Description
In Markowitz model a number of co-variances have to be estimated., Sharpe assumed that the return of a security is linearly related to a single index like the market index.
The Sharpe Index
Model
Need for Sharpe Model
? In Markowitz model a number of co-variances have
to be estimated.
? If a financial institution buys 150 stocks, it has to
estimate 11,175 i.e.,(N
2
–N)/2 correlation
co-efficients.
? Sharpe assumed that the return of a security is
linearly related to a single index like the market
index.
Single Index Model
? Casual observation of the stock prices over a
period of time reveals that most of the stock
prices move with the market index.
? When the Sensex increases, stock prices
also tend to increase and vice – versa.
? This indicates that some underlying factors
affect the market index as well as the stock
prices.
Stock prices are related to the market index and this
relationship could be used to estimate the return of
stock.
R
i
= o
i
+ |
i
R
m
+ e
i
where R
i
— expected return on security i
o
i
— intercept of the straight line or alpha co-efficient
|
i
— slope of straight line or beta co-efficient
R
m
— the rate of return on market index
e
i
— error term
Risk
? Systematic risk = |
i
2
× variance of market index
= |
i
2
o
m
2
? Unsystematic risk= Total variance – Systematic risk
e
i
2
= o
i
2
– Systematic risk
? Thus the total risk = Systematic risk + Unsystematic risk
= |
i
2
o
m
2
+ e
i
2
Portfolio Variance
2
N N
2 2 2 2
p i i m i i
i =1 i =1
? = x ? + x e
(
| | (
o
(
(
|
\ .
(
¸ ¸
¸ ¸
¿ ¿
Where:
?
2
p
= variance of portfolio
?
2
m
= expected variance of market index
e
2
i
= Unsystematic risk
x
i
= the portion of stock i in the portfolio
Example
? The following details are given for x and y companies’
stocks and the Sensex for a period of one year.
Calculate the systematic and unsystematic risk for the
companies stock. If equal amount of money is allocated
for the stocks , then what would be the portfolio risk ?
X stock Y stock Sensex
? Average return 0.15 0.25 0.06
? Variance of return 6.30 5.86 2.25
? ?eta 0.71 0.27
Company X
? Systematic risk = |
i
2
× variance of market index
= |
i
2
o
m
2
= ( 0.71)
2
x 2.25 = 1.134
? Unsystematic risk = Total variance – Systematic risk
e
i
2
= o
i
2
– Systematic risk
= 6.3 – 1.134 = 5.166
? Total risk = Systematic risk + Unsystematic risk
= |
i
2
o
m
2
+ e
i
2
= 1.134 + 5.166 = 6.3
Company Y
? Systematic risk = |
i
2
× variance of market index
= |
i
2
o
m
2
= ( 0.27)
2
x 2.25 = 0.1640
? Unsystematic risk = Total variance – Systematic risk
e
i
2
= o
i
2
– Systematic risk
= 5.86 – 0.1640 = 5.696
? Total risk = Systematic risk + Unsystematic risk
= |
i
2
o
m
2
+ e
i
2
= 0.1640 + 5.696 = 5.86
? ?
2
p
= [ ( .5 x .71 + .5 x .27)
2
2.25 ] + [ ( .5)
2
(5.166) + (.5 )
2
( 5.696) ]
? = [ ( .355 + .135 )
2
x 2.25 ] + [ ( 1.292 + 1.424 ) ]
? = 0.540 + 2.716
? = 3.256
Corner portfolio
? The entry or exit of a new stock in the portfolio
generates a series of corner portfolio.
? In an one stock portfolio, it itself is the corner
portfolio .
? In a two stock portfolio, the minimum risk and the
lowest return would be the corner portfolio.
? As the number of stocks increases in a portfolio, the
corner portfolio would be the one with lowest return
and risk combination.
Expected Return of Portfolio
N
p i i i m
i =1
R = x (? + ? R )
¿
? For each security o
i
and |
i
should be
estimated
? Portfolio return is the weighted average of the
estimated return for each security in the
portfolio.
? The weights are the respective stocks’
proportions in the portfolio.
Portfolio Beta
N
p i i
i =1
= x | |
¿
A portfolio’s beta value is the weighted
average of the beta values of its component
stocks using relative share of them in the
portfolio as weights.
|
p
is the portfolio beta.
Sharpe’s optimal portfolio
i f
i
R R
?
÷
? The selection of any stock is directly related to its
excess return to beta ratio.
