In order to come up with this behavioral asset pricing equation. We start with the individual maximization problem. In words: We are considering an expected utility maximizer. This Maximization Problem is explained by the fact that ππ is the probability of state “k”. πΏπ(ππ ) is defined by the natural logarithm of consumption in state “k”. Which is represented by the concave utility function of the expected utility. This type of utility function states that the individual is overall risk averse. To sum up this up, this individual is maximizing consumption through different possible states of nature that is happing. πΆ0 is referring to the amount of savings that an individual will put into a bank account. Where πΆπ is interpreted as the amount of shares of Arrow-Debreau assets that will pay something in state 1,2,3 etc. This maximization issue is subject to a budget constraint. Where “W” is referring to the given amount of wealth you are having. The left hand side of the equation is the cost of your portfolio. Where π£π is the price of an Arrow-Debreau asset and πΆπ is how many shares of the asset that you have. By solving π£π you’ll get the optimal consumption equation. Optimal consumption: What this says is that you are going to consume ππ , which in this case is the consumption that is equal to the amount of shares you have in a particular Arrow-Debreau assets or savings. ππ is equal to half of the wealth times a ratio of the probability of a particular state happening ,and the price of the asset that pays something in that particular state. It is more intuitive if you think about this as a demand function. Notice that the price of π£π is in the denominator, which tells that the higher price, the lower you demand for that asset. ππ is the probability of that particular state happening, and you will demand more of that asset if you think that particular state is more likely. π£π is the price of an Arrow Debreau asset, and there is two versions of this asset pricing equation. Type 1: The more general one. This equation states, that the price you pay for an asset that will pay something in state k, is directly proportional to how likely you think that state is going to happen, and the marginal utility you expect to have in that state. When the Marginal Utility is high, you are having a low consumption, which results in you wanting to pay more for that asset, because of the utility function being concave. The behavioral component are saying that you are willing to pay more for an asset if it helps me to hedge against a state where you are really poor. And that you are willing to pay a lot for an asset that pays something in a state that I think is likely to happen. I.e. If I expect a market crash is coming, and I believe it is very likely, I will be willing to pay a lot for an asset that hedges me against that risk. Or, if I anticipate that I am going to be very poor in the future, I will pay a lot for an asset that pays me something when that happens. Type 2: ππ is defined as the ratio of ππ /π0. Where ππ is capturing how much your consumption will grow in a particular state of the world. It is measuring how poor or rich you are. If you are rich in a particular state you don’t hedging, because you are already rich in that state. When ππ are high you are willing to pay less for an asset, because you don’t need the money. We have been studying a particular deviation from rationality, which is deviations from Bayesian Updating, which was caused by Representativeness heuristic. This heuristic is a way to update probabilities. An example was that Linda had a profile that looked like some sort of an activist. You are thought that it is more likely that she is a bank teller and active in the feminist movement, because it is representative from what her profile is. From a rational point of view, the second statement is more likely. But Representativeness heuristic makes us believe that she is a bank teller and active in the feminist movement. When it comes to asset pricing. How does representativeness heuristic affect prices? The states are defined by whether the economy is in a strong regime or weak regime. And whether we have observed the market has gone up or down, in the past 6 months for example. One state of nature is that the economy is in a strong regime, and I have observed the stock market going up 6 times in the last 6 months. Here, the number of ups in the last 6 months in the stock market is a signal that we use to infer whether the economy is in a strong or weak regime. Why the probabilities is written in red: It is to emphasize what will happen with the likelihood assessment of these states under representativeness heuristic compared to the Bayesian Updating. Let’s focus on the 4 written in red from 0 - 3 number of ups in the strong regime: These probabilities compared to the same number of ups and regime in the Bayesian Updating, the representativeness probabilites are lower. Why is the reason? When people look at the signal (the stock market going up 0,1,2,3 times), the representativeness heuristic makes them think, this doesn’t seem representative of a strong economy (de føler ikke det repræsenterer en stærk økonomi). What happens is that they will put less probability weight on these states as compared to a Bayesian Updater, and will therefore put more weight on state 4,5,6 (Why? Because when they observe the stock market going up 4,5,6 times this seems representative of a strong regime (economy). When the individual observe a strong signal that the stock market is going 4,5,6 times, they say NO NO NO WAIT A MINUTE, this does not represent a weak regime. These are in red because when you compare it to what a Bayesian Updater will do, the likelihood that those states will happen is lower, and therefore it will put more weight on (number of ups 0,1,2,3). The likelihood that those states will happen is lower. Therefore it will put more weight on these states, that the individual think is more representative of a weak regime. Why? That person will observe the stock market going up 0,1,2,3 times. What we can see is that representativeness heuristic will bias the probabilites depending whether the signal is a weak regime or strong regime. In our asset pricing equation, beliefs affects directly the price of Arrow-Debreau asset. If I think something is more likely ππ , then people would like to pay more for an Arrow-Debreau asset that pays something in that particular state. These to matrixes are displaying the price of Arrow-Debreau assets, since the amount of money people are willing to pay for AD-assets is directly proportional to the likelihood of a given state of nature happening. The lower probabilities will result in a lower asset price that pays something in those states. What happens is that people, in respect to a Bayesian Update, overpay for assets that pays something in those states. This is people overpricing assets when they observe good signals, and underprice assets when people observe bad signals. Asset pricing implications of different assumptions of deciosion making models: In prospect theory people care about the marginal utility of changes in wealth. Where r is a reference level of wealth. Prospect Theory: Gain: If I in any given state of nature experience a small gain in change of wealth, I price assets higher. (Det er min egen forklaring).
