# Managing weather risks: the case of J. League soccer teams in Japan.

INTRODUCTIONWeather-related risk exposures, such as varying temperature and precipitation, are important concerns for companies worldwide. According to the Chicago Mercantile Exchange (CME) Group, (1) one-third of businesses around the world are affected by adverse weather conditions. In particular, 80 percent of U.S. firms and 75 percent of Japanese firms have been exposed to weather-related risks to different extents (Myers, 2008; Yokoyama, 2014). Weather risks are of particular interest to the leisure and entertainment industries, such as ski resorts with snow risks (Brockett, Wang, and Yang, 2005), golf courses and beaches with precipitation risks (Smith, 1993; Leggio, 2007), and as we will focus on in our study, the sports industry with an assortment of different weather risks. Some initial evidence has been provided in the literature on varying effects of adverse weather for sports teams in different types of sports and geographic areas (e.g., Bird, 1982; Bruggink and Eaton, 1996; DeSchriver, 2007).

In most regions of the world, weather risks have been managed more or less by specialized insurance policies or other government-sponsored financial mechanisms (IPCC, 2012). In the United States, these risks are now primarily managed using weather derivative products since they became available on the CME in 1997. The trading volume on CME was $9.4 billion from April 2010 to March 2011 with a large number of trades (466,000 in total) (Weather Risk Management Association [WRMA], Annual Industry Survey, 2011). Although there has been a drop from the high of over $40 billion from April 2005 to March 2006, it is still of paramount significance. Temperature contracts account for the great majority of CME contracts and less than 5 percent are precipitation contracts. Temperature-based weather derivatives are primarily based on indices of heating degree days (HDD) and cooling degrees days (CDD) and are traded for 24 U.S. cities, 6 Canadian cities, 11 European cities, 3 Australian cities, and 3 Japanese cities. Precipitation-based contracts are available for March to October in the form of monthly contracts or seasonal contracts (2 to 8 consecutive months in length). Both futures and options contracts are available and they are based on a precipitation index reflecting the cumulative precipitation amount (measured in inches) for the desired time period. Futures contracts pay $500 times the respective index value, with a tick size of 0.1 inch. Option contracts have exercise prices ranging from 0 to 20 inches for monthly contracts and 0 to 60 inch. for seasonal contracts. The indices are currently available only for 10 U.S. cities and are not available for other countries. In fact, most of the CME contracts are used for weather events in different regions within North America and less than 5 percent are used for regions outside of North America and Europe.

According to WRMA annual industry surveys since 2001, the growing over-thecounter (OTC) market plays an active role in complementing the CME market. Of the 14 surveyed companies (WRMA members in various industries such as banking, insurance, and energy), a total of 998 OTC contracts were written from April 2010 to March 2011, a 160 percent increase from the previous year. These contracts are large with over $2 million notional value on average and in total, represent a volume of $2.5 billion in 2010/2011. This represents the highest total notional value since 2004/2005. Note that the percentage of OTC contracts with counterparties that do not participate in the survey has been consistently high over time (79 percent in the 2011 survey). In contrast to the CME market, precipitation contracts account for at least 20 percent of the OTC contracts and regions outside Europe and North America have a much better representation (over 10 percent in 2011 and over 20 percent in 2010).

Despite the increasing trend of contracts covering precipitation events and Asian markets in the recent years, the weather derivatives market is still greatly dominated by temperature contracts covering North American events. Consequently, existing academic research has focused on temperature-based weather risks and the U.S. market, with few exceptions (e.g., Martin, Barnett, and Coble, 2001; Skees et al., 2001; Leggio, 2007). (2) In this article, we fill in a gap by providing specific evidence on using precipitation-based weather derivatives for managing risk exposures in an international marketplace. Using data from the premier Japanese soccer league, we examine its financial susceptibility to precipitation related risk, propose a weather hedging mechanism for the league, and examine its value.

The Japanese sports industry, in particular soccer, is large and growing. Established in 1993 with only 10 teams in total, J. League now comprises two divisions, the premier division J1 and the lower division J2, with 38 teams in total. With a 2010 attendance of 7,928,976 (J1: 5,638,894; J2: 2,290,082), the total revenue for J. League in 2010 was US$880 million (J1: US$665 million; J2: US$215 million), or an average of US$38 million for J1 teams and US$11 million for J2 teams. Our analysis is mainly based on data from the J1 teams because of their popularity and financial significance.

Adverse weather conditions were not taken into consideration when most soccer stadiums were built in Japan. Of the 49 stadiums that hosted a J. League game in 2011, 9 (18.4 percent) have roofs covering the entire stadium, 18 (36.7 percent) have partial roofs, and 22 (44.9 percent) have no roofs at all. Since J. League never suspends a game because of rain, spectators must watch the game in the rain, and might even have to sit directly on the wet grass as some stadiums do not have seats. (3) Although soccer games are scheduled well in advance and teams are contractually bound to play rain or shine, spectators can and do rely on weather forecasts when making plans to attend a game. This exposes soccer teams to significant financial risks in ticket sales and related revenues that are their primary sources of income. Left unmanaged, inclement weather can be a substantial contributing factor to the average operating loss of US$500,000 for J1 teams in 2010. This volatile financial performance might ultimately threaten continuing corporate sponsorship, which is key to the survival of sports teams worldwide.

Despite significant weather risks, very few teams have taken effective measures to manage these risks. Among the 38 J. League teams, only 3 teams (Teams Shimizu, C Osaka, and Hiroshima) purchased precipitation-based insurance contracts after it was first introduced in Japan in 2002 (Nihon Keizai Shimbun, 2002a, 2002b). However, according to personal interviews conducted by the authors in 2011 with J. League managers, four out of seven showed interest in weather risk management. The interviews also revealed concerns about the currently available risk management methods that are expensive and time consuming. Teams can either build a new stadium with a roof or retrofit an existing stadium, or purchase an insurance policy with high loadings. (4) These findings call for better risk management options designed based on empirical evidence that documents and measures the financial impact of adverse weather. This article examines the use of weather derivatives as a potential risk management solution for the Japanese Soccer League teams.

Using two matched data sets composed of J1 teams' game attendance and corresponding weather conditions between 1993 and 2010, we are able to develop quantified measures of the teams' precipitation-related weather exposures, design a hedging mechanism that caters to the quantified exposure, and carry out subsequent analyses to evaluate this risk management option for the teams. In particular, we use the Wang transform model (Wang, 2000) to arrive at valuations that address the incompleteness of this market. We specifically account for the risk preferences of the decision makers through a survey-based measure. Our regression and simulation analysis results suggest that the J. League soccer teams have a significant precipitation exposure. We also find that the proposed hedging mechanism is valuable based on the estimated managerial risk preferences, even after taking into account transaction costs and premium loadings associated with the derivatives.

This article contributes to the growing literature on capital market solutions to managing physical risks, such as financial product innovations as new or alternative approaches in catastrophic risk, and longevity/mortality risk management. In particular, weather risk products in an incomplete capital market post additional challenges in designing, pricing, and evaluating different risk management solutions. This is one of the first papers in the peer-reviewed literature, to our knowledge, that provides a complete analysis of the impact of hedging weather risk and its financial consequences for a specific potential weather derivative user in a well-defined industry. By focusing on precipitation risk and a less explored industry with a significant risk exposure, we complement the existing literature that considers primarily temperature derivative contracts for the energy and agriculture industries. Although our analysis is done using data from the Japanese soccer industry, it provides insights for the sports industries around the world. Just like in most of these cases, corporate sponsorship plays a major role in J. League clubs' financial viability. Properly managing weather risks and hence reducing the likelihood of unfavorable financial outcomes and the uncertainties therein will help cultivate continued support from the corporate sponsors. In addition, the proposed design and evaluation of the hedging instrument from a managerial perspective are more generally applicable to businesses in various sports and entertainment industries for their weather risk exposures.

The rest of the article is organized as follows. The "Literature Review" section discusses the related literature. The "Quantifying Precipitation Risk Exposure for J. League Teams" section provides a comprehensive set of empirical analyses that document and estimate the weather exposures of J. League teams. In the "Simulation Analyses for the Impact of Precipitation on Financial Performance" section we use simulations to calibrate the impact of the weather exposures on the teams' financial performance based on empirically estimated parameters. In the "Weather Risk Hedging and Its Contribution to the Corporate Value" section we evaluate the contribution of a proposed weather derivative hedging instrument to a team's corporate value using the Wang transform model with survey-based risk aversion parameters. The "Conclusion" section concludes the article.

LITERATURE REVIEW

Starting with Noll (1974), a plethora of papers focus on estimating the demand function for the sports industry, a nice review of which is provided by Borland and MacDonald (2003). Different factors are found to be significant contributors to the demand function, including the winning rate of the team (Noll, 1974; Davis, 2008), the presence of super stars (DeSchriver, 2007) and terrorist risks (e.g., Toohey and Taylor, 2008; Kalist, 2010), and the competitiveness of the games (Coates and Humphreys, 2010). Management of these risk factors is also extensively studied in the literature (see, e.g., Fuller and Drawer, 2004; Lee, Farley, and Kwon, 2010).

However, there have been only limited studies of the effect of weather conditions on the demand for sports teams or its risk management, even though the impact is suspected to be significant and unpredictable. In addition, these studies provide mixed findings. For example, major sport events in the United States such as Major League Baseball (MLB) (Bruggink and Eaton, 1996; Butler, 2002) or National Football League (NFL) games (Welki and Zlatoper, 1999) have been significantly affected by excessive precipitation and extreme temperature. However, U.S. soccer teams do not seem to be affected by weather conditions (DeSchriver, 2007). Significant weather effect was not found for European sports teams in general (Bird, 1982; Hynds and Smith, 1994; Carmichael, Millington, and Simmons, 1999), except for cricket whose attendance is negatively affected by the rain (Schofield, 1983; Morley and Thomas, 2007). In particular, Kawai and Hirata (2008) estimate, for the first time, the demand function for J. League soccer games and report that the teams experience significantly lower attendance when it rains. Their study used one dummy variable based on precipitation observed at the stadium at the beginning of the game, and used data from 1993 to 2005. Using data from 1993 to 2010, we design and examine a comprehensive set of precipitation variables to thoroughly capture the effect weather has on attendance rates and, moreover, measure its impact on the teams' financial outcomes. More importantly, we further investigate how a hedging mechanism can stabilize cash flows, thereby improving the team value.

