D.W.Diamond and R.E.Verrecchia,Price adjustment to private information 287 no trade,there will not be a transaction price equal to p,.Although this is of no relevance to investors,it does imply a censored sample problem for empirical measurement.We discuss this problem in section 5.3. 3.The effect of prohibiting short-sales In this section we consider the effect on informational efficiency of prohibit- ing some short-sales by assuming there are no traders in our economy who face proceeds-restrictions (i.e.,c2=0),and we examine the effect of increasing the fraction of traders prohibited from shorting by reducing the fraction of those who are unconstrained. To introduce this characterization,consider the stochastic process that prices follow beginning at Po=.For simplicity,we use conditional expecta- tions in place of prices when there is no-trade.(The empirical implications of observing only transaction prices are presented in section 5.3.)When the complete (private)information becomes public (or,equivalently,approaches its public revelation asymptotically as a result of trading),the price will be either 1 or 0.Let us define two prices P#and pL,where P#is strictly greater than PL,which serve as benchmarks for how close the trading process comes to reflecting all (private)information.For example,if #andPL= then,by computing the expected number of periods for the price to first exceed PH or fall below PL,we can determine the expected amount of time necessary for the price to reflect (to the uninformed)that the odds are three-to-one in favor of either a value of 1 or a value 0.That is,we define the time of adjustment to private information as the expected number of time periods until the price first passes beyond the (fixed)thresholds of P#or PL conditional upon either =1 or o=0,or unconditionally. Let the random variable N,with realization N,represent the number of time periods that pass until price is first greater than or cqual to pH or less than or equal to p.For convenience,let N and No represent the expected values of the random variable N conditional upon =1 or =0 being the true state-of-nature,respectively:that is,N=E[N=1]and No=E[Nv=0]. The expressions N and No are implicitly functions of the parameters that characterize the economy (e.g..g.a.h).as well as p#and PL.In particular, as we vary the relative proportion of traders who face no short-prohibitions on short-selling (i.e.,c),to those who do (i.e.,c3),we can determine the effect of these prohibitions on the expected number of time periods.Changing the relative proportion implies the following behavior for the expected adjustment time. Proposition 1.When all traders who can sell short can do so costlessly (i.e.,the use of proceeds is not deferred )the expected number of periods required for the (absolute)adjustment of prices to bad news,No,and to good news,N,are both
D. W. Diamond and R. E. Verrecchia, Price adjustment to private informanon 287 no trade, there will not be a transaction price equal to P,. Although this is of no relevance to investors, it does imply a censored sample problem for empirical measurement. We discuss this problem in section 5.3. 3. The effect of prohibiting short-sales In this section we consider the effect on informational efficiency of prohibiting some short-sales by assuming there are no traders in our economy who face proceeds-restrictions (i.e., c2 = 0), and we examine the effect of increasing the fraction of traders prohibited from shorting by reducing the fraction of those who are unconstrained. To introduce this characterization, consider the stochastic process that prices follow beginning at P,, = :. For simplicity, we use conditional expectations in place of prices when there is no-trade. (The empirical implications of observing only transaction prices are presented in section 5.3.) When the complete (private) information becomes public (or, equivalently, approaches its public revelation asymptotically as a result of trading), the price will- be either 1 or 0. Let us define two prices PH and PL, where PH is strictly greater than PL, which serve as benchmarks for how close the trading process comes to reflecting all (private) information. For example, if PH = 2 and P L = i, then, by computing the expected number of periods for the price to first exceed P H or fall below P L, we can determine the expected amount of time necessary for the price to reflect (to the uninformed) that the odds are three-to-one in favor of either a value of 1 or a value 0. That is, we define the time of adjustment to private information as the expected number of time periods until the price first passes beyond the (fixed) thresholds of P H or PL, conditional upon either u = 1 or u = 0, or unconditionally. Let the random variable N, with realization N, represent the number of time periods that pass until price is first greater than or equal to PH or less than or equal to P L. For convenience, let gi and No represent the expected values of the random variable 3 conditiopal upon u = 1 or u = 0 being the true state-of-nature, respectively: that is, Iv, = E[@ ) u = l] and & = WI? ] u = 01. The expressions ZVi and ZV,, are implicitly functions of the parameters that characterize the economy (e.g., g, (I, h), as well as PH and P L. In particular, as we vary the relative proportion of traders who face no short-prohibitions on short-selling (i.e., ci), to those who do (i.e., c,), we can determine the effect of these prohibitions on the expected number of time periods. Changing the relative proportion implies the following behavior for the expected adjustment time. Proposition I. When all traders who can sell short can do so costlessly (i.e., the use of proceeds is not deferred), the expected number of periods required for the (absolute) a#.+ment of prices to bad news, go, and to good news, Iv,, are both
