13% 4.75%DispIncome -0.10% -1.07% -0.03% -0.27% -0.12% -0.19%GDP -0.08% -0.83% -0.07% -0.59% 0.22% 0.33%NegInequality -0.06% -0.62% -0.11% -0.87% -0.24% -0.36%NegCrime -0.03% -0.31% 0.04% 0.32% 0.09% 0.13%NegGenderParity -0.03% -0.26% -0.02% -0.19% -0.24% -0.37%LifeExpect -0.02% -0.24% -0.03% -0.23% -0.02% -0.03%Confidence 0.13% 1.36% 0.18% 1.44% 0.73% 1.11%NegCPI 0.32% 3.26% 0.38% 3.08% 0.19% 0.29%NegUnemploy 0.60% 6.14% 1.02% 8.18% -0.31% -0.46%Sentiment 1.33% 13.69% 1.78% 14.27% 7.93% 12.04%NegUncertainty 3.19% 32.84% 4.01% 32.16% 19.87% 30.18%NegVXO 5.17% 53.30% 6.50% 52.09% 34.63% 52.58%
Asset and investment management firms commonly use stress-testing to determine the resilienceof a given portfolio against possible undesirable financial situations (assess the risk), and thenset in place any hedging strategies necessary to mitigate against possible losses. The intent is toevaluate how well the assets might weather certain market occurrences and external events. Thedetermination of appropriate factors that may contribute to these “stressful” events, is in itself adifficult task.In this section we consider stress testing the
Index with the contemporaneously observed factors trade balance ( Trade ) and legal immigration ( Immig ), acting as stressors. The data on
Trade represents the difference between the total value of US export and import of goods and services,based on the Balance of Payments. The annual data is obtained from the US Census Bureau.
Immig is the total number of persons obtaining lawful permanent resident status, and is retrieved from the14epartment of Homeland Security. We choose these two variables as stress factors because theymay be sensitive to policy makers’ decisions, but they should be a priori neutral in that changesin their values do not straightforwardly imply a cause-effect relationship with the well-being of UScitizens.The top panels of Figure 9 plots the actual values of the stress factor time series, while thebottom panel shows the log-returns of these as well as the
Index , which are all white noise, asexpected. In fact, a Ljung-Box test on all 3 series of returns does not detect any serial correlation,or even dependence (all p-values larger than 0.5). Thus we opted not to put the series through theARMA(1,1)-GARCH(1,1) filter for this analysis, i.e., the ARIMA(0,0,0) with zero mean suggestedby AIC and BIC in Section 4 was fitted to the returns of all three series:
Index , Trade , and
Immig . Trade (actual values)
Time1985 1990 1995 2000 2005 2010 2015 − + − + − + Immigration (actual values)
Time1985 1990 1995 2000 2005 2010 2015
The Index, Trade, and Immigration (returns)
Time1990 1995 2000 2005 2010 2015 − . − . . . IndexTradeImmigration
Figure 9: Time Series Plots of the
Index and stress factors
Trade and
Immig .Starting from these three plausibly iid series, we proceeded by fitting bivariate GH models totheir joint marginal distributions:
Trade vs.
Index , and
Immig vs.
Index . The models suggested byAIC and BIC had values for the tail parameter of approximately λ ≈ .
4, which is somewhat closeto a VG distribution, although the latter, as well as NIG, provided substantially inferior fits. In15rder to compute the systemic risk measures discussed below, 10,000 simulated values were drawnfrom these models. Figures 10a and 10b display the fitted contour plots from each model, overlaidwith the 10,000 simulated values and the 30 observed data points. As is noted in the figures, theempirical correlation coefficients based on the observed data, suggest a weak positive relationshipbetween
Trade and the
Index ( ˆ ρ = 0 . Immig andthe
Index ( ˆ ρ = − . Trade I nde x . . . . . . . . . . . . . . . . . . . . . . . −1.0 −0.5 0.0 0.5 − . − . − . . . . l ll ll lll lll lll l l ll lll l ll l llll ll ll l l ll lll l ll llll ll l lll llll lll ll lll lll ll lll l ll ll ll ll l lllll lll llllll lll l llll ll llll ll llll l ll ll l ll l llll l l ll ll l ll lll llllll ll llll l ll ll lll ll l lll lll ll l l llll l lll l l l lllll l ll l l llll ll ll ll ll ll ll l l ll ll l ll l ll lll ll ll ll ll lll l ll ll lll ll ll l ll ll lll l ll lll l lll l ll l ll lll l lll l ll l ll ll ll lll l ll l lll l l l ll l ll ll l lllll l lll ll lllll lll llll lll ll l ll lll ll ll l ll l l ll l ll l lll lll l l ll ll lll lll lll ll l ll lll ll ll l lll ll ll lll l ll ll lll ll l lll lll l l ll l lll lll lll lll ll lll l ll ll lll ll l ll ll ll ll l l ll lll lll ll lll ll llll ll llll ll l l l llll ll lllll ll ll ll l lll llll l lll lll ll ll lll l ll ll ll lll ll llll lll lllll l llll l l ll lll llll lll l l ll lll ll lll ll l lllll l llll ll l lllll l lll l l llll lll lllll ll llll l lll ll llll ll llll ll l ll lllll l ll ll l ll lll ll lll ll l ll ll llll ll lll ll llll l l ll ll llll l lllll lll l ll lll l ll ll ll ll l lll ll ll ll l lll l l ll l lll lll l ll l ll lll ll ll l lll ll ll l lll l ll l ll lll l llll ll ll ll ll l ll llll ll l lll lll ll lll ll l l ll ll lll ll ll l lll