Chapter 10. Liquidity, Price Discovery and Corporate Policies

Transcription

1 Chapter 10 Liquidity, Price Discovery and Corporate Policies 1

2 10.7 Exercises. Stock market as guide to investment. In the model analyzed in section , we assumed that G<I and found that on average the stock market encourages investment. Consider here the complementary case where G>I: a. Show that in this case the stock market deters inappropriate investment. Explain the intuition behind this result. b. Assume that the market is informative enough for the stock price to a ect investment and derive the new expression for the increase in firm s ex-ante value resulting from the presence of a stock market. 4. Bid and ask prices when investment does not react to stock prices. In the model analyzed in section , assume that the informativeness condition (10.5) does not hold. a. Compute the equilibrium bid and ask prices, and show that in this case too, in equilibrium the informed speculator will buy upon receiving positive information and sell otherwise. b. Compute the bid-ask spread and find out whether in this case it is increasing in the frequency of informed trading π. c. Compare this bid-ask spread with that given by equation (10.1) under the informativeness condition (10.5). d. Finally, compute the midprice and see whether it depends on the frequency of informed trading π, explainingwhy.

3 10.8 Solutions Exercise : a. If G>Ithe project s expected net present value, 1 G 1 I,ispositive,so that the manager has incentive to invest even when he has no information about the quality of the project. It follows that, in the absence of a stock market, the manager does not invest only if he receives private information and if the information is negative. Viceversa, in the presence of a stock market, depending on whether the manager observes a transaction at the ask or at bid, he updates his estimate of the probability of the project being of quality H or L, exactlyasinsection10.4.1: Pr(H p = a) = Pr(L p = b) = 1+π, Pr(H p = b) = Pr(L p = a) = 1 π. Since the unconditional expected net present value of the project is positive, a fortiori the expected net present value conditional on observing a buy order is positive for every possible value of π. Infact: 1+π and π is always greater than I G I+G G 1 π I>0 if π I G I + G given that G>I. π I G I + G. On the contrary, when the manager observes a sell order, the expected net present value of the project conditional on observing a sell order becomes: 1 π G 1+π I 3

4 which is negative if and only if π G I I + G. So, when stock market trading is informative enough, it deters inappropriate investment as we wanted to show. b. As we have done in Section , we compare the outcome of the investment decision with and without a stock market to compute the allocative value of the information coming from the stock market. Again we assume that the informativeness condition we have derived under point a) holds otherwise the stock market would clearly play no role in the investment decision. In the absence of a stock market, the manager will invest both when he receives private information about the project and if the information is positive, which happens with probability γ,andwhenhereceivesnoinformation,giventhattheex-ante expected net present value of the project is positive. Therefore, the ex ante value of the firm, if privately held, is V private = V + γ G +(1 γ) G I {z } NPV On the other hand, if the firm is listed on the stock market, the manager does not invest iin case he observes a transaction at the bid price, provided that the informativeness condition holds. As before, he receives positive private information with probability γ/, in which case again he invests. If he does not receive private information, which occurs with probability 1 γ, he invests only when he observes abuyorderwhichoccurswithprobability1/. Therefore, the ex ante value of the firm when publicly listed is V public = V + γ G + 1 γ % 1+π G 1 π & I 4

5 which is the same we have derived in Section The di erence between V public and V private represents the increase in firm s ex-ante value which comes from the presence of the stock market: V public V private = V + γ G + 1 γ! 1+π G 1 π " # I V + γ $ G I G +(1 γ) = = 1 γ # π G + I G I $. Moreover, we can also notice that the total contribution of the stock market to the investment decision is composed of two e ects: the gain from not investing when investment is inappropriate and the loss from underinvesting when noise trader sells stock. As we can easily see, the net e ect is positive whenever the informativeness condition holds. In fact: π G + I G I 0, π G I I + G. Exercise 4: Theequilibriumaskpriceis: a = V +(1 π) γ G + πγg and the equilibrium bid price is: b = V +(1 π) γ G, so that the bid-ask spread is S = πγg, which is increasing in π. Sincenowweareassumingthatcondition(10.5)doesnot hold, this bid-ask spread will be smaller than [(I G)/(I + G)] γg. In contrast, under the informativeness condition (10.5), the spread given by equation (10.1) 5

6 exceeds [(I G)/(I + G)] γg. Hence, when π is low enough that stock prices are disregarded as a guide for investment, the market is more liquid than when π is large enough that they are used to guide investment. The mid price is m = V + γg, so that it doe not depend on π, becauseinthiscaseinformedtradingdoesnot contribute to the allocation of investment and therefore does not a ect the firm s value. 6