Chapter 5 - Model of Asset Returns Flashcards
Define a multifactor model.
A multifactor model of security returns attempts to explain the observed historical return by an equation of the form:
R_i=a_i +b_i,1l_1 + b_i,2I2 +…_b_i,L*I_L + c_i
R_i - return on security i
a_i - constant part of the component of return unique to security i
c_i - random part of the component of return unique to security i
I_1…I_L - the changes in a set of L factors which explain the variation of R_i about the expected return a_i
b_i,k - sensitivity of security i to K
What are the L factors in the multifactor model?
L are the systematic factors that influence the returns on every security and the corresponding part of the total return.
What is a_i and c_i in the multifactor model?
specific return of security i with a_i being the constant component and c_i being the random component of return unique to security i.
What is E[Ci]?
0
What is cov[c_i,c_j]?
0
What is cov[c_i,l_k]?
0
What is the goal of the builders of a multifactor model?
The goal of the builders of such a model is to find a set of factors which explain as much as possible of the observed historical variation, without introducing too much ‘noise’ into predictions of future returns.
What is a Macroeconomic factor model?
They use observable economic time series as factors. They include the main macroeconomic variables such as interest rates, inflation, economic growth and exchange rates.
Once the factors have been determined, a time series regression is performed to determine the sensitivities of these factors.
What is a Fundamental factor model?
Closely related to Macroeconomic models, but the factors will be company specifics such as Price Earning ratios, liquidity ratios and gearing measurements. The model is the constructed by regression.
What is a Statistical factor model?
the factors are not specific items initially. The method uses principal components analysis and historical returns on stocks to decide upon the factors.
How would we determine how many factors to use in a model?
Start with relatively few, perform the regression and measure the residual (unexplained) variance. An extra factor is then added and the regression repeated. The whole process is repeated until the addition of an extra factor causes no significant reduction in the residual variance. However, too many factors may introduce the issue of multicollinearity
What is Principle Component Analysis?
Principal components analysis is a technique used to investigate the relationship between a set of endogenous (variables that correlate with one another) variables, such as the factors determining the investment return in a multifactor model. It can then be used to:
- determine the relative significance of the various factors by analysing the variance-covariance matrix (between them) to determine which factors have the most influence upon the total variance of security returns
-combine groups of highly correlated factors into single factors or principal components that are much less highly correlated with each other – thereby reducing the number of factors in the model and improving the efficiency of the model.
What is the Single Index Model?
The single index model is a special case of the Multifactor Model that includes only one factor, normally the return on the investment market as a whole.
What is the Single Index Model formula?
R_i = a_i + B_i*R_M + e_i
R_i - return on security i
a_i (constant) - specific return of security i which is independent of the return on the market
B_i (constant) - the sensitivity of returns on security i that is directly related to movements in the market
R_M - return on market
e_i - is a random variable representing the component of Ri not related to the market.
Why should we only expect to be rewarded for an increase in systematic risk rather than specific risk?
Specific risk can be diversified away, where as systematic risk cannot.
What data points do we need to apply the single index model?
N a_i’s
N b_i’s
N V_e_i’s
E_M
and V_M
i.e 3N+2
State the formula for the Expected return, Variance and Covariance for the Single Index Model.
E_i= a_i + B_iEM
V_i=(B_i)^2 + Var(e_i) (recall e_i is a variable not a constant)
Cij = B_iB_j*V_M
What formula do we use for the mean, var and cov of a portfolio that can be modelled using the single index model?
R_p=sum(xi*R_i)
E[R_p] = sum(x_i*E(r_i))
V(R_p) = Var(sum(xiRi))
recall var(aX+bY+cZ) = a^2Var(x)+b^2Var(Y) + c^2Var(Z) + 2cov(x,y) + 2cov(x,z) + 2cov(x,z)
What are the optimal values for Alpha and Beta in the single index model?
They are the values where the sum of the error is 0 and sum of error squared is minimised.
