Jointly Estimating Jump Betas

The current issue and full text archive of this journal is available at www.emeraldinsight.com/1526-5943.htm Jointly estimating jump betas Vassilis Polimenis and Ioannis Papantonis Jointly estimating jump betas Division of Business Administration, School of Law and Economics, Aristotle University of Thessaloniki, Thessaloniki, Greece 131 Abstract Purpose – This paper aims to enhance a co-skew-based risk measurement methodology initially Received 29 July 2013 Revised 20 October 2013 introduced in Polimenis, by extending it for the joint estimation of the jump betas for two stocks. 17 November 2013 Design/methodology/approach – The authors introduce the possibility of idiosyncratic jumps and analyze the robustness of the estimated sensitivities when two stocks are jointly fit to the same set of Accepted 28 November 2013 latent jump factors. When individual stock skews substantially differ from those of the market, the requirement that the individual skew is exactly matched is placing a strain on the single stock estimation system. Findings – The authors argue that, once the authors relax this restrictive requirement in an enhanced joint framework, the system calibrates to a more robust solution in terms of uncovering the true magnitude of the latent parameters of the model, at the same time revealing information about the level of idiosyncratic skews in individual stock return distributions. Research limitations/implications – Allowing for idiosyncratic skews relaxes the demands placed on the estimation system and hence improves its explanatory power by focusing on matching systematic skew that is more informational. Furthermore, allowing for stock-specific jumps that are not related to the market is a realistic assumption. There is now evidence that idiosyncratic risks are priced as well, and this has been a major drawback and criticism in using CAPM to assess risk premia. Practical implications – Since jumps in stock prices incorporate the most valuable information, then quantifying a stock’s exposure to jump events can have important practical implications for financial risk management, portfolio construction and option pricing. Originality/value – This approach boosts the “signal-to-noise” ratio by utilizing co-skew moments, so that the diffusive component is filtered out through higher-order cumulants. Without making any distributional assumptions, the authors are able not only to capture the asymmetric sensitivity of a stock to latent upward and downward systematic jump risks, but also to uncover the magnitude of idiosyncratic stock skewness. Since cumulants in a Levy process evolve linearly in time, this approach is horizon independent and hence can be deployed at all frequencies. Keywords Jump betas, Idiosyncratic skewness, Asymmetric betas, Lévy process Paper type Research paper 1. Introduction Since the discovery of the CAPM, the discussion of whether beta is an appropriate measure of risk is one of the most persistent and ongoing debates in financial economics. The classical beta arises as the proper measure of risk only under very strict conditions that are now under heavy questioning. First of all, it is well accepted that investors care not only about variance, but also about skewness in returns. Since Jiang and Oomen (2008), Fishburn (1977) and Harlow and Rao (1989), co-skewness with market returns is proven to be a significant factor driving asset prices. Moreover, there is evidence of asymmetric correlations among financial returns, which has triggered the focus on asymmetric response betas and especially downside risk[1]. Following Markowitz (1959), academics have increasingly argued in favor of allowing for discontinuities in the price paths of financial assets, thus casting doubts on the validity of asset pricing models (such as the classical model of Black and Scholes (1973)) The Journal of Risk Finance Vol. 15 No. 2, 2014 pp. 131-148 q Emerald Group Publishing Limited 1526-5943 DOI 10.1108/JRF-07-2013-0052 JRF 15,2 132 that assume continuity for the return dynamics. Investors are exposed to jump-risks and hence demand higher compensation for bearing the possibility of an unfavorable extreme event (Newey and West, 1987). More recent studies, like those of Carr and Wu (2004) or Duffie et al. (2000), tried to generalize previous approaches by allowing for jumps in the volatility process as well. Jumps are indeed prevalent in financial asset prices and these jumps are also responsible for introducing asymmetries in the distributions of asset returns (Carr and Wu, 2003)[2]. A recent empirical finding by Carr et al. (2002) indicates that these jumps may not only be large and infrequent Poisson jumps, but also smaller jumps that arrive at an infinite rate[3]. Thus, a natural and parsimonious model, that is also general enough so that it encompasses most empirical findings, is one where: . stock returns are exposed to both diffusive and jump risks; . jump arrivals of various sizes may occur at low and high frequencies; . stock returns may exhibit differential sensitivities to diffuse versus jump market information; and . jump sensitivities are potentially asymmetrical. In this framework, systematic jumps are random points in time where market-wide information arrival