llll ll llll lll ll ll ll llll l l ll lll ll ll ll llll ll lll ll l ll ll lll l l lll lll lll ll ll l ll lll l ll ll lll ll ll llll l ll ll l ll ll llll lll l lll l ll ll lll l lll ll ll ll ll lll ll ll ll ll l lll l ll l llll l ll lll l ll ll ll lll ll l l ll l ll l ll ll l ll l ll l lll l l lll l ll l ll ll l l llll l lll ll l ll l l lll lll ll l lll ll l l lll l ll l llll lll ll ll l ll lll lllll l l ll llll lll llll ll ll l ll l llll l lll l lll ll ll l l llll l lll ll l l ll ll llll lllll ll ll l ll ll lll l ll lll ll ll llll lll ll l ll ll l l l ll llll l l ll lll lll l lll ll l lll l ll ll ll ll ll l ll ll ll l ll llllll l ll lll lll l lll l lll ll lll ll lll ll l ll lll ll ll lllll ll l lll l l l lll l lll ll lll l ll ll lll l l lll ll lll l lll ll l lll lll ll l ll lll ll l ll lll l l lll l l ll ll l ll lll ll l lll l ll l l lll ll l lll lll l lll l ll ll ll l lll l lll lll ll l l l ll l lll ll lll l ll l llll l ll llll ll l lll lll ll l ll ll ll ll ll l lll ll ll ll l ll l l lll l ll ll lll lll l ll l ll ll llll l lll llll lll llll l ll llll lll llll lll l ll l lll lll l ll llll l lll l ll ll ll ll ll l ll l ll ll lll ll llll l ll l ll lll llll l ll llll l l lll ll l ll ll ll llll l ll lll lll ll ll ll l ll lllll l lll l ll ll lll ll l lll l ll lll lll l ll lll l ll llll ll ll ll ll lll lll lll llll l lll ll llll lll l lll lll l ll llll ll ll l llll lll l l ll l l ll ll l l lll l llll ll lll lll l lll lll l ll lll l lll l lll l lll l l l l lll llll ll lll lll ll lll ll ll ll ll lll l ll l lll ll lllll l ll ll ll ll l ll ll llll ll ll l lll l ll ll ll l llll l ll ll lll l ll l lllllll l lll ll l ll l lll llll l lll lll ll l ll ll l ll l ll ll lll ll ll ll ll lll l lll ll l ll ll lll l llll ll ll ll ll lll llllll lll l lll lll ll l lll lll l lll llll llll l ll lll ll l ll lll lll llll lllll l lll ll l ll ll lll llll lll ll l l lll ll l ll ll ll ll l l l lll l l lll ll ll lll lll l ll ll l ll l lll lll l ll l ll lll lll l ll l ll lll l lll l ll ll l lll ll l lll ll l llll l ll ll lllll l ll ll l lll l lll ll lll ll lll ll ll l ll l lll l lllll ll l l ll ll lll ll lll l ll lll ll lll ll ll ll ll l l ll lll ll ll ll ll lll llll lll ll ll ll ll l ll ll ll l ll ll l ll lll lll llll ll lll lll lll lll llll ll ll lll lll ll ll lll llllll l l ll l l lll l lll lll lllllll llll ll l ll l ll lll lllll lll ll llll lll l lll l lll ll lll l ll ll lll lll l llll ll l lll ll lll ll l lll lll ll ll l lll ll l ll lll lll ll llll ll lll llll l lll lll ll lll ll llll ll ll ll lllllll ll ll l ll ll lll ll l lll ll l ll ll l l lllll ll ll ll llll l lll ll ll ll l l l l ll l ll lll ll ll ll lll lll l ll llll ll l l lll lll lll ll l l ll lll ll lll l l l ll ll l l ll lll ll l ll l llllll lll lll ll llll ll ll ll l ll ll ll l ll l lllll l lll l lll lll l lllll lll ll lll l l llll lllll l ll ll lll l ll lll l l lll lll ll ll ll lllll l ll l l ll ll lll l lll ll l ll ll llll llll lll lllll ll ll lllll ll lll l ll ll lllll ll l lll ll ll l lll ll ll lll l l lll ll ll ll ll l ll ll l lll llll ll ll ll ll ll ll ll ll lll lll lll l lll llll l ll ll ll ll lll ll l lll ll lll l ll ll ll l lllllll lll llll lllll ll lll ll ll l l llll ll l ll l lll ll llll ll ll l llll ll l llll l ll l lllll lll l lll l l llll ll ll ll ll ll l ll l ll ll l l lll l llll ll ll lll ll lll ll l ll lll ll lll lll l ll ll ll llll lll l lll l ll ll llll ll ll ll ll ll lll lll l l l l lll l ll ll l l lll llll llll ll lll lll l llll ll ll ll lll lllll l ll l lll l lllll lll llll ll l lll llll l ll l lll ll llllll ll ll lll llll ll ll l lll ll l l ll lll lll lll l ll ll ll ll lll llll lllll lll l lll ll l lll lll lll l lll ll l lll l lll ll ll lll ll l lll l l llll l ll llll ll l lllll l ll lll lllll lll lll l l l ll l lll l ll lll ll l ll l lll ll ll lll llll lllll lllllll lll lll ll ll lllll lll ll l ll ll lll ll ll lll l ll ll l llll l l llll lll ll llll l llll ll ll lll ll l ll ll ll ll l l llll ll l ll llll llll ll lll l llll ll ll lll ll l ll ll lll l lllll l ll l lll ll lll ll l lll lll ll llll l ll llllll ll lll l lll ll l lll lll l ll llll l lll l ll ll ll l l l ll ll ll ll ll ll l l l ll ll l ll ll l lll ll llll llll l lll l ll lll ll l ll l l lll l llll ll ll lll ll ll lll lllll lll l lll l ll l ll ll l ll lll ll lll l ll lll ll ll l l ll ll l l ll ll lll ll ll l l lll l l lllllll ll ll l llll l ll ll ll ll ll ll llll l lll l l lllll l lll lll ll l l l lll ll l l lllll l ll ll lll l l lll ll lll l lll llll ll l llll ll l ll ll ll ll ll ll l ll lll ll l llll l llll lll ll ll l ll l ll l l llll ll l ll l l ll ll ll ll l ll lll ll lll ll lllll ll ll l l lll lll l lll ll l lll l lll ll ll lll ll ll ll l l ll lll ll ll lll l llll llll ll l ll l llll lllll lllll l ll l ll lll l l ll lll l ll l l l lll llll l ll ll l ll llll l lll lll llll lllll ll llll llll ll ll l ll ll lll l lll l lll l ll llll lllll lll l ll lll ll ll llll ll lll llll ll ll l ll lll l l ll llll ll ll ll l l l ll ll lllll l ll ll lll l ll ll l lllll lll lll lll ll