CHAPTER 7
Portfolio Management
What are we going to learn
in this chapter?
Risk Aversion
 You have two stocks you consider for purchase;
which one do you pick? Why?
 50 TL certain – 100/0 coin flip example
 Risk averse
 Risk neutral
 Risk seeking
 Main assumption about risk aversion
Markowitz Portfolio Theory
 Variance of returns and diversification
 The Markowitz model is based on several assumptions regarding investor
behavior:
1. Investors consider each investment alternative as being represented by a
probability distribution of expected returns over some holding period.
2. Investors maximize one-period expected utility, and their utility curves
demonstrate diminishing marginal utility of wealth.
3. Investors estimate the risk of the portfolio on the basis of the variability
of expected returns.
4. Investors base decisions solely on expected return and risk.
5. For a given risk level, investors prefer higher returns to lower returns.
Similarly, for a given level of expected return, investors prefer less risk to
more risk.
Alternative Risk Measures
 Variance & standard deviation of expected
returns
 Range of returns
 Semivariance
Expected Rates of Return
 The expected rate of return for an individual
investment
 What is the expected return for Tortu?
Probability
0.25
Possible Return Rate
0.08
0.25
0.25
0.25
0.10
0.12
0.14
Expected Return
E(R) =
Expected Rates of Return
 The expected rate of return for a portfolio of risky
assets
 What is the expected return for a portfolio of Sütaş,
Arçelik, Merko, Penguan Gıda?
Weight
0.20
Expected Sec. Ret.
0.10
0.30
0.30
0.20
0.11
0.12
0.13
Expected Return
E(R) =
Risk
 Standard deviation of returns for an individual
investment
 What is the standard deviation for Tortu?
Possible
Return Rate
0.08
0.10
0.12
0.14
Expected
Return
Ri – E(Ri)
[Ri – E(Ri)]^2
Pi
([Ri – E(Ri)]^2) * Pi
Risk
 Standard deviation of returns for a portfolio of
risky assets
 Covariance
 Correlation
Covariance
 What is Covariance?
 Prices vs. returns?
 A positive covariance means …….
 A negative covariance indicates ……
 The magnitude of the covariance depends on ……
Covariance
•Tukaş & Ünye Çimento: E(R)=?
Date
Close Price
Dividend
Return R.
Close Price
Dividend
Return R.
Dec 2011
60.938
???
45.688
???
Jan 2012
58.000
???
48.200
???
Feb 2012
53.030
???
42.500
???
Mar 2012
45.160
???
43.100
Apr 2012
46.190
2.28
47.100
9.28
May 2012
47.400
2.62
49.290
4.65
Jun 2012
45.000
-4.68
47.240
Jul
2012
44.600
-0.89
50.370
Aug 2012
48.670
9.13
45.950
Sept 2012
46.850
-3.37
38.370
-16.50
Oct 2012
47.880
2.20
38.230
-0.36
Nov 2012
46.960
-1.55
46.650
Dec 2012
47.150
0.40
51.010
0.18
0.18
0.18
0.18
0.04
0.04
???
-4.08
6.63
0.04
0.05
-8.70
22.16
9.35
Covariance
Covariance
Covariance
 Although the rates of return for the two stocks
moved together during some months, in other
months they moved in opposite directions. The
covariance statistic provides an absolute measure of
how they moved together over time.
 Formulation
 Formulation for 12 monthly returns of 2 assets
 When would covariation be positive/negative?
Covariance
•Tukaş & Ünye Çimento: Covariance?
Date
Ri
Rj
Ri – E(Ri)
Rj– E(Rj)
[Ri – E(Ri)] [Rj– E(Rj)]
Jan 2012
-4.82
5.50
???
???
???
Feb 2012
-8.57
-11.83
???
???
???
Mar 2012
14.50
1.51
???
???
???