There has been only limited research on the design and effectiveness of weather derivatives and most of the studies focus on the temperature exposure in the United States. Vedenov and Barnett (2004) examine the effectiveness of weather derivatives in hedging temperature and precipitation risks affecting agriculture yields in the United States. They use semivariance, value-at-risk, and certainty-equivalent measures to evaluate the reduction of crop yield variations. Golden, Wang, and Yang (2007) and Golden, Yang, and Zou focus on temperature-related weather exposure and examine hedging effectiveness of CME-traded weather derivatives in the U.S. power industry using variance as the measure, while Yang, Li, and Wen study that for three European countries. Perez-Gonzalez and Yun (2013) show a significant and positive relationship between the temperature-based weather exposure of energy firms and the adoption of weather derivatives when they became available in 1997. They also provide evidence that the use of weather derivatives improves firm value. Leggio (2007) and Martin, Barnett, and Coble (2001) study the design of precipitation derivatives for golf courses and the agricultural business in the United States, respectively. Skees et al. (2001) is one of the few papers that propose precipitation risk management in the international market-place. It studies an index-based drought insurance product for cereal producers in Morocco and examines the impact of basis risk. Most of these studies do not focus on assessing the value of weather hedging and most times do not address market incompleteness. The popularity of soccer games in Asian and European countries and the lack of evidence in the international markets led us to take advantage of our unique data sets and focus on the much less explored precipitation risk and its management.

Because weather is not a traded asset, we need to address the incompleteness of the market when evaluating the hedging mechanism. Various valuation approaches have been proposed, including the Esscher transform (Gerber and Shiu, 1994, 1995), the Wang transform (Wang, 2000, 2002), the relative entropy optimization (McLeish and Reesor, 2003), and the indifference pricing method (Brockett et al., 2009). See Brockett, Wang, and Yang (2005) for an overview of these approaches. In this article, we adopt the Wang transform (Wang, 2000, 2002) to evaluate the contribution of a proposed weather hedging mechanism to the corporate value of J. League soccer teams. The Wang Transform provides a universal framework for pricing contingent claims. It has clear economic interpretations that capture the risk preference of the decision-maker without imposing a functional form for the utility. (5) Furthermore, it can recover the Black-Scholes formula and the capital asset pricing model (CAPM) under appropriate assumptions (Wang, 2000). The Wang transform has been applied to many contingent claim pricing scenarios (e.g., Chen and Cox, 2009; Godin, Mayoral, and Morales, 2009), empirical estimation of the risk aversion coefficient (Jones and Zitikis, 2007), and derivation of optimal economic capital for insurers (Hurlimann, 2004). The Wang transform has also been extended to multivariate risk settings (Wang, 2007; Kijima and Muromachi, 2008).

Using the Wang transform model, we calculate the expected cash flows under the distorted probability measure considering the risk preferences of the managers. One challenge is to estimate the risk aversion parameter. We follow the recent literature to use a set of lottery-based questionnaires distributed to the J. League managers to obtain this estimate (see, e.g., Pennings and Smidts, 2000; Donkers, Melenberg, and Soest, 2001 ; Holt and Laury, 2002; Andersen et al., 2008). We discuss the design of our questionnaires and summarize the results in the "Weather Risk Hedging and Its Contribution to the Corporate Value" section.

QUANTIFYING PRECIPITATION RISK EXPOSURE FOR J. LEAGUE TEAMS

Taking advantage of the available data sources, we investigate the weather risk exposures of J. League soccer teams by empirically examining the impact of weather conditions on game attendance. Game attendance is closely related to direct ticket sales and other indirect revenue sources based on popularity (e.g., concessions and other sales in the stadium, advertisement and commercial revenues, etc.). These constitute a significant portion of revenue for the teams. (6) We run a set of regressions for each team using game-by-game J. League attendance data matched with corresponding weather data from 1993 to 2010. Our main interest is in precipitation since preliminary analyses suggest that excessive precipitation is the most plausible weather concern for low attendance. We explore two different types of precipitation measures in our analysis. The first type uses the raw value of the precipitation amount and the second one is a dummy variable with different threshold values of raw precipitation.

Data Description and Preliminary Data Processing

We obtain the game attendance data from J. League and focus only on J1 teams, which have much higher attendance and revenues than J2 teams. If a team is relegated from J1 to J2 in our sample period, we exclude this team from the analysis in the subsequent years. We include the team again if it returns to J1 and, similarly, include the corresponding data points if a new team is upgraded to J1. In our analysis, we focus on teams whose main stadium is "nonroofed," that is, the roof covers no greater than 50 percent of the spectator area, as they are likely to have a higher weather exposure. (7) A team's main stadium is defined by the J. League at the beginning of the 2011 season. Most teams have only one main stadium with the exception of Team Yokohama FM has two (Nippa stadium and Nissan stadium). As such, our attendance data set includes 2,082 games out of a total of 3,984 J1 games during the sample period, 1993-2010. We run time-series regressions for each unique combination of a team and its main stadium, with the exception of Team Yokohama FM, for which we run regressions for two unique combinations.

We obtain corresponding precipitation data from the Japan Metrological Agency (JMA) (http://www.jma.go.jp/jma/index.html). In the JMA data set, precipitation is recorded hourly as "-", "0.0 (mm)," "0.5 (mm)," and hereafter increasing in 0.5-mm increments. In particular, "-" means no rainfall was observed whereas "0.0" means rainfall was observed for less than the measurable amount (i.e., at least 0.5 mm to be measurable). (8) We design a set of level measures based on the raw precipitation amounts. Specifically, we calculate the cumulative amount of precipitation around the game time for the following durations: 1, 2, 3, 4, 6, and 24 hours.

Additionally, we define a set of dummy measures from the raw precipitation amounts to account for any nonlinear effect on game attendance; that is, individuals' willingness to go to a game can be based on a certain cutoff point rather than monotonically related to the amount of rain. Lacking any previous evidence for threshold choices, we have constructed the dummy measures of precipitation in a largely exploratory manner. We set the threshold from 0 to 5 mm for all variables with 0.5 mm increments. We set additional thresholds from 5.5 to 15 mm in 1-mm increments for the 6-hour and above precipitation variables, and additional thresholds of 20, 30, 40, and 50 mm for the 24-hour precipitation variables to accommodate the higher amount of rain anticipated for longer periods. The dummy variable takes value 1 if the amount of precipitation is at least the threshold level, and 0 otherwise. As a result, we have a total of 354 precipitation variables, including both the level measures and the dummy measures. The detailed definitions of these variables are presented in the Appendix.

Empirical Calibration of Precipitation Exposures

We run a set of time-series generalized least squares (GLS) regressions to calibrate precipitation exposure for the J1 teams and to facilitate further team-based simulation analyses and evaluations. In addition to the precipitation measure, we include the following control variables from the sports risk management literature. Distance measures the distance that needs to be traveled by the supporters of the away team and is proxied by the natural log of the train ticket price in 2012. The longer the distance, the less attendance is expected (Garcia and Rodriguez, 2002). Natural log of the Population of the city where the team is located is expected to be positively correlated with game attendance (Hart, Hutton, and Sharot, 1975; Kalist, 2010). To proxy for team popularity, we count the total number of star players, stars_both, in both the home and the away teams. A player is identified as a "star" if he represented the Japanese National Soccer Team in the previous year. (9) We also include lagged average game attendance for the home team (Forrest, Simmons, and Szymanski, 2004; Kawai and Hirata, 2008), h_latt, and that for the away team (Kawai and Hirata, 2008), a_latt.

In addition, we control for a set of dummy variables that can affect game attendance. Rival equals 1 if the game is between two teams from the same prefecture, and 0 otherwise. Like football or baseball in the United States, there is likely an increased interest for games between two rival teams that are often defined by the proximate locations. Opening refers to the opening game of the season, which can generate the greatest interest and hence the highest attendance. Top3 equals 1 if either team was ranked among the top three in the previous season, and 0 otherwise, as more fans will be following the leading teams. Similarly, because fans can be especially concerned if their teams are running the risk of being downgraded to the J2 League, we also include a dummy variable, Worst3, that equals 1 if either team was ranked within the last three in the previous year, and 0 otherwise. Note that we rank as the lowest those teams just promoted to the J1 League from the J2 League this year, and thus variable Worst3 also captures any particular interest in these newly promoted teams. Weekday indicates a game on a working day since fewer spectators are likely to be at the game (Garcia and Rodriguez, 2002; Kawai and Hirata, 2008). Lastly, we also include an April dummy and an October dummy as attendance tends to be higher at the beginning and the end of the soccer season (Kawai and Hirata, 2008). Our multivariate GLS regression model is specified as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [Y.sub.i,n,t] is the natural log of game attendance and [R.sub.i,j,n,t] is jth precipitation measure, j = 1,..., 354, for a unique combination i of a team and its home stadium's game n in soccer season t. (10) While we have run regressions using all 354 precipitation variables, we only present in Table 1 the results using the intuitive dummy measures d0_h3 and d4_h24, taking value 1 if it rains during the 3-hour period right before the game or if there is at least 4-mm precipitation on the game day, respectively. (11) Note that to conserve space, Table 1 only shows the coefficients for the precipitation dummies and the model fit statistics, while full regression results are available in the accompanying online appendix. (12)

As shown in Table 1, precipitation is a significant contributing factor to game attendance for most team/stadium combinations after controlling for other relevant factors. For example, our analysis shows that the Kyoto, Saikyogoku combination is heavily affected by adverse weather conditions. If there is at least 4 mm of rain on the game day, attendance is expected to be 20.5 percent less, other things being equal. Even for a less affected combination, such as Team Yokohama FM in the Nippa Stadium, the effect is still significant. Our results show that their attendance is 9.9 percent less when there is a minimum amount of precipitation during the 3-hour period before the game.

Table 1 also shows that the precipitation effect differs in the magnitude for different teams and stadiums. These variations may be due in part to the joint effect of popularity of the team and the capacity of the stadium. (13) Overall we find an adverse weather effect to different extents for 14 out of 15 unique combinations of teams and stadiums, illustrating that weather risk is an imperative concern for the J. League. (14)

In the next section, we use the team-by-team time-series regression results (Table 1) to further calibrate the impact of precipitation on a team's financial performance and to evaluate the proposed weather hedging mechanism for a J. League team. (15)

SIMULATION ANALYSES FOR THE IMPACT OF PRECIPITATION ON FINANCIAL PERFORMANCE

Design of the Simulation Analyses

Because detailed financial data are not available for the J. League teams for a sufficiently long sample period, (16) we use Monte Carlo simulations to gauge the effect of weather on teams' financial performance. More specifically, we simulate game-day precipitation using parameters estimated from historical weather data and obtain simulated attendance using the precipitation exposures measured in the regression analysis. We then use aggregate financial data to calibrate cash flow sensitivity to attendance and generate team cash flows from simulated attendance. In our simulation analysis, we focus on the intuitive dummy measure of game-day precipitation with the 4-mm threshold value (d4_h24); that is, the precipitation variable takes value 1 if the cumulative precipitation on the game day exceeds 4 mm, and 0 otherwise. This also allows us to make a comparison with the precipitation insurance product offered in Japan designed based on game-day cumulative precipitation. (17)

We illustrate our simulation analyses using a representative team, Kawasaki F, and their home stadium, Todoroki, located in Kawasaki City (population of 1.4 million) in the greater Tokyo area. Team Kawasaki F was established in 1997 and joined the J. League in 1999. The Todoroki stadium has 50 percent roof coverage and a capacity of 25,000 spectators. With an operating profit of 47 million JPY (US$573,000), Team Kawasaki F ranks in the middle among all J. League teams. (18) Additionally, all of its home games are held in the Todoroki Stadium, ensuring a more precise analysis than the teams using multiple home stadiums.

