Download Chapter 12

Chapter 12:
Portfolio Opportunities
and Choice
Objective
To understand the theory of personal
portfolio selection in theory
and in practice
1
Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall
Chapter 12 Contents
• 12.1 The process of personal portfolio
selection
• 12.2 The trade-off between expected
return and risk
• 12.3 Efficient diversification with many
risky assets
2
Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall
Objectives
• To understand the process of personal
portfolio selection in theory and practice
3
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Introduction
• How should you invest your wealth
optimally?
– Portfolio selection
• Your wealth portfolio contains
– Stock, bonds, shares of unincorporated
businesses, houses, pension benefits,
insurance policies, and all liabilities
4
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Portfolio Selection Strategy
• There are general principles to guide
you, but the implementation will depend
such factors as your (and your spouse’s)
– age, existing wealth, existing and target level
of education, health, future earnings
potential, consumption preferences, risk
preferences, life goals, your children’s
educational needs, obligations to older family
members, and a host of other factors
5
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12.1 The Process of Personal
Portfolio Selection
• Portfolio selection
– the study of how people should invest their
wealth
– process of trading off risk & expected return
to find the best portfolio of assets & liabilities
• Narrower dfn: consider only securities
• Wider dfn: house purchase, insurance, debt
• Broad dfn: human capital, education
6
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The Life Cycle
• The risk exposure you should accept
depends upon your age
• Consider two investments (rho=0.2)
– Security 1 has a volatility of 20% and an
expected return of 12%
– Security 2 has a volatility of 8% and an
expected return of 5%
7
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Price Trajectories
• The following graph show the the price
of the two securities generated by a
bivariate normal distribution for returns
– The more risky security may be thought of
as a share of common stock or a stock
mutual fund
– The less risky security may be thought of as
a bond or a bond mutual fund
8
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Security Prices
100000
Stock
Bond
Stock_Mu
Bond_Mu
Value (Log)
10000
1000
100
10
0
5
10
15
20
25
30
Years
9
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35
40
Interpretation of the Graph
• The graph is plotted on a log scale in so
that you can see the important features
• The magenta bond trajectory is clearly
less risky than the navy-blue stock
trajectory
• The expected prices of the bond and the
stock are straight lines on a log scale
10
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Interpretation of the Graph
• Recall the log scale: the volatility
increases with the length of the
investment
• You begin to form the conjecture that the
chances of the stock price being less
than the price bond is higher in earlier
years
11
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Generating More Trajectories
• This was just one of an infinite number
of trajectories generated by the same 2
means, 2 volatilities, and the correlation
– I have not cheated you, this was indeed the
first trajectory generated by the statistics
– the following trajectories are not reordered
nor edited
• Instructor: On slower computers there may be a delay
12
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Security Prices
100000
Stock
Bond
Stock_Mu
Bond_Mu
Value (Log)
10000
1000
100
10
0
5
10
15
20
25
30
Years
13
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35
40
Security Prices
100000
Stock
Bond
Stock_Mu
Bond_Mu
Value (Log)
10000
1000
100
10
0
5
10
15
20
25
30
Years
14
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35
40
…and Lots More!
Security Prices
Security Prices
100000
100000
Stock
Bond
Stock_Mu
Bond_Mu
1000
100
1000
100
10
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
Years
Years
Security Prices
Security Prices
30
35
40
30
35
40
100000
100000
Stock
Bond
Stock_Mu
Bond_Mu
Stock
Bond
Stock_Mu
Bond_Mu
10000
Value (Log)
10000
Value (Log)
Stock
Bond
Stock_Mu
Bond_Mu
10000
Value (Log)
Value (Log)
10000
1000
1000
100
100
10
10
0
5
10
15
20
Years
25
30
35
40
0
5
10
15
15
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20
Years
25
From Conjecture to
Hypothesis
• You are probably ready to make the
hypothesis that
– the probability of the high-risk, high-return
security will out-perform the low-risk, lowreturn increases with time
16
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But:
• I promised to be perfectly frank and
honest (pfah) with you about the
ordering of the simulated trajectories
• The next trajectory truly was the next
trajectory in the sequence, honest!