ll l lll ll ll l lll l lll l l lll lll ll ll l ll ll l l ll ll ll l ll l llll lll l ll lll lll ll ll lll l ll ll ll lll ll l l ll ll l l l llll ll llll ll l l ll l l ll ll ll ll l lll lll l ll ll ll lll lll ll ll ll ll llll l l ll l llll llll ll l lll ll ll l lll l ll ll ll l l lllllll l lll l l llll lll l lll l lll l l lll ll lll l lll ll llllll ll l lll l lll l ll l ll ll l ll l l lll llll l ll l lll l ll lll ll lll ll ll lllllll lll lllll l lll ll ll ll ll l ll ll ll lll llll ll l lll llll lll l l llll lll ll llll ll ll ll ll l llll ll ll l ll lll l l ll lll l l lll l lll ll lll ll l lll ll ll l ll l ll ll ll ll llll llllll ll l l l lll l l lll lll lll ll ll lll ll ll llll ll l lll lll ll l ll lll l lll ll l ll ll l lll l l l ll ll l ll l ll ll lll lll llllll l ll ll ll ll ll lll l lll l ll l l llll l llllll ll l lll lll ll lll lll ll llll ll l llll l ll lll ll ll ll l lll llll l llll ll l ll lll l ll l l llll lll ll l lll lll l ll ll lll l ll l l ll l ll lllll llll llll ll lll ll l ll ll l l lll lll l ll l l ll ll l ll ll ll l lll ll ll ll llll ll l llll l ll lll l l l lll llll ll l ll ll lll ll lllll ll lll lll l llllll l ll lll ll l ll ll ll lllll ll lll ll ll lll ll lll llllll ll lll llllll llll ll ll ll l lllll l ll lll ll ll llll lll l lll lll l llll l ll l llll ll l ll lll l lll ll llll ll ll ll l ll l ll ll ll lll lll l ll l lll l l l llll l ll ll ll ll ll ll l llll lll ll ll lll lll l ll lll lll lllll l l lllll ll ll lll ll ll lll lll ll lll lll l lll ll lll lll l ll l ll ll lll l ll lll l ll ll ll l lllll ll lll ll lll ll llll l lll l l llll ll ll lll l llll llll ll ll l ll l ll l l lll ll l lll ll lll l ll ll l ll ll l ll ll lll l ll llll lll l l ll l lll ll lll llll ll llllllll l lll ll l ll lll llll ll l l l ll lll l llll lll l llll l ll l ll ll l l ll ll lll lll llll ll ll lll ll ll ll llll l ll ll ll l l ll l ll ll llll ll llllll lll ll llll l ll lll llll llll llll l lll l ll ll l ll ll llll l ll l lll l lll l lll l ll lllll llll ll l lll lll ll l lll l ll ll ll l ll llll lll l llll ll l lll ll l ll l ll llll l ll llll l ll ll lll l llll l ll lll lll ll l ll l l ll ll lllll l ll ll ll ll lll lll ll ll l ll l l ll ll ll lllll l l lll l ll ll ll l ll l ll lll lllll ll ll lll lll l lll ll llll l ll l lllll lll lll ll ll ll lll ll l ll lll l l lll ll lll ll llll l ll l ll lll ll ll l ll l ll lll ll lll lll lll l ll l l l ll llll ll l l lll ll l l lll ll l lll lll lll l lll l ll l ll ll l lll ll lll lll l ll l lll llll l lll ll lll ll l l lllll l lll l ll llll l llll ll ll ll lll l ll l lll l l llllllllll l lll lll lll l l l llll ll ll ll l llll llll ll l lll ll l l ll ll llll l l lll lll l l l l ll lll ll l ll llllllll ll llllll ll ll ll ll lll llll ll llll lll ll ll ll ll l lll ll ll lll l ll lll l lll l ll l l ll lll l ll ll l ll l ll lll ll ll l l ll l ll lll lll l ll l ll ll llll l l l lll ll l ll l lll ll lll l ll ll ll l lll ll ll ll ll ll ll ll lll l l lll l ll llll ll ll ll l l ll ll l ll ll lll l l l ll ll ll lll lll l llll ll l llll lll ll ll l ll ll ll ll l lll ll l llll ll ll ll ll ll l ll llll lll llll llll ll l ll l lll lll l ll l ll l l ll l llll llll ll l ll lll l ll lll l ll ll ll ll l ll ll l ll l llll l llll llllll ll l ll ll lll ll l llll l ll ll ll llll l lll ll ll llll l ll lll l ll l ll l l ll llll l ll llll ll lll l ll ll l ll l lll lll llll ll l lll ll ll ll ll ll lll ll l lll ll ll lll l lll ll llll llll ll l l ll lll l lll llll l ll ll l l lll ll l l lll l ll ll ll ll ll ll lllll ll ll l llll l l ll ll ll ll l ll llll lllll llll l ll llll lll lll ll ll l ll ll l lllll lll lll ll l lll l ll ll ll l ll ll ll ll l lll l ll l ll llll ll ll llll l lll l ll llll llll l lll l lll ll lll ll ll l ll lll l l ll ll ll l l llll llll ll ll llll lll lll ll lll lll l llllll ll lll lll lllll l ll lll ll lllll ll llll ll lllll l lll ll l ll ll lll l l lll l lllll ll lll l ll lll llll lll l lll llll l ll l llll l lll ll lll llll ll ll lllll l l ll ll lllll ll ll lll ll l l ll lll ll ll ll ll lll llll lll l ll ll ll llll ll lll ll l lll ll ll l ll ll l lll ll ll l ll lllll l l llll llll l l ll ll ll llll ll l ll l ll ll lll llll l ll lll ll ll ll l lll llll ll lll lll ll ll ll ll l ll lll lll llll l ll l ll ll lll ll lllll ll llllll l lll lll lll l l l lll ll ll ll l ll l ll ll llll l ll l lll ll l llll ll ll l ll ll lll ll ll ll lll lll l lll ll l lll l lll l l ll l l ll lll ll l ll ll l llllll ll ll ll lll lll llll ll l lllll ll lll l l l ll lllll llll lll l ll l lll l ll l lll l l ll ll ll ll l ll l llll l ll ll lll lll lllll l ll l lll lll ll ll l l ll ll ll ll lll l l l l llllll l lll l l ll l ll ll lll lll l ll ll ll l ll l lll l llll l l lll l ll ll ll l llll l llllll ll ll llll l ll lll ll l ll lll llll l l lllll ll lll ll l l ll lll l ll ll lll ll ll l l llll lll l ll ll ll l l ll ll ll lll l lll ll l l lllll ll ll ll ll ll l ll ll llll l l ll l ll ll ll l lll ll ll ll ll l l lll l l llll lll ll llll lll lll ll ll lll ll lll ll ll ll lllll ll l lll lll