The adverse weather impact on Kawasaki F's game attendance is significant: both precipitation measures reported are highly significant at the 1 percent level. The game-day dummy measure with a 4-mm threshold value (d4_h24) contributes to a 21.4 percent reduction in game attendance in the (unreported) univariate analysis. In the multivariate specification, the adverse impact is 20.1 percent.

Game Day Precipitation Simulation and Attendance Estimation

We assume that the occurrence of a rainy game day follows a Bernoulli distribution and use Monte Carlo simulations to estimate the number of rainy day games and non-rainy day games for a soccer season. We estimate the parameter of the probability of a rainy game day from the empirical distribution of precipitation obtained from historical weather data observed at the Yokohama Observatory (the closest to the Todoroki Stadium) from 1968 to 2010. Because the soccer season spans 10 months of the year and there is significant heterogeneity in precipitation during these months, we break the months into different "seasons" before estimating the probability of precipitation. We define five seasons so that within a season there is no significant difference between the months in the likelihood of a rainy game day, whereas between seasons the differences are statistically and economically significant. The combinations are: Season 1: March, April, May, and October; Season 2: June and September; Season 3: July; Season 4: August and November; and Season 5: December. January and February are excluded because there are no games during this time. Table 2 summarizes the mean probability of a rainy game day for each season and the pairwise differences between seasons. We can see that the differences between seasons are significant both economically and statistically.

Using the probability parameters shown in Table 2, we generate for each season one Bernoulli variable for a rainy game day. According to the 2012 game schedule of Kawasaki F, eight home games are held in Season 1, four in Season 2, one in Season 3, and four in Season 4. No game id held in Season 5 in 2012. With a further assumption that precipitation measures between different games are independent, we can simulate the weather conditions for each of the 17 games in a soccer season. Ten thousand simulation paths are run to obtain the weather conditions. Table 3 presents the simulated probabilities of having n rainy game days in a soccer season. The most probable outcome is having 2 rainy days, with a probability of 0.2702. The probabilities of having 10 or more rainy days are very small and are assumed to be zero in further analysis.

We can then use the regression parameter estimates, the simulated game day precipitation data, and the actual values for the control variables from year 2012 to obtain game attendance for a soccer season. (19) We re-estimated the precipitationattendance regression for our prototype team Kawasaki F with a shorter sample period (2007-2010) to be consistent with the available financial data needed for further analysis. The regression results are qualitatively similar to those presented in Table 1. (20) In addition to using the point estimate (-0.143) for the precipitation variable (d4_h24), we also consider its 90 percent confidence interval (-0.226, -0.058) for robustness.

Team Net Cash Flows Calibration

Building on the estimated game attendance, we calculate the net cash flows (NCFs) of the team for each year. We first estimate the sensitivity of revenues and costs to annual game attendance using OLS regressions from 2007 to 2010. (21) The aggregate-level financial data are obtained from the team's financial statements available at the J. League Web site (http://www.j-league.or.jp/eng/). (22) The regression models we use are

ln(revenuet) = [[alpha].sub.revenue] + [[beta].sub.revenue][Attend.sub.t] + [[??].sub.revenue], (1)

ln([cost.sub.t]) = [[alpha].sub.cost] + [[beta].sub.cost][Attend.sub.t] + [[??].sub.cost], (2)

where [revenue.sub.t] is the annual revenue of Kawasaki F in year t, [cost.sub.t] is annual total cost of Kawasaki F in year t, [Attend.sub.t] is the total attendance for Kawasaki F in year t, and [[??].sub.revenue] and [[??].sub.cost] are residuals in each regression model. The estimation results are presented in Table 4.

We calculate team NCFs as

[NCF.sub.Without_hedge] = revenue - cost, (3)

and use the calibrated attendance and parameter estimates from the revenue and cost regressions (Table 4) to obtain simulated cash flows. (23) Recall that we calibrated attendance using the point estimate of the precipitation exposure from the regression analysis, along with the upper and lower bound of the 90 percent confidence interval of the estimate. Descriptive statistics for simulated NCFs are reported in Table 5.

Table 5 demonstrates the significant effect of weather conditions on the variations of NCFs of Team Kawasaki F. Kawasaki F's net income ranges from US$73,000 to US$293,000. Given a standard deviation of US$51,260 (Table 5), it is likely that Kawasaki F can suffer from negative net income taking into account losses from precipitation and possibly other losses and costs/expenses. The losses can be even more severe if the actual impact of adverse weather is higher than the estimated value, such as the estimated numbers using the lower bound of the confidence interval as Table 5 shows.

WEATHER RISK HEDGING AND ITS CONTRIBUTION TO THE CORPORATE VALUE

Based on the simulated NCFs, we propose a hedging mechanism and evaluate its potential contribution to the team's corporate value. We employ the Wang transform for the evaluation to mitigate problems associated with the incomplete market. An estimated range of values for the decision maker's risk aversion parameter is obtained through a survey of J. League managers.

The Wang Transform for the Evaluation

Overview of the Wang Transform. The Wang transform (Wang, 2000, 2002) was developed as a universal framework for asset pricing and can be used in an incomplete market setting pertaining to the weather risk scenarios discussed in this article. Incomplete capital markets, in which such products exist, make the pricing of such capital market products difficult and risk-neutral measures nonunique. Instead of having to estimate the risk-adjusted discount rate, a rather difficult task often required by other valuation models, the Wang transform incorporates the decision maker's risk preference into the valuation by calculating the expected value with a transformed set of probabilities. In particular, decision makers' risk preferences can still be correctly represented when the expected cash flow under the physical probability measure is zero, whereas under the classic discounted cash flow models, zero expected cash flow uniformly leads to a present value of zero, rendering the choice of the discount rate irrelevant.

The Wang transform is formulated as follows:

[F.sub.Q] (x) = [PSI]([[PSI].sup.-1](FP(x)) + [lambda]), (4)

where [F.sup.Q](x) is the cumulative distribution function (CDF) under the risk-neutral probability measure Q, and [F.sup.P](x) is the CDF under the physical probability measure P. As in other option pricing methods, such as the Black-Scholes model, the riskneutral measure stipulates valuing contingent claims as if the decisionmaker were risk neutral and thus the risk-free rate is the appropriate discount rate. [PSI](*) is the CDF of the standard Normal distribution and [[PSI].sup.-1](*) is its inverse function. They are used for transforming the physical probabilities to the risk-neutral probabilities given the risk aversion parameter [lambda].

If the parameter [lambda] is positive, the physical CDF is shifted to the left. It implies that the subjective probabilities of the good scenarios (e.g., higher cash flows) are lower, indicating that the decision maker is risk averse. Similarly, a negative [lambda] indicates that the decision maker is risk seeking and assigns a higher probability to the good scenarios. If x follows the Normal distribution, [lambda] becomes the market price of risk, or the Sharpe ratio, (E(r) - [r.sub.f])/[sigma].

The economic interpretations of the Wang transform are closely related to the Esscher transform and both approaches are widely used in the insurance and finance literature. These approaches have been applied to pricing in a variety of areas including the securitization of longevity and mortality risks (Cox, Lin, and Wang, 2006; Denuit, Devolder, and Goderniaux, 2007; Chen, Zhang, and Zhao, 2010), annuity (Lin, Tan, and Yang, 2009), mortgage (Chen, Cox, and Wang, 2010), weather derivatives (Moridaira, 2010), and options and other derivatives in general (Gerber and Shiu, 1994; Gerber and Landry, 1997; Vyncke et al., 2003; Monfort and Pegoraro, 2012). Connections between the Esscher and the Wang transforms are discussed in detail by Labuschagne and Offwood (2010). In particular, by using a negative exponential or a power utility function with an additional assumption that underlying asset returns follow a Normal distribution, we can recover the Esscher transform from the Wang transform with identical risk aversion parameters. (24)

A Hedging Mechanism for Precipitation Risk. We propose a simple hedging mechanism for precipitation risk with coverage period for a soccer season. As described previously, team cash flows are calibrated from simulated game-day weather conditions and their estimated weather exposures. We then construct the expected payoff of the hedging instrument with an estimated range of reasonable risk loadings. (25)

We continue to use Team Kawasaki F as an illustrative example. For simplicity, we construct the payoff as a complete hedge that covers 100 percent of the expected loss due to precipitation based on the point estimate of precipitation exposure. (26) This payoff structure will result in incomplete hedge if cash flow realizations are estimated based on the upper and lower bounds of the point estimate. The payoff is triggered if a game is on a rainy day.

The expected loss per game from precipitation is calibrated by an OLS regression using the simulated NCFs and the number of rainy day games per soccer season:

Simulated NCF = [[alpha].sub.rainyday] + [[beta].sub.rainyday] Simulated Number of Rainy Day Games Per Season, (5)

where [[alpha].sub.rainyday] captures expected NCF if there is no rainy-day game in a soccer season and [[beta].sub.rainyday] measures the expected cash flow differential, or expected loss, per rainy-day game in a season. The parameter estimates are [[alpha].sub.rainyday] = US$805,218 and [[beta].sub.rainyday] = -US$34,702. Under the assumption of a complete hedge, the payoff of the hedging instrument per rainy game is thus -[[beta].sub.rainyday] = US$34,702. Consequently we obtain the expected pay off from weather hedging per soccer season as

[Payoff From Hedging Per Season] = [N.summation over (n = 1)] [p.sub.n] x n x Payoff From Hedging Per Game = US$88, 783, (6)

where [p.sub.n] is the probability of having n rainy days in a soccer season. The simulated probabilities [p.sub.n] are found in Table 3.

