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What is Multi Factor Model?

Multi Index Model: 
It is a model which includes more than one factor which affects the risk and return of a portfolio. It helps to construct a portfolio. In single index model only one factor affect the securities return that is market return but in this model not only market factor consider but other factors are also consider which help to analyse the portfolio return more accurately. But it is difficult to decide the factors and their quantity. 

Advantages of Multi Index Model:
·         It considers more than one factor because it is assumed that not only one factor affects the securities return other factors also have some effects on the assets return.

Disadvantages of multi Index model:
·         It is based on historical data which does not help to forecast future return accurately.
·         It is difficult to decide the factors and its quantity included to form this model.
·         It is much more complicated than single index model.

Formula:
ERi = RF+ β1RP1 + β2 RP2 + β3 RP3 +ei
Where,
ERi = Expected return on security i
RP= risk premium
RF = risk free rate
ei = error term ; ei = 0

Example: Find out the expected return of given portfolio which consist 3 securities. The risk free rate of given portfolio is 3.8%.
Security
Beta
Beta
Risk premium
A
-0.25
1.87
1.5
B
1.65
0.75
2.3
C
0.47
2.31
3.98
The weight of the securities A, B and C in the portfolio is 40%, 20% and 40% respectively. The risk free premium of factor 1 is 2.45% and 4.20% of factor 2.

Solution: Firstly we calculate the portfolio beta which contains different stocks in a portfolio.
 βp = Ʃni=1wAβA + wBβB + wCβC
Ʃβ1= 0.40*(-0.25) + 0.20*1.65 + 0.40*0.47
= -0.1 + 0.33 + 0.188
= 0.42
Ʃβ2= 0.40*1.87+ 0.20*0.75 + 0.40*2.31
= 0.748 + 0.15 + 0.924
= 1.82
ERi = RF+ β1RP1 + β2 RP2 + β3 RP3
= 3.8 + 0.42*2.45 + 1.82*4.20
= 3.8 + 1.029 + 7.644
= 12.47%

Example: Considering the multi factor model arbitrage pricing theory with two factor model. A stock D has 2.9% risk free rate. The expected return is 8%. The risk premium of factor 1 is 2.6%. The beta 1 and 2 of stock D are 1.6 and 0.56 respectively. Find out the risk premium of factor 2.

Solution: Multi factor arbitrage pricing theory:
ERi = RF+ β1RP1 + β2 RP2
8= 2.9 + 1.6*2.6 + 0.56*RP2
8= 2.9 + 4.16 + 0.56RP2
8 – 7.06 = 0.56RP2
0.94/0.56 = RP2
RP2= 1.68%

Example: Consider the multi factor arbitrage pricing theory with 2 factors. There are two stocks i.e. stock A and stock B. The expected return of Stock A and B are 7% and 12% respectively. The risk free rate is 3.5%. The beta of stock A is 2.3 and -1.8 respectively. The beta of stock B is 1.4 and 0.24 respectively. Find out the risk premium of factor 1 and 2.

Solution: ERi = RF+ β1RP1 + β2 RP2
Stock A:
7 = 3.5 + 2.3 RP1 + (-1.8) RP2 ----equation 1
Stock B:
12= 3.5 + 1.4 RP1 + 0.24 RP2 ----- equation 2
Multiply equation 1 by 1.4 and equation 2 by 2.3
4.9 = 3.5 + 3.22 RP1 + (-2.52) RP2 ------equation 1
19.55 =3.5 + 3.22 RP1 + 0.552 RP2 -------equation 2
Subtracting equation 1 from equation 2 we get
14.65 = 3.07RP2
RP2 = 4.76%
Now put the value of factor 2 in equation1
7 = 3.5 + 2.3 RP1 + (-1.8)*7.44
3.5 = 2.3 RP1 + (-13.39)
16.89 = 2.3 RP1
RP1 = 7.34%





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