An Application of Extreme Value Analysis to U.S. Movie Box Office Returns

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1 A Applicatio of Extreme Value Aalysis to U.S. Movie Box Office Returs Bi, G. ad D.E. Giles Departmet of Ecoomics, Uiversity of Victoria, Victoria BC, Caada Keywords: Movie reveue, extreme values, geeralized Pareto distributio, value at risk EXTENDED ABSTRACT This study uses extreme value theory (EVT) to model U.S. weeked movie box office returs. Most Hollywood movies ope i theaters o a weeked, as the majority of audieces watch movies durig the weeked. The weeked box office reveue therefore accouts for a major part of the total box office reveue. Weeked box office returs i.e., the percetage chage i reveue from oe weeked to the ext - have empirical fluctuatios that led themselves aturally to beig modeled by EVT. I this paper we use the Peaks over Threshold method ad maximum likelihood estimatio to fit the Geeralized Pareto Distributio (GPD) to the tails of the distributios for both extreme positive ad egative returs i box office returs. We use these results to calculate value at risk (VaR) ad expected shortfall (ES) measures of retur risk. whe it does the average fall 73.2%. That is, if the first weeked s box office reveue is $ millio, there is a oe percet probability that the reveue will decrease to $3.28 millio ext week ad the correspodig expected reveue for all possible reveues less tha $3.28 millio is $26.79 millio. These estimates are useful for film distributors i determiig the umber of film prits, ad as a referece for potetial ivestors i the movie idustry. We fid that we are able to model the tails of the distributios for both positive ad egative returs satisfactorily with the GPD. Our estimates of VaR ad ES for positive retur idicate that, with probability %, the reveue icrease from oe weeked to the ext could exceed 8.4%, ad that whe it does, the average icrease is 96.32%. Also, with probability %, box office reveue could drop 68.72% from oe weeked to the ext, ad that 2652

2 . INTRODUCTION After beig adjusted for the effects of seasoality, U.S. weeked box office reveue is domiated by high budget movies. Accordig to the Iteret Movie Database (IMDB), amog 36 blockbusters with gross box office icome of over $ millio durig their theatrical rus, 29 movies, or about 8%, had budgets above $6 millio. I most cases, the distributio of box office reveue is domiated by these high budget movies. However, this is ot always the case. Some high budget movies sustai losses at the box office. Based o absolute loss o worldwide gross, for example, each of the top five moey losers had budgets of over $ millio but lost over $9 millio. Whe these movies were released, they dragged the weeked box office returs dow. I cotrast, some low budget movies are box office wiers. For example, the most profitable movie based o retur o ivestmet, The Blair Witch Project, had a budget of oly $35, but worldwide gross earigs of $248 millio. Blockbusters ad losers that appear i the distributio of the weeked box office ca be take as extreme evets which ca be aalyzed via Extreme Value theory (EVT). EVT has bee applied i may areas where disasters occur, such as earthquakes, floods, ad eve terrorism attacks (e.g., Jekiso, 955, Embrechts et al., 999, ad Reiss ad Thomas, 997). May studies have aalyzed the variatios i fiacial markets with EVT. The tail behavior of fiacial returs series has bee discussed by Koedijk et al. (99), Logi (996), Daielsso ad de Vries (2), Neiftci (2), McNeil ad Frey (2), Geçay et al. (23) ad Geçay ad Selçuk (26), for example. We use the Peaks over Threshold method ad maximum likelihood estimatio to fit the Geeralized Pareto Distributio to the tails of the distributios for both extreme positive ad egative box office returs. We also calculate value at risk ad expected shortfall measures of retur risk. 2. EXTREME VALUE THEORY The pricipal result of extreme value theory relates to the asymptotic distributio of block maxima i.e., the maximum values of blocks, or sapshots, of data from a ukow uderlyig distributio. The Fisher ad Tippet (928) Theorem tells us that if these maxima are suitably ormalized, they coverge i distributio to oe of oly three forms Gumbel, Fréchet, or Weibull. This is a extreme value aalogue to covetioal cetral limit theory. These three distributios ca be ecompassed by a sigle oe the Geeralized Extreme Value (GEV) family of distributios. (Coles, 2, pp ). If idividual data values {X, X 2, }, rather tha blocks, are available the it is iefficiet to artificially block them ad estimate a GEV distributio. The detailed data iformatio ca be used more efficietly by modelig the distributio, F u, of values that are extreme (i.e., exceed some high threshold value,. The Coditioal Excess distributio fuctio is defied as: F u ( y) = P( X y X > y x u, F where X is a radom variable, u is a particular threshold value, y = x u are the excesses (or exceedaces ), ad x < is the right edpoit of the ukow populatio distributio, F. So, F 2653