Because precipitation-based weather derivative products are not currently traded in the Japanese market and thus price information is not available, we assume that the embedded premium is calculated as (27)

Premium of the Hedging Instrument = (1 + Safety Loading) x Expected Payoff. (7)

We investigate a range of safety loadings from 0 to 20 percent for the proposed hedging instrument. This choice of range is consistent with the literature and the market data. Previous literature assumes a loading of 0 to 20 percent in a weather derivative contract. For instance, Golden, Wang, and Yang (2007) assume that weather forward prices are actuarially fair. Vedenov and Barnett (2004) assume that a safety loading is 0, 5 percent, or 10 percent. According to information provided by MDA Weather Services (a wholly owned subsidiary of MacDonald, Dettwiler and Associates Ltd., Richmond, Canada.), a U.S. company providing comprehensive data sets and consulting services related to weather risks, the temperature option premium traded in the United States is between 10 percent and 20 percent (Climetrix, 2010). Chinacarini (2011) finds a 5 percent safety loading using CDD (i.e., cooling degree days) forward contracts for Atlanta, Chicago, Cincinnati, Dallas, and New York from 1999 to 2004. (28)

In addition, we benchmark our safety loading assumption against that of the weather insurance contracts, an alternative mechanism to manage weather risks. Weather insurance contracts are used by three J. clubs: Teams Shimizu, Hiroshima, and C Osaka. (29)

Assuming again that the occurrence of a rainy-day game follows the Bernoulli distribution, we can back out the implied safety loadings in the insurance contracts. We find that the implied safety loadings for these contracts are at least 60 percent. (30) In the subsequent evaluation analysis, we add an insurance-implied loading of 60 percent to the range of loadings from 0 to 20 percent.

We can now calculate the NCFs when a hedging instrument is adopted as

[NCF.sub.with_hedge] = [NCF.sub.without_hedge] - premium + payoff,

where the payoff and the premium of the hedging instrument are defined in Equations (6) and (7) above.

Table 6 presents descriptive statistics of simulated cash flows with hedging for team Kawasaki F, with two alternative assumptions of safety loading at 10 percent and 20 percent, respectively. We present these for each of the three cases of cash flow calibrations before hedging, based on the point estimate of cash flow sensitivity to precipitation and its lower and upper bounds, respectively. For comparison purposes, we also include again the simulated cash flows without hedging from Table 5. Intuitively, in the point estimate-based and the lower bound estimate-based cases, the standard deviation of cash flows is dramatically reduced when hedging is used and the ranges of probable cash flows are also much smaller, whereas the expected values of cash flows are reduced for the cost of hedging. Interestingly for the upper bound estimate-based case, the standard deviations are larger when hedging is used. This is because of the overhedging scenario when the cash flow sensitivity to precipitation is actually lower than expected, upon which the payoff is based.

Evaluation of the Hedging Instrument. Within the framework of the Wang transform, we gauge the value of the hedging instrument for precipitation by calculating the expected NCF differentials under the risk-neutral probability measure. We apply the Wang transform (4) to obtain q-probabilities from the p-probabilities for NCFs with and without hedging as previously described. We discuss in the next section the determination of an appropriate risk aversion parameter [lambda] for the team manager in order to obtain the risk neutral q-probabilities. The expected NCF when the soccer team does not use a hedging mechanism is:

[E.sup.Q] ([NCF.sub.without_hedge]) = [I.summation over (i = 1)][NCF.sub.without_hedge,i],[f.sup.Q]([NCF.sub.without_hedge,i])

[f.sup.Q] ([NCF.sub.without_hedge,i]) = [F.sup.Q]([NCF.sub.without_hedge,i]) - [F.sup.Q]([NCF.sub.without_hedge,i-1]),

where i indicates the ith smallest NCF among the simulation paths.

Similarly, we evaluate the expected NCF when the manager chooses to adopt such a hedging mechanism. The expected NCF with hedging is derived as

[E.sup.Q]([NCF.sub.with_edge]) = [I.summation over(i = 1)] [NCF.sub.with_hedge,i][f.sup.Q] ([NCF.sub.with_hedge,i-1]) [f.sup.Q] ([NCF.sub.with_hedge,i]) = [F.sup.Q] ([NCF.sub.with_hedge,i]) - [F.sup.Q] ([NCF.sub.with_hedge,i-1]).

Finally, we obtain the value of the hedging mechanism by calculating the firm value differential under the risk-neutral q-probability measure, that is,

V = [E.sup.Q] ([NCF.sub.with_hedge]) - [E.sup.Q] ([NCF.sub.withou_hedge]). (8)

This method of evaluation not only takes into account a decision maker's risk preference, but controls for her wealth level, a factor that tends to influence the risk preference.

Estimation of the Risk Aversion Parameter [lambda]

To obtain a range of risk aversion coefficient [lambda] appropriate for the J. League team managers, we follow the economics literature to obtain a direct measure by creating and distributing four sets of questionnaires that ask interviewees to choose from paired hypothetical lotteries. We adopt the methodology proposed in the recent literature (Pennings and Smidts, 2000; Andersen et al., 2008) to design our questionnaires. This approach is easier to implement than estimating the risk preference in a natural experiment setting (Kahneman and Tversky, 1979).

Table 7 summarizes the questionnaires distributed to 10 current J. League team managers. (31) The questionnaires are designed based on data from an average J1 club, Iwata. Iwata's revenue in 2010 ranked 10th among 18 J1 clubs and is the closest to the average revenue of all J1 clubs in the 2010 fiscal year. Panels A and B reflect the manager's preference between two risky scenarios where one is relatively risker than the other. Panel A includes questions regarding attendance while Panel B includes questions regarding dollar amounts. Panels C and D reflect the manager's preference between a risk-free scenario and a risky scenario. Again, Panel C includes attendance-related questions and Panel D includes dollar-amount-related questions.

In Panels A and B, Scenario A is relatively risker than Scenario B. For Panel A (B), the higher value in Scenario A is calculated as the mean attendance (revenue) of Iwata in 1994-2010 plus two standard deviations. Similarly, the lower value in Scenario A equals the mean attendance (revenue) of Iwata in 1994-2010 minus two standard deviations of attendance (revenue). The higher value in Scenario B is calculated as the mean attendance (revenue) of Iwata in 1994-2010 plus one standard deviation in Panel A (B). Similarly, the lower value in Scenario B equals the mean attendance (revenue) of Iwata in 1994-2010 minus one standard deviation. In Panels C and D, we set Scenario B's attendance so that the implied interval of [lambda] is the same as in Panels A and B for each question. The actual distributed questionnaires do not include expected values, their differences, or the implied intervals of [lambda] (all in italics in Table 7).

We asked the J. League managers to evaluate the pair of scenarios in each question given the probabilities and the outcomes (in the number of attendance or monetary values), and indicate their preferred scenario for each pair. By noting the question at which their choice of preferred scenario switches from Scenario B (less risky) to Scenario A (more risky), we can obtain the implied interval of her risk aversion parameter l. For example, if the manager indicates that she prefers Scenario B to A for Q1-Q5 and then switches to preferring A to B for Q6-Q9, her implied [lambda] ranges from 0 to 0.25. We collected 7 out of 10 sets of questionnaires that were distributed and the results are summarized in Table 8.

We calculate the most plausible range of parameter [lambda] for each manager as follows. First, we obtain the range of [lambda] implied from each set of questions in the questionnaires as presented in Panels A-D of Table 7. Second, for each team manager, we determine the most frequently represented range of [lambda] in the four sets. For example, if the implied interval of estimated [lambda] is 0.25-0.52 in Set 1 (Panel A), 0.52-0.84 in Set 2 (Panel B), 0.25-0.52 in Set 3 (Panel C), and 0-0.25 in Set 4 (Panel D), the most frequently represented range of implied [lambda] is 0.25-0.52. In Table 8, we note that Manager 4 has inconsistent risk preferences because instead of switching from Scenario B to A, his preferred scenario changes from A to B in all panels. Lastly, we find the most plausible range of risk aversion parameter [lambda] for an average J. club manager to be 0.25-0.52 since this particular range appears the most times (four out of seven) among all managers' estimated ranges.

Contribution of the Weather Hedging Instrument to Team Value

Building on the empirically calibrated precipitation exposure, simulated NCFs, estimated safety loading, and estimated risk aversion parameter, we evaluate the hedging instrument as a risk management option for our prototype Team Kawasaki F according to (8). To calculate the expected values under the Q-measure, we use the simulated cash flows in Table 6 for the three cases presented (the point estimate, the upper bound estimate, and the lower bond estimate), and transform the physical P-measure into the risk-neutral Q-measure with a set of risk aversion coefficient [lambda] ranging from 0.25 to 0.52. Note that using a Monte Carlo simulation with 10,000 iterations, the physical CDF is obtained by simply assuming equal probabilities of each simulation path, that is, 1/10,000. Results are shown in Table 9.

Table 9 shows the extent to which the team's corporate value increases with a hedging instrument as the manager's risk aversion increases. Panel A presents our baseline case built on the point estimate of precipitation exposure. When the safety loading for the hedging instrument is 10 percent, the manager will find risk management valuable as long as her risk aversion coefficient [lambda] is at least 0.2, thus including our entire estimated range of risk aversion for J. League managers. Even when hedging is more costly with a 20 percent safety loading, it will be valuable to managers with a [lambda] of 0.4 or higher, still within the range of our estimated most plausible risk aversion parameters. (32) Intuitively, when the impact of precipitation is higher than expected, proxied by the lower bound of the 90 percent confidence interval of the point estimate, the value of the hedging instrument and the acceptable safety loadings are even higher. The results are presented in Panels C and D. Interestingly, in the case when cash flow sensitivity is much lower than expected, proxied by the upper bound case (Panels B and D), managers have to incur an unnecessarily high cost to overhedge, or end up speculating rather than hedging. Therefore, the value of weather derivatives is negative for the risk-averse managers. The more risk-seeking managers will also find the hedging instrument valuable because it provides them with induced uncertainties due to overhedging. (33)

Our results show that the empirically calibrated risk management instrument, combined with a reasonable risk premium in the marketplace, can be employed to help J. League clubs improve their corporate value. When the safety loading becomes prohibitively large at 60 percent as is estimated from the precipitation insurance product in Japan, the J. League managers cannot easily opt for managing this significant risk exposure. This is perhaps why only three teams in the J. League have purchased weather insurance despite the wide-spread concern about this risk exposure. Our results also suggest the need for establishing a weather derivatives market in Japan (and other countries) to facilitate effective and feasible weather risk management for the sports and many other industries.

CONCLUSION

Weather risks are suspected to have significant impact on the sports industry. They not only entail lower attendance and hence reduce team profits, but also threaten the retention of corporate sponsorships that are critical to the teams' survival and success. In the recent economic recession, many sports club sponsors have pulled their support due to the uncertain and sometimes unfavorable financial outcomes of these sports teams. Our empirical analysis documents significant weather exposures for sports teams. Although many other factors can contribute to the fluctuations in revenues, weather risk, specifically precipitation, is one that can be affordably managed with a hedging instrument as illustrated in this article. Our analysis and results provide insights for the international markets in general in understanding the impact of precipitation risk, assessing its financial consequences, and evaluating weather derivatives as a risk management option from a managerial perspective. Our results also point to the need for introducing a weather derivatives market in Japan and other countries to allow the use of such risk management options.