? where R
i
= the expected return on stock i
R
f
= the return on a risk less asset
|
i
= Systematic risk
Optimal Portfolio
? The steps for finding out the stocks to be
included in the optimal portfolio are as:
? Find out the ?excess return to beta? ratio for each
stock under consideration
? Rank them from the highest to the lowest
? Proceed to calculate C
i
for all the stocks
according to the ranked order using the following
formula
N
2
i f i
m
2
i =1
ei
i
2
N
2
i
m
2
i =1
ei
(R R )?
?
?
C
?
1+ ?
?
÷
¿
¿
o
m
2
= variance of the market index
o
ei
2
= stock’s unsystematic risk
Cut-off point
? The cumulated values of C
i
start declining
after a particular C
i
and that point is taken
as the cut-off point and that stock ratio is
the cut-off ratio C.
? o
ei
2 - Unsystematic risk
(Ri – Rf) / ?i – Excess return
to Beta
Example
? Data for finding out the optimal portfolio are given below
Security number Mean return Excess return Beta o
ei
2 (Ri – Rf) / ?i
? Ri Ri – Rf
?
?
1 19 14 1.0 20 14
?
2 23 18 1.5 30 12
?
3 11 6 0.5 10 12
?
4 25 20 2.0
40 10
?
5 13 8 1.0
20 8
?
6 9 4 0.5 50 8
?
7
14 9 1.5 30 6
? The riskless rate of intrest is 5% and the market variance is 10.
Determine the cut – off point .
?
(Ri – Rf) / ?i [(Ri – Rf) x ?i
]/ ?
2
ei
?
[(Ri – Rf) x ?i
]/ ?
2
ei
?
14 .7 .7
?
12 .9 1.6
?
12 .3 1.9
?
10 1.0 2.9
?
8 .4 3.3
?
8 .04 3.34
?
6 .45 3.79
?
?i2/
?
2
ei
?
?i2/
?
2
ei
?
.05 .05
?
.075 .125
?
.25 .15
?
.1 .25
?
.05 .3
?
.005 .305
?
.075
.38
? C1 = (10 x .7)/ [ 1 + ( 10 x .05)] = 4.67
? C2 = (10 x 1.6)/ [ 1 + ( 10 x .125)] = 7.11
? C3 = (10 x 1.9)/ [ 1 + ( 10 x .15)] = 7.6
? C4 = (10 x 2.9)/ [ 1 + ( 10 x .25)] = 8.29* Cut-off point
? C5 = (10 x 3.3)/ [ 1 + ( 10 x .3)] = 8.25
? C6 = (10 x 3.34)/ [ 1 +( 10 x .305)] = 8.25
? C7= (10 x 3.79)/ [ 1 + ( 10 x .38)] = 7.90
? The highest Ci value is taken as the cut-off
point i.e C*.
? The stocks ranked above C* have high
excess returns to beta than the cut off Ci and
all the stocks ranked below C* have low
excess return to beta.
? Here the cut off rate is 8.29.
? Hence the first four securities are selected.
? If the number of stocks is larger there is no
need to calculate ci values for all the stocks
after the ranking has been done.
? It can be calculated until the C* value is found
and after calculating for one or two stocks
below it, the calculations can be terminated.
Construction of the optimal portfolio
? After determining the securities to be
selected, the portfolio manager should find
out how much should be invested in each
security.The percentage of funds to be
invested in each security can be estimated as
follows .
? Zi = (?i / ?
2
ei
) x [ (Ri – Rf / ?i) – C* ]
? Zi = (?i / ?
2
ei
) x [ (Ri – Rf / ?i) – C ]
? Z1 = (1/20) x ( 14 – 8.29) = 0.285
? Z2 = (1.5 / 30) x ( 12 – 8.29) = 0.186
? Z3 = (0.5 / 10) x ( 12 – 8.29) = 0.186
? Z4 = (2 / 40) x ( 10 – 8.29) = 0.086
? Total = .743
? X1 = 0.285/ .743 = 0.38
? X2 = 0.186 / .743 = 0.25
? X3 = 0.186 / .743 = 0.25
? X4 = 0.086 / .743 = 0.12
? So the largest investment should be made in
security 1 ( 0.38%) and the smallest in
security 4 ( 0.12%).
? The characteristics of a stock that make it
desirable can be determined before the
calculations of an optimal portfolio is begun.
