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# Rational Bubble Theory Review

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Basing on the empirical researches, bubble could be divided into many categories. However, in this paper, for better illustrate the bubbles in stock market, we regard that bubble could be distinguish by irrational bubble and rational bubble. In rational bubble theory, the real price of assets is consist of intrinsic value and rational bubble. In irrational bubble theory, the stock market is not always efficient, the information is not complete, the expectation among investors are different. Thus, the behaviour of investors will be irrational.

## Rational bubble

According to the research of Blanchard and Watson (1982), the solution of rational bubble could be found by building rational expectation equation and solving difference equation, which is under the premise of arbitrage equilibrium. After the further study, Brunnermeier and Pedersen (2005) confirmed that there is no negative rational bubble. Because when we assume that the stock market has negative rational bubble at beginning, the absolute value of negative rational bubble would rise to infinity with the increase of time, which means it will exceed its intrinsic value and makes share price negative at some point in the future. Therefore, this is not realistic.

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Basing on the empirical researches, the most widely recognized solution set of rational bubble today was derived by Granger and Swanson (1997) from normal martingale process model. In this paper, the existence of the bubble could be summarized as the equation below:

P_t=(E_t [P_(t+1)+D_(t+1)])/(1+r_f ) (1a)

Where

P_t is the stock price at time t

r_f means risk-free rate

D_(t+1) is dividend on stocks from period t-1 to period t

E_t denotes expectation

Basing on the equation 1a, P_t could be represented as: Therefore, we get the equation 3a:

P_t=P_t^f+B_t (3a)

Generally, the fundamental value of stock could be expressed as the discounted sum of expected dividends in the future (Summer 1986). Hence, bubble B_t represented the balance between stock price and fundamental value. Efficient market theory holds that stock price will not deviate from their fundamental value for a long time, which means the temporary deviation of stock price from fundamental value will return as the arbitrage opportunities leads to change in behaviour of investors. From equation 3a, if stock price equal to its fundamental value (P_t=P_t^f), the bubble not exist (B_t = lim┬(j→∞)⁡〖(E_t P_(t+j))/〖(1+r_f)〗^j 〗=0). In the contrary, B_t=P_t-P_t^f≠0, the bubble exist. Thus, the bubble should satisfied:

E_t B_(t+1)=(1+r_f ) B_t (4a)

To prove the equation 4a, substitute it into equation 1a. The equation 5a as shown: it is simply known that P_t^f=(E_t P_(t+1)^f)/(1+r_f )+(E_t D_(t+1))/(1+r_f ) . Furthermore, we get the equation 6a: P_t=P_t^f+(E_t B_(t+1))/(1+r_f ) (6a)

Combine equation 3 and 6, P_t=P_t^f+B_t=P_t^f+(E_t B_(t+1))/(1+r_f ) ⟺E_t B_(t+1)=(1+r_f ) B_t . As a result, E_t B_(t+1)=(1+r_f ) B_t is true and rational bubble conforms to the explosive increase in expectation. Hence, rational investors still prefer the high valuation stocks as well as expect to sell them with a high price to profit, so stock prices continue to deviate from their fundamental value. In sum up, the rational bubble not only break the efficient market theory, but also support the econometric test of stock price bubbles.

## Intrinsic bubble

Froot and Obstfeld (1991) proposed the intrinsic bubble theory which is different from the other rational bubbles. Intrinsic bubble exists in a non-linear way and influenced by fundamentals merely. Basing on a simply condition, when current yield is a constant, the real stock price time series is related to the real dividend time series. Assuming that P_t is the real price per share from time t, D_t is the actual dividend per share paid during time period t, r denotes constant real interest rate.

P_t=(E_t (D_t+P_(t+1)))/e^r 1d

E_t denotes the market expectation from time t.

P_t^PV presents the present value of P_t,

P_t^PV=∑_(s=t)^∞▒〖e_t^(-r(s-t+1) ) E_t (D_t ) 2d〗

Equation 2d is a particular solution of equation 1d, which is equal to the discount value of stock prices to expected future dividends. Simultaneously, suppose that P_t^PV always exist, so continuously compounded rate of expected dividends lower than r. d_(t+1)=μ+d_t+δ_(t+1) 3d

Where,

μ is trend growth in dividends

d_t is log value of dividends in time t

δ_(t+1) is a random variable

In equation 3d, conditional mean is 0 and variance is σ^2.