To the best of our knowledge, this is among the first papers to calibrate the weather exposure of the sports industry and analyze the value of a hedging instrument with risk preference estimates obtained from surveying actual managers while addressing the incompleteness of the market. With detailed attendance data from Japan's premier soccer league, J. League, and matching weather data, we calibrate the precipitation exposure of J. League teams through a comprehensive set of multivariate GLS regressions. We then use simulation analysis to obtain the cash flows for a prototype J. club and propose a simple hedging instrument. Based on survey results, we explore different managerial risk preferences and show that team value can be significantly increased by hedging. In fact, with a reasonable range of safety loadings (less than 10 percent), hedging always contributes positively to team value for the estimated risk preferences of J. League managers.

In this article, we have focused on precipitation risk for Japanese soccer teams to take advantage of our detailed data sets. Future research can look into these issues in other business applications and other international markets. It will be interesting to examine the joint impact of multiple weather effects (such as temperature and precipitation) and design effective risk management strategies taking into account their dependence. We can also relax the assumption of a complete hedge to determine the optimal hedge ratio given the premium and the risk aversion. Overall, sports teams have many other risks affecting the entertainment value of the team, such as the risk of player injuries and delegation risks. These, along with the weather risk, can all be incorporated into an enterprise risk management framework in future studies.

APPENDIX: DETAILED DEFINITIONS OF THE PRECIPITATION VARIABLES

We define a set of level and dummy measures of precipitation for the regression analysis to examine its impact on game attendance. The level measures are defined based on the raw cumulative precipitation amount and the dummy measures are defined using different threshold values. For the level measures, we examine six different durations: 1, 2, 3, 4, 6, and 24 hours. The 1-hour variables comprise each of the 9 hours before the game, each of the 2 hours during the game, and the 1 hour after the game, resulting in a total of 12 different measures. The 2-hour precipitation variable refers only to the cumulative amount of precipitation during the game period. The 3-hour precipitation variables comprise the 2 hours during the game plus the 1 hour before or after the game, the 3 hours immediately before the game, the 3-hour period beginning 6 hours before the game, and the 3-hour period beginning 9 hours before the game. The 4-hour and the 6-hour precipitation variable refer only to the 4 and 6 hours prior to the game period, respectively. Finally, the 24-hour precipitation variables include the cumulative precipitation on the game day, on the day before the game day, on the day 2 days before the game day, and on the day 3 days before the game day.

Lacking any previous evidence for threshold choices, we have constructed the dummy measures of precipitation in a largely exploratory manner. We set the threshold values from 0 to 5 mm for all variables with 0.5-mm increments. We set additional thresholds from 5.5-mm to 15 mm in 1-mm increments for the 6- and 24hour precipitation variables, and additional thresholds of 20 mm, 30 mm, 40 mm, and 50 mm for 24-hour precipitation variables to address the higher amount of rain anticipated for longer periods. The dummy variable takes value 1 if the amount of precipitation is at least the threshold amount, and 0 otherwise. In total, we have 354 level and dummy measures of precipitation. The definitions are summarized in Table A1.

Table A1 Definitions of Level and Dummy Measures of Precipitation Number of Length of Level Threshold for Variables Period Timing Measure Dummy Measures Defined 1 hour Each of the 9 Yes 0.0-5.0 mm in 144 hours before 0.5 mm increments the game, Each of the 2 hours during the game, the 1 hour after the game 2 hours The 2-hour game Yes 0.0-5.0 mm in 12 period 0.5-mmin crements 3 hours The 1 hour Yes 0.0-5.0 mm in 60 before the game 0.5-mm increments plus 2-hour game period The 1 hour Yes 0.0-5.0 mm in after the game 0.5-mm increments plus 2-hour game period The 3 hours Yes 0.0-5.0 mm in before the game 0.5-mm increments From 6 hours Yes 0.0-5.0 mm in before the game 0.5-mm increments to 3 hours before the game From 9 hours Yes 0.0-5.0 mm in before the game 0.5-mm increments to 6 hours before the game 4 hours The 4 hours Yes 0.0-5.0 mm in 12 before the game 0.5-mm increments 6 hours The 6 hours Yes 0.0-5.0 mm in 22 before the game 0.5-mm increments 6.0-15.0 mm in 1.0-mm increments 24 hours On the game day Yes 0.0-5.0 mm in 104 1, 2, and 3 0.5-mm increments days before the game day 5.0-15.0 mm in 1-mm increments 20.0-50.0 mm in 10-mm increments Total: 354

DOI: 10.1111/jori.12071

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Haruyoshi Ito is Assistant Professor of Finance, at the Graduate School of International Management, International University of Japan, 777 Kokusai-cho, Minami Uonuma-shi, Niigata 979-7277, Japan. Ito can be contacted via e-mail: haru0416@gmail.com. Jing Ai is Associate Professor of Risk Management and Insurance, at the Department of Financial Economics and Institutions, Shidler College of Business, The University of Hawaii at Manoa, Honolulu, HI 96822. Ai can be contacted via e-mail: jing.ai@hawaii.edu. Akihiko Ozawa is Auditor, Hitachi Capital NBL Corporation, Nishi Shimbashi Square 9F, 1-3-1, Nishi Shimbashi, Minato-Ku, Tokyo 105-0003, Japan. Ozawa can be contacted via e-mail: aozawaletter@yahoo. co.jp. We would like to thank participants at the 2013 Asian Pacific Risk and Insurance Association Annual Meeting, three anonymous referees, the editor, and Patrick Brockett for their valuable comments. All errors are ours.

(1) Available at http://www.cmegroup.com/trading/weather/(accessed September 20, 2012).

(2) In untabulated analysis, we find that the correlations among temperature (HDD and CDD) and precipitation derivatives are very low most times so the more popular temperature derivatives cannot be used to hedge precipitation risks.

(3) Only typhoon, thunderstorm, eruption of volcano, and catastrophic earthquake have caused suspension of a game.

(4) The cost to build a new stadium with a roof is approximately US$244 million, and adding a roof to an existing stadium costs approximately US$73 million or US$730,000 assuming a durable period of 100 years and 0 real interest rate. Assuming an average attendance of 20,000 per game, the average profit per attendance of US$37, an average of 3 rainy day games, and assuming a 60 percent insurance loading (implied in the currently available insurance contracts), the annual premium of weather insurance would be US$186,146. As we will show later, the estimated annual cost of the weather derivative (even at the high 20 percent loading) is much lower ($88,783 from equation (6)*1.2 = $106,540). Annual revenues of J1 teams range from US$20 million to US$85 million.

(5) Most alternative evaluation approaches proposed for the incomplete market require an assumption on the utility function. For example, the Esscher transform requires an exponential or power utility function. Brockett et al. (2009) and Golden, Wang, and Yang (2007) propose the indifference pricing method for weather derivatives assuming a mean-variance utility function.

(6) The other sources of revenue include other advertisement income and dividend distributions from the J. League association. However, data are not available at the game-by-game level that we need for our analysis.

(7) Roof coverage can include none, 25 percent, 50 percent, 75 percent, 100 percent of the spectator area, or domes that cover 100 percent of the spectator area and the entire soccer field. We define a nonroofed stadium as one with a significantly large ([greater than or equal to] 50 percent) exposure in the spectator area. Because we do not study the roofed stadiums in this article, the conclusions drawn and the insights derived are not directly applicable to these teams.

(8) We use data from the manned weather station closest to the stadium even though there might be an unmanned station closer to the stadium. This is because a manned station provides more detailed precipitation information, including a distinction between "--" (no rain) and "0.0" (minimum rain), and a more refined unit of increments at 0.5 mm instead of 1mm as for unmanned stations.

(9) One limitation of this definition is that we are potentially excluding foreign star players' impact on game attendance. Because of the recent decline in the percentage of foreign players on J. League teams and the difficulty in comparably defining foreign star players as well as data availability, we do not attempt to include foreign star players here in the analysis. However, in our experiments (not reported), the inclusion of these players yields qualitatively similar results.

(10) While we have searched for and included as controls most variables motivated by the literature or practice, there are possibly other variables not included due to data availability or difficulty in measurement, for example, pollution, market sentiment, and so on.

(11) All univariate and multivariate regression results are available upon request. Among all 354 precipitation measures, 177 are significant at the 1 percent or 5 percent level for Team Shimizu, which is mostly affected by adverse weather condition, while only 6 measures are significant at the 1 percent or 5 percent level for Yokohama FM, Nippa, which is least affected. On average, 89 variables are significant at the 1 percent or 5 percent level.

(12) Most control variables have signs that are consistent with our expectation and the previous literature, as discussed in the article. Surprisingly, Population is negatively significant or nonsignificant for most teams. This is perhaps due to the negative growth of Japanese population in some of the cities where the teams locate, although the popularity of soccer games still prevails.

(13) In fact (untabulated) analysis shows moderate correlation between the significance of the precipitation effects and the game attendance scaled by capacity of the stadiums. For instance, in the case of Team Yokohama FM, the effect of weather is stronger when they play at the Nissan stadium than at the Nippa stadium, although Nippa has no roof while Nissan has a roof covering half of the spectators. Nippa stadium's capacity is only 15,454 compared to a capacity of 73,327 in the Nissan stadium. The average attendance for all Yokohama FM matches was 25,686 in the 2010 season. Therefore, it is plausible that tickets for Nippa are more desirable than for Nissan, making it difficult for fans to give up tickets for a game in the Nippa stadium even on a rainy day.

(14) The combination Kofu, Kose does not have significant results under any of our precipitation definitions for unique reasons. The team focuses mainly on the regional market and there is no other professional sports team close to the Kofu area, resulting in more robust attendance. We did not control for this factor in the regressions because it is unique to this one team.

(15) We have also run panel regressions with team and year fixed effects, using the same set of precipitation measures and control variables. Instead of estimating team-specific precipitation exposures, the panel regressions estimate the average precipitation exposure for a J1 team. Consistent with the team-specific regressions, we find that the average impact of precipitation is highly significant. The control variables also have signs and significance levels similar to the time-series regressions. The panel regression results using the two precipitation measures d0_h3 and d4_h24 are presented in the accompanying online appendix.

(16) Financial information is only available annually for years 2005-2010 and is aggregated at the team level.