occurs, while price continuity is preserved by very frequent noise trading (Black, 1986). Hence, if the market treats asymmetrically price discontinuities from smooth variation, such a decomposition among continuous and discontinuous components is of utmost importance for asset pricing. There is a broad research program that aims to explore the fundamental idea that jumps incorporate the most valuable information for stock prices. Properly quantifying a stock’s exposure to jump events is crucial not only from a financial risk management, but also from a portfolio construction and an option pricing perspective, since jump exposure imposes skew that is known to generate variance risk premia (Pan, 2002; Bakshi et al., 2003). Yet, there is still no simple statistical procedure for estimating jump betas[4]. A recent and extensive strand of literature focuses on analysing high-frequency data using powerful econometric techniques in attempting to isolate jumps. In a significant contribution to this field, Barndorff-Nielsen and Shephard (2004a, b) study two natural measures of realized within-day price variance, the well known day-t realized variance, that measures the summed quadratic log-returns, and the novel realized bi-power variation, that measures summed neighboring log-return cross-products[5]. The significant and intuitive finding is that, under reasonable assumptions about the stock price dynamics and when the fineness of the partition tends to infinity, these two measures have different limits, and this difference allows one to quantify the jump activity. There are numerous studies concerned with trying to decompose the total price variation into its continuous and jump components (Aı̈t-Sahalia, 2004). Using theoretical and Monte Carlo analysis, Hogan and Warren (1974) identify a pitfall in applying the asymptotic approximation over an entire sample and find that microstructure noise limits power and biases the tests against detecting jumps. In order to correct this bias, they propose a simple lagging strategy. Huang and Tauchen (2005) propose a test for jumps in asset returns that is based on variance swap rates, and derive its power from the impact of jumps on higher moments. Lee and Hannig (2010) introduce a new jump detection technique to resolve jump identification problems and provide a model-free tool that helps estimate differential jump dynamics in individual equity and index returns. Lee (2012) detect small jump arrivals by assessing the probability of their presence with a test designed to compare realized test statistics with synthetic data generated from the asymptotic null distribution of the test statistic. Lee and Mykland (2008) propose another empirical test that preprocesses price data for the purpose of de-noising by local averaging. Kraus and Litzenberger (1976) analyze the predictability of jumps in individual stock returns, and uses macroeconomic and firm-specific news in order to achieve a decomposition of individual stock jumps into systematic and idiosyncratic. Generally, the high and ultra-high frequency methods are promising and are quite attractive among researchers and practitioners. Yet, and despite their attractiveness, there are several practical issues related to their implementation that limit our ability to calibrate them to real data and challenge their true practicality and robustness. On the one hand, the power of tests is seemingly improved when the frequency of observations over a fixed time interval is increased. Faster sampling allows for easier jump detection but at the same time it contaminates the data with market microstructure noise[6]. On the other hand, slower sampling takes care of a large part of this noise but increases the need to make assumptions about the distribution and the intensity of the underlying jumps. In contrast to the high-frequency methods discussed above, low-frequency[7] techniques do not rely on the use of high-frequency data and are thus immune to microstructure noise[8]. In an effort to develop a practical and academically sound framework, Polimenis (2003) proposes the use of a set of co-skew based moment conditions for disentangling and estimating asymmetric sensitivities towards systematic jump risks. This approach boosts the “signal-to-noise” ratio by utilizing co-skew moments, so that the diffusive component is filtered out through higher-order cumulants. Since cumulants in a Lévy process evolve linearly in time, this approach is horizon-independent and hence can be deployed at all frequencies. In this paper, we build on this approach by focusing on a joint GMM estimation, at the same time introducing the possibility of idiosyncratic stock skewness. Since skew and co-skew estimates are highly sensitive to extreme realizations, the robustness of the above technique can be significantly enhanced by capturing information from more than one stocks jointly. Allowing for idiosyncratic skews relaxes the demands placed on the estimation system, and hence improves its explanatory power by focusing on matching systematic skew that is more informational. Besides, allowing for stock-specific jumps that are not related to the market is a realistic assumption. There is now evidence that idiosyncratic risks are priced as well, and this has