chapter 8 and 10 powerpoints Flashcards
two drawbacks of the Markowitz Procedure
Requires a huge number of estimates to fill the covariance matrix
Provides no guideline to the forecasting of the security risk premium to construct the efficient frontier of risky asset
advantages of index models
simplify the estimation of the covariance matrix
Enhance the analysis of security risk premiums
Decompose risk into relevant sources of security risk called factors (like systematic risk) versus firm diversifiable risk
Are as accurate as the Markowitz algorithm
Simplify estimates to those common forces that affect most firms
arbitrage opportunity
Whenever a wide set of securities is mispriced and investors can exploit this opportunity
when has arbitrage happened
Whenever a wide set of securities is mispriced and investors earned a risk-free economic profit
arbitrage
involves the simultaneous purchase and sale of equivalent securities in order to profit from discrepancies in their price relationship
The basic principle of capital market theory
In equilibrium securities are properly priced
why does the capital market theory not support arbitrage
If securities are mispriced then strong pressure on security prices will restore the equilibrium (proper equilibrium prices)
As a result, in equilibrium, capital markets satisfy the no arbitrage condition
Arbitrage Pricing Theory or APT
when we want to capture countless economy wide affecting risk factors in a model that explains securitiesβ returns
we obtain a multifactor version of the security market line in which each factor is a separate source of risk with its own risk premium
A single factor APT assumes
securities are affected by a single common risk factor
what do index models assume
one risk factor, the market factor, affects all security prices
how can we price the holding period return on a security using the single factor APT method?
the single factor model
ri = ERi + π½π Β· mi + ei
ri: represents the holding period return that can be earned on the security
ERi: the expected return on the security as of the beginning of the holding period
mi: the unanticipated return achieved on the security caused by unanticipated movements (shocks) in the risk factor
ei: the unanticipated return achieved on the security caused by unanticipated movements (shocks) in the firm itself
the single factor model: the mi and ei assumptions
The expected value of both mi and ei is zero because we cannot expect unanticipated events
The correlation between mi and ei is assumed to be zero: shocks to the risk factor are uncorrelated with firm specific shocks
If our risk factor is a good proxy for the whole economy (or capital market) then our risk factor is called the market factor
in the single factor model, what is the risk when investing in security i
Shocks in the market (mi)
Shocks in the firm itself (ei)
Ο^2 for i in the single factor model
Ο^2 (for i) = π½^2π Β· Ο^2m + Ο^2ei
the covariance between two securities using the single factor model
cov (ri, rj) = π½^π Β· π½^j Β· Ο^2m
the conclusion for the returns of firms with similar market exposure (betas) single factor model framework
should lead to the same expected return
not same realized return
the regression equation
π π(π‘) = πΌπ + π½ππ π(π‘) + ππ(π‘)
Ri(t): is called the dependent variable
RM(t): is called the independent variable
πΌ and π are called the coefficient of the regression
ei(t): is called the residual of the regression
π π(π‘) with a hat above is the estimated dependent variable
the single index model on a portfolio
π p(π‘) = πΌp + π½pπ π(π‘) + πp(π‘)
π½p = (Eπ½t) /n
πΌp = (EπΌt) /n
πp = (Eπt) /n
π^2π = π½^2π Β· π^2π + π^2πp
π^2πp = (π^2π_) / n
As n gets large, π^2πp becomes negligible
how do obtain the estimated dependent variable (π π(π‘) with a hat above)
by evaluating our regression model for every observation of the independent variable
Security Characteristic Line
π π(π‘) = πΌπ + π½ππ π(π‘) + ππ(π‘)
how do we estimate the Security Characteristic Line
- we need to calculate the excess return of both our security βiβ and the market βMβ
- we regress (on EXCEL for this course), the excess return of our security βiβ on the excess market return βMβ
β> The intercept of this line is πΌπ
β> The slope of this line (coefficient of RM(t) ) is our π½π
- Check whether the intercept and the slope are significant
- Check the explanatory power of our SCL for this particular security
We need to check whether the intercept and the slope are statistically significant
how do we check whether the intercept and the slope are statistically significant in the
Security Characteristic Line
when we are saying that the slope and the intercept are statistically significant, it means that they are different from zero
we will rely on the t-statistics to measure the significance of our estimated alpha and beta
If the estimated t-statistics is greater than 1.96 then we can say that we are 95% confident that our estimated coefficient is different from zero
we are satisfied with a 95% confidence level
How to obtain the t-statistics for our estimated slope (beta) and intercept
(alpha):
π‘π½^ = (π½^ - 0) / Standard error π½^
The standard error measures the level of imprecision in estimating our coefficient
(alpha, or beta
the the higher the standard error relative to the estimated coefficient, the more the likelihood that our result would be less than 1.96, the less we are confident that our estimated coefficient is statistically different from zero
R-square
measures how much variation in the dependent variable (the stock in our case) is explained by independent variable (the market)
measured as the ratio of the total variation in the estimated dependent variable to the total variation in the actual observed dependent variable
Adjusted-R-square
similar to R-Square but it adjusts R-Square for the fact that we are conducting our test on a subset of the data (sample) and not the whole entire data available
considered more reliable than R-Square in small sample regression estimation
What are the crucial assumptions needed to build a M-V frontier relying on a single index model?