(17) Note that in simulating the impact of precipitation on teams' financial outcomes, we focus on attendance as this is the main channel on which precipitation takes effect. J. League teams do have other sources of income that are not directly related to game attendance. Notably, a significant percentage of the ticket sales are in the form of season tickets (40 percent according to the J. League Fun Survey in 2009, a summary of which is available from the J. League Web site at http://www.j-league.or.jp/aboutj/document/spectator-survey.html as of March 18, 2014). In addition, advertising income contributes greatly to teams' revenues. However, attendance (and hence precipitation) can still impact these financial sources in other indirect ways, such as concessions and other sales in the stadium, advertisement and commercial revenues based on popularity, and so on. Because we lack the data and precise measures to account for the indirect impact, our analysis mainly explains the income variations due to reduced gate receipt sales and related financial losses.

(18) In fiscal year 2010, J1 teams had an average operating income of -41million JPY (-US$0.5 million), and those that had positive operating income had an average of 133 million JPY (US$1.63 million).

(19) Available at the J. League Web site (http://www.j-league.or.jp/eng/), last accessed in March 2012.

(20) We use the same multivariate regression model to calibrate the impact of precipitation on game attendance, but rerun it with a shorter sample period for two reasons. First, only 6 years of financial data (2005-2010) are available for use as compared to the 18 years of attendance data used for the weather exposure estimation. Second, a structural change between 2006 and 2007 was identified using the Chow test in the relationship between game attendance and adverse weather conditions. OLS regression is used since no autocorrelation is identified in the residuals. The coefficient for precipitation variable (d4_24) is -0.143 with a p-value of 0.008 and standard deviation of 0.052. Full estimation results are shown in the accompanying online appendix.

(21) We cannot run regressions of financial information on the weather conditions directly due to data availability. We only have a very short period of financial information for each team and if we run a pooled regression, the effect of weather on the financials are contaminated because the effect of weather on attendance is varied among teams as we have previously shown. Attendance is closely related to financial outcomes for the pooled data as the significance levels of the coefficients for attendance in the pooled revenue regression and cost regression are 0.04 and 0.02, respectively. Adjusted [R.sup.2]s of the revenue and the cost regression are 86 percent and 94 percent, respectively.

(22) Summary statistics for game attendance and key financial data for team Kawasaki F are provided in the accompanying online appendix.

(23) In calculating the estimated revenues and costs, we add an extra term, the exponential of the squared standard deviations of residuals from the revenue and cost regressions divided by 2, to adjust for the bias resulting from the logarithm transformation. More specifically E[cost] = exp{[[alpha].sub.cost] + [[beta].sub.cost] simulated attendance}exp([[phi].sup.2.sub.[epsilon]]/2), and E[revenue] = exp{[[alpha].sub.revenue] + [[beta].sub.revenue] simulated attendance}exp([[sigma].sup.2.sub.[epsilon]]/Z). This is because E[cost] [not equal to] exp{[[alpha].sub.cost] + [[beta].sub.cost] simulated attendance}. In fact, cost = exp{[[alpha].sub.cost] + [[beta].sub.cost] simulated attendance + [[??].sub.cost]} = exp{[[alpha].sub.cost] + [[beta].sub.cost] simulated attendance}exp([[??].sub.cost]). Thus, E[cost] = E[exp{[[alpha].sub.cost] + [[beta].sub.cost] simulated attendance}]E[exp([[??].sub.cost])]. By assuming error term [[??].sub.cost] follows normal distribution, we have E[cost] = exp{[[alpha].sub.cost] + [[beta].sub.cost] simulated attendance}exp([[sigma].sup.2.sub.[epsilon]]/2). See Goldberger (1968).

(24) Pelsser (2008) argues that the Wang transform has limited applicability in financial pricing because it is not consistent with arbitrage free pricing in the general context. Since the weather derivative market we are working with is typically not an arbitrage-free economy and there is not a uniform method for pricing in this case, the Wang transform seems to be an appropriate approach for us to adopt here.

(25) In order to focus on the effect of the weather-related exposure and due to data limitations, we do not take into consideration interest rate risk, credit risk, basis risk, and others.

(26) Although we are considering a complete hedge ex ante, the linear payoff structure of the derivative is based on the expected loss from the precipitation exposure instead of the actual loss determined after the games. Therefore, the weather derivative product we consider is not to be confused with a weather insurance policy. In addition, we perform most analyses based on the complete-hedge contract design only for simplicity. Similar analysis can be easily implemented for contracts that provide only a partial hedge ex ante, such as those developed from the lower bounds of precipitation exposure estimates discussed in this article.

(27) Here we design a weather hedging instrument that pays off even when there is only one rainy day per season. In some real-life scenarios, the firm might want to use a similar future contract or option contract whose payoff is positive only if the number of rainy days is above the average or a set threshold value. However, we do not have sufficient information on the prices of these precipitation contracts because they are not traded for the Japanese market. We attempted to estimate the bounds using limited information from the U.S. market as a proxy but were not able to obtain meaningful results. In addition, we do not have the risk preference of market participants to calibrate the premium under the risk neutral measure. This is why we resort to the current design of the hedging instrument. The insights, however, should be largely similar.

(28) Chinacarini (2011) estimates the median temperature forward premium to be 5.76 percent. The forward premium is defined as E([F.sub.t] - [S.sub.t+1])/[F.sub.t], where [F.sub.t] is forward price at time t and [S.sub.t+1] is underlying weather index at time t + 1. If we calculate the safety loading by [F.sub.t],/E([S.sub.t+1]) - 1, the 5.76 percent premium translates to a safety loading of 5 percent.

(29) We summarize the policies for Teams Shimizu and Hiroshima in the accompanying online appendix (details only available for 2001). For example, Team Shimizu's insurance policy stipulates that the insurance company pays the team US$2,957 (242,500 JPY) if there is at least 10 mm of precipitation on the day of the insured game. The details of the contract for Team C Osaka are not available.

(30) We describe our estimation for the implied safety loading in detail in the accompanying online appendix.

(31) The teams are Kobe, Tokyo V, Yokohama FM, C Osaka, Nigata, Okayama, Sapporo, Ehime, Tochigi, and Tokushima.

(32) There are possibly other types of transaction costs in addition to the premium loading considered in the article. Our results on the evaluation of weather hedging instrument is fairly robust to reasonable transaction costs. For example, Jewson (2002) and Chen, Roberts, and Thraen (2006) cite a 5 percent or 6 percent transaction cost for weather derivatives. Even for a high premium loading of 20 percent, the hedging instrument is still valuable to managers whose risk preference falls within the estimated bounds in our article.

(33) Overhedging will be a rare case in practice because a complete hedge is unlikely due to the cost of hedging.

