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Time-Inconsistency: Problems and Mathematical …
time - inconsistency : Problems and Mathematical TheoryJiongmin Yong( university of Central Florida)December 28, 2015 Outline1. Introduction: time -Consistency2. time -Inconsistent Problems3. Equilibrium Strategies4. Open Problems1. Introduction: time -ConsistencyContinuous Compound Interest Exponential (0) initial principalr annual interest rateP(t) =P(0)ert Amount at the end oft-th year(compounded continuously)For any given future timesT>t>0, fromP(T) =P(0)erT,P(t) =P(0)ert,one hasP(T) =P(t)er(T t),0<t<T,or, equivalently,P(t) =P(T)e r(T t),0<t< is the value (price) attof a payoffP(T) r(T t) exponential any 0<t1<t2<T, one haser(T t1)P(t1) =P(T) =er(T t2)P(t2).Therefore,possessP(t1) att1 possessP(t2) att2 This is called theTime-Consistencyofexponential discountingPreferred Choice:Assume that annual rate isr= 10%Option(A): Get $100 today (December 28, 2015).Option(B): Get $105 (>100(1 +r12)) on January 28, (A ): Get $110 (= 100 ) on December 28, (B ): Get $ (>110(1 +r12)) on January 28, atime-consistentperson,(A) (A ),(B) (B ),(B) (A),(B ) (A ).
Time-Inconsistency: Problems and Mathematical Theory Jiongmin Yong (University of Central Florida) December 28, 2015
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