17
Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall
Security Prices
100000
Stock
Bond
Stock_Mu
Bond_Mu
Value (Log)
10000
1000
100
10
0
5
10
15
20
Years
25
18
30
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35
40
Explanation
• The bond and the stock end up at about
the same price, when the expected prices
are more than a magnitude apart
• There is either a very good explanation
for this, or there is a very high probability
that I have been much less than perfectly
frank and honest with you
19
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Another View of the Model
• A little mathematics, and we are able to
generate the following price distributions
for the stock and the bond for 2, 5, 10,
and 40 years into the future
20
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Probability of Future Price
0.035
Prob_Stock_2
Prob_Bond_2
Prob_Stock_5
Prob_Bond_5
Prob_Stock_10
Prob_Bond_10
Prob_Stock_40
Prob_Bond_40
Probability Density
0.030
0.025
0.020
0.015
0.010
0.005
0.000
0
50
100
150
Value
200
21
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250
300
• There is a lot going on here, so we will
further constrain our view
• First look at stock prices over a period of
10 years
• The prices are distributed according to
the lognormal distribution
22
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Probabilistic Stock Price Changes Over Time
0.020
Stock_Year_1
Stock_Year_2
Stock_Year_3
Stock_Year_4
Stock_Year_5
Stock_Year_6
Stock_Year_7
Stock_Year_8
Stock_Year_9
Stock_Year_10
0.018
Probability Density
0.016
0.014
0.012
0.010
0.008
0.006
0.004
0.002
0.000
0
200
400
Price
600
23
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800
Note
– the scale is $0 to $800
– the distribution diffuses and drifts towards
higher prices with time
– the diffusion is more pronounced in the
earlier years than in the later years
– you may see that the mode, median, and
mean appear to drift apart with time
24
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Bond in Time
• You will recall that if you invest in a 5year default-free pure discount bond for
5 years, the return is known with
certainty
• To avoid this effect, assume we invest in
short term bonds, and roll them over as
they mature
25
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Probabilistic Bond Price Changes over Time
0.045
Bond_Year_1
Bond_Year_2
Bond_Year_3
Bond_Year_4
Bond_Year_5
Bond_Year_6
Bond_Year_7
Bond_Year_8
Bond_Year_9
Bond_Year_10
0.040
Probability Density
0.035
0.030
0.025
0.020
0.015
0.010
0.005
0.000
0
100
200
Price
300
26
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400
Note
– the scale is now $0 to $400 (not $0 to $800
as in the case of the stock)
– we observe the same kind of diffusion and
drift behavior, and there is less of each
• (remember to adjust for the scale)
27
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Contrast of Trajectories and
Distributions
• The price distributions and the
trajectories were generated from the
same distribution. But
• They do not seem to agree
– The distributions appear to produce much
lower averages (expected returns) than the
trajectories
28
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Meaty Tails
• The resolution is that the distributions
have much meatier tails than your
intuition allows, pushing the median and
mean further and further from the mode
with time
• The region where the left tail appears to
have drifted into insignificance has a
profound affect on the mean
29
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Stock and Bonds Distributions
Compared at the Same Times
• The next sequence of slides contrasts the
distribution of stock and bond prices at 1,
2, 5, 10, and 40 into the future
• Some of the slides have different
measures of central tendency indicated
• Note the behavior of these statistics as
time increases
30
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Mode =104
Median=104
1-Year Out
Mode =106
Mean =104
0.0450
0.0400
Median=111
Stock_1_Year
Bond_1_Year
0.0350
Mean = 113
Density
0.0300
0.0250
0.0200
0.0150
0.0100
0.0050
0.0000
0
20
40
60
80
100
120
140
Price
31