3 F u u + y) x) ( y) = =. As the realizatios of the radom variable X lie maily betwee ad u, the estimatio of F i this iterval is usually quite straightforward. However, the estimatio of the portio, F u, which is of iterest here, ca be difficult due to the fact that the umber of observatios above the large eough threshold might be quite limited. The followig asymptotic result is a atural geeralizatio of the GEV result for block maxima: Theorem (Pickads, 975; Balkema ad de Haa, 974): For a large class of uderlyig distributio fuctios F the Coditioal Excess Distributio fuctio F u (y), for u large, is well approximated by F u ( y) G, ( y), u, where G ξ, σ ξ σ ξ ( + y) ( y) = σ y / e σ / ξ if if ξ ξ = ξ for y [,( xf ] if ξ, ad y [, ] if σ ξ <. Gξ, σ is the so-called Geeralized Pareto Distributio (GPD). 3. THE PEAK OVER THRESHOLD METHOD The Peak over Threshold (POT) method is used to obtai the distributio of exceedaces above a certai threshold. The POT method ivolves the followig steps: select the threshold u; fit the GPD fuctio to the exceedaces over u; compute estimates for various risk measures. The selectio of the threshold u is the key factor that decides the fractio of data belogig to the tail, ad therefore affects the results of the MLE of the parameters of the GPD fuctio. The value of u should be high eough to satisfy the Theorem i sectio 2, but the higher the threshold the fewer observatios are left for the estimatio of the parameters. There is a trade-off, ad the determiatio of the threshold is complicated. Previous research (e.g., Daielsso et al., 2; Dupuis, 998) has attempted to deal with this issue, but there is o uambiguous method for selectig the threshold. Graphical tools are usually adopted (e.g., Gilli ad Këllezi, 26). We use two tools - the Sample Mea Excess (SME) plot ad the Shape Parameter (SP) plot - to determie the threshold, u. Defiig x = u + y, the GPD ca be writte as a fuctio of x: G ξ, σ ξ ( + ( x ) ( x) = σ ( x / σ e / ξ if ξ if ξ = where u is the threshold, ξ is the shape parameter ad σ is the scale parameter. Maximum Likelihood Estimatio (MLE) ca be used to estimate the parameters of the GPD after selectig a appropriate threshold u. The we ca fit the GPD to the exceedaces. The details follow. The (populatio) mea excess fuctio of the GPD with parameter ξ < is e ( z) = E( X z X > z) = ( σ + ξ z) /( ξ ), σ + ξ z >. This gives the average value of the excesses of X coditioal o a value for the threshold, z. The SME-plot is defied by the poits: ( u, e ( ) ; x < < x where e ( is the sample mea excess fuctio defied as: i k i e ( = ( k + ) = ( x ; 2654