been a major drawback and criticism in using CAPM to assess risk premia[9]. One of the key benefits of our methodology here of jointly estimating the jump betas for two stocks is that it allows us to identify and quantify the existence of idiosyncratic skews in individual stock return distributions. Results of fitting Google and Yahoo versus the NASDAQ reveal that both stocks may exhibit a significantly higher exposure to upward jumps, even during a strongly bearish period for the market. In this way the negative skewness of the market index may naturally co-exist with a positive skew for an individual stock, hence providing further insights to the skewness preference behavior of investors. Furthermore, the joint estimation procedure uncovers a significant positive idiosyncratic skew component for Yahoo that would not be Jointly estimating jump betas 133 JRF 15,2 134 observable under the single-stock system, a finding that also explains to some extent why investors may be willing to hold stocks of poor mean-variance profiles. The robustness of the model is verified in two different ways. We first replace the benchmark market index with S&P500 in order to see how the system calibrates to different values of the latent skew parameters. We observe that the resulting jump betas of the stocks exhibit a similar pattern as before and are also comparable in magnitude. Then, we perform a Monte Carlo simulation that provides further evidence in favor of the validity of the model. We find that the estimators for the jump betas of a stock are unbiased and we practically demonstrate that the classical CAPM-beta coefficient carries no information regarding the asymmetric behavior of an individual stock, which again underlines the importance of capturing asymmetric betas within a jump-diffusion framework. The structure of the paper is as follows. In the next section we present the model framework. We then explain about the estimation procedures for the single-stock and the joint cases. Then, we discuss on the results and highlight the importance of a joint estimation approach for uncovering idiosyncratic skews. Finally, we examine the robustness of the model and summarize the key contributions of our technique. 2. Framework Let yðtÞ and zðtÞ denote two latent market information factors (i-factors), respectively, for positive and negative market news, that are significant enough to produce price jumps. Also, define g and d to be the sensitivities to these jumps, which are incorporated in an individual asset’s returns through: Ri ðdtÞ ¼ ai dt þ bi qm dW t þ gi yðdtÞ 2 di zðdtÞ þ qi dW i;t þ ui ðdtÞ; where a is a deterministic drift, b a classical (CAPM-like) systematic diffuse coefficient and ðW ; W 1 ; . . . ; W N Þ a standard Brownian motion vector with independent components. The stock price includes jumps introduced either systematically by the pure jump processes y and z or individually by the idiosyncratic corporate jump risk u. Jumps in y are counted via the Poisson random measure my and arrive at a rate ly ðdyÞ, similarly for jumps in z and u. All jump arrivals are independent. Our interest here is primarily in the jump coefficients g and d, that regulate the transmission of the systematic jump risk factors y and z. Market returns lack an idiosyncratic component and are driven by: Rm ðdtÞ ¼ am dt þ qm dW t þ yðdtÞ 2 zðdtÞ: Total risk decomposition into its three latent components is not directly observable. Jumps cannot be safely disentangled from the continuous counterpart; thus properly quantifying the magnitude and significance of the various systematic risk components is not a trivial task, especially if we are unwilling to impose a heavy distributional structure on the jump dynamics. Since all higher-order cumulants of the diffuse part are zero, one can easily filter out the diffusive component of the individual stock price variation (Polimenis, 2012). In an environment of no idiosyncratic skew (i.e. ui ¼ 0), we can derive the following higher-order system of equations by utilizing the affine-transformation property of cumulants: 2 s mmm 6 6 s imm 6 6 6 s iim 4 s iii 3 2 s yyy 2 s zzz 3 7 7 6 7 6 gi s yyy 2 di s zzz 7 7 7 6 7 ¼ 6 g 2 s 2 d2 s 7 7 6 i yyy i zzz 7 5 5 4 g 3i s yyy 2 d3i s zzz ð1Þ 135 where s yyy and s zzz are the skew cumulants[10] of the latent positive and negative jump factors, respectively. The notation s imm and s iim refers to the two co-skews that relate stock returns with market returns. Finally, s iii is the individual stock skew and s mmm denotes market skew. As we will demonstrate in a following section, by relaxing the fourth moment condition of the system above, we will be essentially allowing for idiosyncratic stock skews. 2.1 The joint GMM estimation framework It is a fact that skew and co-skew estimates are notoriously sensitive to extreme return realizations, which may significantly limit the robustness of the estimated parameters of the single-stock system. We are indeed witnessing an irreconcilable wedge between the implied latent skew parameters when the system is estimated separately for two different stocks (Table II). This spurs the necessity for an enhanced estimation procedure that will be able to capture information from multiple assets jointly and drive the system to a more robust solution. We may naturally expand system (1) to incorporate an additional stock by slightly increasing the dimensionality of the problem. Rearranging yields the following set of non-linear sample moment conditions: 2 ðRm;t 2 mm Þ3 2 ðs yyy 2 s zzz Þ Jointly estimating jump betas 3 7 6 6 ðRi;t 2 mi ÞðRm;t 2 mm Þ2 2 ðgi s yyy 2 di s zzz Þ 7 7 6 7 6 6 ðRi;t 2 mi Þ2 ðRm;t 2 mm Þ 2 ðg 2i s yyy 2 d2i s zzz Þ 7 7 6 7 6 7 6 ðRi;t 2 mi Þ3 2 ðg 3i s yyy 2 d3i s zzz Þ E½h t ðu0 Þ
