Empirical Asset Pricing Theory - Assignment Example

Empirical Asset Pricing Theory Introduction Empirical pricing theories relating to the assets are used as benchmarks for valuation of an asset. Stochastic discount factor method of pricing is itself a criterion through which assets are valued. This paper will outline how this method of valuing an asset fits to be a factor pricing model. This will entail discussing the assumptions relating to the form of stochastic discount model as well as how the factor method is related to the acquiring of equilibrium risk premium. In other words, the paper will look at the negative covariance of SDF and excess returns. The paper will also outline the Fama-French factors. This will include entailing how these factors work, and the motives behind choosing or selecting of models. Finally, the paper will discuss how the technique used by Pastor and Stambaugh differ from the ones used by Fama-French factors. Stochastic Discount Factor Pricing Model SDF as a Factor Pricing Model According to Fama and French (25 - 30) this model helps in the formulating of n econometric analysis that is used in the pricing of assets. The methods included this model include the capital asset pricing model that was proposed by Sharpe in 1964 and other as well as the consumption based inter-temporal capital asset pricing models (CCAPM). Stochastic discount factor (SDF) uses both of the approaches that are used in asset pricing. This includes the absolute and the relative pricing of asset. The absolute pricing of asset involve the pricing of an asset relative to the sources that expose it to the macroeconomic risks. The relative pricing of asset entails pricing assets according to how other assets are priced. The pricing equation that is used to estimate the stochastic discount factor is normally assumed. The limitations that are imposed on the behavior relating to the stochastic model are assumed to be standard. Based on the pricing equation assumptions the model, the price of n asset which is denoted as ‘t’ is calculated through discounting the value of the assets in the period of paying off. The equation for determining the price of the asset is: Pt=ET (Mt+sXt+s). The assets pay off is represented by Xt+s while the discounting factor is represented by Mt+s. the part denoted as ET represents the expectation given the information that is available at a given time t. The discounting factor represents the stochastic variable (Renault and Hansen 3-15). The assets that can be priced using this model include a stock that pays a dividend of DT+1. This stock should also have a resale value and a pay off period. A treasury bill is also applicable if only it pays only one unit of goods or a good being consumed. This equates the payoff to 1. A bond whose coupon payment is constant and yet can be sold is applicable for pricing using this model. This model can also price bank deposits that pay the risk free return rate and equate the pay off period to 1+ rft. Finally the call option whose price is Pt and gives the holder of the option the right of purchasing any stock at the price exercised (Renault and Hansen 12-21). Assumptions Relating to the Form of SDF In the development of the stochastic estimator, there are four assumptions that are taken into considerations. The first assumption is that the pricing equation 2 always holds. This equation is equivalent to the law of one price. The assumption here is that all the securities that have the same pay off should bear the same price. There are no choices of the preference. The second assumption states that the stochastic discounting factor labels Mt to be greater than zero. The same applies even to mimicking portfolio. The implication here is that no arbitrage opportunities exist. The third assumption states that the risk free rate exists. The risk free rate is measurable relative to sigma-algebra. The conditioning set that is also used in the computation of the conditioning moments generates this algebra. The existence of this rate allows for payoff space that is needed when one asset pays a specific amount despite the nature state (Fama and French 28-35). The fourth assumption states that RT is an N * 1 vector. This means that it stacks all the returns from every asset in the economy. There is hence a stationary covariance between the process of vector and the finite first as well as the second moments. This assumption regulates the degree of dependence based on the cross section and the time series of a given data (Fama and French 28-35). SDF and Equilibrium Risk Premium With reference to the stochastic discount factor, excess returns are expected to be perpendicular. The assets that are expected to pay high risk premiums relative to the excess returns are those that have a negative covariance with stochastic discount factor (Renault and Hansen 18-22). Etmt+1Et (rj, t+1 ? rt) = ?Covt (mt+1, rj, t+1) The equation above represents the relationship that exists between stochastic discount factor and the equilibrium risk premium. The conditional risk premium is normally zero while the unconditional one is equal returns, hence equilibrating it (Renault and Hansen 18-22). Fama-French Factors Description of the Factors In the capital asset pricing model, the only factor that was