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160
180
200
Two Years Out
0.035
0.030
Stock_2_Year
Bond_2_Year
Density
0.025
0.020
0.015
0.010
0.005
0.000
0
20
40
60
80
100
120
140
Price
32
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160
180
200
5-Years Out
Mode = 122
0.020
0.018
Stock_5_Year
Bond_5_Year
0.016
Median=
126
Mean = 128
Density
0.014
Mode = 135
0.012
0.010
Median=
165
Mean = 182
0.008
0.006
0.004
0.002
0.000
0
100
200
300
Price
400
33
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500
10-Years Out
0.012
0.010
Stock_10_Year
Density
0.008
Bond_10_Year
0.006
0.004
0.002
0.000
0
200
400
600
800
Value
34
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1,000
40 Years Out
Mode =503
0.002
Median=650
0.001
Mean =739
0.001
Stock_40_Year
Bond_40_Year
Mode =1,102
Density
0.001
0.001
Median=5,460
0.001
Mean =12,151
0.000
0.000
0.000
0
5,000
10,000
15,000
20,000
Value
35
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25,000
30,000
Slide Sequence Summary
• The next table summarizes the drifts of
the measures of central tendency
• Note that the means do in fact tie back
to the trajectories
• The last (anomalous?) trajectory not an
uncommon occurrence, and I was pfah
with you
36
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Value of Central Tendency Statistics for the LogNormal
1_Year
2_Years
5_Years
10_Years 40_Years
Assume: Sig = 0.20, Mu = 0.12
mode
$106.18
$112.75
median
$110.52
$122.14
mean
$112.75
$127.12
$134.99
$164.87
$182.21
$182.21 $1,102.32
$271.83 $5,459.82
$332.01 $12,151.04
Assume: Sig = 0.08, Mu = 0.05
mode
$104.12
$108.42
median
$104.79
$109.81
mean
$105.13
$110.52
$122.38
$126.36
$128.40
$149.78
$159.68
$164.87
mode
median
mean
The most probable price
50% of prices are equal or lower that this
The expected or average price
37
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$503.29
$650.13
$738.91
Implication for Investors
– If you are older, the average remaining life of
the investment is relatively short, and there
is a larger probability that an investment in
the risky security will result in a loss
– This is not serious if you have substantial
assets, in which case you can afford to take
the risk, and enjoy higher expected returns
38
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Implication for Investors
– If you are younger, the average remaining
life of retirement investment is longer, and
there is only a small probability that an
investment in the risky security will be less
than the “safer” one
– Investing in the less risky security will almost
always result in a significantly smaller
retirement income
39
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Implication for Investors
– Relatively early during a typical life cycle,
there may be a need to liquidate some
invested funds, perhaps for a house deposit,
a child’s education, or an uninsured medical
emergency
– In the case where liquidating an investment
early may damage long-term goals, some
precautionary funds should be kept in lowerrisk securities
40
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Time Horizons
– Planning horizon
• The total length of time for which one plans
– Decision horizon
• The length of time between decisions to
revise a portfolio
– Trading horizon
• The shortest possible time interval over which
investors may revise their portfolios
41
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Computing Life Expectancy
• Mortality tables may be organized as
three columns: actuary age, deaths/year
per 1000 live births, and remaining life
expectation. Note:
• if you survive from 60 to 65, for example,
the expected date of your death
advances by 3 to 4 years
• young women have a higher life expectation
than men, but this is lost42 with advancing age
Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall
Useful Internet Address
• The Society of Actuaries maintain a web
site that provides detailed mortality
tables, interactive computer models,
mortgage experiences, career
information, and current research papers
• www.soa.org
43
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Mortality Table
Male
Female
