08_Dynamic_Present_Value

# 08_Dynamic_Present_Value - Econ 251a Fall 2006 Price...

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Econ 251a Fall 2006 Price Dynamics of Long-Lived Assets The "zero" prices π t are known to all serious market participants. These enable participants to calculate forward rates 1+ i F t . In the last lecture we assumed that these were stationary: 1+ i F t =1+ i. In this lecture we drop stationarity. If there is no uncertainty in the world, then the forward rates will be completely reliable predictors of the future interest rates; 1+ i F t =1+ i t . Since the present values of securities with purely nominal (money) payo f s are determined entirely by interest rates and those cash f ows, it follows that if we can predict future interest rates and future cash f ows, then we can also predict the future present values of the securities. Without uncertainty, present values and prices will be the same. Hence the whole evolution of security prices can be foreseen from the zero prices π t . Putting this another way, an economist who does not know much about the world but who can see the the zero prices π t can deduce what the traders think the future will be, provided the traders themselves see no uncertainty in the world. In this chapter we examine the evolution of prices for some simple securities, assuming various simple scenarios of forward interest rates and security payo f s. The easiest and most important scenario is when the forward rate i F t =0 (implying that traders expect a zero interest rate between time t and time t +1) and the security payo f is 0 at time t +1 . This scenario is not far-fetched at all. If the length of time between period t and t +1 is very short, the interest rate will indeed usually be very low, and the security might well not pay any dividends. The implication, if traders face no uncertainty about the future, is that they must be expecting the security pr iceatt ime t +1 tobethesameasatt ime t. Thus we could not be in a rational world if every trader was sure the price of some asset (like housing or oil) was going to go up over a very short time interval. On the other hand, over longer periods, if the interest rate is non-zero or if dividends are not zero (and even f uctuate), we should often expect prices to change. We give examples including discount and premium bonds for which the price must change over time, which for some reason surprises many people. 1P r e s e n t V a l u e We denote the price today (at time 0) of \$1 at time t by π t = price today for \$1 at time t π t = 1 1+ R t 1 π t =1+ R t = price at time t of \$1 today We say that the present value of a dollar at time t is π t . Sometimes π t is called the

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1.1 Interest Rate Since people are impatient (and since money is storable) it is always the case that π t 1 Anybody who gives up \$1 today can expect to get more than \$1 tomorrow, 1 π t 1 . Many people
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## This note was uploaded on 12/08/2009 for the course ECON 251 at Yale.

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08_Dynamic_Present_Value - Econ 251a Fall 2006 Price...

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