We have properly estimated the betas of our securities
We have properly estimated the next period market premium (RM βRF)
Alpha Transpose
the process of building an investment strategy that has an expected return equal to alpha (like πΌP for having the πΌ of portfolio P) without being exposed to the market risk (systematic market risk)
The process of searching for alphas is one major role of hedge funds
we long portfolio P with a positive alpha, and short portfolio Q with no alpha (properly priced)
β> πΈ(ππΆ) = πΌπ and ππΆ = πΌπ + πP
when using the Alpha Transpose, if f portfolio βPβ is well diversified, then why should eP be small?
what happens to portfolio Q
because it represents unsystematic risk
Portfolio βQβ should be well priced (alphaQ = 0) and having only market exposure, BetaQ
β> βQβ should also be well diversified
how to create portfolio βQβ+
Alpha Transpose
We can create portfolio βQβ with BetaQ by investing in an Index Fund and RF
If BetaP = 1.5 then we should form a portfolio Q with BetaQ
= 1.5
We can invest 150% of the money we have in a market index fund and borrow 50% of our money from RF
β> The resulting portfolio has a beta = 1.5
two factor index model
can be estimated using the same techniques used for the single factor index model
we rely on regression technique to evaluate our multifactor model
Single vs. Multi- Factor Model
In the single index model, we assume that there is one systematic risk factor that is affecting our economy
β> All risk factors in the economy can be aggregated in one risk-factor (the market factor)
β> we are assuming that all
our securities have the same exposure (Beta) to all the risk factors forming our market factor
In multifactor models, we managed to find more than one systematic risk factor that is affecting our stocksβ returns
β> In multifactor models, we give our securities the flexibility to have different
sensitivities (betas) to every risk factors
β> it may be justified to have negative factor risk premium
why may it be justified to have negative factor risk premium in the multi factor model?
if the exposure to this risk factor is considered a positive thing (good news) by investors and may accept a lower return for it
who developed the Arbitrage Pricing Theory (APT) model?
Stephen Ross
Propositions that lead to the derivation of APT models
Whenever an arbitrage opportunity exists, investors will invest a big amount of money in it (high volume) independent of their Risk Aversion
β> This implies that Arbitrage Opportunities, once found, will not last for long periods
There are sufficient securities to drive away idiosyncratic risk
Securities can be explained in a factor model
No market imperfections: no taxes, no transaction costs and no restrictions on short sales
whenever an Arbitrage Opportunity exists, investors, regardless of their risk aversion, will try to take infinite positions in them and this will quickly push prices up or down until this opportunity disappears
β> markets should satisfy
the no-arbitrage opportunities
The Law of One Price
states that if two assets are equivalent in all economically relevant respects, then they should have the same market price
what happens when the Law of One Price is not respected
an Arbitrage Opportunity exists and investors will buy the cheap asset and sell the expensive one until the law of one price is established
the difference between the critical properties between APT and CAPM/ICAPM
in APT, whenever an Arbitrage Opportunity exists, investors, regardless of their risk aversion, will try to take infinite positions in them and this will quickly push prices up or down until this opportunity disappears
β> markets should satisfy
the no-arbitrage opportunities
β> prices will adjust quickly to their proper prices.
in CAPM/ICAPM, whenever there is a mispricing in securities, then investors, each according to his/her risk aversion, will adjust his/her own portfolio in order to include more of this mispriced security
β> This will push prices to gradually adjust to their proper values
Uncertainty in asset returns has which two sources?
A common or systematic or macroeconomic factor
A firm-specific or idiosyncratic or microeconomic factor
β> In well diversified portfolios, the firm-specific factor is reduced to zero
which risk is the only one that requires a risk premium?
market risk
the APT and arbitrage opportunity implies that we bear zero risk, this basically means that firm specific risk is null even if we solely rely on on security only
does this mean that its better than diversification?
naaah booooooy
we cannot get rid of firm specific risk by APT strategy when have a single security
this means we will apply it to well diversified portfolios
β> If a portfolio is mispriced, then arbitrage opportunity exists and an arbitrage strategy can take place if economically profitable
β> If a one or very few securities are mispriced then the effect on well diversified portfolios is small, and an Arbitrage Opportunity will not be economically profitable
β> For this reason we conclude that in an APT world, we can face few securities that may violate the arbitrage pricing theory but we cannot find a lot of these mispriced security
APT vs. CAPM
- The APT stresses the fact that investors should require premium for non-diversifiable risk and nothing for diversifiable ones (that is why, we tolerate the existence of few securities that may violate the APT models)
The CAPM model requires that all securities follow the CAPM
- The APT yields an SML where the factors can be any type of risk factors that investors deem important
CAPM is derived based on the unobserved market factor and what possible other factors may affect our investment opportunity set (ICAPM)
- CAPM is based on the mean-variance efficiency
- CAPM specifies that the relevant factor is the market factor
β> ICAPM adds factors affecting our opportunity set to the market factor
APT does not specify what are the factors! It does not provide any economic justification for the chosen factors! Consequently it leaves the door open for the selection and choice of the factors (that may be deemed to affect securities returns)!
Fama and French three factor model
big ass formula
SMB (small minus big): the return of a portfolio of small stocks in excess of the return of a portfolio of large stocks
HML (high minus low): the return of a portfolio of stocks with a high book-to-market ratio in excess of the return of a portfolio of stocks with a low book-to-market ratio