Age MDePm MExLife FDePm FExLife
60
16.08
17.51
9.47
21.25
61
17.54
16.79
10.13
20.44
65
25.42
14.04
14.59
17.32
70
39.51
10.96
22.11
13.67
75
64.19
8.31
38.24
10.32
80
98.84
6.18
65.99
7.48
85 152.95
4.46
116.1
5.18
90 221.77
3.18 190.75
3.45
95 329.96
1.87 317.32
1.91
44
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Deaths Per Thousand M & F
350
300
MDePm
Deaths / 1000
250
FDePm
200
150
100
50
0
60
65
70
75
80
85
Age
45
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90
95
•Life Expectancy
•Remaining Expected Life
•25
•20
•MExLife
•15
•FExLife
•10
•5
•0
•60
•65
•70
•75
•80
•85
•Age
46
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•90
•95
Risk Tolerance
• Your tolerance for bearing risk is a major
determinant of portfolio choices
– It is the mirror image of risk aversion
– Whatever its cause, we do not distinguish
between capacity to bear risk and attitude
towards risk
47
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Role of Professional Asset
Managers
• Most people have neither the time nor
the skill necessary to optimize a portfolio
for risk and return
– Professional fund managers provide this
service as
• individually designed solutions to the precise
needs of a customer ($$$$)
• a set of financial products which may be used
together to satisfy most customer goals ($$)
48
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12.2 Trade-Off between
Expected Return and Risk
• Assume a world with a single risky asset
and a single riskless asset
• The risky asset is, in the real world, a
portfolio of risky assets
• The risk-free asset is a default-free bond
with the same maturity as the investor’s
decision (or possibly the trading) horizon
49
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Trade-Off between Expected
Return and Risk
• The assumption of a risky and riskless
security simplifies the analysis
50
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The Risk-Reward Trade-Off Line
51
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Combining the Riskless Asset
and a Single Risky Asset
• Assume that you invest W1 proportion of
your wealth in security 1 and proportion
W2 of your wealth in security 2
• You must invest in either 1 or 2, so
W1+W2 = 1
• Let 2 be the riskless asset, and 1 be the
risky asset (portfolio)
52
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Combining the Riskless Asset
and a Single Risky Asset
• Your statistics background tells you how
to determine the expected return and
volatility of any two-security portfolio
– 1. Form a new random variable, the return
of the portfolio,RP, from the two given
random variables, R1 and R2
RP = W1*R1 + W2*R2
53
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Combining the Riskless Asset
and a Single Risky Asset
– The expected return of the portfolio is the
weighted average of the component returns
mp = W1*m1 + W2*m2
mp = W1*m1 + (1- W1)*m2
54
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Combining the Riskless Asset
and a Single Risky Asset
– The volatility of the portfolio is not quite as
simple:
sp = ((W1* s1)2 + 2W1* s1* W2* s2
+ (W2* s2)2)1/2
55
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Combining the Riskless Asset
and a Single Risky Asset
– We know something special about the
portfolio, namely that security 2 is riskless,
so
s2 = 0, and sp becomes:
sp = ((W1* s1)2 + 2W1* s1* W2* 0 +
(W2* 0)2)1/2
sp = |W1| * s1
56
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Combining the Riskless Asset
and a Single Risky Asset
– In summary
sp = |W1| * s1,
And:
mp = W1*m1 + (1- W1)*rf , So:
If
W1>0, mp = [(rf -m1)/ s1]*sp + rf
Else
mp = [(m1-rf )/ s1]*sp + rf
57
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Reflection
• The risk-free rate, rf, the risky security’s
expected rate of return, m1, and volatility,
s1, are constants, so we have a “ray”
that “reflects” from the expected return
axes at mp = rf
58
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Illustration
• Consider the set of all portfolios that may
be formed by investing (long and or
short) in
– a risky security with a volatility of 20% and
an expected return of 15%