¼ 6 7¼0 7 6 6 ðRj;t 2 mj ÞðRm;t 2 mm Þ2 2 ðgj s yyy 2 dj s zzz Þ 7 7 6 7 6 6 ðRj;t 2 mj Þ2 ðRm;t 2 mm Þ 2 ðg 2 s yyy 2 d2j s zzz Þ 7 j 7 6 5 4 3 3 3 ðRj;t 2 mj Þ 2 ðg j s yyy 2 dj s zzz Þ ð2Þ whereu ¼ ðs yyy ; s zzz ; gi ; di ; gj ; dj Þ is an L-parameter vector that satisfies the orthogonality conditions and h t ðuÞ is a K £ T matrix that represents the sample moment errors for each of the K moment conditions. Denote by g T ðuÞ the K £ 1 vector of the average P sample moment errors for an arbitrary vector u of parameters, i.e. g T ðuÞ ¼ 1=T Tt¼1 h t ðuÞ, with h t ðuÞ being a stationary and ergodic process. The key concept of the GMM lies within the minimization of the objective function u^ðWÞ ¼ argminu J ðu; WÞ, where J ðu; WÞ ¼ Tg T ðuÞ0 Wg T ðuÞ and W is a K £ K symmetric and positive definite weighting matrix. According to Friend and Westerfield (1980) an efficient GMM estimator results if we set W ¼ S^ 21 , where S^ 21 represents the inverse of a consistent estimate of the JRF 15,2 136 asymptotic variance-covariance matrix S ¼ avarðgÞ of the average sample moment ^ error vector g T ðuÞ, such that SpS. If the specified moment conditions are valid, then as T ! 1, the objective value J should follow a x 2 distribution J , x 2 ðK 2 LÞ, the degrees of freedom being equal to the number of over identifying restrictions K 2 L. Note that, unlike the single-stock case, system (2) in the joint GMM estimation will be over identified since K . L. To account for the possibility of heteroscedasticity and serial correlation in the sample moment error vector we estimate the long-run variance-covariance matrix with the HAC estimation procedure suggested by Merton (1976). The estimator of S has the form: ^0þ S^ ¼ V m X ^ j ðu^Þ þ V ^ 0 ðu^ÞÞ; wj ðV j ^ j ðu^Þ ¼ 1=T with V j¼1 T X 0 h^ t ðu^Þh^ t2j ðu^Þ; t¼jþ1 where wj is a kð · Þ continuous weighting function (i.e. a kernel) used to weight lagged variance-covariance matrices and m is the smoothing parameter (bandwidth). According to the Bartlett kernel specification the weights are defined as kð j; mÞ ¼ 1 2 ½ j=ðm þ 1Þ
[11]. 3. Discussion on the results 3.1 Single-stock estimations In order to compare with Polimenis (2012) we use the same three-year sample (2006-2008) of adjusted daily returns for Google, Yahoo and the NASDAQ. We start by estimating the exact-identified system for each individual stock separately. As expected, the estimated parameters perfectly match the moment conditions. For Google, the estimated skew factors for the upside and downside jumps are s^yyy ¼ 0:11 £ 1026 and s^zzz ¼ 1:02 £ 1026 , respectively. The estimated coefficients to these factors are g^ ¼ 4:13 and d^ ¼ 0:98, which suggests a significantly asymmetrical exposure to systematic jumps. These asymmetries are even more pronounced for Yahoo, where we estimate g^ ¼ 15:74 and d^ ¼ 1:85, the corresponding implied skew parameters being s^yyy ¼ 0:011 £ 1026 and s^zzz ¼ 0:924 £ 1026 . Results are also included in Table II. We observe that despite their negative returns, both stocks (and especially Yahoo) are highly exposed to upward market jumps, even if these jumps appear to be less powerful than the negative ones (Table I). The basic idea of our approach is to relax the underlying assumption in equation (1) that all stock skew is systematic, and allow for idiosyncratic stock skews. In this context, our analysis can be thought as a minimization of the unexplained idiosyncratic stock skewness. Thus, we maintain the exact fit for the market skew and the co-skew Table I. Descriptive statistics Market Google Yahoo mI (bps) si (%) siii a siii /si3 simm a siim a 24.67 24.60 216.05 1.71 2.50 3.40 2 0.9132 6.6406 36.0750 20.1826 0.4249 0.9220 20.5556 21.5379 0.8585 2 0.4926 Note: aAll third raw cumulants have been scaled up by 106 moments, since market jumps are by definition systematic, but relax the fit for the individual stock skew to include both a systematic and an idiosyncratic component. We allow the jump cumulants to vary and plot the resulting (sub-optimal) J-values[12]. Technically, by imposing these constraints, we damage the overall fit of the system by canceling three degrees of freedom. Yet, and since the original system in equation (1) is exactly identified, the optimal results of the restricted estimation are identical to those of the unrestricted model that we mentioned above. However, this exercise provides a better insight to the sensitivity of the coefficients to changes in the underlying systematic skew parameters. Figure 3 reveals that the objective J-value is very sensitive to changes in s yyy , which is due to the highly nonlinear nature of the system equations. Figures 1 and 2 reveal that both jump betas (i-betas) g and d are strictly positive, with the i-gammas being negatively sloped with a positive