used to define the returns that a portfolio generated was beta. Fama and French, who were professors at that time, formulated two more variables to enable explanation of the cross section of returns relating to stocks (Fama and French 25 - 46). These include book to market variables and size. The variable relating to size comprises of both small minus big as well as high minus low. Small minus big variable measures the historical additional or extra returns those investors received as a result of investing in the stocks of different companies, taking into consideration the small market capitalization. These extra returns are normally referred to as size premium. The high minus low on the other hand measure value premiums paid to investors for putting their investments in companies that have high book-market values. The value is normally placed as the ratio with reference to the value of other public markets (Fama and French 427- 429). Technique Employed The factors are built on the combination of value weight portfolios (Fama and French 430- 444). Small minus big is the difference between the return on average of given small portfolios and the return on average of given big portfolios. The high minus low is the difference between the return on average of two value portfolios and two growth portfolios. The book to market ratio is normally a contrast of the value stocks and growth stocks. With the inclusion of the Fama and French model, the capital pricing model translates to the following formula: The coefficients bs and bv are calculated using the linear regressions. The values of the two coefficients can either be positive or negative. The technique the two used is that of a diversified portfolio. The explanation is given on 90% of the given diversified portfolio returns. A comparison was made with the average 70% that was formulated by CAPM. The conclusion made was based on the coefficient’s signs. The signs suggest that the portfolios with higher expected return are those that had small capitalization and value. They were also found to have the highest expected risk compared to those that had large cap and growth portfolios (Fama and French 450-462). Rationale for Choosing the Factors The reason why the two concluded on the two factors is because they believe that size and market to book ratio are what caused most and biggest variation in the stock returns. Their analysis was done in US. The two variables are believed to compensate the investors for holding riskier but less profitable stocks (Fama and French 45-46). Pastor and Stambaugh (JPE 2003) Studies Description of the Liquidity Factor The belief of these two is that liquidity is the pricing factor for assets. They suggest the liquidity of an asset generates a factor that is separate from the other pricing factors. The two used just an empirical implementation and not a formal model as compared to Fama and French. The technique they used was that some investors in different institutions cannot resist the constraints caused by solvency. This hence forces them to liquidate their positions in the market when it is not favorable (Pastor and Stambaugh 643-651). Technique Employed If the aggregate liquidity is termed low and the liquidation is an option, the investors will definitely prefer those assets whose returns are less affected by aggregate liquidity. This will hence help them avoid forced liquidation. The implication is that a premium is usually paid by those assets whose returns have a positive covariance with the aggregate liquidity. The investors are said to have earned well in the past years before the crisis hit that led liquidation in the down market (Pastor and Stambaugh 649-658). rei, d+1 = 0i + 0iri;d + risign (rei,d) vi,d + E,d+1 The formula above represents the daily stock level as proposed in this model. The sign volume denoted as risign is a major cause of pressure on a given day. This pressure is what reverts on the next given day. The stronger the reversion of the stock, the more illiquid this stock will be. The daily return is hence expected to be negative. Systematic variations in the stock liquidity are evidenced by the co movements of the daily returns. Portfolios here are formed on the basis of the liquidity factor betas. These betas are either approximated or previous betas used (Pastor and Stambaugh 669-721). Conclusion Based on the above study, conclusion drawn is that stochastic discount factor comprises of two methods of asset pricing, that is, CAPM and CCAPM. These are all related to the Fama and French three factor model since the only alteration they do is including two more variables to the model. The paper has analyzed all the three cases including Pastor and Stambaugh (JPE 2003) studies that look at liquidity as the pricing factor. Work cited Fama and French. “The Capital Asset Pricing Model: Theory and Evidence.” Journal of Economics Perspectives, 18. 3 (2004): 25 - 46. Print. Fama and French. “The Cross-Section of Expected Stock Returns”. The Journal 0f Finance, 47. 2 (1992): 427- 465. Print. Pastor and Stambaugh. “Liquidity Risk and Expected Stock Returns.” Journal of Political Economy, 111.3 (2003): 643. Print. Renault and Hansen. “Pricing Kernels and Stochastic Discount Factors.” University of Chicago: USA, 2009.PP 3. Read More