– a riskless security with a volatility of 0% and
a known return of 5%
59
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A Portfolio of a Risky and a Riskless Security
0.30
0.25
0.20
Return
0.15
0.10
0.05
0.00
0.00
-0.05
0.10
0.20
0.30
-0.10
-0.15
-0.20
Volatility
60
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0.40
0.50
Sub-Optimal Investments
• Investments on the higher part of the
line are always preferred (by normal folk)
to investments on the lower part of the
line, so for our current purposes we may
ignore the lower line
• That is, we will not sell the risky asset
short and invest the proceeds in the
riskless security
61
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Capital Market Line
0.30
100%
Risky
0.25
Long risky and short risk-free
Return
0.20
0.15
100% RiskLong both risky and risk-free
less
0.10
0.05
0.00
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Volatility
62
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0.40
0.45
0.50
Observations
– An investor with a low risk tolerance may
invest in a portfolio containing a small % of
risky securities, and a correspondingly higher
% of riskless securities
– An investor with a high tolerance for risk
may sell risk-free securities he does not own,
and invest the proceeding in the risky
investment
– They both use the same two securities
63
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Observations
– The graph has been labeled the “capital
market line” a little prematurely
• We will soon discover that if
– the risky security is the market portfolio of risky
securities
– investors have similar expectations and time
horizons
• All investors will invest (long or short) in the
market portfolio and risk-free security
– The line joins the capital markets for risky
and risk-less securities 64
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Achieving a Target Expected
Return (1)
• Your boss has just read an ad’ that
included the data for the Janus Twenty
Fund (Scientific American, Sept 1998, page 6)
• “You beat them, or I’ll find another
portfolio manager”, she quips
• “Wrong way to compute return?” you
venture, as you rush for the door
65
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Mutual Fund Average % Total
Returns
YTD
1-Yr
3-Yrs
5-Yrs
10-Yrs Life
14.81 30.40 15.87 14.15 16.53 16.96
66
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To obtain a 20% Return
• You settle on a 20% return, and decide
not to pursue on the computational issue
– Recall:
mp = W1*m1 + (1- W1)*rf
– Your portfolio: s = 20%, m = 15%, rf = 5%
– So:
W1 = (mp - rf)/(m1 - rf)
= (0.20 - 0.05)/(0.15 - 0.05) = 150%
67
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To obtain a 20% Return
• Assume that you manage a $50,000,000
portfolio
• A W1 of 1.5 or 150% means you invest
(go long) $75,000,000, and borrow
(short) $25,000,000 to finance the
difference
• Borrowing at the risk-free rate is moot
68
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To obtain a 20% Return
• How risky is this strategy?
sp = |W1| * s1 = 1.5 * 0.20 = 0.30
• The portfolio has a volatility of 30%
69
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Important Observation
• It doesn’t require much skill to leverage a
portfolio; stockbrokers will let most
investors trade “on margin”
• When evaluating an investment’s
performance, you must examine both
the risk and the expected return
70
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Returning to the Example
• Advertisements for mutual funds do not
generally disclose a quantifiable measure
of risk, and Janus is no exception
– The advertised “Janus Twenty Fund” returns
are completely meaningless from a financial
point of view
– More information is needed
71
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Returning to the Example
• You can leverage the funds expected
returns up or down
• If you want an expected returns of 10%,
or, 20%, 30%, 40%, 50%, 60%… you
can have it (under the condition you can
continue to borrow at the risk-free rate)
72
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How Should my Boss Judge
my Fund’s Performance?
