12: Chapter Portfolio Selection and Diversification
To understand the theory of personal portfolio selection in theory and in practice
1 Copyright Prentice Hall Inc. 1999. Author: Nick Bagley
Objective
Chapter 12 Contents
12.1 The process of personal portfolio selection 12.2 The tradeoff between expected return and risk 12.3 Efficient diversification with many risky assets
2
Objectives
To understand the process of personal portfolio selection in theory and practice
3
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
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
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
12.1.1 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
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
Security Prices
100000 Stock Bond Stock_Mu Bond_Mu
10000 Value (Log)
1000
100
10 0 5 10 15 20 Years
9
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 navyblue stock trajectory The expected prices of the bond and the stock are straight lines on a log scale
10
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 bond price is higher in earlier years
11
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
Security Prices
100000 Stock Bond Stock_Mu Bond_Mu
10000 Value (Log)
1000
100
10 0 5 10 15 20 Years
13
25
30
35
40
Security Prices
100000 Stock Bond Stock_Mu Bond_Mu
10000 Value (Log)
1000
100
10 0 5 10 15 20 Years
14
25
30
35
40
...and Lots More!
Security Prices
100000 Stock Bond Stock_Mu Bond_Mu 100000 Stock Bond Stock_Mu Bond_Mu
Security Prices
10000 Value (Log)
10000 Value (Log)
1000
1000
100
100
10 0 100000 Stock Bond Stock_Mu Bond_Mu 5 10
Security Prices 15 20 25
Years
10 30 35 40 100000 Stock Bond Stock_Mu Bond_Mu 0 5 10
Security Prices 15 20 25
Years
30
35
40
10000 Value (Log)
10000 Value (Log)
1000
1000
100
100
10 0 5 10 15 20 Years 25 30 35 40
10 0
15
5
10
15
20 Years
25
30
35
40
From Conjecture to Hypothesis
You are probably ready to make the hypothesis that
the probability of the highrisk, highreturn security will outperform the lowrisk, low return increases with time
16
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
Security Prices
100000 Stock Bond Stock_Mu Bond_Mu
10000 Value (Log)
1000
100
10 0 5 10 15 20 Years 25
18
30
35
40
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
19
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
20
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 longterm goals, some precautionary funds should be kept in lower risk securities
21
12.1.2 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
22
12.1.3 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
23
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 ($$$$) a set of financial products which may be used together to satisfy most customer goals ($$)
24
12.1.4 Role of Professional Asset Managers
12. 2 TradeOff 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 riskfree asset is a defaultfree bond with the same maturity as the investor's decision (or possibly the trading) horizon
25
TradeOff between Expected Return and Risk
The assumption of a risky and riskless security simplifies the analysis
26
Combining the Riskless Asset and a Single Risky Asset
Assume that you invest w proportion of your wealth in a risky security and (1w) proportion of your wealth in a riskless security Let rs and rf be the returns on the risky security and the riskless security, respectively.
