Finance

1. Course: Securities Analysis and Portfolio Management
Faculty: Dr. Sonia Garg (Email: sonia.garg@thapar.edu)
Session 20: Single Index Models
Duration: 75 mins
Slides: 11
CLEANLINESS IS NEXT TO GODLINESS
LM THAPAR SCHOOL OF MANAGEMENT,
THAPAR INSTITUTE OF ENGINEERING & TECHNOLOGY
Masters of Business Administration

2. Session Objectives
Understand a single index model
– Derivation of single index model
– Characteristics of single index model
– Estimating Beta
– Bias Correction techniques of Beta
Next Session: Multi Index Models
21-12-2022 Single Index Models 2

3. Single index models
If market goes up most stocks tend to increase in price and
vice versa, thus return on stock can be Ri = ai + βiRm
ai - security’s return independent of market’s performance (a random
variable)
βi - constant that measures expected change in Ri given a change in Rm
Rm - rate of return on market index (a random variable)
Let αi be the expected value of ai and ei represent the
random element of ai
Thus Ri = αi + βiRm + ei
21-12-2022 Single Index Models 3

4. Assumptions
1) ei and Rm are both random variables, it is
convenient to assume that they are
independent
1) Securities are only related through common
response to market
21-12-2022 Single Index Models 4
cov(ei
RM
) = E[(ei
-0)(Rm
- Rm
)]= 0
cov(ei
ej
) = E[(ei
-0)(ej
-0)] = E(ei
ej
) = 0

5. Derivation of single index model
By construction: Mean of ei
By definition: Variance of ei
Variance of Rm
Mean return:
21-12-2022 Single Index Models 5
E(ei
) = 0
E(ei
)2
= sei
2
E(Rm
- Rm
)2
=sm
2
E(Ri
) = E[ai
+ bi
Rm
+ ei
]
= E(ai
)+ E(bi
Rm
)+ E(ei
) = ai
+ bi
Rm

6. Variance of return of a security
21-12-2022 Single Index Models 6
si
2
= E(Ri
- Ri
)2
= E[(ai
+ bi
Rm
+ ei
) - (ai
+ bi
Rm
)]2
= E[bi
(Rm
- Rm
) + ei
]2
= b2
i
E(Rm
- Rm
)2
+ 2bi
E[ei
(Rm
- Rm
)]+ E(ei
)2
= b2
i
E(Rm
- Rm
)2
+ E(ei
)2
= b2
i
sm
2
+sei
2

7. Covariance between two securities
21-12-2022 Single Index Models 7
= bi
bj
sm
2
sij
= E{[(ai
+ bi
Rm
+ ei
) - (ai
+ bi
Rm
)]
[(a j
+ bj
Rm
+ ej
)- (a j
+ bj
Rm
)]
= bi
bj
E(Rm
- Rm
)2
+ bj
E[ei
(Rm
- Rm
)]
+bi
E[ej
(Rm
- Rm
)]+ E(ei
ej
)
= E{[bi
(Rm
- Rm
)+ ei
][bj
(Rm
- Rm
)+ ej
]}

8. Characteristics of single index model
21-12-2022 Single Index Models 8
bP
= Xi
bi
i=1
N
å aP
= Xi
ai
i=1
N
å RP
= aP
+ bP
Rm
s 2
P
= Xi
2
i=1
N
å b2
i
sm
2
+ Xi
2
i=1
N
å sei
2
+ Xi
j=1
j¹i
N
å
i=1
N
å X j
bi
bj
sm
2
s 2
P
= Xi
j=1
N
å
i=1
N
å X j
bi
bj
sm
2
+ Xi
2
i=1
N
å sei
2
s 2
P
= ( Xi
bi
i=1
N
å )( X j
bj
j=1
N
å )sm
2
+ Xi
2
i=1
N
å sei
2
= bP
2
sm
2
+ Xi
2
i=1
N
å sei
2
Can be reduced
to zero
Only
relevant risk

9. Estimation of Beta
Use historical data for security and market returns and regress security
returns against market returns to find alpha and beta in the following
equation
Problems with predictive power of historical beta
– The beta of security might change over time
– Beta is measured with random error, larger the
random error less the predictive power
21-12-2022 Single Index Models 9
Ri
= ai
+ bi
Rm

10. Adjusting historical beta: Blume’s
Technique
Assumption: Adjustment in one period is a good estimate of
adjustment in the next period
The equation obtained by regressing betas of period 2 against betas of
period 1 as shown below
gives the adjustment required for betas in period 3. This equation
lowers high values of beta and raises low values.
This technique results in continued extrapolation of trend, if that is not
desirable then adjustments should be made such that mean beta is
same as historical mean
21-12-2022 Single Index Models 10
bi,t
= a + bbi,t-1

11. Adjusting historical beta: Vasicek’s
Technique
Adjust each beta towards the average beta using bayesian
estimation technique
individual beta for security i in period t-1
standard error of estimate of beta of security i in period t-1
average beta across a sample of stocks in period t-1
standard error of estimate of average beta across a sample
of stocks in period t-1
High beta stocks have larger standard errors and thus are adjusted
more than lower beta stocks
21-12-2022 Single Index Models 11
bi,t
= (
sbi,t-1
2
sbt-1
2
+sbi,t-1
2
bt-1
)+ (
sbt-1
2
sbt-1
2
+sbi,t-1
2
bi,t-1
)
bi,t-1
sbi,t-1
bt-1
sbt-1