? The desirability of any stock is solely a
function of its excess return to beta ratio.
doc_208287484.ppt
In Markowitz model a number of co-variances have to be estimated., Sharpe assumed that the return of a security is linearly related to a single index like the market index.
The Sharpe Index
Model
Need for Sharpe Model
? In Markowitz model a number of co-variances have
to be estimated.
? If a financial institution buys 150 stocks, it has to
estimate 11,175 i.e.,(N
2
–N)/2 correlation
co-efficients.
? Sharpe assumed that the return of a security is
linearly related to a single index like the market
index.
Single Index Model
? Casual observation of the stock prices over a
period of time reveals that most of the stock
prices move with the market index.
? When the Sensex increases, stock prices
also tend to increase and vice – versa.
? This indicates that some underlying factors
affect the market index as well as the stock
prices.
Stock prices are related to the market index and this
relationship could be used to estimate the return of
stock.
R
i
= o
i
+ |
i
R
m
+ e
i
where R
i
— expected return on security i
o
i
— intercept of the straight line or alpha co-efficient
|
i
— slope of straight line or beta co-efficient
R
m
— the rate of return on market index
e
i
— error term
Risk
? Systematic risk = |
i
2
× variance of market index
= |
i
2
o
m
2
? Unsystematic risk= Total variance – Systematic risk
e
i
2
= o
i
2
– Systematic risk
? Thus the total risk = Systematic risk + Unsystematic risk
= |
i
2
o
m
2
+ e
i
2
Portfolio Variance
2
N N
2 2 2 2
p i i m i i
i =1 i =1
? = x ? + x e
(
| | (
o
(
(
|
\ .
(
¸ ¸
¸ ¸
¿ ¿
Where:
?
2
p
= variance of portfolio
?
2
m
= expected variance of market index
e
2
i
= Unsystematic risk
x
i
= the portion of stock i in the portfolio
Example
? The following details are given for x and y companies’
stocks and the Sensex for a period of one year.
Calculate the systematic and unsystematic risk for the
companies stock. If equal amount of money is allocated
for the stocks , then what would be the portfolio risk ?
X stock Y stock Sensex
? Average return 0.15 0.25 0.06
? Variance of return 6.30 5.86 2.25
? ?eta 0.71 0.27
Company X
? Systematic risk = |
i
2
× variance of market index
= |
i
2
o
m
2
= ( 0.71)
2
x 2.25 = 1.134
? Unsystematic risk = Total variance – Systematic risk
e
i
2
= o
i
2
– Systematic risk
= 6.3 – 1.134 = 5.166
? Total risk = Systematic risk + Unsystematic risk
= |
i
2
o
m
2
+ e
i
2
= 1.134 + 5.166 = 6.3
Company Y
? Systematic risk = |
i
2
× variance of market index
= |
i
2
o
m
2
= ( 0.27)
2
x 2.25 = 0.1640
? Unsystematic risk = Total variance – Systematic risk
e
i
2
= o
i
2
– Systematic risk
= 5.86 – 0.1640 = 5.696
? Total risk = Systematic risk + Unsystematic risk
= |
i
2
o
m
2
+ e
i
2
= 0.1640 + 5.696 = 5.86
? ?
2
p
= [ ( .5 x .71 + .5 x .27)
2
2.25 ] + [ ( .5)
2
(5.166) + (.5 )
2
( 5.696) ]
? = [ ( .355 + .135 )
2
x 2.25 ] + [ ( 1.292 + 1.424 ) ]
? = 0.540 + 2.716
? = 3.256
Corner portfolio
? The entry or exit of a new stock in the portfolio
generates a series of corner portfolio.
? In an one stock portfolio, it itself is the corner
portfolio .
? In a two stock portfolio, the minimum risk and the
lowest return would be the corner portfolio.
? As the number of stocks increases in a portfolio, the
corner portfolio would be the one with lowest return
and risk combination.
Expected Return of Portfolio
N
p i i i m
i =1
R = x (? + ? R )
¿
? For each security o
i
and |
i
should be
estimated
? Portfolio return is the weighted average of the
estimated return for each security in the
portfolio.
? The weights are the respective stocks’
proportions in the portfolio.
Portfolio Beta
N
p i i
i =1
= x | |
¿
A portfolio’s beta value is the weighted
average of the beta values of its component
stocks using relative share of them in the
portfolio as weights.
|
p
is the portfolio beta.
Sharpe’s optimal portfolio
i f
i
R R
?
÷
? The selection of any stock is directly related to its
excess return to beta ratio.
