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This study demonstrates that when the length of the excess earnings period is not known with certainty, all rational expectations pricing models result in some degree of overpricing when compared ex post facto to perfect foresight models. This study examines the time paths of price under existing valuation models such as Baek et al. [1] and Ohlson and Jeuttner-Nauroth [2] under the following stylized facts: we assume that we are dealing with an all-equity firm with opportunity cost of equity of r , and with a proprietary technology which enables it to achieve a marginal return on equity of R > r for approximately N periods, after which a Schumpeterian event S is predicted to occurs and the marginal return on equity is expected to revert to the opportunity cost of equity r . Our study demonstrates a deviation of the predicted rational expectations price from the perfect foresight price and demonstrates that such deviation may become extreme near the end of the excess earnings period, resulting in a catastrophic price adjustment when that period comes to an end.

In his original formulation of the rational expectations (RE) hypothesis, Muth [

Recent valuation studies have addressed the problem that all firms will eventually exhaust their opportunities for positive NPV projects. One such study, Feltham and Ohlson [

Baek et al. [

Following Baek et al. [

Under such conditions of perfect foresight, where the length of the excess earning period is known with perfect foresight, Baek et al. [

P 0 = α e ( R − r ) N r

Our study does not challenge the substance of Baek et al. [

P 0 * = α r ∫ 0 ∞ e ( R − r ) x f ( x ) d x

For simplicity, we begin with the stylized facts where we do not know the exact length of the excess earnings period, but only the constant probability m = 1/N that it will end in any one year.

f ( x ) 1 − F ( x ) = m

We assume that our estimate of the length of the excess earnings period is unbiased.

∫ 0 ∞ x f ( x ) d x = N

This results in the constant stopping rate or m= 1/N and the following marginal distribution functions, f(x), is used.

f ( x ) = m e − m x

Thus, our rational expectations price can be summarized by the following formula.

P 0 * = α r ∫ 0 ∞ m e − m x e ( R − r ) x d x = α r ( m r − R + m )

where the sufficient condition of convergence is that R − r − m < 0 . This of course assumes that state S t ∈ { 0 , 1 } is uncorrelated with the rate of returns on the market M_{t}. In other words, it assumes that

C o v ( d S t , M t ) = 0

where S t ∈ { 0 , 1 } . The first observation which we make is that rational expectations pricing is biased v is a v is perfect foresight pricing. To illustrate, we assume an initial earnings per share of 2, a marginal return on equity of .10, and an opportunity cost of equity of 0.06, with an expected excess earnings period of 10 years. Under those assumptions, we have a perfect foresight share price P_{0} = 49.73 versus a rational expectation share price P 0 ∗ = 55.56 . Further, we can generalize these results to include all distributions f(x) with mean N. We begin with the insight that

P * ( N ) = ∫ 0 ∞ f ( x ) P ( x ) d x

where f(x) has a mean of N. Further, we can simplify P(x) to read

P ( x ) = k a x

where k = α e R t r , and a = e ( R − r ) . This greatly simplifies the extraction of first and second derivatives. Notably,

∂ P ∂ x = k a x ln a

∂ 2 P ∂ x 2 = k a x ( ln a ) 2 > 0

Thus, we demonstrate that P(x) is convex in x. Further, we demonstrate by Jensen’s inequality that

P t * ( N ) > P t (N)

for all t ≥ 0, all N > 0 and for all distributions f(x) with mean of N.

Thus, we are able to demonstrate that there will always be some “overpricing” under rational expectations and under these stylized facts, so long as there exists any uncertainty regarding the time remaining in the excess earnings period; and this will continue to hold even when there is some partial updating of our estimate of the time remaining.

Having established the fact of rational expectations “overpricing”, we now provide evidence of the degree of such “overpricing”. To do so, we examine the further divergence of the rational expectations price from the perfect foresight price as we move forward through the excess earnings period.

Under our perfect foresight model, infinitesimal price increase is proportional to the opportunity cost of equity r.

P t = α e R t r e ( R − r ) ( N − t )

∂ ( ln P t ) ∂ t = r

By contrast, our rational expectations model produces infinitesimal price increase is proportional to the marginal return on equity R.

P t * = α e R t r ( m r − R + m )

∂ ( ln P t * ) ∂ t = R

Of course, this cannot persist forever. Eventually, Schumpeterian event Swill occur which would bring the excess earnings period to a close, leading to a post-event price of P**, where

P t * * = α e R t r

and this would represent a potentially catastrophic price decline, where

P t * * − P t * P t * = − ( R − r ) N

To illustrate, we return to our numerical assumptions where we assumed initial earnings per share of 2, a marginal return on equity of 0.10, an opportunity cost of equity of 0.06, an expected excess earnings period of 10 years. As we demonstrated above, the initial price under perfect foresight was 49.73; and the initial price under rational expectations was 55.56.

Moving forward, the expected price increases under perfect foresight would be precisely proportional to 0.06, or the opportunity cost of equity. By contrast, the expected price increases under the rational expectations model would be proportional to 0.10, causing the two price estimates to diverge even further as we move forward through the excess earnings period.

Further, the PE ratios under the two models will continue to diverge. Under perfect foresight, the PE ratio will gradually decline from 24.87 to 16.67 over the 10-year excess earnings period, while the PE ratio will remain fixed at 27.78 under rational expectations until S occurs. At that time, P t + 1 * * / P t * will be a catastrophic 0.6, reflecting a price decline of almost 40 percent, as the PE ratio declines from 27.78 to 16.67 in a single period.

Clearly, we have departed from the special case of the rational expectation (RE) hypothesis, as originally formulated by Muth [

But such is not the case in the example we have provided. We have demonstrated that none of the proposed rational expectations models will help us to anticipate catastrophic price adjustments, given the stylized facts of this study.

Therefore, let us examine our stylized facts to see if they represent the true state of the economy. We assumed that the length of the excess earnings could only be estimated. In the first model we presented, we assumed that the only information available to the investor was the probability that S would occur at the end of any given period. Is that a realistic picture of our economy?

Actually, in most cases we cannot even determine the direction from which an S event will break upon us, much less calculate the time of its arrival. So that aspect of our stylized facts appears to be realistic.

With increasing interrelationships among technologies, where each new technology is expected to have an impact―either negative or positive―upon a host of related technologies, we could expect Schumpeterian events to impact entire sectors of the economy, and not just one or two firms. And because of this possibility, we might no longer be able to claim that S is uncorrelated with the market as a whole; and that leads to problems of establishing a rational expectations price which is beyond the scope of this study.

Further, our stylized facts are consistent with the observation that PE ratios remain relatively fixed for long periods of time, until catastrophic declines occur.

Fama and French [

The authors declare no conflicts of interest regarding the publication of this paper.

Fogelberg, L. and Baek, C. (2018) Equity Pricing: Perfect Foresight versus Rational Expectations. Theoretical Economics Letters, 8, 3353-3360. https://doi.org/10.4236/tel.2018.815206