Fin 501:Asset Pricing ISlide 07-1Multi-period ModelLecture 07: Multi-period ModelProf. Markus K. BrunnermeierLecture 07Fin 501:Asset Pricing ISlide 07-2Multi-period ModelIntroduction• accommodate multiple and even infinitely many periods.• several issues: how to define assets in an multi period model, how to model intertemporal preferences, what market completeness means in this environment, how the infinite horizon may the sensible definition of a budget constraint (Ponzi schemes), and how the infinite horizon may affect pricing (bubbles).• This section is mostly based on Lengwiler (2004)Lecture 07Fin 501:Asset Pricing ISlide 07-3Multi-period Model0 123many one period modelshow to model information?Lecture 07Fin 501:Asset Pricing ISlide 07-4Multi-period Model0 123 0 1 2 3F1F2;[ [Events Ai,tStates sLecture 07Fin 501:Asset Pricing ISlide 07-5Multi-period Modelfrom static to dynamic…asset holdings Dynamic strategy (adapted process)asset payoff x Next period’s payoff xt+1+ pt+1Payoff of a strategyspan of assets Marketed subspace of strategiesMarket completeness a) Static completeness (Debreu)b) Dynamic completeness (Arrow)No arbitrage w.r.t. holdings No arbitrage w.r.t strategiesState prices q(s) Event prices qt(At(s))Lecture 07Fin 501:Asset Pricing ISlide 07-6Multi-period Model…from static to dynamicState prices q(s) Event prices qt(A(s))Risk free rate r Risk free rate rtvaries over timeDiscount factor from t to 0t(s)Risk neutral prob.*(s) = q(s) rRisk neutral prob.*(At(s)) = qt(At(s)) / t(At)Pricing kernelpj= E[kqxj]1 = E[kq] rPricing kernelktptj= Et[kt+1(pjt+1+ xjt+1)]kt= rt+1Et[kt+1]_ ____Lecture 07Fin 501:Asset Pricing ISlide 07-8Multi-period ModelAssets in many periodsLecture 07Fin 501:Asset Pricing ISlide 07-9Multi-period ModelMultiple period uncertaintyA0A1A2A3A4A5A6t=1t=0 t=2 We recall the event tree that captures the gradual resolution of uncertainty. This tree has 7 events (A0to A6).(Lengwiler uses e0to e6) 3 time periods (0 to 2). If A is some event, we denote the period it belongs to as (A). So for instance, (A2)=1, (A4)=2. We denote a path with as follows…0(A4)1(A4)2(A4)Lecture 07Fin 501:Asset Pricing ISlide 07-10Multi-period ModelMultiple period uncertaintyA0A1A2A3A4A5A6t=1t=0 t=2 Last period events have prob.,3-6.  The earlier events also have probabilities. To be consistent, the probability of an event is equal to the sum of the probabilities of its successor events. So for instance, 1= 3+4.Lecture 07Fin 501:Asset Pricing ISlide 07-11Multi-period ModelMultiple period assetsA typical multiple period asset is a coupon bond:.)( if0,)( ifcoupon1,)(0 ifcoupon:***tAtAtArAThe coupon bond pays the coupon in each period & pays the coupon plus the principal at maturity t*.A consol is a coupon bond with t* = 1; it pays a coupon forever.A discount bond (or zero-coupon bond) finite maturity bond with no coupon. It just pays 1 at expiration, and nothing otherwise.Lecture 07Fin 501:Asset Pricing ISlide 07-12Multi-period ModelMultiple period assets• create STRIPS by extracting only those payments that occur in a particular period. STRIPS are the same as discount bonds.• More generally, arbitrary assets (not just bonds) could be striped.Lecture 07Fin 501:Asset Pricing ISlide 07-14Multi-period ModelTime preferences with many periodsLecture 07Fin 501:Asset Pricing ISlide 07-15Multi-period ModelTime preferenceu(y0) + t(t) E{u(yt)}• Discount factor (t) number between 0 and 1 • Assume (t)> (t+1) for all t.• Suppose you are in period 0 and you make a plan of your present and future consumption: y0, y1, y2, …• The relation between consecutive consumption will depend on the interpersonal rate of substitution, which is (t).• Time consistency (t) = t (exponential discounting)Lecture 07Fin 501:Asset Pricing ISlide 07-16Multi-period ModelPricing in astatic dynamic modelLecture 07Fin 501:Asset Pricing ISlide 07-17Multi-period ModelA static dynamic model• We consider pricing in a model that contains many periods (possibly infinitely many)…• …and we assume that information is gradually revealed (this is the dynamic part)…• …but we also assume that all assets are only traded "at the beginning of time" (this is the static part).• There is dynamics in the model because there is time, but the decision making is completely static.Lecture 07Fin 501:Asset Pricing ISlide 07-18Multi-period ModelMaximization over many periods• vNM exponential utility representative agentmax{t=0tE{u(yt)} | y-w 2 B(p)}• If all Arrow securities (conditional on each event) are traded, we can express the first-order conditions as,u'(y0) = , (A)Au'(wA) = qA.Lecture 07Fin 501:Asset Pricing ISlide 07-19Multi-period ModelMulti-period SDF• The equilibrium SDF is computed in the same fashion as in the static model we saw before• We call MAthe "one-period ahead" SDF and MAthe multi-period SDF (“state-price density”).)(')('0)(wuwuqAAAA)(')(')(')(')(')(')()()(0)(1)(121AAAAAAwuwuwuwuwuwu)(1AM)(2AM)(1)(AAMAM:Lecture 07Fin 501:Asset Pricing ISlide 07-20Multi-period ModelThe fundamental pricing formula• To price an arbitrary asset x, portfolio of STRIPed cash flows, xj= x1j+x2j++x1j, where xtjdenotes the cash-flows in period t.• The price of asset xjis simply the sum of the prices of its STRIPed payoffs, sotjttjxEp }{M• This is the fundamental pricing formula.• Note that Mt =tif the repr agent is risk neutral. The fundamental pricing formula then just reduces to the present value of expected dividends, pj= tE{xtj}.Lecture 07Fin 501:Asset Pricing ISlide 07-21Multi-period ModelDynamic completionLecture 07Fin 501:Asset Pricing ISlide 07-22Multi-period ModelDynamic trading• In the "static dynamic" model we assumed that there were many periods and information was gradually revealed (this is the dynamic part)…• …but all assets are traded "at the beginning of time" (this is the static part).• Now consequences of re-opening financial markets. Assets can be traded at each instant.• This has deep implications.  allows us to reduce the number of assets available at each instant through dynamic completion.  It opens up some nasty possibilities (Ponzi schemes and bubbles), Lecture 07Fin 501:Asset Pricing ISlide 07-23Multi-period ModelCompletion with short-lived assets• If the horizon is infinite, the number of events is also infinite. Does that imply that we need an infinite number of assets