? where R
i
= the expected return on stock i
R
f
= the return on a risk less asset
|
i
= Systematic risk
Optimal Portfolio
? The steps for finding out the stocks to be
included in the optimal portfolio are as:
? Find out the ?excess return to beta? ratio for each
stock under consideration
? Rank them from the highest to the lowest
? Proceed to calculate C
i
for all the stocks
according to the ranked order using the following
formula
N
2
i f i
m
2
i =1
ei
i
2
N
2
i
m
2
i =1
ei
(R R )?
?
?
C
?
1+ ?
?
÷
¿
¿
o
m
2
= variance of the market index
o
ei
2
= stock’s unsystematic risk
Cut-off point
? The cumulated values of C
i
start declining
after a particular C
i
and that point is taken
as the cut-off point and that stock ratio is
the cut-off ratio C.
? o
ei
2 - Unsystematic risk
(Ri – Rf) / ?i – Excess return
to Beta
Example
? Data for finding out the optimal portfolio are given below
Security number Mean return Excess return Beta o
ei
2 (Ri – Rf) / ?i
? Ri Ri – Rf
?
?
1 19 14 1.0 20 14
?
2 23 18 1.5 30 12
?
3 11 6 0.5 10 12
?
4 25 20 2.0
40 10
?
5 13 8 1.0
20 8
?
6 9 4 0.5 50 8
?
7
14 9 1.5 30 6
? The riskless rate of intrest is 5% and the market variance is 10.
Determine the cut – off point .
?
(Ri – Rf) / ?i [(Ri – Rf) x ?i
]/ ?
2
ei
?
[(Ri – Rf) x ?i
]/ ?
2
ei
?
14 .7 .7
?
12 .9 1.6
?
12 .3 1.9
?
10 1.0 2.9
?
8 .4 3.3
?
8 .04 3.34
?
6 .45 3.79
?
?i2/
?
2
ei
?
?i2/
?
2
ei
?
.05 .05
?
.075 .125
?
.25 .15
?
.1 .25
?
.05 .3
?
.005 .305
?
.075
.38
? C1 = (10 x .7)/ [ 1 + ( 10 x .05)] = 4.67
? C2 = (10 x 1.6)/ [ 1 + ( 10 x .125)] = 7.11
? C3 = (10 x 1.9)/ [ 1 + ( 10 x .15)] = 7.6
? C4 = (10 x 2.9)/ [ 1 + ( 10 x .25)] = 8.29* Cut-off point
? C5 = (10 x 3.3)/ [ 1 + ( 10 x .3)] = 8.25
? C6 = (10 x 3.34)/ [ 1 +( 10 x .305)] = 8.25
? C7= (10 x 3.79)/ [ 1 + ( 10 x .38)] = 7.90
? The highest Ci value is taken as the cut-off
point i.e C*.
? The stocks ranked above C* have high
excess returns to beta than the cut off Ci and
all the stocks ranked below C* have low
excess return to beta.
? Here the cut off rate is 8.29.
? Hence the first four securities are selected.
? If the number of stocks is larger there is no
need to calculate ci values for all the stocks
after the ranking has been done.
? It can be calculated until the C* value is found
and after calculating for one or two stocks
below it, the calculations can be terminated.
Construction of the optimal portfolio
? After determining the securities to be
selected, the portfolio manager should find
out how much should be invested in each
security.The percentage of funds to be
invested in each security can be estimated as
follows .
? Zi = (?i / ?
2
ei
) x [ (Ri – Rf / ?i) – C* ]
? Zi = (?i / ?
2
ei
) x [ (Ri – Rf / ?i) – C ]
? Z1 = (1/20) x ( 14 – 8.29) = 0.285
? Z2 = (1.5 / 30) x ( 12 – 8.29) = 0.186
? Z3 = (0.5 / 10) x ( 12 – 8.29) = 0.186
? Z4 = (2 / 40) x ( 10 – 8.29) = 0.086
? Total = .743
? X1 = 0.285/ .743 = 0.38
? X2 = 0.186 / .743 = 0.25
? X3 = 0.186 / .743 = 0.25
? X4 = 0.086 / .743 = 0.12
? So the largest investment should be made in
security 1 ( 0.38%) and the smallest in
security 4 ( 0.12%).
? The characteristics of a stock that make it
desirable can be determined before the
calculations of an optimal portfolio is begun.
? The desirability of any stock is solely a
function of its excess return to beta ratio.
doc_208287484.ppt