• It is a little early to answer this question
– If the risky security is the market portfolio,
then given your portfolio’s risk, consistent
returns above the CML line may appear
appealing
73
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Portfolio Efficiency
74
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Portfolio Efficiency
• An efficient portfolio is defined as the
portfolio that offers the investor the
highest possible expected rate of return
at a specific risk
• We now investigate more than one risky
asset in a portfolio
75
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12.3 Efficient Diversification
with Many Risky Assets
• We have considered
– Investments with a single risky, and a single
riskless, security
– Investments where each security shares the
same underlying return statistics
• We will now investigate investments with
more than one (heterogeneous) stock
76
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Portfolio of Two Risky Assets
• Recall from statistics, that two random
variables, such as two security returns,
may be combined to form a new random
variable
• A reasonable assumption for returns on
different securities is the linear model:
rp  w1r1  w2 r2 ; with w1  w2  1
77
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The Risk-Reward Trade-Off Curve:
Risky Assets Only
78
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Equations for Two Shares
• The sum of the weights w1 and w2 being
1 is not necessary for the validity of the
following equations, for portfolios it
happens to be true
• The expected return on the portfolio is
the sum of its weighted expectations
m p  w1 m 1  w 2 m 2
79
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Equations for Two Shares
• Ideally, we would like to have a similar
result for risk
s p  w1s 1  w 2s 2 (wrong)
– Later we discover a measure of risk with this
property, but for standard deviation:
2
2 2
2 2
p
1 1
1 2 1 2 1, 2
2 2
s  w s  2w w s s   w s
80
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Mnemonic
• There is a mnemonic that will help you
remember the volatility equations for two
or more securities
• To obtain the formula, move through
each cell in the table, multiplying it by
the row heading by the column heading,
and summing
81
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Variance with 2 Securities
W1*Sig1
W1*Sig1
W2*Sig2
1
Rho(1,2)
W2*Sig2 Rho(2,1)
1
s  w s  w s  2w1w2s1s 2 1,2
2
p
2 2
1 1
2
2
2
2
82
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Variance with 3 Securities
W1*Sig1 W2*Sig2 W3*Sig3
W1*Sig1
1
Rho(1,2) Rho(1,3)
W2*Sig2 Rho(2,1)
1
Rho(2,3)
W3*Sig3 Rho(3,1) Rho(3,2)
1
s  w s  w s  w s  2w1w2s1s 2 1,2 
2
p
2 2
1 1
2
2
2
2
2
3
2
3
2w1w3s1s 3 1,3  2w2 w3s 2s 3 2,3
83
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Note:
• The correlation of a with b is equal to the
correlation of b with a
• For every element in the upper triangle,
there is an element in the lower triangle
– so compute each upper triangle element
once, and just double it
• This generalizes in the expected manner
84
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Correlated Common Stock
• The next slide shows statistics of two
common stock with these statistics:
–
–
–
–
–
–
–
mean return 1 = 0.15
mean return 2 = 0.10
standard deviation 1 = 0.20
standard deviation 2 = 0.25
correlation of returns = 0.90
initial price 1 = $57.25
initial price 2 = $72.625
85
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2-Shares: Is One "Better?"
0.16
0.14
Expected Return
0.12
0.1
0.08
0.06
0.04
0.02
0
0
0.05
0.1
0.15
0.2
Standard Deviation
86
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0.25
0.3
Observation
• The statistics indicate that one security
appears to totally dominate the other
– Security 1 has a lower risk and higher return
than security 2
– In an efficient market:
• Wouldn’t everybody short 2, and buy 1?
• Wouldn’t supply and demand then cause the
relative expected returns to “flip”?
87
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Does it Happen?
• The purpose of selecting two shares with
this paradoxical form is to illustrate an
important point later
• This kind of relationship does occur in
the real world
88
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A Pair of Price Trajectories
• The next graph shows a trajectory of two
share prices with the statistics we have
selected
89
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Share Prices
350
Value (adjusted for Splits)
300
