Transcript Ch12part1

Chapter 12: Portfolio
Selection and Diversification
Objective
To understand the theory of personal
portfolio selection in theory
and in practice
Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley
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
Objectives
• To understand the process of personal
portfolio selection in theory and practice
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
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
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
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%
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
Security Prices
100000
Stock
Bond
Stock_Mu
Bond_Mu
Value (Log)
10000
1000
100
10
0
5
10
15
20
Years
25
30
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
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
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
Security Prices
100000
Stock
Bond
Stock_Mu
Bond_Mu
Value (Log)
10000
1000
100
10
0
5
10
15
20
Years
25
30
35
40
Security Prices
100000
Stock
Bond
Stock_Mu
Bond_Mu
Value (Log)
10000
1000
100
10
0
5
10
15
20
Years
25
30
35
40
…and Lots More!
Security Prices
Security Prices
100000
100000
Stock
Bond
Stock_Mu
Bond_Mu
Stock
Bond
Stock_Mu
Bond_Mu
10000
Value (Log)
Value (Log)
10000
1000
100
1000
100
10
10
0
5
10
Security
15
20Prices
25
30
35
40
0
5
10
Security
15
20Prices
25
Years
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)
30
Years
1000
1000
100
100
10
10
0
5
10
15
20
Years
25
30
35
40
0
5
10
15
20
Years
25
Odd Behavior
• The top slide has some odd behavior
between years 20 and 25
– The price of the stock and bond track each
other quite closely, and then they separate, and
both end up at 40-years close the their expected
prices
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
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!
Security Prices
100000
Stock
Bond
Stock_Mu
Bond_Mu
Value (Log)
10000
1000
100
10
0
5
10
15
20
Years
25
30
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
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
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
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
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
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
Bond in Time
• You will recall that if you invest in a 5-year
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
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
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)
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
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
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
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
Price
120
140
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
Price
120
140
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
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
Value
800
1,000
40 Years Out
0.002
Median=650
0.001
Mean =739
0.001
Stock_40_Year
Bond_40_Year
Mode =1,102
0.001
Density
Mode =503
0.001
Median=5,460
0.001
Mean =12,151
0.000
0.000
0.000
0
5,000
10,000
15,000
Value
20,000
25,000
30,000