second derivative. On the other hand i-deltas exhibit a positive relation with the upward skew parameter, the second derivative being negative. The explanation of these plots is not trivial; the key to understand them is to realize that we need to calibrate the system to values that can translate a negative market skew into a strongly positive stock skew. In the absence of idiosyncratic skew, the strain of switching the skew sign is placed on the informational sensitivities g and d. This means that for small upside jump risks, we need high i-gamma sensitivities to match the strong positive stock skews, given the fact that the market is negatively skewed (Figures 3 and 4). The analysis of Yahoo reveals a higher sensitivity to the level of jump risks, since its coefficients to the i-factors change sharply with changes of the latent skew parameters. The objective J-value for Yahoo is minimized at s^yyy ¼ 0:011 £ 1026 , the remaining implied parameter values also being the same as in the unrestricted estimation. Note that the error in fitting individual stock skewness (Figure 4) is a positively sloped and concave function with respect to s yyy . This is important because by underestimating the magnitude of positive jump risks in the market, we may end up severely overestimating individual stock skewness. Jointly estimating jump betas 137 12 10 i-gamma 8 6 4 2 0 0 0.5 1 1.5 2 2.5 σyyy 3 3.5 4 4.5 5 x 10–7 Figure 1. Google i-gamma JRF 15,2 1.3 1.2 1.1 i-delta 138 1 0.9 0.8 0.7 Figure 2. Google i-delta 0 0.5 1 1.5 2 2.5 σyyy 3 2.5 σyyy 3 3.5 4 4.5 5 x 10–7 0.25 0.2 J-value 0.15 0.1 0.05 Figure 3. Google J-values 0 0 0.5 1 1.5 2 3.5 4 4.5 5 x 10–7 3.2 A joint estimation for two stocks We observe a wedge between the values of the implied market skew parameters when equation (1) is estimated by the two different stocks. When fitting Yahoo, the positive skew cumulant is approximately s^yyy ¼ 0:011 £ 1026 , smaller than the respective quantity we get by fitting Google. This becomes clear in Figure 5, where we see that the Yahoo plot attains its minimal value far to the left of the Google plot. The magnitude of this difference is the reason why a joint estimation is required. 5 Jointly estimating jump betas × 10–5 4 idiosyncratic skew 3 139 2 1 0 –1 –2 –3 –4 –5 0 0.5 1 1.5 2 2.5 σyyy 3 3.5 4 4.5 5 x 10–7 Figure 4. Google idiosyncratic skew 0.5 0.45 0.4 0.35 J-value 0.3 0.25 0.2 0.15 0.1 0.05 0 0 0.5 1 1.5 2 σyyy 2.5 × 10–7 Notes: The solid line is for the joint case; the dashed line for Google; the dash-dot line for Yahoo Results from an unrestricted joint GMM estimation[13] suggest that the upward jump skew factor is equal to s^yyy ¼ 0:10 £ 1026 . Similarly, the estimated optimal skew for downward jumps is s^zzz ¼ 0:73 £ 1026 . As expected, the unrestricted joint model is unable to capture total market skewness. The implied market skew based on the estimated parameters is s^mmm ¼ 20:63 £ 1026 . This means that a level of 20:29 £ 1026 of systematic skew Figure 5. Joint case; J-values JRF 15,2 140 remains unexplained. It is reasonable to find a more pronounced risk factor associated with negative news than for positive news, since during the period that we are studying the market was dominated by downward jumps that induced a significantly negative skewness in its return distribution. The estimated exposures of Google (stock i ) to the upward and downward jumps are g^i ¼ 4:25 and d^i ¼ 0:85. These values are close to those estimated in the exactly identified single-stock case, but this is not the case for Yahoo (stock j ). We find g^j ¼ 7:50 and d^j ¼ 2:89, due to the implied upward skew parameter being higher than when estimated solely by Yahoo. The fitted skew of Google is 7:35 £ 1026 , which means that the model slightly overestimates the skewness of this stock by 0:71 £ 1026 , implying a slightly negative idiosyncratic skew component. For Yahoo, the skew fitted by the system is 25:14 £ 1026 , hence a level of þ10:94 £ 1026 of skew remains unexplained by the systematic factors, indicating a significantly positive idiosyncratic jump risk factor for this stock. In order to capture total systematic market skew and further focus on idiosyncratic skew, we apply a restricted estimation. We require an exact match for the market skew and all co-skews with the market, thus forcing all fitting to be accomplished by the individual stock skews s iii and s jjj [14]. The restricted optimization of the system shows that the upward skew slightly drops to 0.084£1026 , hence demanding a higher level for the downward jump factor of approximately 1026 . Results for Google are close to those estimated in the single-stock case, and specifically the values for g^ and d^ are 4.56 and 0.94. The optimized parameters for Yahoo are 6.91 and 2.12, slightly lower compared to their unrestricted estimates. Results are less sensitive for Google, which is visible if we compare the J-statistic functions for both individual and joint cases (Figures 5 and 6). 