ll ll lll l lll ll ll ll llll lll ll ll lllll l l l ll ll ll lll ll ll ll lll ll ll lll ll l l ll lll ll ll ll lll llll ll l lll ll lll lll l ll l llll l lll ll ll l ll ll ll ll llll l l lll lll l l lll ll lll llll ll ll ll lll ll ll ll lll l l ll lll l ll lll ll ll lll ll ll lll lll ll ll l ll lll ll l lll l lll ll lll llll ll llll lll ll ll ll l ll lll lll l l ll lll l ll ll llll lll ll l l lll lll ll ll l lll l lll ll l lll ll ll l l ll llll ll ll ll ll ll ll l ll ll ll llll ll l ll l ll l lll ll lll l l ll lll l ll lll lll llll l lll lll llll ll lll ll ll ll lll lll l lll ll ll l ll ll llll l ll ll ll ll l ll ll llll l lll l lll l ll ll lll llll ll l l lll l lll lll l ll lll ll ll l ll l ll l l ll llll lllll l ll l lll ll ll l lll l l ll llllll l l lll ll l ll ll lll lll llll ll ll l lll l lll lll l lll lll ll l ll ll lll l llll lll lll l l ll l ll ll ll l ll ll ll llll l l lll l ll l l llll l ll l ll lll l ll lll ll ll ll ll ll lll lll lll ll ll l ll l lll ll ll l ll ll l llll lll l lll ll lll lll ll ll l lll l ll l llll ll llll l l l lll ll ll llll ll l l lll ll ll ll ll ll ll ll lllll ll lll l lll lll l ll ll l llll lllll lll lll lll ll lll lll l lll ll lll l ll lll l l llll lll lll ll l lll ll lll ll lll lll ll l l ll ll ll ll ll ll l ll ll l llll l lll ll ll ll ll l l ll l lll l llll lll l lll ll llll lll ll lll llll l ll lll llllll lll l ll ll ll lll ll lll ll lll l ll llll l l l ll llllll l llll l lll lll ll l l lll ll ll ll l ll lll ll lll lll llll l ll lll l lllll l ll l lll ll lll ll lll ll lll l lll l l ll ll lll l ll l ll ll ll lll l ll llll ll lll ll ll l ll l lllll l lll ll l lll ll l lll ll l l lll l lll l lll ll ll l ll ll l ll ll lllll ll lll ll ll l lll l l l ll lllll ll l lll ll l ll lll llll lll ll lll lll lll ll ll lll lll l l llll l ll l llll llllll lll ll l ll ll ll l ll ll lll ll ll lll llll lll ll ll ll ll ll lll ll l ll ll ll llll lll l ll l ll ll lll ll ll l ll llll l llllll l lll ll ll ll ll llllll lll l ll lll l ll ll ll lll llll ll lll ll ll llll l ll lll ll l l llll ll lllll lllll l ll llll ll lll l llll l ll l lll lll l ll lll l llll lllll lll ll ll ll ll ll ll lll ll llll lll l lll l ll llllll ll lll ll llllll llll l l llllll ll lll l llll l lll ll ll ll ll l ll ll l l ll ll l l lll lll ll l l l lll ll lll lllll llll l ll l l lll ll ll ll l ll ll l ll lll l lll ll ll ll ll ll l l llll ll l lll l ll lll lll l l l l ll ll ll ll lll ll l ll l lll ll llll ll llll ll llll lll (a) Trade vs.
Index (ˆ ρ = 0 . Immigration I nde x . . . . . . . . . . . . . . . . . . . . . . . . . . −1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 − . − . − . . . . l ll llll l ll llllll lll l lll ll ll ll ll ll l l ll l ll lll l llll ll lll ll ll l l ll l lll lll ll lll l lll ll lll ll l ll lll lll llll ll lll l lllll l ll llll ll llll l ll ll ll l lllll l lll ll ll ll lll l lllll lll l l ll lll ll ll lll ll l ll l llll lll ll ll l lllll l ll ll llll llll l lll l lll ll ll lll l lll ll ll l ll ll lll llll l llll lll l ll l lll ll ll l ll l ll llll l ll lll ll lll ll l lll lll ll ll lll l ll ll lll l l ll llll ll l lll ll l ll ll llll llll llll ll l ll l ll lll l ll ll l ll l l lll llll l l ll ll ll ll lll lll lll lll ll l ll l lll llll lll ll llll ll lll lll l lll lll ll ll ll ll ll llll ll ll ll llll lll ll lll ll l ll ll ll ll llll ll l ll l llll l ll ll ll ll ll llll l lllllll ll llll ll l lll l lll lll ll l lll lll ll l l lll ll lll ll ll l ll llll ll ll l ll ll l ll ll l ll ll ll lll ll ll ll l ll lll ll lll lll l ll ll ll l l ll l llll llllll l ll l l llllll lll ll ll ll l llll lll ll llll ll lll ll ll ll ll llll lll l ll llll lll ll llll lll lllll ll ll l lll lll l ll ll llll l ll ll lll l ll ll l l ll ll lll ll ll llll ll ll l lll ll llll ll ll ll ll l ll lll ll ll lll l ll ll l ll l l ll l l l ll l ll l lll l ll ll l llll l llll ll lll ll l lllll ll l l ll ll ll llll ll ll ll l lll l ll lll ll l l ll l ll l ll l l l ll lll ll lll ll ll l ll lll l ll l l ll ll lll l l lll ll l lll ll ll ll l ll ll ll ll ll lll l ll ll l ll ll lll l l ll lll ll ll lll l l lll ll l l ll ll l ll ll l l lllll ll ll ll lll lll l ll llll lll l ll ll l llll ll l lll l ll l ll lll ll lll l lll l lll lll ll l lll ll l ll l ll ll l ll ll ll l l l l ll l l lll lll l ll l l l lllll ll ll lllll ll ll ll l l ll lll ll llll lll l lll llllll ll l l lll l l llll l lll l ll l l ll ll l ll ll l l ll lll lll lll lll lll ll ll lll lll lll ll l l ll l l ll llll lll ll ll ll ll l l ll llllll l ll lll lll l lll ll ll ll ll ll ll l ll lll ll l l ll lll ll lllll ll ll l ll l llll ll l l ll ll lll ll lll ll l ll l lll ll ll llll l ll l l lll l lllll l l lll l l lll ll ll ll ll l l ll ll ll ll l llll l lll l lllll l l ll ll l llll ll l l llll l ll ll l lll lll ll llll lll l lll ll l ll l l lll ll lll ll ll l lll llllll llll l l l l llll l ll lll l lll lll l ll lll ll l ll ll l lll ll l lll l lll l ll l ll lll l ll ll ll lll ll l l ll ll llll ll ll ll ll lll ll l ll lll ll lll lll l l llllll l ll llll