27
Combining the Riskless Asset and a Single Risky Asset
Your statistics background tells you how to determine the expected return and volatility of any twosecurity portfolio Form a new random variable, the return of the portfolio, rP, from the two security return variables, rs and rf
rP = w*rs + (1w) rf
28
Combining the Riskless Asset and a Single Risky Asset
The expected return of the portfolio is the weighted average of the expected returns on the securities:
E(rP)= w*E(rs) + (1w)*rf or, E(rP)= rf + w*(E(rs) rf)
29
Combining the Riskless Asset and a Single Risky Asset
The volatility of the portfolio is not quite as simple:
Variance = P2 = (w * s)2 + 2w*(1w) s* f+ ((1w)* f)2 Std. dev. = P = ((w * s)2 + 2w*(1 30 w) s* f+ ((1w)* f)2)1/2
Combining the Riskless Asset and a Single Risky Asset
We know something special about the portfolio, namely that riskless security has zero standard deviation or f = 0, and P becomes: P = ((w * s)2 + 2w*(1w) s*0 + ((1w)* 0)2)1/2 P = |w| * s and w = P/ s
31
Combining the Riskless Asset and a Single Risky Asset
Case 1: w > 0 Substituting w = P/ s from the previous slide into the last equation on slide 29 we get:
E(rP)= rf + [(E(rs) rf)/ s] P
32
Combining the Riskless Asset and a Single Risky Asset
Case 2: w < 0 Substituting w = P/ s from slide 31 into the last equation on slide 29 we get:
E(rP)= rf [(E(rs) rf)/ s] P
33
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%
34
A Portfolio of a Risky and a Riskless Security
0.30 0.25 0.20 0.15 Return 0.10 0.05 0.00 0.00 -0.05 -0.10 -0.15 -0.20 Volatility
35
0.10
0.20
0.30
0.40
0.50
SubOptimal Investments
Investments on the higher part of the line (i.e., case 1 on slide 32) are always preferred to investments on the lower part of the line (i.e., case 2 on slide 33) so for our current purposes we may ignore the lower line. That is, we will sell not the risky asset short and invest the proceeds in the riskless security
36
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 riskfree securities he does not own (also called short selling), and invest the proceeding in the risky investment They both use the same two securities
37
Achieving a Target Expected Return (1)
Suppose you want an expected return of 20% on your portfolio. What should be the allocation between the risky security and the riskless security? Assume E(rs) = 15%, s = 20%, and rf = 5% Compute w and 1w
38
Achieving a Target Expected Return (1)
Rearranging the last equation on slide 29 we get: w =
(E(rP) rf)/(E(rs) rf)
w = (0.20 0.05)/(0.15 0.05) = 150% 1w = 100% 150% = 50%
39
Assume that you manage a $50,000,000 portfolio A w 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 riskfree rate is required
Achieving a Target Expected Return (1)
40
Achieving a Target Expected Return (1)
How risky is this strategy?
= |w| * s = 1.5 * 0.20 = 0.30
P
The portfolio has a volatility of 30%
41
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
42
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 riskfree rate)
43
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
44
12.3 Efficient Diversification with Many 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
45
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
46
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
p = w11 + w2 2
47
Equations for Two Shares
Ideally, we would like to have a similar result for risk
p = w11 + w2 2 (wrong)
= w + 2w w + w
48
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
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
49
Variance with 2 Securities
W1* Sig1 W1* Sig1 1 W2* Sig2 Rho(1,2) 1
W2* Sig2 Rho(2,1)
= w + w + 2w1w2 1 2 1, 2
2 p 2 1 2 1 2 2 2 2
50
Variance with 3 Securities
W1* Sig1 W2* Sig2 W3* Sig3 W1* Sig1 1 Rho(1,2) Rho(1,3) 1 Rho(2,3) 1
2 3
W2* Sig2 Rho(2,1)
W3* Sig3 Rho(3,1) Rho(3,2)
2 p 2 1 2 1 2 2 2 2 2 3
= w + w + w + 2w1w2 1 2 1, 2 +
2w1w3 1 3 1,3 + 2w2 w3 2 3 2,3
51
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
52
Correlated Common Stocks
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
53
2-Shares: Is One "Better?"
0.16 0.14 0.12 Expected Return 0.1 0.08 0.06 0.04 0.02 0 0 0.05 0.1 0.15 Standard Deviation
54
0.2
0.25
0.3
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
55
Portfolio of Two Securities
0.25 Efficient 0.20 Expected Return
Share 1 Share 2
0.15
Minimum Variance
0.10 Suboptima l
0.05
0.00 0.15
0.17
0.19
0.21
0.23
56
0.25
0.27
0.29
Standard Deviation
Observation
Shorting the highrisk, lowreturn stock, and reinvesting in the lowrisk, high return stock, creates efficient portfolios
Shorting highrisk 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)
57
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 (minvar), W2 (minvar)
(Formulae given later)
Use ...