3 × 10–5 idiosyncratic skew 2 1 0 –1 –2 –3 Figure 6. Joint case; idiosyncratic skews 0 0.2 0.4 0.6 0.8 1 σyyy 1.2 1.4 1.6 1.8 2 × 10–7 Notes: The solid line is for the joint case; the dashed line for Google; the dash-dot line for Yahoo The requirement that the entire positive Yahoo skew is systematic, while the skew of the market is negative, places an exorbitant strain on the single stock system for Yahoo. This requirement can only be met at a very high g value for Yahoo and a very low positive market skew (s yyy ). Once we relax this exact match requirement, and essentially allow for the presence of idiosyncratic stock skews, the system calibrates to a more robust solution (Table III(a)). The model still overestimates the skewness of Google by 0.46£1026 (lower than the overestimation found in the unrestricted case), but this is only in the expense of capturing an even lower percentage of the total skewness of Yahoo in order to improve the fit for Google. Generally, the idiosyncratic jump risk factor is much more intense for the case of Yahoo, since there is a significantly non-zero residual skew term of 17:95 £ 1026 that cannot be explained by the two systematic jump risk factors. We should also note that the extremely low value for the positive jump factor needed to calibrate Yahoo’s skew in the single-stock estimation (Table II) is another strong indication that a significant part of this skew is idiosyncratic. This idiosyncratic skew is uncovered during the restricted GMM estimation procedure. By perfectly fitting the market skew and co-skew cumulants we are actually able to utilize the unexplained idiosyncratic skew information of a multi-asset space in order to optimize the explanatory power of the system. So, we may think of the two idiosyncratic skew values of the restricted and unrestricted cases as the high and low estimates of the idiosyncratic skew of a specific stock, respectively. Our results also reveal that some stocks with more volatile and positively skewed returns, like Yahoo, may exhibit increased exposure to jumps, and especially to positive ones. During the three-year period of our study the index was loosing approximately 4.67 basis points on average per day. The stock price of Yahoo was declining rapidly in value, shrinking by a daily average of 16.05 basis points, with a daily volatility of 3.4 per cent, which was twice the volatility of the market returns. Theoretically, in a standard mean-variance world, it would be suboptimal for a risk-averse investor to hold stocks with low Sharpe ratios. Yet, we find that even in a strongly bearish market, a stock like Yahoo may possess a higher exposure to an unexpected upward market jump. This indicates that theories about skewness preference (Jiang and Oomen, 2008) may provide alternative insights to the reasons why investors are willing to hold stocks of poor mean-variance profile in their portfolios. Our newly proposed framework is likely to provide an alternative explanation about investors’ behavior. Implied non-linearities in stochastic discount factor may allow for a potentially risk-seeking domain (Polimenis, 2012) in the utility optimization scheme faced by investors. (a) vs NASDAQ (b) vs S&P500 g^i 4.13 3.14 d^i 0.98 0.95 (a) vs NASDAQ (b) vs S&P500 g^j 15.74 15.26 d^j 1.85 1.32 Google s^yyy s^zzz 0.11 1.02 0.25 1.50 Yahoo s^yyy s^zzz 0.01 0.92 0.01 1.26 Note: All third raw cumulants have been scaled up by 106 s^mmm 2 0.91 2 1.25 s^iii 6.64 6.64 e s^mmm 0 0 s^iiie 0 0 s^mmm 2 0.91 2 1.25 s^jjj 36.08 36.08 e s^mmm 0 0 s^jjje 0 0 Jointly estimating jump betas 141 Table II. Single-stock estimations JRF 15,2 142 3.3 Robustness tests It is crucial to verify the robustness of the results that we presented in the previous sections. Since we are dealing with a highly non-linear system of equations in order to capture the underlying jump betas, this task will not be trivial because the output of the model may vary significantly even with minor changes to the estimated parameter values. We differentiate the panel of data that we are using by replacing the index that proxies the market with S&P500, but at the same time keeping Google and Yahoo in our dataset in order to achieve comparability among the results. A quick glimpse at the data reveals that the S&P500 index was falling by 4.51 basis points and with a volatility of 1.65 per cent per day. It also appears to be even more negatively skewed than the NASDAQ, with a significant skew of 21:25 £ 1026 , which is again explained by the fact that the market was dominated by negative jumps during that period (Table II). We start by applying the system separately for Google and Yahoo, this time versus the S&P500 (panel (b)). The results are summarized and compared to those of the previous estimations (panel (a)) in Table II. In this panel, the downward jump risk factor s ^zzz is slightly larger than before, which is directly caused by the more pronounced negative skewness of the S&P500 index. The robustness of the model is indicated by the fact that even though we used a different market index (i.e. with different latent skew factors), the estimated jump sensitivities g^ and d^ for the two stocks are of similar magnitude with those of the first panel. For the purposes of the table we use s^iii for the estimated skewness by the model and e s^zzz for the “idiosyncratic” part of skewness that cannot be explained by the two latent 3 systematic skew factors. Hence, we have s^iii ¼ g^i3 s^yyy 2 d^i s^zzz for stock i and s^mmm ¼ s^yyy 2 s^zzz for the market. It follows that the idiosyncratic skew