View
219
Download
0
Category
Preview:
Citation preview
8/8/2019 Portfolio Analysis - Efficient Frontiers
1/23
1
Portfolio AnalysisPortfolio Analysis Efficient FrontierEfficient Frontier
Alok Kumar
8/8/2019 Portfolio Analysis - Efficient Frontiers
2/23
2
Efficient FrontierEfficient Frontier
Two Conditions
1) Offer Maximum Return for varying levels of Risk,and
2) Offer Minimum Risk for varying levels of expected
return
All the feasible sets are not efficient unless it passes throughthis test
8/8/2019 Portfolio Analysis - Efficient Frontiers
3/23
3
Efficient Sets and Feasible SetsEfficient Sets and Feasible Sets
Feasible Sets
A
D
C
B
8/8/2019 Portfolio Analysis - Efficient Frontiers
4/23
8/8/2019 Portfolio Analysis - Efficient Frontiers
5/23
5
How to form Efficient Frontier ?How to form Efficient Frontier ?
8/8/2019 Portfolio Analysis - Efficient Frontiers
6/23
6
2 Stock Case2 Stock Case
Stocks Expected Return Standard Deviation
A 5% 20%
B 15% 40%
8/8/2019 Portfolio Analysis - Efficient Frontiers
7/23
7
FormulaFormula
Expected Return of Portfolio = Xiri,where i range from 0to n.
and X is Proportion of total investment in ith security and riis expected return of the security.
Standard deviation of Portfolio =( Xi Xj ij)1/2
where i and j vary from 0 to n, and ij is covariance of iand j securities.
ij = iji j,where i & j is standard deviation of i and jrespectively.
8/8/2019 Portfolio Analysis - Efficient Frontiers
8/23
8
Expected Return for PortfoliosExpected Return for Portfolios
Portfolios Proportion in X Proportion in Y Return
A 1 0 5.00%
B 0.83 0.17 6.70%
C 0.67 0.33 8.30%
D 0.5 0.5 10.00%
E 0.33 0.67 11.71%F 0.17 0.83 13.30%
G 0 1 15.00%
8/8/2019 Portfolio Analysis - Efficient Frontiers
9/23
9
Standard Deviation of PortfolioStandard Deviation of Portfolio
Portfolios Lower Bound Upper Bound No relationship
A 20.00% 20.00% 20.00%
B 10.00% 23.33% 17.94%
C 0.00% 26.67% 18.81%
D 10.00% 30.00% 22.36%
E 20.00% 33.33% 27.60%
F 30.00% 36.67% 33.37%
G 40.00% 40.00% 40.00%
8/8/2019 Portfolio Analysis - Efficient Frontiers
10/23
10
Efficient FrontierEfficient Frontier
Upper and Lower Bounds to Portfolios
0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
12.00%
14.00%
16.00%
0.00% 5.00% 10.00% 15.00% 20.00% 25.00% 30.00% 35.00% 40.00% 45.00%
Standard Deviations
E
x
ected
Return
8/8/2019 Portfolio Analysis - Efficient Frontiers
11/23
11
Market
Models
Market
Models
8/8/2019 Portfolio Analysis - Efficient Frontiers
12/23
12
Market ModelMarket Model
ri = iI + iI rI + iI
Where, ri = return on security i for given period
iI = intercept form
iI = slope form
rI = return on market index I for the same period
iI =random error
8/8/2019 Portfolio Analysis - Efficient Frontiers
13/23
13
Graphical Presentation ofMarket ModelGraphical Presentation ofMarket Model
ri = iI + iI rI
8/8/2019 Portfolio Analysis - Efficient Frontiers
14/23
14
BetaBeta
iI = iI
I2
iI = Covariance
I2 = Variance of Market Index
8/8/2019 Portfolio Analysis - Efficient Frontiers
15/23
15
Random ErrorRandom Error
Security A Security B
Intercept 2% -1%
Actual Return
on the Market
index X beta
10% X 2% = 12% 10% X 8% = 8%
Actual Return
on Security
9% 11%
Random Error 9%- (2% + 12%)
= -5%
11%- (-1% +8%) =
4%
8/8/2019 Portfolio Analysis - Efficient Frontiers
16/23
16
Graphical Presentation ofMarket ModelGraphical Presentation ofMarket Model
Infotech versus S&P 500: 1992-1996
-6.00%
-4.00%
-2.00%
0.00%
2.00%
4.00%
6.00%
8.00%
-15.00% -10.00% -5.00% 0.00% 5.00% 10.00% 15.00% 20.00%
8/8/2019 Portfolio Analysis - Efficient Frontiers
17/23
17
Securitys Total RiskSecuritys Total Risk
i2 =iI
2X I2 + i
2
Where ,
i2= variance of security i
iI2
X I2
= Market risk of security ii
2= Unique risk of security i
8/8/2019 Portfolio Analysis - Efficient Frontiers
18/23
18
Portfolios ReturnPortfolios Return
rp = Xi riWhere i range from o to n. and
Xi = proportion of investment in security i.ri = expected return of security i.
Also,
ri = iI + iI rI + iI
Hence rp = Xi (iI + iI rI + iI)
.....continued
8/8/2019 Portfolio Analysis - Efficient Frontiers
19/23
19
.....continued.....continued
rp = Xi (iI + iI rI + iI)
= Xi iI + (Xi iI ) rI + XiiI
= pI + pI rI + pI
Where i range from o to n.
Intercept Slope X independent
VariableRandom Error
8/8/2019 Portfolio Analysis - Efficient Frontiers
20/23
20
Portfolio RiskPortfolio Risk
2p =2pI
2I +
2p
Where ,
2pI = [Xi iI] 2 ----- Systematic Risk
2p = Xi2 2i ----- Unique Risk
8/8/2019 Portfolio Analysis - Efficient Frontiers
21/23
21
Risk and DiversificationRisk and Diversification
Unique Risk Market Risk
Total Riskp
N
8/8/2019 Portfolio Analysis - Efficient Frontiers
22/23
22
CalculationsCalculations
Stock
Portfolio
Weight Beta
Expected Return of
Stock
Variance of
Stock
A 0.25 0.5 0.4 0.07
B 0.25 0.5 0.25 0.05
C 0.5 1 0.21 0.07
Variance of
Market 0.06
8/8/2019 Portfolio Analysis - Efficient Frontiers
23/23
23
QuestionsQuestions
Residual Variance of each of the stocks?
Beta of the portfolio?
Variance of the Portfolio?
Expected Return on the portfolio?
Portfolio Variance on teh basis of Markowitz Variance Covariance
formula.
Covariance (A,B) = 0.020
Covariance (A,C) = 0.035
Covariance (B,C) = 0.035
Recommended