llll ll l lll l lll ll lll l l ll l lll ll llll l lll ll ll ll ll ll lll l ll l ll l lll ll lll l lll l lll l lll ll ll ll lll l ll ll l ll ll lll lll ll l l ll l l llll ll ll l ll l ll llll ll l lll l ll lll lll ll lllll ll lll llll lll ll llll ll l llllll lll ll l llllll l l ll lll ll l lll lll ll lllll ll l ll ll ll lll l lll l ll ll ll lll lll lll l ll l l lll l lll l lll l ll llllll ll l llll lll ll ll lll ll ll lll ll ll ll lll l lll lll ll ll llll llll l ll lll ll lll llll lll l l ll l ll ll ll l llll l ll ll llll ll l lll l ll l l lll llll lll ll ll lllll lll ll l ll l ll ll l lll ll ll lll ll ll ll ll l ll l l ll lll l ll ll lll ll lll lll ll ll l ll ll l ll llll lll l ll lll l l llll lll lll l ll l ll ll ll ll l ll ll ll ll l llllll l llll ll lll l l lll l ll lllll l lll llll l l ll ll l l lll ll lll ll l l llll ll ll lll lll lll l l lll l lll ll l lll l ll lll lll ll lll l lll ll ll ll ll l ll ll l lll ll l lll ll l lll l ll ll l ll ll l llll llll llll lll ll lll ll ll lll ll l ll llll l llll lll l lll ll llll l lll l ll lll ll lll llll ll ll l lll ll l ll ll ll lll ll ll ll ll lll ll l ll ll ll l ll ll lll l ll llll l ll l ll ll ll lll lll llll ll lll l llll ll l ll ll llll l lll l l ll llll ll ll l ll ll l l ll lll l ll ll l llll l ll ll ll l ll ll l lll ll lll llll ll ll l lll llllll lll lll lll l lll ll ll ll ll ll l ll l ll lll ll lll l lllllll lll lll l ll ll llll ll lll l lll ll l lll l llll lll ll lll ll llllll ll ll llllll l ll l ll llll lll ll l ll lll lll lll ll l l llll ll ll l lllll ll l l ll l llll lll l l ll lll ll ll lll ll lll llll l ll l lll l l ll lll l ll ll l l ll ll l ll lll ll ll ll ll l lll l ll ll l ll ll lll lll lll l l ll ll ll ll l ll l ll ll ll llll ll lll l l lll lll lll l ll llll ll l llll ll ll lllll l ll ll l ll ll ll ll llll lll l l ll l l l ll l ll lll ll ll l ll llllll llll ll ll llll lll ll ll lll l ll ll ll lll ll l lll llll lll ll ll ll ll llll ll ll l lll llll l l ll lll ll ll ll lll ll ll l lll l ll l lll ll ll ll ll ll ll l lll l ll lll l lll lll l ll lll l lll lll ll llll ll lll lll ll ll lllllll lll llll l llllll lll l lll ll l lll ll l lll llll lll ll ll ll l lll l ll lll lll ll ll llllll lll l lll ll l llll ll ll ll ll ll l ll ll ll ll lll l llll l l llll l lll l ll l ll lllll lll lll l ll l ll ll llll l ll l l ll lll l l l lll ll llll ll ll ll l ll l ll l l lll l l ll ll ll lll l ll lll ll ll l l ll l l lll ll l l lll l ll ll ll l llll lll lllll llll lll ll ll ll l lll lll ll ll ll l ll ll ll llll ll ll llll llll l llll l lll ll l l ll ll l lll lll lll ll ll ll l lll ll ll ll llll llll ll ll ll l lll ll lll l l ll ll l ll l l ll ll ll ll ll ll lll l ll l l lll l lll llll l l lll l l lll ll l ll ll l lll l ll l l ll ll llll lllll lll l l lll lll l lll ll lll ll lllll l llllll lll l ll ll ll l llll ll ll ll l ll llll l ll llll ll lll lll ll ll ll lll ll l ll lll l lll l ll ll lllll l ll ll ll l ll llll ll lll lll ll ll llll ll llll l ll l ll ll lll ll l l ll l ll ll ll lll ll l ll lll ll ll ll lll llll ll ll lll ll ll ll llll ll lll l lll lll l ll lll lll lll ll l lll llllll l l lll ll lll ll lll l ll lll l ll l ll l lll lll ll ll l l llll l lll ll lll ll l lll ll lll lllll ll lllll l l lll ll l ll lllll ll ll lll l ll lll ll ll l lll llll l lll llll l ll ll ll l l llll ll ll l ll lll lll ll lll ll ll l l ll llll ll lll l ll ll l l lll ll llllll l lllll ll ll lllll l l ll l l ll lll l l l ll l lll l ll l l lll ll l ll lll l lllll lllll lll lll ll ll l ll ll l l ll ll ll ll lll l l lll ll l ll ll lll ll ll lll ll ll lll ll l lll lll l lll lll l ll ll ll ll l ll ll ll lll ll l ll llll lll l lll lll l lll ll l ll ll ll ll ll ll lll ll l lll l lll llll lll ll l ll l lll ll l lll ll lllll lll l ll llll l ll l lll ll l lll ll ll ll l ll ll ll l ll lll ll ll lll l lll l lll lll lll lll lll l llll lll l ll lll l lll ll l l l lll lll l l ll ll ll ll llll l llll lllllll ll ll llll ll lll lll llll ll ll lll lll l ll ll ll ll l l lllll lll llll ll ll lll l ll lll lllll l lll l llll llll l lll ll l lllll ll lll ll l ll l llll ll l ll ll l l lll lll lll l l llllll lll lll lll ll l ll l ll lll lll l lll lll l ll ll l ll ll ll lllll llll ll ll ll ll l llll l ll l ll lll ll ll l l ll ll l ll ll lll ll lll ll llllll ll llll lll ll l llll l l ll l ll ll ll ll lllll l l l ll l ll ll l lll l lll ll ll l lll lll ll l llll lll llll ll ll l lll l lll l ll lll lll ll l lll l lll ll l ll l lll l ll lll ll lll ll llll llll l ll llll ll lll ll ll ll lll l lll ll lll l lll ll l l ll ll l l llll l ll lll lll l ll l lll ll ll ll lll llll lll ll lll l lll ll l lll ll l ll l ll l llll l ll llllll ll l l ll l llll llll llll lll lll lll l ll l ll lll l l ll l lll lll lll ll llll ll ll l ll lll lll ll lll l lll ll l lll ll ll lll l ll lll lll ll ll l ll ll ll llll ll lll l lll ll ll lll l ll l ll l l llll l lll ll ll l lll lll lll ll llll lll l lll ll l lllll ll ll lll ll ll lll lll l lll l l l ll ll l l l lll l ll l l llll l ll l ll ll l ll l lll ll l lll ll l l