estimate for stock i equals s^iiie ¼ siii 2 s^iii . Note once more that in all single-stock estimations, the system perfectly matches the moment conditions, producing no residual idiosyncratic term. In order to capture more accurately the level of the latent systematic jump risk factors and uncover information about the idiosyncratic skewness in individual stock return distributions, we need to deploy the joint GMM estimation techniques. Results regarding the unrestricted (U) and the restricted (R) GMM estimations can be found in Table III. As expected, in the joint cases the system calibrates somewhere in between the two stocks, hence capturing information from a multi-asset space. The values of the estimated upward and downward jump factors always lay within the band that is determined by the single-stock cases of Google and Yahoo separately, the downward skew parameter always being larger in order to explain the negative skew of the index. Starting from the g^i Table III. Joint estimations d^i g^j Panel (a): NASDAQ U 4.25 0.85 7.50 R 4.56 0.94 6.91 Panel (b): S&P500 U 3.58 0.85 4.92 R 3.79 0.86 5.16 d^j s^yyy s^zzz s^mmm s^iii s^jjj e s^mmm s^iiie s^jjje 2.89 2.12 0.10 0.08 0.73 1.00 20.63 20.91 7.35 7.10 25.14 18.12 20.29 0 20.71 20.46 10.94 17.95 1.75 1.63 0.17 0.15 1.33 1.40 21.16 21.25 7.16 7.46 13.55 15.03 20.09 0 20.52 20.82 22.53 21.05 Note: All third raw cumulants have been scaled up by 106 unrestricted case we find that s^yyy ¼ 0:17 £ 1026 and s^zzz ¼ 1:33 £ 1026 . The positive (and negative) jump sensitivities for the two stocks are 3.58 (and 0.85) for Google and 4.92 (and 1.75) for Yahoo, respectively. As with the use of NASDAQ in (a), we see that in the joint case the jump coefficients of Yahoo drop significantly compared to its single-stock case. After imposing the condition that the system perfectly explains market skewness, we find that the results of the restricted GMM (R) are very close to the unrestricted ones (U). There is only a minor increase in the level of the downward skew factor, accompanied by a small increase in the gammas of the two stocks, i.e. 3.79 and 5.16, respectively, for Google and Yahoo. The system slightly overestimates the skew of Google by a small amount of 0:82 £ 1026 , as indicated by the residual skew term s^iiie . On the other hand, it seems that there is a very significant part of the skewness of Yahoo that remains unexplained by s^yyy and s^zzz . The result for s^jjje suggests that there exists a strongly positive idiosyncratic skew component equal to 21:05 £ 1026 embedded in the returns distribution of Yahoo. Overall, we see that in spite of the highly non-linear nature of the system, the estimated parameter values among the two panels are directly comparable in magnitude and also exhibit similar patterns, which is the case not only for the single-stock cases but also for the joint GMM estimations. 3.4 A Monte Carlo simulation To further examine the properties of the skew-based single stock estimator, we perform a simulation where 10,000 random paths (each consisting of 10,000 days) are generated for the latent jump and noise components. We generate Gamma-distributed market jumps (since Gamma is a commonly used distribution to model jumps in finance), where y , Gð0:91; 3:93 £ 1023 Þ and z , Gð80:83; 1:85 £ 1023 Þ, while the noise is Gaussian with 1 , Nð0; 113:67 £ 1026 Þ. Stock returns are simulated using the single-stock Google estimated values g^ ¼ 4:1 and d^ ¼ 1, which implies that 38 per cent (44 per cent) of the stock variance is generated by positive (negative) market news, while 18 per cent is idiosyncratic. Since the classical CAPM beta b ¼ ðsi;m =sm2 Þ is a weighted average of the jump loadings b ¼ ðsy2 =sm2 Þg þ ðsz2 =sm2 Þd with sm2 ¼ sy2 þ sz2 , then for this particular simulation the classic beta equals b^ ¼ 1:15. The estimated g^, d^ and b^ values are unbiased, but an interesting finding that merits further investigation is that, since g^ and d^ depend on skew and co-skew estimates, they are sensitive to extreme realizations. On the contrary, b^ carries no information about such extreme return realizations. This is evidenced by the path-by-path variation for g, d and b that equals 6.6 per cent, 0.4 per cent and only 1 basis point, respectively. This is also depicted in Figure 7. To the extent that one believes that the biggest market movements are responsible for (asymmetrically) transmitting the most significant stock price information, it is clear that the traditional beta is devoid of such information. 