Empirical finance.

1. Basic explanation of the central concept of modern asset pricing theory,

The basic background

 the law of one price: two assets with identical payoffs in every state must have the same price    this imply   (1) where  and . If we multiply and divide equation (1) by the objective probability of each state,   where M(s) is the stochastic discount factor or SDF in state s and is The asset price as the expected product of the asset’s payoff and the SDF also called The Fundamental Equation of asset pricing. Now let Consider a riskless asset with payoff X(s) = 1 in every state. The price   So the riskless interest rate   So the SDF must be fairly close to one and the conditional mean of the SDF is the reciprocal of the gross riskless interest rate [pic 1][pic 2][pic 3][pic 4][pic 5][pic 6][pic 7][pic 8]

Now let Consider a price-taking investor who chooses initial consumption   and consumption in each future state  to maximize time-separable utility of consumption and that  For each investor the subjective state probabilities coincide with the objective probabilities  and investor have rational expectations  then the investors maximization problem is  [pic 9][pic 10][pic 11][pic 12]

Under constraint  , the FO of this problem is [pic 13][pic 14]

For each state Witch imply that  [pic 15]

In words, the SDF is the discounted ratio of marginal utility tomorrow to marginal utility today. This representation of the SDF is the starting point for the large literature on equilibrium asset pricing, which seeks to relate asset prices to the arguments of consumers

The point above assumes that all investors have rational expectations and thus assign the same probabilities to the different states of the world. In general, we must also allow for diffirences in the utility function across investors, adding a subscript to marginal utility as well. A similar observation applies to the SDF, the ratio of state price to objective probability:  Volatility of the SDF across states may correspond either to volatile deviations of investor subjective probabilities from objective probabilities, or to volatile marginal utility across states. The usual assumption that investors have homogeneous beliefs rules out the first of these possibilities, while the behavioral finance literature embraces it.[pic 16][pic 17][pic 18]

        By using this equation   witch state that the price of any asset is its expected payoff, discounted at the riskless interest rate, plus a correction for the conditional covariance of the payoff with the SDF and  and some other mathematics rearrangement we get the equation of risk premium:    , the risk premium on any asset is the gross riskless interest rate times the covariance of the asset’s excess return with the SDF.[pic 19][pic 20][pic 21]

By using the fact that the correlation between the SDF and any excess return must be greater than minus one, we can calculate a lower bound of this sort from a single risky asset return and the return on a riskless asset. The standard deviation of the SDF, divided by its mean  must be at least as great as the mean of the risky asset.s excess return divided by its standard deviation, that is, the Sharpe

ratio of the risky asset :  .[pic 22]

One last point to take into account is that  One can take unconditional expectations of the  conditional asset pricing equation, to obtain an unconditional asset pricing formula:  . The ability to take unconditional expectations without altering the form of the asset pricing formula is a strength of the SDF approach to asset pricing. [pic 23]

market efficiency

When money is put into the stock market, the goal is to generate a return on the capital invested. Many investors try not only to make a profitable return, but also to outperform, or beat, the market.
FAMA define a Market efficiency as “A market in which prices always fully reflect Available information is called efficient”. But a problem arises with this definition because the definitional statement is so general that it has no empirically testable implication.

FAMA in the paper “Fama survey” states the joint hypothesis problem witch say that market efficiency implies a zero conditional mean for asset returns measured relative to the riskless interest rate and the equilibrium compensation for risk. This is the rub in tests of market efficiency. Any test is simultaneously a test of efficiency and of assumptions about the characteristics of market equilibrium.

To tackle the joint hypothesis problem, it was common in the earliest finance literature to assume that expected returns on assets were constant over time, although they could vary across assets.

The modern event study

Event studies examine the reaction of asset prices to public news events like merger, acquisition etc . Market efficiency implies that there should be no tendency for systematically positive or negative returns after news events, except to the extent that the events alter assets compensated risk exposures. If, as traditionally assumed, events have no e¤ect on such risk exposures, the price reaction at the time of the news event is an estimate of the change in fundamental value of the asset  implied by the news release.The enormous empirical literature using event studies finds that many events have immediate impacts with no subsequent tendency for drift in prices. Some events, however, do seem to be followed by drifts in prices in the same direction.

One of the most common uses of event studies is to measure the value consequences of various events. If the market correctly incorporates the new information, the value effects of a particular event, such as a corporate decision, a macroeconomic announcement or a regulatory change, can be measured by averaging the abnormal returns across a large number of such events for different assets and time periods. This method has become commonly used to test predictions from various economic theories, in particular in corporate finance

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