ll lll ll llll llll llll l ll ll l l lll ll ll lll ll ll lll l ll ll ll lll ll ll l ll lll ll l l ll l llll l ll ll lll l lllll ll llll l ll lll l lll ll l l llll l lll llll l ll l l ll ll l ll llll ll ll lll lll l ll lllll l ll l lll lll lll lllll llll ll lll ll llll lll lll l ll lll l ll l lllll l lll llll lll l l llllll l lll ll llll l lll ll llll l ll ll l ll l ll ll ll llll ll l ll ll l lll lll lll lll l ll lll ll ll l ll ll llll ll ll ll ll ll lll lll ll lll ll ll l ll lllll lll ll lll ll ll lll ll l lll l ll l lll lll llll lll ll l llll lll ll l ll l ll l ll ll l ll lll ll lll l ll lll ll lll l lll l lll l lll ll ll lll ll ll llll ll ll l lll ll l lll l lll ll l ll ll l lll l ll lll ll l lll ll ll l ll ll ll lllll lllllll ll lll lll ll lll llll lll llll ll l ll lll l lll ll lllllll l lll ll ll ll lll l ll l ll ll ll l lll lll lll l l lll ll llll l ll ll ll lll ll l ll ll ll ll ll ll ll llll ll lllll ll lll ll ll ll l ll lll lll l lll ll l l lll l lll l lll lll llll l ll l l lll l lll l llll ll llll llll ll l ll ll lll lll lll lll l ll l lll l lllll llll l ll l lll lll ll l ll lllll llll lll l llll ll ll lll l ll ll ll ll l lll ll l ll lll l ll lll ll lll ll lll l lll lll lll lll ll ll ll lll ll l lll l ll llll ll lll l lll l ll llll lll ll lll ll lll lll ll lll ll lll lll ll ll lll l lll l ll ll ll ll ll ll l l ll lll l lll l ll lll ll lll ll ll l lll ll llll ll l l ll l ll llll lll lll llll lll l lll lllll l ll ll ll ll ll lll l l lll lll lll l ll l ll ll l ll ll llll ll lll lllll l lll l lll l lll llll ll l llllll l lll l ll ll lll l llll l lllllllll l ll ll lll l lll lll lll ll lll ll lll ll l l l llll lll l ll ll lll l ll ll ll l l l lllll llll l llll l llll lll ll ll ll lll ll lll l lll l lll ll lll lll lllll l llllll ll ll l llll ll lll llll l llll lll l lllll lll lll llll ll lll lll l ll lll lll ll lll l lll lll ll lll ll l ll lll l ll l ll ll l llll l l lll ll llll l ll ll ll llll lllll llll ll ll ll l lll ll l ll lll l ll ll lll l ll ll l l llll lll ll ll ll l l llll ll ll lll ll llll l ll l lll ll l l ll ll l ll lll lll ll ll l lllll l llll ll l ll ll l ll l llll ll l l lll ll lll ll l lllll l ll l lll l ll l lll ll l l lll lll ll l ll l ll lll llll l llll llll ll l lll lllll llll lll ll l ll l ll l l lllll ll llll l ll ll ll ll l ll lll l ll lll llll lll lll lll lll l lll lllll lll l ll lll l lll lll ll l l ll lll l lll l ll l lll l ll ll llll lllll llll ll ll l lll l lll ll llll l ll ll lll lll ll l lll l lll ll llll l ll ll l ll lll l lll l lll l lll ll l ll ll lll l l ll ll lllll lll l ll lll l lll lllll ll lllll ll l ll l l l ll ll l llll lll l ll ll l lll ll lll llll l ll l l ll ll ll ll ll ll llll ll lll l l ll l ll llll l lll ll ll ll ll l ll l lll l ll ll l ll llllll l ll ll lll l lll ll ll l l ll ll ll l lll l ll ll ll l ll ll lll ll llll ll ll l lll l lll l lll l l lll ll ll lll ll l ll ll ll l lll l l ll lll l lllll l lll llll ll l llll ll ll ll llll ll ll l ll lll l ll lll l lll l ll l ll ll ll l ll ll llll lll lll lll llll ll l lllll llll ll l ll llll ll ll ll lll lll ll ll lll ll l l lll ll ll ll l l l ll ll l lll ll llll l lll llll l l llll llll lll ll llll l l lllll lll ll ll ll lll ll ll ll llll l lll ll ll lll llll l l l l ll ll ll lll lll llll ll lll lll l ll ll ll ll lll lll l lllll l l ll ll lll ll lllllll ll ll ll lllll llll lll ll lll llll ll lll ll llll l l ll ll lll l ll ll ll l ll llll ll l ll lll llll l lll ll lll ll ll lll ll llll l ll llll ll ll lll lll l ll ll l lll lll ll l ll l ll lll llll ll l llll l l llll l lll llll l llll lll lll llll ll l ll llll l l ll ll ll ll l ll l ll ll l lll l lll lll lll l lllll l ll l l ll lllll ll l lll lll llll ll l ll l lll lll l l lll ll lll ll ll lll llll l ll lll lll lll ll lll llll ll lll lll lll l llll l lll l ll l l l l ll lll l ll lll ll lll ll ll ll l ll lllll l ll ll l lll ll lllll ll lll ll ll ll llll l ll l lll ll ll l llll l ll lll llllll ll l l l ll l ll ll ll ll ll ll ll ll lll ll lll l lll ll l lll lll l ll ll ll ll ll ll lll l l lll lllll ll l l l ll lll ll ll ll lll ll l lll ll l ll l llll lll llll ll l l llll lll l l ll ll lll lll l l llllll ll l lll ll lll lll l ll ll lll lll ll l ll l l lll llllll lll ll ll ll l ll llll lll ll ll l ll ll ll ll l ll l ll ll l ll ll llllll llllll ll l l lll lll lll ll ll l l ll ll lll ll ll ll ll l l ll lll l ll lll l l lll ll ll ll lll l ll ll lll l ll llll ll l lll l ll lll lll ll ll ll lll l ll l ll l lll ll lll lll ll ll lll l ll ll lll ll l ll lll lll l ll lll lllll ll l lllllll ll ll ll llll l ll ll ll ll ll ll l ll ll l llllll ll l l l lll l lllll lll l ll lll l ll l lll ll lll ll ll llll lll ll ll ll l lll ll lll lll l ll l llllll l lll ll lll l ll ll ll ll l ll ll l ll ll l llll ll lll l ll ll lll llll l ll llll l llll l l ll lll l ll llll l l l lllll ll lllll l l lll l ll ll ll l ll l l llll ll ll ll l l ll ll l lllll llll ll ll ll lll ll l lll lll ll lll ll ll l lll lll lll l lll lll l lll llll llllll ll lllll l ll l llll l ll ll l ll lll ll