3.5 A final note on the discovery of the asymmetric jump betas In order to present an alternative approach to our estimation procedure and further examine the observability of the jump betas we deploy a simple linear regression approach. We utilize a common cut-off criterion to detect market jumps, where the jumps are determined as the returns that are observed in the highest and lowest quantiles of the empirical distribution of the market returns. We consider a regression model of the following form: Jointly estimating jump betas 143 JRF 15,2 6,000 positive jump coefficient (γ) 5,000 negative jump coefficient (δ) CAPM beta (b) 4,000 144 3,000 2,000 1,000 Figure 7. Simulation joint histogram for g, d and CAPM-b 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 2 2 Ri;t ¼ c þ bi Rm;t þ gi R2m;t 1þ þ et; {Rm;t $qaþ } 2 di R m;t 1{Rm;t #q2 a} 2 where the two dummy variables 1þ indicate a positive or negative {Rm;t $qaþ } and 1{Rm;t #q2 a} jump, respectively, for the cases where the realization of the market return exceeds the critical cut-off value that is defined from the highest and lowest quantiles qaþ and qa2 of the distribution of Rm . Just for illustration and comparison purposes with the results above we focus again on the case of Google versus the NASDAQ. We allow the level of alpha for the quantiles to vary from 5 to 20 per cent. By increasing the level of alpha it actually becomes easier to detect jumps in the data. The results of the regression estimations are summarized in Table IV. We can see that the beta coefficient is close to one, as expected. The sensitivity to the positive jumps turns out to be higher than the sensitivity to the negative jumps, but both values are slightly smaller than those estimated by our skew-based methodology. This is due to the highly non-linear nature of our estimation procedure, which exploits information that cannot be captured by a regression approach and takes into further consideration the idiosyncratic stock skewness. 4. Conclusions We expand on the Polimenis (2003) technique by applying it jointly for two stocks, at the same time allowing for idiosyncratic stock skew. Introducing the possibility of a Table IV. Regression results 0.05 0.08 0.10 0.12 0.15 0.20 qaþ q2 a Nj c bi gi di 0.0229 0.0186 0.0148 0.0134 0.0116 0.0088 2 0.0253 2 0.0216 2 0.0186 2 0.0167 2 0.0130 2 0.0094 76 114 150 180 226 303 2 0.000113 2 0.000113 2 0.000102 2 0.000107 2 0.000119 2 0.000099 0.964469 0.954094 0.938986 0.935632 0.935223 0.931804 1.201313 1.362240 1.607842 1.678973 1.711015 1.681466 0.083686 0.341695 0.721759 0.773302 0.738053 0.878304 idiosyncratic jumps helps to avoid the restrictive assumption that the individual skews can only be explained by systematic skew factors. We argue that, once we relax this restrictive requirement, the system calibrates to a more robust solution in terms of not only uncovering the true magnitude of the latent parameters of the model, but also revealing information about the level of idiosyncratic skews in individual stock return distributions. Generally, we aim to explore the fundamental idea that jumps in stock prices incorporate the most valuable information, and this can have many practical ramifications since the quantification of a stock’s exposure to jump events is important for: . financial risk management; . portfolio construction; and . option pricing. In particular, measurement of the jump betas is important for the proper valuation of stock options, since jump exposure imposes skew that is known to significantly affect benchmark volatility such as VIX. Notes 1. Lee and Mykland (2012), Harvey and Siddique (2000), Bawa and Lindenberg (1977), Estrada (2007), Hansen (1982), Ang et al. (2006), Post and Levy (2005), Eraker (2004) and others have suggested the use of lower-partial moments and, as a special case, semivariance, in order to proxy downside risk. A more detailed presentation of downside risk measures can be found in Post and Van Vliet (2006). 2. Carr et al. (2002) extract information about the return process for an underlying asset by utilizing option prices. 3. Aı̈t-Sahalia and Jacod (2009) draw a similar conclusion by analyzing the degree of jump activity in high-frequency data series. 4. Sortino and Satchell (2001) utilize the framework of Barndorff-Nielsen and Shephard (2004) in order to disentangle the systematic continuous and discontinuous parts and estimate the sensitivities of individual stocks towards these risk factors. 5. In this context, time is measured in daily units for integer t and the within-day log-returns are determined for an N-partition of the t th day. 6. Such noise arises from bid-ask bounce, price discreteness, non-synchronicity in trades, etc. 7. Or better, frequency-neutral. 8. Skewness and jumps are generally related to illiquidity and market frictions and hence high-frequency techniques will introduce complications in exactly the situations where higher moments are expected to heavily impact asset prices. 9. Ahern et al. (2011) propose to circumvent this problem by using ARCH techniques to estimate the equity risk premium for public utilities based on the entire stock risk. 10. We use the sijk notation for the third cumulant, where sijk ¼ E½ðRi 2 mi ÞðRj 2 mj ÞðRk 2 mk Þ
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Jointly estimating jump betas 147 JRF 15,2 148 Post, T. and Van Vliet, P. (2006), “Downside risk and asset pricing”, Journal of Banking & Finance, Vol. 30 No. 3, pp. 823-849. Sortino, F.A. and Satchell, S. (2001), Managing Downside Risk in Financial Markets, Butterworth-Heinemann, Oxford. Further reading Todorov, V. and Bollerslev, T. (2010), “Jumps and betas: a new framework for disentangling and estimating systematic risks”, Journal of Econometrics, Vol. 157 No. 2, pp. 220-235. Corresponding author Vassilis Polimenis can be contacted at: polimenis@econ.auth.gr To purchase reprints of this article please e-mail: reprints@emeraldinsight.com Or visit our web site for further details: www.emeraldinsight.com/reprints