Apr 2012
2.28
9.28
4.09
7.81
31.98
May 2012
2.62
4.65
4.43
3.18
14.11
Jun 2012
-4.68
-4.08
-2.87
-5.54
15.92
Jul
2012
-0.89
6.63
0.92
5.16
4.76
Aug 2012
9.13
-8.70
10.94
-10.16
-111.16
Sept 2012
-3.37
-16.50
-1.56
-17.96
27.97
Oct 2012
2.20
-0.36
4.01
-1.83
-7.35
Nov 2012
-1.55
22.16
0.27
20.69
5.52
Dec 2012
0.40
9.35
2.22
7.88
17.47
Covariance & Correlation
 What does the covariance figure mean?
 Standardization
 Correlation
 Correlation boundaries and the interpretation
Correlation
•Tukaş & Ünye Çimento: Correlation?
Date
Ri – E(Ri)
[Ri – E(Ri)]^2
Rj – E(Rj)
[Rj– E(Rj)]^2
Jan 2012
-3.01
???
4.03
???
Feb 2012
-6.76
???
-13.29
???
Mar 2012
-12.69
???
0.04
???
Apr 2012
4.09
16.75
7.81
61.06
May 2012
4.43
19.64
3.18
10.13
Jun 2012
-2.87
8.24
-5.54
30.74
2012
0.92
0.85
5.16
26.61
Aug 2012
10.94
119.64
-10.16
103.28
Sept 2012
-1.56
2.42
-17.96
322.67
Oct 2012
4.01
16.09
-1.83
3.36
Nov 2012
0.27
0.07
20.69
428.01
Dec 2012
2.22
4.92
7.88
62.08
Jul
Standard Deviation of a Portfolio
 What does the standard deviation of a
portfolio indicate?
 Formal formulation
Standard Deviation of a Portfolio
 EXAMPLE:
E(Ra) = 0.20 Stn. Dev. = 0.10 Wa = 0.50
E(Rb) = 0.20 Stn. Dev. = 0.10 Wb = 0.50
Correlation = 0.10
Risk = ?
 The lesson learnt?
Standard Deviation of a Portfolio
 EXAMPLE:
E(Ra) = 0.20 Stn. Dev. = 0.10 Wa = 0.50
E(Rb) = 0.20 Stn. Dev. = 0.10 Wb = 0.50
Correlation = 0.05
Risk = ?
 The lesson learnt?
Standard Deviation of a Portfolio
 EXAMPLE:
E(Ra) = 0.20 Stn. Dev. = 0.10 Wa = 0.50
E(Rb) = 0.20 Stn. Dev. = 0.10 Wb = 0.50
 If Correlation = 0.00 Stn. Dev. ?
 If Correlation = -0.50 Stn. Dev. ?
 If Correlation = 0.00 Stn. Dev. ?
Standard Deviation of a Portfolio
 EXAMPLE:
If Correlation is 1.00; what is the risk of the
portfolio?
Asset
1
2
E(Ri)
0.10
0.20
W(i)
0.50
0.50
Variance(i)
0.0049
0.0100
Stn. Dev. (i)
???
???
Standard Deviation of a Portfolio
 What if the correlations were:
+0.50
0.00
-0.50
-1.00
Three Asset Portfolio
 Correlations:
Asset Classes
Stocks
Bonds
Cash Equivalents
r S,B = 0.25
r S,C = -0.08
r B,C = 0.15
E(Ri)
0.12
0.08
0.04
Expected return = ?
Standard deviation = ?
Stn. Dev. (i)
0.20
0.10
0.03
Weight (i)
0.60
0.30
0.10
The Efficient Frontier
 If we examined different two-asset
combinations and derived the curves
assuming all the possible weights, we would
have a graph like:
The Efficient Frontier
The Efficient Frontier
 Efficient frontier
 Comparisons of portfolios A, B and C
 As an investor, you will target a point along the efficient
frontier based on your utility function and your attitude
toward risk.
 No portfolio on the efficient frontier can dominate any
other portfolio on the efficient frontier. All of these
portfolios have different return and risk measures, with
expected rates of return that increase with higher risk.
The Efficient Frontier
Efficient Frontier & Investor Utility
 Slope of the efficient frontier
 An individual investor’s utility curves
 Two investors will choose the same portfolio from
the efficient set only if their utility curves are
identical.
 The optimal portfolio
Efficient Frontier & Investor Utility
END OF CHAPTER