llll l lll ll ll ll ll ll ll l ll ll lll ll lll l ll llll l ll lll lll ll ll l ll ll ll ll ll ll lll ll l lll l ll ll lll llll ll ll l ll l ll l ll lllll ll llll lll l ll ll ll ll l llll ll l ll l ll ll l ll l ll ll l lll l llll lll lll ll lll lll llll ll lll l ll ll lll l lll ll l llll ll ll l ll ll ll ll l lll llll ll l lll ll ll ll l ll l lll lllll l lll ll ll ll l ll ll l ll ll l ll lll ll ll l ll ll lll l l ll lll lll lll l ll lll ll l ll lll ll l l ll ll ll lllll ll lll ll l ll l ll ll l ll ll ll llll l l ll llll ll l lll l ll lllll l ll ll l lll ll ll lll ll ll llll l ll l lllll l lll lll l ll ll ll l l l ll lllllll l llll ll l lll llll l ll lll ll ll l l lll ll ll ll ll l ll l lll ll l lll ll ll ll lll l ll lll llll l l ll l lll ll ll lll ll l ll lll llll ll lll ll l l llll l ll lllll l lll l llll ll ll lll ll ll lll lll ll l ll l ll l ll lllll ll lll lllll ll ll l llll lll l l lll ll lll lll l ll l ll ll ll ll ll lll lll l lll ll ll ll ll ll lllll l ll l l ll l lll l lll l ll llll ll lll lllll lll llllll llll ll l ll l ll ll ll l lllll l ll ll ll lll l ll ll ll ll l l llll ll ll l l l ll lll l lll ll l lll lll l llll lll llll ll ll llll ll l lll ll lll lll l lll lll l l llll llll l l ll ll l l ll lll ll ll ll l lll ll l lll lll ll l ll l l ll llll lll ll lll ll lll lll ll ll lll llll l llll l l llll ll ll lll ll l lll l ll ll l ll lll l ll llll l l l lll lllll llll l l ll l l lll lllll ll ll ll l ll ll llll ll l l l ll ll llll ll lll ll lll llll ll ll lllll ll l ll ll ll ll llll l l llll lll lll l llll llll lll lll ll l (b) Immig vs.
Index (ˆ ρ = − . Figure 10: Contour plots of fitted bivariate GH models to the joint log-returns of
Trade and
Index (left panel), and
Immig and
Index (right panel). The observed data and the 10,000 simulated valuesfrom the fitted model are displayed as black and grey dots, respectively.The assessment of how extremely negative events on the stress factors, i.e., values on theirleft tail, impact the
Index , can be done with the various measures of systemic risk currently invogue, all of which target the left tail of the conditional distribution of the random variable ofinterest, Y , given that the stressor, X , is at or below some low quantile. This notion of ConditionalValue-at-Risk (CoVaR), was originally introduced by Adrian and Brunnermeier (2016).Here, and due to the better properties pointed out by Mainik and Schaanning (2014), we opt totake the variant of CoVaR developed by Girardi and Ergun (2013). If F Y | X denotes the conditionaldistribution of Y given X , each with respective distribution functions F Y and F X , then we denoteby ξ q the CoVaR at level q , or CoVaR q , which is defined to be ξ q := CoVaR q := F − Y | X ≤ F − X ( q ) ( q ) = VaR q ( Y | X ≤ VaR q ( X )) , (10)where, using fairly standard notation from the risk modeling literature, VaR q ( X ) := F − X ( q ) denotesthe VaR of X at level q , which is simply the q -quantile of X . In line with this concept, the analogousvalue for the closely associated Expected Shortfall (ES), defined as the tail mean beyond VaR, isgiven by (Mainik and Schaanning, 2014)CoES q := E ( Y | Y ≤ ξ q , X ≤ VaR q ( X )) . (11)A variation on this proposed by Biglova et al. (2014) isCoETL q := E ( Y | Y ≤ VaR q ( Y ) , X ≤ VaR q ( X )) . (12)16he stress-testing results are presented in Table 3. The general pattern seems to be that forthe same level, stress on Trade appears to have a marginally larger impact on the
Index than doesstress on
Immig ; a finding which is also consistent with the sign of the correlation coefficients notedabove. However, at the highest stress level of 1%, the results are mixed.Table 3: Left-tail systemic risk measures on the
Index at different levels, based on stressing thefactors
Trade and
Immig .Stress Stress Risk Measure on
Index (left tail)Factor Level CoES CoVaR CoETL
Trade
10% -0.8403 -0.5448 -0.49805% -1.2377 -0.8414 -0.66681% -1.6367 -1.2331 -0.9728
Immig
10% -0.7027 -0.4370 -0.47485% -0.9902 -0.7288 -0.66451% -1.3788 -1.2682 -1.0084
We proposed an annual well-being index (SWBI) constructed as the log-returns of an equally-weighted linear combination of several socioeconomic factors, in order to dynamically measure themood of US citizens. The data, publically available from reliable government sources, spans the30-year period from 1986 to 2016. Although the SWBI exhibits no apparent serial dependence, wefitted an ARMA-GARCH time series model in order to appropriately capture the marginal (or cross-sectional) distribution which was consistent with a member of the generalized hyperbolic family, aspredicted by contemporary best-practices financial theory (Massing, 2019). This procedure was alsoa prerequisite step to generating valid option prices and performing risk budgeting for the SWBI.The resulting values reveal the relationships among time to maturity, strike price, and option price,enabling the construction and valuation of insurance-type financial instruments.To