4 i > k = mi{ i x u}, ad (-k+) is the umber of observatios exceedig the threshold. As a estimate of the mea excess fuctio, the sample mea excess fuctio should be liear. This property ca be used as a criterio for the selectio of the threshold, u. The selected u should be that located at the begiig of a portio of the sample mea excess plot that is roughly liear ad slopig up (Agelii, 22). This ivolves a subjective choice i practice. The shape parameter (SP) plot graphs the estimates of the shape parameter ξ o the vertical axis as a fuctio of icreasig thresholds u o the horizotal axis. For a sample y = {y,, y } of exceedaces, the log-likelihood fuctio for the GPD is: ξyi logσ ( + ) log( + ) ; ξ i= L = ξ σ logσ (/ σ ) yi ; ξ = i= We ca compute the MLE s of the parameters for oe sample of exceedaces, y, defied by the observatios exceedig a sigle threshold u. After geeratig a series of thresholds ad repeatig the process of computig estimates o the basis of equatio (6), a series of estimates for ξ ad σ ca be computed. Estimates of ξ are plotted agaist the associated thresholds to fid a rage over which the estimates are relatively stable. After selectig a threshold u, usig the above tools the correspodig MLE s of the parameters are used with the sample of exceedaces, y, to costruct a series of values for ( y). Plottig both the theoretical ad G ˆ ξ, ˆ σ the empirical distributio fuctio, we ca observe if the GPD provides a reasoable fit to the exceedaces above the threshold. 4. RISK MEASURES Two typical risk measures are the Value at Risk (VaR) ad the Expected Shortfall (ES). Value at Risk is the retur sufficiet to cover, i most istaces, gais or losses over a fixed umber of weekeds. Suppose a radom variable X, with cotiuous distributio fuctio F, models positive or egative returs over a certai time horizo. The VaR p is the p-th. quatile of the distributio F such that VaR p = F ( p), where F - is the p th quatile fuctio. The expected shortfall is defied as the expected size of a retur that exceeds VaR p : ES = E X X > VaR ). p ( p Assumig a GPD fuctio for the tail distributio, the VaR p ad ESp ca be expressed i terms of the GPD parameters: VaR ˆ ES ˆ p ˆ σ = u + ˆ ξ = VaR ˆ N u p ξˆ VaR ˆ σ ˆ p ˆ ξu + e( z) = + ˆ ξ ˆ ξ p p. 5. DATA Weeked box office total reveues of the top 2 movies per week, from 8 Jauary 982 to 5 September 26, have bee obtaied from the olie movie database Box Office Mojo ( Weekly returs i reveue are calculated as logarithmic differeces. As the distributios of the positive ad egative are asymmetric, we model them separately (as is usually the case with fiacial returs). Our sample comprises 62 positive returs ad 668 egative returs. 2655

5 Growth Rate Date Figure. U.S. weeked box office returs. 6. RESULTS Our results were obtaied by writig program code for EViews 5.. The R package, PoT, was used to verify the MLE results. Figure 2 shows the SME plots for the positive ad egative returs. Values of u=.35 (.28) for the positive (egative) returs locate the begiig of a portio of the SME plot that is approximately liear ad slopig up. Mea Excess Threshold ( Shape Parameter (P=.95) Threshold ( Figure 3. Top (Bottom): Shape parameter estimates for positive (egative) returs as a fuctio of the threshold. Figure 3 provides the SP plots with 95% cofidece itervals. For the positive returs we choose u =.3529 (8 exceedaces). For the egative returs we choose u =.28 (53 exceedaces). These thresholds closely match those suggested by the SME plots. Table gives the MLE results. Table. Parameter estimatio of GPD. Positive Negative returs returs u =.3529 u =.28 ξˆ (Std. Error) (.36) (.69) σˆ (Std. Error) (.275) (.2285) Mea Excess Threshold ( X Exceedace Figure 2. Top (Bottom): ME plot for positive (egative) returs. Shape Parameter (P=.95) Threshold ( X Exceedace Figure 4. Top: Positive returs. GPD fitted to 8 exceedaces above u = Bottom: GPD fitted to the tail exceedaces. 2656

6 X- -X Exceedace Exceedace Figure 5. Top: Negative returs. GPD fitted to 53 exceedaces above u =.28. Bottom: GPD fitted to the tail exceedaces. A series of values for ( y) ca the be G ˆ ξ, σ ˆ computed by substitutig these estimates ito the GPD fuctio. Figures 4 ad 5 show the GPD fits, which are relatively satisfactory. Table 2 gives the estimates of VaR ad ES for p =.,., with 95% asymptotic cofidece itervals (from the Delta method). Table 2. Estimated values of VaR ad ES. Positive Returs Probability VaR ES = % Cofidece Iterval.32~.3.222~.75 = % Cofidece Iterval.32~ ~.89 Negative Returs Probabilty VaR ES = % Cofidece Iterval.584~ ~.22 = % Cofidece Iterval.373~ ~.965 The estimates of VaR ad ES for positive retur idicate that, with % probability, the returs from oe weeked to the ext could exceed 8.4%, ad that the average returs above this level will be 96.32%. So, if a weeked s box office reveue is $ millio, there is oe percet probability that reveue will icrease to $8.4 millio ext weeked, ad the expected value for all reveues over $8.4 millio is $96.32 millio, etc. Such estimates ca be used i differet ways. For example, the VaR results imply that, give the same amout of ivestmet the possibility of loss for a ivestmet i the movie idustry is relatively lower tha the possibility of gai. I additio, the differece betwee the VaR ad ES for the positive returs is bigger tha that for the egative returs. This meas that the expected gai over the VaR uder the situatio of gai is more tha the expected loss over the VaR uder the situatio of loss. The risk measures could also help movie producers forecast the required umber of prits of ew movies. By calculatig VaR ad ES with a give probability ad estimatig the total box office reveue at the releasig weeked, the producer could the divide the estimated reveue by the average ticket price to get the total audiece, which could be used to estimate the umber of prits of the ew movie. This would be most useful for high budget movie producers as these movies usually domiate the box office upo release. 7. CONCLUSIONS We illustrate how extreme value theory ca be used to model the tails of the distributio for weeked box office returs i the U.S.. Oe implicatio of our estimates of Value at Risk ad Expected Shortfall is that the possibility of loss for a ivestmet i the movie idustry is 2657