Table 1 Precipitation Effects on Game Attendance Shimizu Nihon Daira G Osaka Banpaku d0_h3 -0.103 *** -0.127 *** (0.027) (0.043) d4_h24 -0.108 *** -0.084 * (0.028) (0.049) AIC -75.337 -75.315 124.558 130.587 BIC -19.713 -19.692 181.343 187.372 N 239 239 257 257 Kofu Kose Iwata Yamaha d0_h3 0.013 -0.075 *** (0.086) (0.027) d4_h24 -0.045 -0.064 *** (0.091) (0.028) AIC -8.764 -9.114 -129.601 -126.824 BIC 11.756 11.406 -75.086 -72.309 N 32 32 223 223 Kyoto Saikyogoku C Osaka Nagai d0_h3 -0.168 *** -0.062 (0.049) (0.073) d4_h24 -0.229 *** -0.097 (0.053) (0.085) AIC 63.548 56.645 135.925 135.354 BIC 108.095 101.192 183.218 182.647 N 144 144 142 142 Kashiwa Kashiwa Omiya NACK d0_h3 -0.074 ** -0.141 * (0.029) (0.073) d4_h24 -0.105 *** -0.085 (0.031) (0.069) AIC -109.325 -114.518 -17.931 -15.205 BIC -60.734 -65.926 12.978 15.704 N 154 154 51 51 Nagoya Mizuho Rikujo Hiroshima Hiroshima d0_h3 -0.085 ** -0.176 *** (0.039) (0.048) d4_h24 -0.169 *** -0.125 ** (0.041) (0.057) AIC -2.421 -14.637 126.912 135.238 BIC 45.965 33.749 176.084 184.41 N 186 186 196 196 Kawasaki F Todoroki Yokohama FM Nissan d0_h3 -0.181 *** -0.114 ** (0.050) (0.047) d4_h24 -0.224 *** -0.156 *** (0.051) (0.055) AIC 8.892 2.396 42.366 40.408 BIC 52.95 46.454 91.866 89.908 N 116 116 163 163 Yokohama EM Nippa d0_h3 -0.104 ** (0.047) d4_h24 -0.068 (0.057) AIC -22.109 -18.135 BIC 16.398 20.373 N 82 82 Shonan Hiratsuka d0_h3 -0.098 * (0.057) d4_h24 -0.133 ** (0.063) AIC 1.745 0.117 BIC 41.382 39.754 N 88 88 d0_h3 d4_h24 AIC BIC N Notes: This table presents parameter estimates from GLS regressions examining the impact of precipitation on game attendance for each unique team/home stadium combination. (The natural log of) game attendance is regressed on precipitation, as measure by a dummy variable d0_h3 or d4_h24, respectively, taking value 1 if there is any precipitation in the 3 hours before the game or at least 4-mm precipitation during the game day, and a set of control variables. AIC (Akaike information criteria) and BIC (Bayesian information criteria) measures are presented for model fit. N is the total number of observations for each regression: the number of games held in each J1 team's home stadium during the sample period.***, **, and * indicate 1 percent, 5 percent, and 10 percent significance level, respectively. Full regression results are available in the accompanying online appendix. Table 2 Mean Probability of a Rainy Day Game in Each Season and Pairwise Differences Between Seasons Season Mean 1 2 3 4 5 Probability 1 0.157 0 2 0.193 0.036 *** 0 3 0.137 0.020 * 0.056 *** 0 4 0.103 0.054 *** 0.090 *** 0.034 *** 0 5 0.064 0.093 *** 0.129 *** 0.073 *** 0.039 *** 0 Notes: This table presents mean probabilities of rainy days for each season and pairwise differences between seasons. Season 1 includes March, April, May, and October; Season 2 includes June and September; Season 3 includes July; Season 4 includes August and November; and Season 5 includes December. January and February are excluded since no game is held. A rainy day is defined as a day with at least 4-mm precipitation as in the variable d4_h24. These probabilities are calculated from the weather data observed at the Yokohama Observatory from 1968 to 2010. *** and * indicate 1 percent and 10 percent significance levels, respectively. Table 3 Simulated Probability of Having n Rainy Game Days in a Soccer Season Number of Rainy Game Days (n) 0 1 2 3 4 5 Probability 0.0596 0.1897 0.2702 0.2390 0.1411 0.0657 Number of Rainy Game Days (n) 6 7 8 9 Probability 0.0261 0.0071 0.0014 0.0001 Notes: Probabilities of having n rainy days in a soccer season are simulated using the mean probabilities of rainy days in each of the five seasons as shown in Table 2. The probabilities of having 10 or more rainy days are ignorable. Ten thousand simulation paths are used. TABLE 4 Parameter Estimates for Financial Performance of Team Kawasaki F Dependent ln(revenue) ln(cost) Variable Intercept 6.540 *** (-0.339) 6.598 *** (-0.221) Attendance 5.167 x [10.sup.-6] *** 4.932 x [10.sup.-6] *** (1.101 x [10.sup.-6]) (7.175 x [10.sup.-7]) sd([??]) 0.020 0.013 Notes: This table presents the parameter estimates and standard errors (in parentheses) from regressions of ln(revenue) and ln(cost) on attendance, respectively, for team Kawasaki F. Also presented are the standard deviations of the residuals in the regression models. *** indicates 1 percent significance level. Table 5 Descriptive Statistics for Simulated Net Cash Flows for Team Kawasaki F (in Thousands of U.S. Dollars; Number of Simulation Paths: 10,000) Statistic Lower Bound Point Estimate Upper Bound Expected value 670.93 716.44 768.38 Median 686.65 727.45 773.39 Standard deviation 75.86 51.26 22.47 Maximum 807.72 807.56 807.72 Minimum 373.12 508.69 674.14 Skewness -4.51 -5.09 -5.73 Notes: This table presents descriptive statistics of simulated net cash flows for team Kawasaki F. We first run simulations to generate game-day precipitation as presented in Table 3 and estimate the attendance using regression coefficients as described in footnote 20. In addition to using the point estimate for the precipitation exposure (-0.143), we also use its 90 percent confidence interval (-0.226, --0.058), that is, the "lower bound" and "upper bound" estimates. We estimate the cost and revenue based on the calibrated attendance and parameters presented in Table 4. NCF is then calculated as revenue--cost. Table 6 Simulated Net Cash Flows With Hedging for Team Kawasaki F in (Thousands of U.S. Dollars) Lower Bound No Hedging Hedging Hedging (10%) (20%) Expected value 670.93 662.05 653.17 Median 686.65 662.47 653.59 Standard deviation 75.86 25.60 25.60 Maximum 807.72 710.06 701.18 Minimum 373.12 575.78 566.90 Skewness -4.51 -2.31 -2.31 Point Estimate No Hedging Hedging Hedging (10%) (20%) Expected value 716.44 707.56 698.68 Median 727.45 708.09 699.22 Standard deviation 51.26 4.95 4.95 Maximum 807.56 731.60 722.72 Minimum 508.69 688.34 679.46 Skewness -5.09 -4.96 -4.96 Upper Bound No Hedging Hedging Hedging (10%) (20%) Expected value 768.38 759.50 750.62 Median 773.39 751.10 742.22 Standard deviation 22.47 28.73 28.73 Maximum 807.72 888.80 879.92 Minimum 674.14 710.06 701.18 Skewness -5.73 6.39 6.39 Notes: This table presents summary statistics for simulated net cash flows with hedging for the three estimates of precipitation exposure (the point estimate, the lower bound of the point estimate, and the upper bound). Net cash flows with hedging are calculated as [NCF.sub.with_hedge] = [NCF.sub.without hedge] - premium + payoff, where payoff is the expected loss based on the point estimate and premium is expected payoff plus loadings. Net cash flows without hedging are from Table 5. Table 7 Summarized Questionnaires for Risk Aversion Estimation of J. League Managers Scenario A P Attendance P Attendance Panel A: Greater Difference in Outcomes (Scenario A) Versus Lesser Difference (Scenario B), Unit: One Person 01 10% 19,007 90% 7,449 02 20% 19,007 80% 7,449 03 30% 19,007 70% 7,449 04 40% 19,007 60% 7,449 05 50% 19,007 50% 7,449 06 60% 19,007 40% 7,449 07 70% 19,007 30% 7,449 08 80% 19,007 20% 7,449 Q9 90% 19,007 10% 7,449 Scenario A P Revenue P Revenue Panel B: Greater Difference in Outcomes (Scenario A) Versus Lesser Difference (Scenario B), Unit: Thousand JPY Q10 10% 37,577 90% 14,727 Qll 20% 37,577 80% 14,727 Q12 30% 37,577 70% 14,727 Q13 40% 37,577 60% 14,727 Q14 50% 37,577 50% 14,727 Q15 60% 37,577 40% 14,727 Q16 70% 37,577 30% 14,727 Q17 80% 37,577 20% 14,727 Q18 90% 37,577 10% 14,727 Scenario B E(A) E(B) P Attendance P Attendance Panel A: Greater Difference in Outcomes (Scenario A) Versus Lesser Difference (Scenario B), Unit: One Person 01 10% 16,118 90% 10,338 8,605 10,916 02 20% 16,118 80% 10,338 9,761 11,494 03 30% 16,118 70% 10,338 10,916 12,072 04 40% 16,118 60% 10,338 12,072 12,650 05 50% 16,118 50% 10,338 13,228 13,228 06 60% 16,118 40% 10,338 14,384 13,806 07 70% 16,118 30% 10,338 15,540 14,384 08 80% 16,118 20% 10,338 16,695 14,962 Q9 90% 16,118 10% 10,338 17,851 15,540 Scenario B E(A) E(B) P Revenue P Revenue Panel B: Greater Difference in Outcomes (Scenario A) Versus Lesser Difference (Scenario B), Unit: Thousand JPY Q10 10% 31,865 90% 20,438 17,012 21,581 Qll 20% 31,865 80% 20,438 19,297 22,723 Q12 30% 31,865 70% 20,438 21,582 23,866 Q13 40% 31,865 60% 20,438 23,867 25,009 Q14 50% 31,865 50% 20,438 26,152 26,152 Q15 60% 31,865 40% 20,438 28,437 27,294 Q16 70% 31,865 30% 20,438 30,722 28,437 Q17 80% 31,865 20% 20,438 33,007 29,580 Q18 90% 31,865 10% 20,438 35,292 30,722 E(A)--E(B) Implied Interval of [lambda] Panel A: Greater Difference in Outcomes (Scenario A) Versus Lesser Difference (Scenario B), Unit: One Person 01 -2,311 -[infinity] -1.28 02 -1,733 -1.28 -0.84 03 -1,156 -0.84 -0.52 04 -578 -0.52 -0.25 05 0 -0.25 0 06 578 0 0.25 07 1,156 0.25 0.52 08 1,733 0.52 0.84 Q9 2,311 0.84 1.28 E(A)--E(B) Implied Interval of [lambda] Panel B: Greater Difference in Outcomes (Scenario A) Versus Lesser Difference (Scenario B), Unit: Thousand JPY Q10 -4,569 -[infinity] -1.28 Qll -3,426 -1.28 -0.84 Q12 -2,284 -0.84 -0.52 Q13 -1,142 -0.52 -0.25 Q14 0 -0.25 0 Q15 1,143 0 0.25 Q16 2,285 0.25 0.52 Q17 3,427 0.52 0.84 Q18 4,570 0.84, 1.28 Scenario A Scenario B P Atten- P Atten- P Atten- E(A) dance dance dance Panel C: Risky Scenario (Scenario A) Versus Deterministic Scenario (Scenario B), Unit: One Person Q19 50% 19,007 50% 7,449 100% 17,851 13,228 Q20 50% 19,007 50% 7,449 100% 16,695 13,228 Q21 50% 19,007 50% 7,449 100% 15,540 13,228 Q22 50% 19,007 50% 7,449 100% 14,384 13,228 Q23 50% 19,007 50% 7,449 100% 13,228 13,228 Q24 50% 19,007 50% 7,449 100% 12,072 13,228 Q25 50% 19,007 50% 7,449 100% 10,916 13,228 Q26 50% 19,007 50% 7,449 100% 9,761 13,228 Q27 50% 19,007 50% 7,449 100% 8,605 13,228 Q28 50% 19,007 50% 7,449 100% 7,449 13,228 Scenario A Scenario B P Revenue P Revenue P Revenue E(A) Panel D: Risky Scenario (Scenario A) Versus Deterministic Scenario (Scenario B), Unit: Thousand JPY Q29 50% 37,577 50% 14,727 100% 35,292 26,152 Q30 50% 37,577 50% 14,727 100% 33,007 26,152 Q31 50% 37,577 50% 14,727 100% 30,722 26,152 Q32 50% 37,577 50% 14,727 100% 28,437 26,152 Q33 50% 37,577 50% 14,727 100% 26,152 26,152 Q34 50% 37,577 50% 14,727 100% 23,867 26,152 Q35 50% 37,577 50% 14,727 100% 21,582 26,152 Q36 50% 37,577 50% 14,727 100% 19,297 26,152 Q37 50% 37,577 50% 14,727 100% 17,012 26,152 Q38 50% 37,577 50% 14,727 100% 14,727 26,152 Risk Free Implied Interval of [lambda] B--E(A) Panel C: Risky Scenario (Scenario A) Versus Deterministic Scenario (Scenario B), Unit: One Person Q19 4,623 -[infinity] -1.28 Q20 3,467 -1.28 -0.84 Q21 2,312 -0.84 -0.52 Q22 1156 -0.52 -0.25 Q23 0 -0.25 0 Q24 -1156 0 0.25 Q25 -2,312 0.25 0.52 Q26 -3,467 0.52 0.84 Q27 -4,623 0.84 1.28 Q28 -5,779 1.28 [infinity] Risk Free Implied Interval of [lambda] B--E(A) Panel D: Risky Scenario (Scenario A) Versus Deterministic Scenario (Scenario B), Unit: Thousand JPY Q29 9,140 -[infinity] -1.28 Q30 6,855 -1.28 -0.84 Q31 4,570 -0.84 -0.52 Q32 2,285 -0.52 -0.25 Q33 0 -0.25 0 Q34 -2,285 0 0.25 Q35 -4,570 0.25 0.52 Q36 -6,855 0.52 0.84 Q37 -9,140 0.84, 1.28 Q38 -11,425 1.28 [infinity] Notes: This table summarizes the four sets of questionnaires distributed to J. League managers in which they select the preferred scenario in each question. The questionnaires are designed based on data from the average J1 club, Iwata. In Panels A and B, Scenario A is relatively risker than Scenario B. The higher value in Scenario A is calculated as the mean attendance (revenue) of Iwata from 1994 to 2010 plus two standard deviations in Panel A (B). The lower value in Scenario A is similar but with two standard deviations below the mean instead. The higher and lower value in Scenario B is similarly calculated as in Scenario A but with one standard deviation above or below the mean instead of two standard deviations. In Panels C and D, we set Scenario B's attendance so that the implied interval of A is the same as in Panels A and B for each question. The implied interval of A for each respondent corresponds to a switch of preferred choice from Scenario B to A in all panels. The columns E(A), E(B), Risk Free--E(A), and Implied Interval of A were not included in the actual distributed questionnaires. Revenues in the questionnaire were denoted in lapanese yen (JPY). In Panel B, JPY 37,577 thousand, 14,727 thousand, 31,865 thousand, and 20,438 thousand are equivalent to US$458,000, US$180,000, US$389,000, and US$249,000, respectively (as of March, 2012 when we distributed the surveys). Table 8 Summary of the Most Plausible Ranges for the Implied Risk Aversion Parameter [lambda] Manager Min [lambda] Max [lambda] 1 0.25 0.52 2 0.25 0.52 3 0 0.52 4 Inconsistent Inconsistent 5 0.84 Infinity 6 0 0 7 0.25 0.52 All 0.25 0.52 Notes: This table presents the most plausible ranges for J. League managers' risk aversion parameter [lambda]. Min and Max [lambda] indicate the lower and upper bounds of the most plausible range of l for each manager. We calculate the most plausible range of parameter [lambda] for each manager as follows. First, we obtain the range of [lambda] implied from each set of questions in the questionnaires as presented in Table 7, Panels A-D. Second, for each team manager, we determine the most frequently represented range of [lambda] in the four sets. We note that Manager 4 has inconsistent risk preferences because his preferred scenario changes from A to B in all panels. Lastly, we find the most plausible range for an average J. club manager as the range that appears the most times among all managers. Table 9 The Wang-Transform-Based Evaluation of the Weather Hedging Instrument Safety Loading With Hedge [lam- No 0% 5% 10% 15% 20% 60% bda] Hedge Panel A: Expected Net Cash Flows Under Q-Measure and Value of Hedging, Based on Point Estimate (Thousands of U.S. Dollars) 0 716.44 716.44 712.00 707.56 703.12 698.68 663.17 0.1 711.34 715.94 711.50 707.06 702.62 698.18 662.67 0.2 706.17 715.44 711.00 706.56 702.12 697.68 662.17 0.25 703.56 715.19 710.75 706.31 701.87 697.43 661.92 0.3 700.92 714.93 710.49 706.05 701.61 697.17 661.66 0.4 695.61 714.42 709.98 705.54 701.10 696.66 661.15 0.5 690.23 713.90 709.46 705.02 700.58 696.14 660.63 0.52 689.14 713.79 709.35 704.91 700.47 696.03 660.52 0.6 684.79 713.37 708.93 704.49 700.05 695.61 660.10 0.7 679.30 712.84 708.40 703.96 699.52 695.08 659.57 0.8 673.76 712.30 707.86 703.42 698.98 694.55 659.03 0.9 668.18 711.76 707.32 702.88 698.44 694.01 658.49 1 662.56 711.22 706.78 702.34 697.90 693.46 657.95 Panel B: Expected Net Cash Flows Under Q-Measure and Value of Hedging, Based on the Upper Bound (Thousands of U.S. Dollars) 0 768.38 768.38 763.94 759.50 755.06 750.62 715.11 0.1 766.15 765.59 761.15 756.71 752.27 747.83 712.32 0.2 763.87 762.86 758.42 753.98 749.54 745.10 709.59 0.25 762.72 761.52 757.08 752.64 748.20 743.76 708.25 0.3 761.57 760.19 755.75 751.31 746.87 742.44 706.92 0.4 759.22 757.59 753.15 748.71 744.27 739.83 704.32 0.5 756.85 755.05 750.62 746.18 741.74 737.30 701.78 0.52 756.37 754.56 750.12 745.68 741.24 736.80 701.29 0.6 754.44 752.59 748.15 743.71 739.27 734.83 699.32 0.7 752.01 750.20 745.76 741.32 736.88 732.44 696.93 0.8 749.55 747.89 743.45 739.01 734.57 730.13 694.62 0.9 747.07 745.65 741.21 736.77 732.34 727.90 692.38 1 744.56 743.50 739.06 734.63 730.19 725.75 690.23 Panel C: Expected Net Cash Flows Under Q-Measure and Value of Hedging, Based on the Lower Bound (Thousands of U.S. Dollars) 0 670.93 670.93 666.49 662.05 657.61 653.17 617.66 0.1 663.39 668.38 663.94 659.50 655.06 650.62 615.11 0.2 655.75 665.80 661.37 656.93 652.49 648.05 612.54 0.25 651.89 664.51 660.07 655.63 651.20 646.76 611.24 0.3 648.01 663.22 658.78 654.34 649.90 645.46 609.95 0.4 640.18 660.62 656.18 651.74 647.30 642.87 607.35 0.5 632.28 658.02 653.58 649.14 644.70 640.26 604.75 0.52 630.68 657.50 653.06 648.62 644.18 639.74 604.23 0.6 624.29 655.41 650.97 646.53 642.10 637.66 602.14 0.7 616.25 652.81 648.37 643.93 639.49 635.05 599.54 0.8 608.15 650.21 645.77 641.33 636.89 632.45 596.94 0.9 600.00 647.61 643.17 638.74 634.30 629.86 594.34 1 591.82 645.03 640.59 636.15 631.71 627.28 591.76 Panel D: Maximally Acceptable Safety Loadings 0 0.00% 0.00% 0.00% 0.1 5.18% -0.63% 5.61% 0.2 10.44% -1.14% 11.32% 0.25 13.10% -1.36% 14.21% 0.3 15.78% -1.55% 17.13% 0.4 21.18% -1.84% 23.02% 0.5 26.66% -2.02% 29.00% 0.52 27.76% -2.04% 30.20% 0.6 32.19% -2.09% 35.05% 0.7 37.78% -2.04% 41.18% 0.8 43.41% -1.87% 47.38% 0.9 49.09% -1.59% 53.63% 1 54.81% -1.19% 59.94% Safety Loading Value of Hedging Instrument [lam- No 0% 5% 10% 15% 20% 60% bda] Hedge Panel A: Expected Net Cash Flows Under Q-Measure and Value of Hedging, Based on Point Estimate (Thousands of U.S. Dollars) 0 716.44 0.00 -4.44 -8.88 -13.32 -17.76 -53.27 0.1 711.34 4.60 0.16 -4.28 -8.72 -13.16 -48.67 0.2 706.17 9.27 4.83 0.39 -4.05 -8.49 -44.00 0.25 703.56 11.63 7.19 2.75 1.69 6.13 -41.64 0.3 700.92 14.01 9.57 5.13 0.69 -3.75 -39.26 0.4 695.61 18.81 14.37 9.93 5.49 1.05 -34.46 0.5 690.23 23.67 19.23 14.79 10.35 5.91 29.60 0.52 689.14 24.65 20.21 15.77 11.33 6.89 -28.62 0.6 684.79 28.58 24.14 19.70 15.26 10.82 -24.69 0.7 679.30 33.54 29.10 24.66 20.22 15.78 -19.73 0.8 673.76 38.54 34.11 29.67 25.23 20.79 -14.72 0.9 668.18 43.59 39.15 34.71 30.27 25.83 -9.68 1 662.56 48.66 44.22 39.78 35.34 30.90 -4.61 Panel B: Expected Net Cash Flows Under Q-Measure and Value of Hedging, Based on the Upper Bound (Thousands of U.S. Dollars) 0 768.38 0.00 -4.44 -8.88 -13.32 -17.76 -53.27 0.1 766.15 -0.56 -5.00 -9.44 -13.87 -18.31 -53.83 0.2 763.87 -1.02 -5.45 -9.89 -14.33 -18.77 -54.29 0.25 762.72 1.21 -5.65 10.09 -14.52 18.96 -54.48 0.3 761.57 1.37 -5.81 10.25 14.69 19.13 -54.64 0.4 759.22 1.63 -6.07 10.51 -14.95 19.39 -54.90 0.5 756.85 1.79 -6.23 10.67 15.11 19.55 -55.06 0.52 756.37 1.81 -6.25 10.69 15.13 -19.57 -55.08 0.6 754.44 -1.85 -6.29 -10.73 -15.17 -19.61 -55.12 0.7 752.01 -1.81 -6.25 -10.69 -15.13 -19.57 -55.08 0.8 749.55 -1.66 -6.10 -10.54 -14.98 -19.42 -54.93 0.9 747.07 -1.41 -5.85 -10.29 -14.73 -19.17 -54.68 1 744.56 -1.06 -5.50 -9.94 -14.38 -18.82 -54.33 Panel C: Expected Net Cash Flows Under Q-Measure and Value of Hedging, Based on the Lower Bound (Thousands of U.S. Dollars) 0 670.93 0.00 -4.44 -8.88 -13.32 -17.76 -53.27 0.1 663.39 4.98 0.54 -3.90 -8.34 -12.77 -48.29 0.2 655.75 10.05 5.61 1.17 -3.27 -7.70 -43.22 0.25 651.89 12.62 8.18 3.74 0.70 -5.14 -40.65 0.3 648.01 15.21 10.77 6.33 1.89 -2.55 38.06 0.4 640.18 20.44 16.00 11.56 7.12 2.68 -32.83 0.5 632.28 25.74 21.30 16.87 12.43 7.99 -27.53 0.52 630.68 26.81 22.37 17.94 13.50 9.06 -26.46 0.6 624.29 31.12 26.68 22.24 17.80 13.36 -22.15 0.7 616.25 36.56 32.12 27.68 23.24 18.80 -16.71 0.8 608.15 42.06 37.62 33.18 28.74 24.30 -11.21 0.9 600.00 47.61 43.18 38.74 34.30 29.86 -5.65 1 591.82 53.22 48.78 44.34 39.90 35.46 -0.05 Notes: Panels A, B, and C show the expected net cash flows with and without hedging and the value of the hedging instrument in the three cases when cash flow sensitivity to precipitation is calibrated using the point estimate, the upper bound of the point estimate, and the lower bound, respectively. The expected net cash flows are calculated using the Wang Transform given the risk aversion [lambda] and the safety loading. The value of the hedging instrument is calculated as the difference between the expected net cash flows with and without hedging. Panel D presents the maximally acceptable safety loading, that is, zero expected value for the hedging instrument. The estimated range of [lambda] is 0.25-0.52. These results are shown in bold.

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Author: | Ito, Haruyoshi; Ai, Jing; Ozawa, Akihiko |
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Publication: | Journal of Risk and Insurance |

Article Type: | Report |

Geographic Code: | 9JAPA |

Date: | Dec 1, 2016 |

Words: | 17816 |

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