complete the rational finance-based valuation, we performed risk budgeting for an equallyweighted portfolio, and stress-tested the index by examining the effect of the exogenous butpolitically-sensitive variables, trade imbalance and amount of legal immigration, on the SWBIsystemic risk. Among the component series of the SWBI, the VXO volatility measure of the stockmarket is the greatest contributor to tail risk. Among the external factors, it appears that thelevel of trade imbalance tends to be associated with a larger impact on negative well-being thandoes immigration; a conclusion that mirrors the empirical finding that trade imbalance is positivelycorrelated with SWBI, while immigration is negatively correlated.The main intent of the SWBI is for it to provide an early-warning mechanism for downturns inthe mood of US citizens. Coupled with the proper valuation of SWBI financial instruments, it ishoped that this will alert investors to potential future crises assess and aid them in setting in placehedging strategies necessary to mitigate against possible losses.17 eferences
Adrian, T. and Brunnermeier, M. K. (2016). CoVaR.
American Economic Review , 106(7):1705–41.Baker, S. R., Bloom, N., and Davis, S. J. (2016). Measuring economic policy uncertainty.
TheQuarterly Journal of Economics , 131(4):1593–1636.Barndorff-Nielsen, O. (1977). Exponentially decreasing distributions for the logarithm of particlesize.
Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences ,353(1674):401–419.Barndorff-Nielsen, O. (1997). Processes of normal inverse gaussian type.
Finance and Stochastics ,2.Barndorff-Nielsen, O. (2007). Normal inverse gaussian distributions and stochastic volatility mod-elling.
Scandinavian Journal of Statistics , 24:1–13.Biglova, A., Ortobelli, S., and Fabozzi, F. (2014). Portfolio selection in the presence of systemicrisk.
Journal of Asset Management , 15:285–299.Blaesild, P. (1981). The two-dimensional hyperbolic distribution and related distributions with anapplication to johannsens bean data.
Mathematical Finance , 68:251–263.Blanchflower, D. D. and Oswald, A. J. (2004). Well-being over time in britain and the usa.
Journalof Public Economics , 88(7-8):1359–1386.Boudt, K., Carl, P., and Peterson, B. G. (2013). Asset allocation with conditional value-at-riskbudgets.
Journal of Risk , 15(3):39–68.Chorro, C. (2012). Option pricing for garch-type models with generalized hyperbolic innovations.
Quantitative Finance , 12(7):1079–1094.Chow, G. and Kritzman, M. (2001). Risk budgets.
Journal of Portfolio Management , pages 56–60.Di Tella, R., MacCulloch, R. J., and Oswald, A. J. (2003). The macroeconomics of happiess.
Reviewof Economics and Statistics , 85(4):809–827.Duan, J. (1995). The garch option pricing model.
Mathematical Finance , 5:13–32.Duffie, D. (2001).
Dynamic Asset Pricing Theory . Princeton University Press, Princeton, 3rdedition.Ferrer-i Carbonell, A. and Frijters, P. (2004). How important is methodology for the estimates ofthe determinants of happiness.
The Economic Journal , 114(497):641–659.Gerber, H. U. and Shiu, E. S. W. (1994). Option pricing by esscher transforms.
Transactions ofthe Society of Actuaries , 46:99–191.Girardi, G. and Ergun, A. T. (2013). Systemic risk measurement: Multivariate garch estimationof coVaR.
Journal of Banking & Finance , 37(8):3169–3180.18ohansen, S. (1988). Statistical analysis of cointegration vectors.
Journal of economic dynamicsand control , 12(2-3):231–254.Jorgensen, B. (1982).
Statistical properties of the generalized inverse Gaussian distribution , vol-ume 9 of
Lecture Notes in Statistics . Springer-Verlag.Krueger, A. B. (2009).
Measuring the Subjective Well-Being of Nations . The University of ChicagoPress, 1 edition.Litterman, R. B. (1996). Hot spots and hedges.
Journal of Portfolio Management , pages 52–75.Luethi, D. and Breymann, W. (2016). ghyp: A Package on Generalized Hyperbolic Distributionand Its Special Cases . R package version 1.5.7.Maillard, S., Roncalli, T., and Teiletche, J. (2010). On the properties of equallyweighted riskcontributions portfolios.
Journal of Portfolio Management , pages 52–75.Mainik, G. and Schaanning, E. (2014). On dependence consistency of covar and some other systemicrisk measures.
Statistics & Risk Modeling , 31(1):49–77.Massing, T. (2019). What is the best l´evy model for stock indices? a comparative study with aview to time consistency.
Financial Markets and Portfolio Management , 33(3):277–344.McLean, D. (2014). National and international indices of well-being: A critical analysis.
Journalof the Indiana Academy of the Social Sciences , 17(1):5.Paolella, M. S. (2007).
Intermediate Probability: A Computational Approach . John Wiley & Sons.Peterson, B. and Boudt, K. (2008). Component var for a non-normal world.
Journal of Risk , pages78–81.Pflug, G. and Werner, R. (2007).
Modeling, measuring and managing risk . World Scientific.Schmidt, T. (2007). Coping with copulas. In Rank, J., editor,