1. CHAPTER SEVEN
THE PORTFOLIO SELECTION
PROBLEM

2. INTRODUCTION
• Portfolio is a collection of securities.
• With a given amount of wealth and
securities, an investor can design
innumerable portfolios.
• THE BASIC PROBLEM:
– given uncertain outcomes, what risky securities
should an investor own?
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3. INTRODUCTION
• THE BASIC PROBLEM:
– The Harry Markowitz Approach
• assume an initial wealth
• a specific holding period (one period)
• a terminal wealth
• diversify
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4. INTRODUCTION
• Initial and Terminal Wealth
• recall one period rate of return
we wb
rt
wb
where rt = the one period rate of return
wb = the beginning of period wealth
we= the end of period wealth
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5. INITIAL AND TERMINAL WEALTH
• DETERMINING THE PORTFOLIO RATE
OF RETURN
– similar to calculating the return on a security
– FORMULA
w1 w0
rp
w0
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6. INITIAL AND TERMINAL WEALTH
• DETERMINING THE PORTFOLIO RATE
OF RETURN
w1 w0
Formula: rp
w0
where w0 = the aggregate purchase
price at time t=0
w1 = aggregate market value at
time t=1
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7. INITIAL AND TERMINAL WEALTH
• OR USING INITIAL AND TERMINAL
WEALTH
w1 1 rp w0
where
w0 =the initial wealth
w1 =the terminal wealth
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8. THE MARKOWITZ APPROACH
• MARKOWITZ PORTFOLIO RETURN
– portfolio return (rp) is a random variable
– defined by the first and second moments of the
distribution
• expected return
• standard deviation
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9. THE MARKOWITZ APPROACH
• MARKOWITZ PORTFOLIO RETURN
– defined by the first and second moments of the
distribution
• expected return (mean returns)
• standard deviation (dispersion of returns about
mean)
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10. THE MARKOWITZ APPROACH
• MARKOWITZ PORTFOLIO RETURN
– First Assumption:
• Non-satiation: investor always prefers a higher rate of
portfolio return/higher terminal wealth.
• This leads to a conclusion “Given two portfolios with similar
risk, investor would prefer the portfolio with higher returns.
Preferable
Portfolio Portfolio Returns Risk
A 12% 10%
B 8% 10%
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11. THE MARKOWITZ APPROACH
• MARKOWITZ PORTFOLIO RETURN
– Second Assumption
• Risk aversion: assume a risk-averse investor will choose a
portfolio with a smaller standard deviation
Portfolio Returns Risk
Preferable
A 12% 10%
Portfolio
B 12% 08%
• in other words, these investors when given a fair bet (odds
50:50) will not take the bet, i.e. $5 if head, and $-5 if tail. Note
expected return on this is 0=(5*0.5)+(-5*0.5).
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12. THE MARKOWITZ APPROACH
• MARKOWITZ PORTFOLIO RETURN
– INVESTOR UTILITY
– DEFINITION: is the relative satisfaction derived by
the investor from the economic activity- work,
consumption, investment.
– It depends upon individual tastes and preferences-One
individual may not seek same satisfaction/utility from
same activity.
– It assumes rationality, i.e. people will seek to maximize
their utility
– Utility wealth function: shows relationship between
utility and wealth.
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13. THE MARKOWITZ APPROACH
• MARGINAL UTILITY
– each investor has a unique utility-of-wealth
function
– incremental or marginal utility differs by
individual investor and depends upon the
amount of wealth one already possesses.
– Richer investor value marginal $ less than a
poor investor does.
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14. THE MARKOWITZ APPROACH
• MARGINAL UTILITY
– Assumes
• diminishing characteristic: As one has more of
wealth, additional/marginal unit of wealth will add
positive utility but on decreasing rate i.e. utility
derived from marginal unit will keep on decreasing
with successive units.
• An investor with diminishing marginal utility is risk
averse and such an investor rate certain investment
higher than riskier one.