250
200
ShareP_1
ShareP_2
150
100
50
0
0
1
2
3
4
5
6
90
7
Years as Prentice Hall
Copyright © 2009 Pearson Education, Inc. Publishing
8
9
10
Observation
• If you were to “cut a piece” from one
trajectory, re-scale it for relative price
differences, and slide it over the other,
you would observe that both trajectories
behave in a broadly similar manner, but
each has independent behavior as well
• Quick confirmation is seen in the region 1
to 4 years where prices are close
91
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Correlation
• The two shares are highly correlated
– They track each other closely, but even
adjusting for the different average returns,
they have some individual behavior
92
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Portfolio of Two Securities
0.25
Efficient
Share 1
Expected Return
0.20
Share 2
0.15
Minimum
Variance
0.10
0.05
Suboptima
l
0.00
0.15
0.17
0.19
0.21
0.23
0.25
Standard Deviation
93
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0.27
0.29
Observation
• Shorting the high-risk, low-return stock,
and re-investing in the low-risk, highreturn stock, creates efficient portfolios
– Shorting high-risk by 80% of the net wealth
crates a portfolio with a volatility of 20% and
a return of 19% (c.f. 15% on security 1)
– Shorting by 180% gives a volatility of 25%,
and a return of 24% (c.f. 10% on security 2)
94
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Observation
• In order to generate a portfolio that
generates the same risk, but with a
higher return
– Compute the weights of the minimum
portfolio, W1 (min-var), W2 (min-var)
• (Formulae given later)
– Use the relationship
• Wi (sub-opt) +Wi (opt) = 2 * Wi (min-var)
95
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Observation
– Another way to generate the two securities is
to form two portfolios consisting of a risky
and a riskless security that each meet the
efficient frontier
– Result: two portfolios that are long the risky
security, and short the riskless security
– Short one of the portfolios and invest in the other
to generate one of the desired efficient portfolios
– Repeat to generate the other
• Prove that the riskless security becomes
96
irrelevant
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Optimal Combination of Risky
Assets
• The following slides are samples of the
computations used to generate the
graphs
97
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Fragments of the Output
Table
-0.30
-0.20
Data For two securities
-0.10
This data has been constructed
to produce the mean-varience paradox 0.00
0.10
mu_1
15.00%
0.20
mu_2
10.00%
0.30
sig_1
20.00%
sig_2
25.00%
rho
90.00%
1.30
1.40
1.50
w_1
w_2
Port_Sig Port_Mu
1.60
-2.50
3.50
0.4776
-0.0250
1.70
-2.40
3.40
0.4674
-0.0200
1.80
-2.30
3.30
0.4573
-0.0150
1.90
-2.20
3.20
0.4472
-0.0100
2.00
2.10
-2.10
3.10
0.4372
-0.0050
2.20
-2.00
3.00
0.4272
0.0000
2.30
-1.90
2.90
0.4173
0.0050
2.40
-1.80
2.80
0.4074
0.0100
2.50
-1.70
2.70
0.3976
0.0150
1.30
1.20
1.10
1.00
0.90
0.80
0.70
0.2723
0.2646
0.2571
0.2500
0.2432
0.2366
0.2305
0.0850
0.0900
0.0950
0.1000
0.1050
0.1100
0.1150
-0.30
-0.40
-0.50
-0.60
-0.70
-0.80
-0.90
-1.00
-1.10
-1.20
-1.30
-1.40
-1.50
0.1953
0.1949
0.1953
0.1962
0.1978
0.2000
0.2028
0.2062
0.2101
0.2145
0.2194
0.2247
0.2305
0.1650
0.1700
0.1750
0.1800
0.1850
0.1900
0.1950
0.2000
0.2050
0.2100
0.2150
0.2200
0.2250
98
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Sample of the Excel Formulae
=w_1*mu_1 + w_2*mu_2
w_1
-2.5
=A14+0.1
=A15+0.1
=A16+0.1
w_2
=1-A14
=1-A15
=1-A16
=1-A17
Port_Sig
=SQRT(w_1^2*sig_1^2
=SQRT(w_1^2*sig_1^2
=SQRT(w_1^2*sig_1^2
=SQRT(w_1^2*sig_1^2
+
+
+
+
2*w_1*w_2*sig_1*sig_2*rho
2*w_1*w_2*sig_1*sig_2*rho
2*w_1*w_2*sig_1*sig_2*rho
2*w_1*w_2*sig_1*sig_2*rho
+
+
+
+
w_2^2*sig_2^2)
w_2^2*sig_2^2)
w_2^2*sig_2^2)
w_2^2*sig_2^2)
Port_Mu
=w_1*mu_1
=w_1*mu_1
=w_1*mu_1
=w_1*mu_1
+
+
+
+
w_2*mu_2
w_2*mu_2
w_2*mu_2
w_2*mu_2
=SQRT(w_1^2*sig_1^2 + 2*w_1*w_2*sig_1*sig_2*rho + w_2^2*sig_2^2)
99
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Formulae for Minimum
Variance Portfolio
2
s 2  1, 2s 1s 2
*
w1  2
2
s 1  2 1, 2s 1s 2  s 2
s  1, 2s 1s 2
w  2
2
s 1  2 1, 2s 1s 2  s 2
2
1
*
2
 1 w
*
1
100
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Selection of the Preferred Portfolio
101
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Formulae for Tangent Portfolio
w1tan 
m
m
2


r
s
1
f
2  m 2  r f 1, 2s 1s 2
2
2






r
s

m

r

m

r

s
s

m

r
s
2
f
1
1
f
2
f
1, 2 1 2
1
f
2
w2tan  1  w1
2


0
.
