TOC
1. Model
1.1 Setting
Motivation
CAPM model and AD model are both static models. CAPM model is one-period model. AD model is multi-period but static: all decisions, including security trades, take place at date zero. Dynamic model requires that new information is available, decisions are made sequentially and today’s decisions influence future.
Assumptions
Assumption: Agents
There are identical infinitely lived consumers:
- infinite lifetimes: No liquidation action and Altruistic.
- identical agents: the representative agent’s utility function is a weighted average of the utilities of the various agents in the economy.
Assumption: Output
There is one, perfectly divisible share, which represents market portfolio. Output is exogenous and random, but in a stationary fashion. In another word, the transition matrix is fixed
Assumption: Economy
This is a rational expectations economy:
- Agent’s expectations is on average correct
- This economy has been functioning for a long enough time
- The agents know output distribution
- Accumulating such knowledge is in agent’s own interest
Agent’s Utility
Agents act to maximize the expected present value of discounted utility of consumption
- is the consumption
- is the period utility function with and
Budget Constraints
Financial market:
- Supply of the security is 1
- is the beginning of period shares: Holding entitles the corresponding fraction of outputs
Consumption Market
- is the economy output
- is the price of security in terms of consumption (the price of consumption is 1)
The optimal problem
FOC:
thus
Left side of the equation: the utility loss associated with the purchase of an additional unit of the security.
Right side of the equation: the expected discounted gain in utility associated with buying the extra unit of security.
1.2 Equilibrium
Equilibrium 1: the agents obtain their optimal given prices
Equilibrium 2: the representative agent owns the entire security
Equilibrium 3: Ownership of the entire security entitles the agent to all the economy’s output
Example
Suppose
Transition matrix
FOC:
In equilibrium, as , thus
Denote the security price when output is as .
thus
Equilibrium Price
1.3 Bubble
From above , substitute into , we have
From the tower rule ,
The last term is called bubble term. A tabular example is shown as below
T=0, Market is in equilibrium. T=1,2, there is a bubble. T=3, the bubble bursts.
The reason for bubbles: Keynes (1936): beauty contest — “Average opinion expects average opinion to be”.
Explanation: Rational Model
Price of a risky asset is given by
where is the fundamental value and is the bubble component. The expected growth of a rational bubble equals the long run average return
Explanation: Disagreement-Based Model
Investors disagree about the value of the stock. With a short-sale constraint, disagreement leads to overpricing: stocks are sold to more optimistic investors.
Equilibrium Price
After eliminating bubble term, stock price is the sum of all expected discounted future dividends.
Special case: discounted cash flow model: define
2. CCAPM
2.1 Stochastic Discount Factor
Define stochastic discount factor (SDF): , thus the fundamental pricing equation can be written as
To price the returns
that is, the price of return is 1.
For risk-free rate
SDF and Risk Premium
Assume asset has a positive correlation with consumption
Actually, many asset payoffs when good time occurs which means in equilibrium, their expected returns should above . However, assets which payoff when bad time occurs may be more desirable in real world.
An example of quadratic utility function is shown as below:
thus
2.2 CCAPM
Denote the portfolio most highly correlated with consumption by index
thus
therefore
If it is possible to construct a portfolio such as its beta with is one
3. Application
3.1 Risk-free rate
Assume , then , thus
In the deterministic case
where
- : people’s patience
- : consumption growth
- : risk-aversion coefficient
since
denote , thus
For normal variable , , thus
therefore,
the new term capture precautionary saving.
3.2 Equity Premium Puzzle
From CCAPM
As , the sharpe ratio of the frontier
Assume CRRA utility , for a frontier portfolio, its Sharpe ratio
Denote , then , where .
Furthermore, assume is normally distributed, then
thus
Investors are more reluctant to take on the extra risk of holding risky assets if
- The economy is risker (if the consumption is more volatile)
- Or if investor are more risk averse
In U.S. market, , thus the historical Sharpe ratio is
The volatility of aggregate consumption has an average rate of 1% annually
However, the empirical evidence: . Mehra and Prescott: the CCAPM is completely unable, once reasonable parameter values are inserted in the model, to replicate such a high observed equity premium.
Explanations
- People are a lot more risk averse than we might have thought
- The stock returns were largely good luck rather than an equilibrium compensation for risk
- Something is deeply wrong with the model, including the utility function and use of aggregate consumption data
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