• nonsatiation
• Concave utility-of-wealth function
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15. THE MARKOWITZ APPROACH
UTILITY OF WEALTH FUNCTION
Utility Risk
Utility of premium
Uc Wealth
Ur
Certainty
equivalent
Wealth
103 110
100 105
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16. Conclusions
• Uc=Utility from certain investment
• Ur=Utility from risky investment
• Uc > Ur
• The amount of positive utility derived from an
additional $1 < the amount of negative utility
(disutility) resulted from loss of $1.
• Note: This is evident from the slope of utility
wealth function which is increasing on decreasing
rate. At any point on curve slope towards right is
lower than the slope to left (Concavity).
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17. Understanding Certainty Equivalents and Risk Premiums
• Suppose you are given two options A and B for investing
$100.
A: that you will earn Rs.105 with certainty.
B: that you will earn either Rs.110 or nothing, probability
of both events is 50:50.
Note: Both options have same expected pay off i.e. Rs.105.
• Which option would you choose?
• Your decision depends upon your attitude to risk. You are:
Risk indifferent, if both options are equally attractive to
you.
Risk averse: if you choose option A, preferring safe $ to
risky $.
Risk taker: if you choose plan B.
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18. Understanding Certainty Equivalents and Risk Premiums
• A risk averse investor will choose option B only if:
~ ceteris paribus, he receives lesser pay off in riskless investment e.g. Rs.101
~ ceteris paribus, he receives even higher pay off in risky investment (Option
B) e.g. Rs.120.
Note there must be an amount, where the investor regard both investments
equally. For example in the Option A, if instead of certain $105, you are
offered $103 and as a result you now regard both certain and risky investments
equal, i.e. you derive same level of expected utility from both options. We call
$103 Certainty Equivalent (CE). And the difference between expected
payoff and CE is called Risk Premium (RP), a compensation to investor for
additional risk taking.
The more risk averse you are the higher risk premium you demand and hence the
lower CE, you have.
Risk averse have positive RP, risk neutral have zero risk premium, and risk takers
have negative risk premium.
Expected payoff (EP)= Risk Premium (RP) + Certainty Equivalent (CE)
CE = EP-RP
RP = EP-CE
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19. INDIFFERENCE CURVE ANALYSIS
• INDIFFERENCE CURVE ANALYSIS
– DEFINITION OF INDIFFERENCE CURVES:
• a graphical representation of a set of various risk
and expected return combinations that provide the
same level of utility
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20. INDIFFERENCE CURVE ANALYSIS
• INDIFFERENCE CURVE ANALYSIS
– Features of Indifference Curves:
• no intersection by another curve
• “further northwest” is more desirable giving greater
utility
• investors possess infinite numbers of indifference
curves
• the slope of the curve is the marginal rate of
substitution which represents the nonsatiation and
risk averse Markowitz assumptions
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21. Indifference Curves Analysis
Return further northwest
A risk averse investor will
A choose Portfolio A, which
offers highest returns, with
B C
relatively lower risk.
Risk
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22. PORTFOLIO RETURN
• CALCULATING PORTFOLIO RETURN
– Expected returns
• Markowitz Approach focuses on terminal wealth
(W1), that is, the effect various portfolios have on
W1
• measured by expected returns and standard
deviation
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23. PORTFOLIO RETURN
• CALCULATING PORTFOLIO RETURN
– Expected returns:
• Method One:
rP = w1 - w0/ w0
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24. PORTFOLIO RETURN
– Expected returns:
• Method Two:
N
rp X i ri
t 1
where rP = the expected return of the portfolio
Xi = the proportion of the portfolio’s initial
value invested in security i
ri = the expected return of security i
N = the number of securities in the
portfolio
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25. Expected returns
• Portfolio expected return is a weighted
average of expected returns of its
constituents securities, i.e. each security
contributes to portfolio by its expected
return and its proportion in portfolio.
