Does U.S. monetary policy react to asset prices? Implications of stock market bubbles, volatility and productivity

Indian Journal of Economics and Business, Dec, 2007 by Rajeev Sooreea

Attempts to explain fundamental values can be traced back to Gordon (1962). However, one of the most influential works on stock market bubbles is that of Froot and Obstfeld (1991). They develop a model of intrinsic bubbles for stock prices. Expanding on this intrinsic bubble framework, this section extracts a measure of the fundamental value of stocks and a measure of stock price misalignments. According to Froot and Obstfeld (1991), an intrinsic bubble is a rational bubble (defined below) that depends exclusively -albeit nonlinearly-on dividends. It is called intrinsic because it derives all of its variability from exogenous fundamental determinants of asset prices (that is dividends). One important feature of intrinsic bubbles is that they are deterministic functions of dividends alone. Thus, this class of bubbles predicts that stable and highly persistent fundamentals lead to stable and highly persistent over- or under-valuations. In addition, these bubbles can cause asset prices to "overreact" to changes in fundamentals. Froot and Obstfeld (1991) start out with a simple present value condition where the real stock price is equal to the present discounted value of real dividend payment plus the real stock price next period. So the real stock price can be written as

[P.sub.t], = [e.sub.-r] [E.sub.t], ([D.sub.t] [P.sub.t 1]) (1)

where [P.sub.t] is real price of a share at the beginning of period t, [D.sub.t] is real dividends per share paid out over period t, r is the constant real discount rate, and [E.sub.t](.) is the market's expectation conditional on information known at the start of period t. The present value solution for P,, denoted [P.sup.PV.sub.t], is then

[P.sup.PV.sub.t] = [[infinity].summation over (s=1)] [E.sup.-r(s-t 1).sub.t] [E.sub.t]([D.sub.s]). (2)

Define a rational bubble [B.sub.t] as one that satisfies

[B.sub.t] = [E.sup.-r] [E.sub.t] ([B.sub.t 1]). (3)

Then, equation (1) can be written as

[B.sub.t] = [P.sup.PV.sub.t] [B.sub.t] (4)

which can be thought of as the sum of the present-value solution and a rational bubble. Rational bubbles are sometimes viewed as being driven by variables extraneous to the valuation problem (6) However, Froot and Obstfeld (1991) argue that some bubbles may depend only on the exogenous fundamental determinants of asset value. An intrinsic bubble is constructed by finding a nonlinear function of fundamentals that satisfies (3). For this, it is assumed that log dividends are generated by the geometric martingale

[d.sub.t 1] = [mu] [d.sub.t], [[xi].sub.t 1] (5)

where [mu] is the drift in dividends, [d.sub.t] is the log of dividends at time t, and [[xi].sub.t 1] is a normal random variable with conditional mean zero and variance oz. Using (5) and assuming that period-t dividends are known when Pt is set, the present value stock price in (2) is directly proportional to dividends:

[P.sup.PV.sub.t] = [kappa][D.sub.t] (6)

where [kappa] = 1/([e.sup.r] - [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]/2).

Equation (6) is essentially a stochastic version of Gordon's (1962) model of stock prices, which predicts that [P.sup.PV.sub.t] = [D.sub.t]/([e.sup.r] - [e.sup.[mu]]) under certainty. The assumption that the sum in (2) converges implies that r > [mu] [[sigma].sup.2]/2. Now, an intrinsic bubble is defined as