Combining the equation 2d and 3d, assume the dividend in time t is known. We could get equation 4d. P_t^PV=〖kD〗_t 4d

The parameter k=(e^r-e^(μ+ σ^2/2) )^(-1) , if equation 2d convergence, r > μ+ σ^2/2 . Assuming the bubble function is

B(D_t )=cD_t^λ 5d

Where

C is constant

λ is positive root of equation 6d

λ^2 σ^2/2+λμ-r=0 6d

Above all, plus P_t^PV and B(D_t ) the bubble model is shown as equation 7d, which proof that the intrinsic bubble is exist.

P(D_t )=P_t^PV+B(D_t ) 7d

## Evans-type bubble

Periodically collapsing bubble is one of the most representative rational bubble (Bohl, 2003). According to the research of Evans (1991), the classic bubble theory suppose that expansion coefficient is constant. Conversely, in the periodically collapsing bubble, expansion coefficient is related to the actual size of bubble, simultaneously, the probability of bubble burst will increase with the rising size of bubble. As the insufficient that linear model is unable to identify the whole process of periodic bubble burst in test, Evans (1991) use nonlinear model to observe the bubble formation process to avoid the mistake. Furthermore, nonlinear model proved that the process of periodically collapsing bubble is repeatedly random expansion and contraction, which means it is close to real bubble.

Above all, Evans bubble is regarded as the standard model to test bubble.

B_(t+1)={((1+R) B_t u_(t+1) B_t≤α@[δ+π^(-1) (1+R) θ_(t+1) (B_t-(1+R)^(-1) δ)] u_(t+1) B_t>α) 1b┤

Where,

δ, R, α are constants, in addition, they satisfied that 0 < δ < (1+R) α.

u_t denotes independent identically distributed random variable which obey u_t≥0, E_t

u_(t+1)=1.

θ_t means independent random variables which is Bernoulli distribution, the probability that θ_t equal to 1 is π, equal to 0 is 1- π.

Link to the equation 4a, It could be determine that Evans’s periodically collapsing bubble model conforms to the conditions of rational bubble in the expectation sense: E_t B_(t+1)=(1+r_f ) B_t .

When B_t≤α , bubble increases with the average growth rate 1+R, on the contrary, if B_t> α , bubble will go up rapidly with a higher growth rate ((1+R))/(π-1)>R with the burst probability of 1-π. In addition, α is a constant bigger than 0, so bubble will not burst to 0 when B_t> α, analogously, bubble rises in growth rate R once bubble burst below α when B_t≤α, these, subject to the periodically movement called periodically collapsing bubble.

## Irrational bubble

Combining the researches of Minsky (1982) and Kindleberger (1987), irrational bubble is defined as irrational rapid upward fluctuation in price which is caused by psychological factors. Therefore, one of the important reasons that irrational bubble exists is the irrational decision-making and trading behaviors of investors which lead to a deviation between stock price and intrinsic basic value (Monies and Pouget, 2009). Specifically, irrational bubble explain the mechanism and reason of bubble formation from the perspective of behaviour finance and greater fool theory. In this paper, irrational bubble will be analysis in two ways: the behaviour and psychological of investors.

Herd behaviour (add sth of paper noise trader and herding) and DSSW model (behaviour of investors)

According to Brunnermeier (2001), the asymmetric information influence the behaviour of investors which may cause volatility in stock market. Meanwhile, under efficient market theory, which assume that all the information is available, the market value is equal to the intrinsic value, otherwise, there would be a gap between price and fundamental value. Because the information from various sources affect the decisions of investors on the purchase or sale, this, further influence the stock market. In this paper, we mainly elucidate how Herd behaviour and noise trader stimulate bubbles generation in the stock market.

