Eco525 Financial Economics I Lecture 07 Multi period Model Prof Markus K Brunnermeier 20 27 Lecture 07 Multi period Model Slide 07 1 Eco525 Financial Economics I Introduction 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 20 27 Lecture 07 Multi period Model Slide 07 2 Eco525 Financial Economics I 0 1 2 3 many one period models how to model information 20 27 Lecture 07 Multi period Model Slide 07 3 Eco525 Financial Economics I 0 1 2 3 0 1 2 3 States s 20 27 Lecture 07 Multi period Model F1 F2 Events A Slide 07 4 i t Eco525 Financial Economics I from static to dynamic asset holdings Dynamic strategy adapted process asset payoff x Next period s payoff xt 1 pt 1 Payoff of a strategy span of assets Marketed subspace of strategies Market completeness a Static completeness Debreu b Dynamic completeness Arrow No arbitrage w r t holdings No arbitrage w r t strategies State prices q s Event prices qt At s 20 27 Lecture 07 Multi period Model Slide 07 5 Eco525 Financial Economics I from static to dynamic State prices q s Risk free rate r Risk neutral prob s q s r Pricing kernel pj E kq xj 1 E kq r 20 27 Lecture 07 Event prices qt A s Risk free rate rt varies over time Discount factor from t to 0 t s Risk neutral prob At s qt At s t At Pricing kernel kt ptj Et kt 1 pjt 1 xjt 1 kt rt 1 Et kt 1 Multi period Model Slide 07 6 Eco525 Financial Economics I Assets in many periods 20 27 Lecture 07 Multi period Model Slide 07 7 Eco525 Financial Economics I Multiple period uncertainty A3 We recall the event tree that captures the gradual resolution of uncertainty 1 A4 A1 0 A4 2 A4 This tree has 7 events A0 to A6 Lengwiler uses e0 to e6 A4 A0 3 time periods 0 to 2 A5 If A is some event we denote the period it belongs to as A A2 A6 t 0 t 1 20 27 Lecture 07 So for instance A2 1 A4 2 We denote a path with as follows t 2 Multi period Model Slide 07 8 Eco525 Financial Economics I Multiple period uncertainty A3 A1 A4 A0 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 A5 A2 t 0 t 1 20 27 Lecture 07 t 2 A6 So for instance 1 3 4 Multi period Model Slide 07 9 Eco525 Financial Economics I Multiple period assets A typical multiple period asset is a coupon bond coupon if 0 A t rA 1 coupon if A t 0 if A t The 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 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 20 27 Lecture 07 Multi period Model Slide 07 10 Eco525 Financial Economics I Multiple 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 20 27 Lecture 07 Multi period Model Slide 07 11 Eco525 Financial Economics I Time preferences with many periods 20 27 Lecture 07 Multi period Model Slide 07 12 Eco525 Financial Economics I Time preference u 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 20 27 Lecture 07 Multi period Model Slide 07 13 Eco525 Financial Economics I Pricing in a static dynamic model 20 27 Lecture 07 Multi period Model Slide 07 14 Eco525 Financial Economics I A 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 20 27 Lecture 07 Multi period Model Slide 07 15 Eco525 Financial Economics I Maximization over many periods vNM exponential utility representative agent max t 0 t E u yt y w 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 20 27 Lecture 07 Multi period Model Slide 07 16 Eco525 Financial Economics I Multi period SDF The equilibrium SDF is computed in the same fashion as in the static modelA we saw before qA A A u w u w0 u w 1 A u w 2 A u w A L A 1 A 0 u w A 1 u w u w M 1 A M 2 A L M A 1 A M A We call MA the one period ahead SDF and MA the multi period SDF state price density 20 27 Lecture 07 Multi period Model Slide 07 17 Eco525 Financial Economics I The fundamental pricing formula To price an arbitrary asset x portfolio of STRIPed cash flows x j x1j x2j L x j where xt j denotes the cash flows in period t The price of asset x j is simply the sum of the prices of its STRIPed payoffs so p j E M t xt j t This is the fundamental pricing formula Note that Mt t if the repr agent is risk neutral The fundamental pricing formula then just reduces to the present value of expected dividends pj t E xtj 20 27 Lecture 07 Multi period Model Slide 07 18 Eco525 Financial Economics I Dynamic completion 20 27 Lecture 07 Multi period Model Slide 07 19 Eco525 Financial Economics I Dynamic 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 …