10
*
0
.
25
 0.05 * 0.90 * 0.20 * 0.25
tan
w1 
0.05 * 0.20 2  0.10  0.05 * 0.90 * 0.20 * 0.25  0.10 * 0.252
w1tan  2 23
w2tan  1 23
102
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Example: What’s the Best
Return given a 10% SD?
m tan  w1tan m1  w2tan m 2
8
5
3
3
 0.2333
m tan  0.15  0.10
m tan
s
2
tan

 s  w  s
tan 2
1
 w
2
2
s tan
s tan
2
1
tan 2
2
2
2
 2 w1tan w2tans 1s 2 1, 2
2
8
 5
 8  5 
2
2
   0.20     0.25  2    * 0.2 * 0.25 * 0.90
3
 3
 3  3 
 0.2409
m tan  r f
0.2333  0.05
m
s  rf 
0.10  0.05  0.1261
s tan
0.2409
103
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Selecting the Preferred
Portfolio
• The procedure is as follows
– Find the portfolio weights of the tangent
portfolio of the line (CML) through (0, rf)
– Determine the standard deviation and
expectation of this point
– Construct the equation of the CML
– Apply investment criterion
104
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Achieving the Target Expected
Return (2): Weights
• Assume that the investment criterion is
to generate a 30% return
m criterion  m tangent w1  rf 1  w1 
m criterion  rf
0.30  0.05
w1 

 1.3636
m tangent  rf 0.2333  0.05
• This is the weight of the risky portfolio on
the CML
105
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Achieving the Target Expected
Return (2):Volatility
• Now determine the volatility associated
with this portfolio
s  w1s tangent  1.3636 * 0.2409  0.3285
• This is the volatility of the portfolio we
seek
106
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Achieving the Target Expected
Return (2): Portfolio Weights
COMPUTATION
WEIGHT
RISKLESS
-0.3636
-0.3636
ASSET 1
1.3636*2.6667
ASSET 2
1.3636*(-1.6667)
3.6363
TOTAL
-2.2727
1.0000
107
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Investment Strategies
• We have examined two strategies in
detail when
– the volatility is specified
– the return is specified
• Additionally, one of the graphs indicated
an approach to take when presented with
investor’s risk/return preferences
108
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Portfolio of Many Risky Assets
• In order to solve problems with more
than two securities requires tools such as
quadratic programming
• The “Solve” function in Excel may be
used to obtain solutions, but it is
generally better to use a software
package such as the one that came with
this book
109
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Chapter Assumptions
• The theory underlying this chapter is
essentially just probability theory, but
there are financial assumptions
110
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– We do not have to assume that the
generating process of returns is normal, but
we do assume that the process has a mean
and a variance. This is may not be the case
in real life
111
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– We assumed that the process was generated
without inter-temporal correlations. Some
investors believe that there is valuable
information in old price data that has not
been incorporated into the current price--this
runs counter to many rigorous empirical
studies.
112
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– There are no “hidden variables” that explain
some of the noise
113
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– Investors make decisions based on meanvariances alone
• statistics such as skewness & kurtosis have
been ignored
114
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• We have made the assumption the we
can lend at the risk-free rate, and that
we can “short” common stock
aggressively
115
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Summary
• There is no single investment strategy
that is suitable for all investors; nor for a
single investor for his whole life
• Time makes risky investments more
attractive than safer investments
• In practice, diversification has somewhat
limited power to reduce risk
116
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