Moral hazard was firstly proposed by the economist (who) in 1980, which defined as people engaged in economic activities maximize their own utilities while costing benefits of others. As an example, the listed companies and securities investment institutions in the stock market may be driven by benefits to hide information or fabricate false information to mislead investors. Once occurs the moral hazard, some institutions manipulate the stock market, drive up the stock price and corporate with listed companies to create false information, which leads to inflated stock prices. As the behaviours departure from intrinsic value, undermines the effectiveness of the market, therefore, the bubble happened.

Generally, the investment decision-making of investors depend on the information they hold. However, the investment decision-making will be influenced which leads to adverse selection, once the investors could not get information in time, completely and really. Due to the asymmetric information, the economic resource could not be allocated according to the principle of efficiency first, so-called adverse selection. With the exposure of some listed company fraud cases, the investors beginning to suspect the authenticity of the reports of listed companies. Moreover, the asymmetric information like these not only cause adverse selection, but also deviate from the intrinsic value of stock, which lead to price distortion and aggravate the falsity of the stock market. Consequently, the bubble is stimulated constantly.

As a significant theory to explain the mechanism of bubble formation, “Herd Effect” is depended on imperfect rational expectation and speculative behaviour. There always some naive participants in the stock markets who have no exact information source, rely on imitating the behaviour of other people to make their own investment decision. Accordingly, under the environment of asymmetric information, the investors have no exact information, obtaining the information by the purchase and sale acts of other prople, this is, the burst of “Herd Effect” and followed with the bubble.

DSSW model (behaviour and psychological)

According to De Long, Shleifer, Summers and Waldmann (1990), who analysis why asset bubbles exist through research the behavior of investors, find that the model of noise trader risk could explain some abnormal phenomenon in financial. Furthermore, the DSSW model explains the persistent deviation of stock price from fundamental value in the perspective of micro-psychology and behavior. The irrational expectation of noise traders increase the risk to asset price, while the risk aversion and short-term of arbitrageurs limit their ability to correct mispricing. In addition, noise traders may overreact to information or adopt positive feedback trading strategies, which will lead to drastic fluctuations in asset prices, causing asset prices to continuously deviate from the fundamental value, thus leading to bubbles.

Fads model and DSV model (psychological of investors)

Fads was first suggested by Shiller (1984) which gave the new thought of the reason why bubble burst in the perspective of fads. Shiller holds the points that the reason of the bubble formation and burst could be explain with behaviour finance and investor psychology . Basing on this opinion, stock price is easily influenced by fads and social dynamics, because the behaviour of conformity among the investors will lead to fads in financial market which generates the bubble. In the equation 1c, it seems that real price equal to the present discounted value of expected future dividends, which means the price could forecast the dividends that stock will pay in the future. However, return could not be forecasted, in the model there only showed that return is not very forecasted.

P_t=∑_(k=0)^∞▒(E_t D_(t+k))/〖(1+ϑ)〗^(k+1) 1c

Where,

E_t is mathematical expectation at time t.

ϑ denotes a constant, at the same time ϑ=E_t R_t

R_t means stock real rate of return between time t and t+1, which is equal to (P_(t+1)-P_t+D_t)/P_t .

P_t is real price

D_t is real dividend

Therefore, equation 2c is structured to replace the old one and add fads in it. Comparing equation 1c and 2c, when some information is disclosed, there must be influence in returns. P_t=∑_(k=0)^∞▒(E_t D_(t+k)+δE_t Y_(t+k))/〖(1+μ+δ)〗^(k+1) 2c

Where,

μ+δ is discount rate

Y_(t+k) is the total value of stock demanded per share by the investors.

### DSV model

In the model of investors sentiment which suggested by Baeberis, Shleifier and Vishny (1998), there will be systematic errors when investors use disclosed information to predict the future cash flow, which leads to the deviation of stock price from fundamental value, thus the bubble happen. There are two kinds of error in the model, conservatism and representativeness.

Conservatism means that investors pay more attention to the old information instead of the new information, representativeness express that investors estimate the overall characteristics of the company from small sample data. Whereas, the stock are prone to misjudgement by investors when a company transitions from loss to profit, which affects the stock price. When investors make a profit from stocks, they believe that the future profit of the company will also increase. However, this may not be conform to the actual development of the company, leading to the occurrence of bubbles.

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