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26. PORTFOLIO RISK
• CALCULATING PORTFOLIO RISK
– Portfolio Risk:
• DEFINITION: a measure that estimates the extent
to which the actual outcome is likely to diverge
from the expected outcome
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27. PORTFOLIO RISK
• CALCULATING PORTFOLIO RISK
– Portfolio Risk:
1/ 2
N N
P Xi X j ij
i 1 j 1
where ij = the covariance of returns
between security i and security j
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28. PORTFOLIO RISK
• CALCULATING PORTFOLIO RISK
– Portfolio Risk:
• COVARIANCE
– DEFINITION: a measure of the relationship between two
random variables
– possible values:
» positive: variables move together
» zero: no relationship
» negative: variables move in opposite directions
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29. PORTFOLIO RISK
CORRELATION COEFFICIENT
– rescales covariance to a range of +1 to -1
Note: Covariance between two
ij ij i j securities i and j = correlation
between i and j x Standard deviation
of I x Standard deviation of j.
where
ρ i j = +1: denotes perfectly positive relationship
between i and j’s returns, implying that as returns
ij ij / i j
of i increase so does j’s.
ρ i j = -1: denotes perfectly negative relationship.
ρ i j = 0: indicate no identifiable relationship.
Note:
-1 ≤ ρ i j ≤ +1
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31. Graphical representation of correlation
B’s return
B’s return
A’s return A’s return
a) Perfectively Positively b) Perfectively negatively
correlated returns correlated returns
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32. Calculating Portfolio Risk
Exp: Given the following variance-covariance matrix for
three securities A, B, and C, as well as the percentage of
the portfolio for each security, calculate the portfolio’s risk
(standard deviation σp.
Variance-covariance Matrix
Security A Security B Security C
(50%) (30%) (20%)
Security A 459 -211 112
Security B -211 312 215
Security C 112 215 179
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33. Calculating Portfolio Risk
1/ 2
N N
Solution: We know PF risk equals P Xi X j ij
i 1 j 1
(.5x.5x459) = (.5x.3x-211)= (.5x.2x112)=
114.75 -31.65 11.2
(.3x.5x-211)= (.3x.3x312)= (.3x.2x215)=
-31.65 28.08 12.9
(.2x.5x112)= (.2x.3x215)= (.2x.2x179)=
11.2 12.9 7.16
Note: This reduces to
((.5x.5x459) + (.3x.3x312) + (.2x.2x179) +
½ ½
2 (.5x.3x-211) + 2 (.5x.2x112) + 2 (.3x.2x215)) = (134.89) = 11.61%
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34. Calculating Portfolio Risk
% of PF in each stock
Sec A 0.5
Variance-covariance Matrix Sec B 0.3
Sec A Sec B Sec C Sec C 0.2
Sec A 459 -211 112
Sec B -211 312 215
Sec C 112 215 179
Some important points about
Solution:
Variance-covariance Matrix:
114.75 -31.65 11.2
-31.65 28.08 12.9
11.2 12.9 7.16 1. It is Square Matrix, having N2
elements for N securities.
2. Variance appear on the
Portfolio Variance = 134.9 diagonal of matrix.
3. The matrix is symmetric.
Portfolio SD = 11.61 %
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35. Risk-seeking Investor
• Risk seeking investor will prefer:
– a gamble when presented a choice.
– Large gambles over small gambles, because utility
gained from winning is greater for him than disutility
gained from loosing.
– on indifference curve position of Farthest northeast
• Risk seeking investors utility functions will be
convex and their indifference curves will be
negatively sloped.
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36. Risk-neutral investors
Risk neutral investors:
• are indifferent to risk.
• Have horizontal indifference curves (IC).
• Will prefer farthest north position on IC.
Note that as risk-neutral
investor just for 1% additional
Return Preferable
A IC expected returns (from portfolio
15% B Portfolio A compared to B) is willing to
14% take 10% additional risk. Such
an investor consider the return
factor only, ignoring risk
altogether.
10% 20% Risk
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