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## Wednesday, April 13, 2016

### Single and Multiple Factor Model

Factor Models
In portfolio management computation of expected return, risk and covariance for every security included in the portfolio is crucial process and it proves a bit difficult too. Factor models relatively make the process easier as security return is assumed to be in correlation with one factor(s) or other(s). Factor models captured macro economic factors that systematically influence prices of securities. Any aspect of a security’s return unexplained by the factor model is taken as security specific. Factor models are otherwise known as index models. Security return when assumed to be related to return on a market index, such model is called as market index model. Similar to return on market index, other factors to which security returns stand related can be modeled and used to estimate returns as securities. Similarly portfolio returns as related to identified factors can be found and used in portfolio management. And this will ease the problem of computing returns. One factor (say Market Return, Growth rate of gross Domestic Product or Inflation rate), two factors (any two of macro economic factors) and multi-factors models can be through of.
Single Factor Model
CAPM is base on the single factor model. According to this model, the asset price depends on a single factor, say gross national product or industrial productions or interest rates, money supply and so on. In general, a single factor model can be represented in the equation form as follows:
R = E + Bf + e
Where, E = Uncertain return on security
B = Security’s sensitivity to change in the factor
f = The actual return on the factor

e = error term
Thus, this model only state that the actual return on a security equals the expected return plus sensitivity times factor movement plus residual risk.

Multiple Factor Model
The Arbitrage pricing theory based model aims to do away with the limitations of one factor model (CAPM) that different stocks will have different sensitivities to different market factors which may be totally different from any other stock under observation. In layman terms, one can say that not all stocks can be assumed to react to single and same parameter always and hence the need to take multifactor and their sensitivities. The formula includes a variable for each factor, and then a factor beta for each factor, representing the security’s sensitivity to movements in that factor. A two-factor version of the arbitrage pricing theory would look like as:
r = E(r) + B1F1 + B2F2 + e
r = return on the security
E(r) = expected return on the security
F1 = the first factor
B1 = the security’s sensitivity to movements in the first factor
F2 = the second factor
B2 = the security’s sensitivity to movements in the second factor
e = the idiosyncratic component of the security’s return
As the formula shows, the expected return on the asset/stock is a form of liner regression taking into consideration many factors that can affect the price of the asset and the degree to which it can affect it i.e. the asset’s sensitivity to those factors.

If one is able to identify a single factor which singly affects the price, the CAPM model shall be sufficient. If there are more than one factor affecting the price of the asset/stock, one will have to work with a two factor model or a multi factor model depending on the number of factors that affect the stock price movement for the company.