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How to calculate Single Index Model?

Single Index Model
Assumptions of Single index model:
·         The first assumption of single index model is that the security return is affected by only one factor i.e systematic risk.
·         The investors have homogeneous interest.
·         The investors hold security for fixed period to estimate the risk and return on security.

Advantages of Single Index Model:
·         With the help of this model the number of input is decreases to calculate expected return on security.
·         The model helps to determine the systematic risk and unsystematic risk on security.
·         It makes easier to calculate the expected return of security.

Example: Mr. Sharma has invested in a portfolio in which 4 different securities included that is A, B, C and D. His friend Nikhil tells about some portfolio term like alpha and beta which help him to determine the expected return of that portfolio that is, the alpha of security shows the security return which is not affected to market return and the beta of security shows the sensitivity of a security in relation to market. So, Mr Sharma wants to know the alpha and beta of his investment portfolio with the help of given weight of security in a portfolio, alpha, beta of each security.
Security in a portfolio
Weight of security
Alpha
Beta
A
0.3
1.2
0.26
B
0.1
-1.56
1.38
C
0.4
2.69
0.52
D
0.2
0.78
1.98

Solution:
Alpha of given portfolio:
 αp = Ʃni=1wiαi
= 0.3*1.2 + 0.1*-1.56 + 0.4*2.69 + 0.2*0.78
= 0.36 + (-0.156) + 1.076 + 0.156
= 1.436

Beta of given portfolio:
βp = Ʃni=1wiβi
= 0.3*0.26 + 0.1*1.38 + 0.4*0.52 + 0.2*1.98
= 0.078 + 0.138 + 0.208 + 0.396
= 0.82

Example: Mr. Verma wants to invest in portfolio which consists 6 securities are P, Q, R, S, T and U. But he doesn’t know how much risk factor affect his portfolio return. So, he is not able to take a decision to invest in that portfolio or not. You have to help him by calculating the portfolio risk with the help of given information: (standard deviation of market return 16)

Security
Weight of security in portfolio
Alpha
Beta
Standard deviation of error  term( σ2ei)
P
0.1
-0.27
1.72
252
Q
0.2
2.25
1.20
495
R
0.1
1.87
0.47
125
S
0.3
1.64
0.33
652
T
0.2
0.99
1.89
322
U
0.1
-1.01
1.44
106

Solution:
αp = 0.1 * (-0.27) + 0.2*2.25 + 0.1*1.87 + 0.3*1.64 + 0.2*0.99 + 0.1*(-1.01)
= (-0.027) + 0.45 + 0.187 + 0.492 + 0.198 + (-0.101)
= 1.199

βp = 0.1*1.72 + 0.2*1.20 + 0.1*0.47 + 0.3*0.33 + 0.2*1.89 + 0.1*1.44
 = 0.172 + 0.24 + 0.047 + 0.099 + 0.378 + 0.144
= 1.08

Portfolio variance: σ2p = β2p σ2m + Ʃni =1 w2i  σ2ei
  =1.082*162 + 0.12*252 + 0.22*495 + 0.12*125 + 0.32*652 + 0.22*322 + 0.12*106
= 1.1664*256 + 2.52 + 19.8 +1.25 + 58.68 + 12.88 + 1.06
=298.60 + 96.19
= 394.79

Example: From the above example find out the expected return of given portfolio if the market return is 18%.

Solution: Erp = αp + βpErm
Erp = Expected return of portfolio
Erm = Expected market return
= αp + βpErm
= 1.199 + 1.08*18
= 20.639


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