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A Mathematical Investigation of the Financial Performance of Movie Sequels

Ron Buckmire, Occidental CollegeDavid Edwards, University of DelawareJacob Ortega-Gingrich ‘13, Occidental College

A Preliminary Report

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Outline

Introduction to Cinematic Box-Office Dynamics– Important variables, concepts and equations– Graphs of typical movie data

The Edwards-Buckmire Model (EBM) Drawbacks of EBM Modified EBM Preliminary Numerical Results Future Work

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Introduction to Cinematic Box-Office Dynamics

Important variables G(t) : cumulative gross receipts of the movie S(t) : number of screens movie is exhibited A(t) : normalized weekly revenue ($ per screen

average) t : time in number of weeks

Important concepts A and S have quasi-exponential profiles

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Actual Movie Data: The Expendables (2010)

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Actual Movie Data: Taken (2009)

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The EBM (Edwards & Buckmire, 2001)

SA

dtdG

ASdtdS

AGS

SdtdA

~

~~

where

*

~SM )1(

1~

DS

A

SPASH

max*

2%~

0)0( G 0)0( AA 0)0( SS

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EBM Parameters

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Typical numerical solutions of EBM

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Reasons Why EBM Needs Modifying

Fails to allow H% to vary with time Movie-specific parameter ( ) estimates

are difficult to make and somewhat arbitrary Most movies have a contract period (2-4

weeks) in which screens is constant, i.e. S’=0 S and A actual data more erratic than first

thought; G is relatively smooth

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Modifying the EBM (J. Ortega-Gingrich)

Uses an Economics-inspired “demand” model where G’(t)=μ(S)D(t) and assumes D(t) decreases exponentially over time

Incorporates fixed contract periods when screens are constant

Greatly modifies the A equation Modified EBM has 3 parameters, 2 of

which are movie-independent (or global)

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Modified EBM (EBM 2)

a=1/T,T is total number of movie theaters in North America (~4,000) The function μ(S) should satisfy the following conditions:

Screen Availability Function μ(S)

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Numerical Calculations with EBM 2

Analyzed119 movies from 2005-2010 (minimum final gross $50m)

All dollars adjusted for inflation to 2005 Used Mathematica to generate numerical

solutions to the modified EBM Found “global” values of

parameters that would minimize the difference between computed G∞ and actual G∞ while also minimizing standard deviation

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Numerical Results: (N=119)

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Distribution of G Computed/G Actual as Histogrammean=1.0389, std. dev.=0.158

Numerical Results: (N=119)

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Numerical Results: Using Global ParametersThe Expendables (2010)

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Numerical Results: Using Global ParametersTaken (2009)

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Numerical Results: Using Chosen ParametersThe Expendables (2010)

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Numerical Results: Using Chosen ParametersOpen Season (2006)

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Open Questions (Future Work)

“The Holy Grail”: Predict the opening weekend gross before the movie is released

The sequel problem: predict the gross of a sequel based on characteristics of the parent film (considered an easier version of the a priori Holy Grail problem if one can find a relationship between sequel and parent films)