288 D.W.Diamond and R.E.Verrecchia,Price adjustment to private information increasing functions of the proportion of traders who are prohibited from short- selling.Furthermore,the ratio of the expected adjustment times of prices to bad news relative to good news,No/N,is also increasing. Proof.See appendix. Note that Proposition 1 also implies that the unconditional expectation of the time for price adjustment to private information,E[N],increases.Proposi- tion 1 shows that short-prohibitions reduce informational efficiency with respect to both good and bad news,but especially to bad news.Although both the expected number of periods required for the (absolute)adjustment of prices to good and bad news increases,the former increases relatively more slowly than the latter.This means that the effect of prohibiting short-sales on the impoundment of private information into price is relatively more pro- nounced in the case of bad news versus good news.This result is of special interest in our discussion of testable implications below,since relative com- parisons are typically easier to measure than absolute.Before developing the intuition behind this result,a useful related result is presented. Faster adjustment to private information(lower expected N)suggests that at any date the price is higher when there is good news (v=1)and lower when there is bad news (v=0)(while the reverse is suggested by slower adjustment). Corollary 1 states a result of this type in terms of log(P/(1-P)).This log transform of the likelihood ratio is increasing in P and is more facile because it follows a random walk. Corollary 1.When all traders who can sell short can do so costlessly,increasing the proportion of traders who are prohibited from short-selling decreases Eflog(P/(1-P))Iv=1]and increases Eflog(P/(1-P.))I=0]for all t. Proof.See appendix. Corollary 1 roughly suggests the following observable result.When private information about the value of the asset is released to the public,the price will be eventually 1 if good news is released and 0 if bad news is released. Therefore when private information is made public,price adjustments that follow in the presence of short-selling prohibitions are larger in magnitude than those which occur in the absence of short-selling prohibitions. For example,if the value of becomes public information after t=1,there would be a larger average absolute value of change in P,in response to that information when short-sales are prohibited (i.e.,c;=1)than when short-sell- ing is unconstrained.The increase in the expected absolute value of this
288 0. W. Diamond and R. E. Verrecchia, Prrce adjustment to pncate informarion increasing functions of the proportion of traders who are prohibited from shortselling. Furthermore, the ratio of the expected adjustment times of prices to bad news relative to good news, g&g,, is also increasing. Proof. See appendix. Note that Proposition 1 also implies that the unconditional expectation of the time for price adjustment to private information, E[fl], increases. Proposition 1 shows that short-prohibitions reduce informational efficiency with respect to both good and bad news, but especially to bad news. Although both the expected number of periods required for the (absolute) adjustment of prices to good and bad news increases, the former increases relatively more slowly than the latter. This means that the effect of prohibiting short-sales on the impoundment of private information into price is relatively more pronounced in the case of bad news versus good news. This result is of special interest in our discussion of testable implications below, since relative comparisons are typically easier to measure than absolute. Before developing the intuition behind this result, a useful related result is presented. Faster adjustment to private information (lower expected Is) suggests that at any date the price is higher when there is good news (v = 1) and lower when there is bad news (v = 0) (while the reverse is suggested by slower adjustment). Corollary 1 states a result of this type in terms of log(P,/(l - P,)). This log transform of the likelihood ratio is increasing in P, and is more facile because it follows a random walk. Corollary 1. When all traders who can sell short can do so costlessly, increasing the proportion of traders who are prohibited from short-selling decreases EM PA1 - P,))lv = l] and increases qlog( P/(1 - P,))lv = 0] for all t. Proof. See appendix. Corollary 1 roughly suggests the following observable result. When private information about the value of the asset is released to the public, the price will be eventually 1 if good news is released and 0 if bad news is released. Therefore when private information is made public, price adjustments that follow in the presence of short-selling prohibitions are larger in magnitude than those which occur in the absence of short-selling prohibitions. For example, if the value of u’ becomes public information after t = 1, there would be a larger average absolute value of change in P, in response to that information when short-sales are prohibited (i.e., c3 = 1) than when short-selling is unconstrained. The increase in the expected absolute value of this