7 lower tha the possibility of gai. These measures ca also be used to estimate the umber of prits of movies that are likely to be eeded, thus avoidig surpluses or shortages of film copies for high budget losers ad low budget wiers at the box office. Ogoig research cosiders returs for idividual compaies i the idustry, ad returs based o et, rather tha gross, reveue. 8. REFERENCES Balkema, A.A. ad L. de Haa (974), Residual life time at great age, Aals of Probability, 2, Coles, S. (2), A Itroductio to Statistical Modelig of Extreme Values, Spriger-Verlag, Lodo. Daielsso, J. ad C.G. de Vries (2), Value-at-risk ad extreme returs, Aales d Ecoomic et de Statistique, 6, Daielsso, J., C.G. de Vries, L. de Haa ad L. Peg (2), Usig a boot-strap method to choose the sample fractio i tail idex estimatio, Joural of Multivariate Aalysis, 76, Dupuis, D. J. (998), Exceedaces over high thresholds: a guide to threshold selectio, Extremes,, Embrechts, P., C. Kläuppelberg ad T. Mikosch (999), Modellig extremal evets for isurace ad fiace, Applicatios of Mathematics, Spriger, Berli. Fisher, R.A. ad L.H.C. Tippett (928), Limitig forms of the frequecy distributio of the largest or smallest member of a sample, Proceedigs of the Cambridge Philosophical Society, 24, 8-9. Geçay, R. ad F. Selçuk (26), Overight borrowig, iterest rates ad extreme value theory, Europea Ecoomic Review, 5, Geçay, R., F. Selçuk, ad A. Ulugülyağcı (23), High volatility, thick tails ad extreme value theory i value-at-risk estimatio. Isurace: Mathematics ad Ecoomics, 33, Gilli, M. ad E. Këllezi (26), A applicatio of extreme value theory for measurig fiacial risk, Computatioal Ecoomics, 27, -23. Jekiso, A.F. (955), The frequecy distributio of the aual maximum (miimum) values of meteorological evets, Quarterly Joural of the Royal Meteorological Society, 8, Jodeau, E. ad M. Rockiger (999), The tail behaviour of stock returs: emergig versus mature markets, mimeo., HEC ad Baque de Frace. Koedijk, K.G., M. Schafgas ad C.G. de Vries (99), The tail idex of exchage rate returs. Joural of Iteratioal Ecoomics, 29, Logi, F.M. (996), The asymptotic distributio of extreme stock market returs, Joural of Busiess, 69, McNeil, A.J. ad R. Frey (2), Estimatio of tail-related risk measures for heteroscedastic fiacial time series: a extreme value approach, Joural of Empirical Fiace, 7, Neiftci, S.N. (2), Value at risk calculatios, extreme evets, ad tail estimatio, Joural of Derivatives,, Pickads, J.I. (975), Statistical iferece usig extreme value order statistics, Aals of Statistics, 3,9-3. Reiss, R. D. ad Thomas, M. (997). Statistical Aalysis of Extreme Values with Applicatios to Isurace, Fiace, Hydrology ad Other Fields. BirkhÄauser Verlag, Basel. 2658

AN APPLICATION OF EXTREME VALUE ANALYSIS TO U.S. MOVIE BOX OFFICE RETURNS

Ecoometrics Workig Paper EWP0705 ISSN 485-644 Departmet of Ecoomics AN APPLICATION OF EXTREME VALUE ANALYSIS TO U.S. MOVIE BOX OFFICE RETURNS Guag Bi & David E. Giles* Departmet of Ecoomics, Uiversity