Transcript Risk and Return for a Portfolio

Risk and Returns
Return Basics
– Holding-Period Returns
– Return Statistics
Risk Statistics
Return and Risk for Individual Securities
Return and Risk for Portfolios
– Efficient Set for Two Assets
– Efficient Set for Many Securities
Riskless Borrowing and Lending
Capital Asset Pricing Model
Chapters 9 & 10 – MBA504
Returns
Dollar Return = Dividend + Change in Market Value
dollar return
percentage return 
beginning market val ue
dividend  change in market val ue

beginning market val ue
 dividend yield  capital gains yield
Chapters 9 & 10 – MBA504
Example
Suppose you bought 100 shares of Wal-Mart (WMT)
one year ago today at $25. Over the last year, you
received $20 in dividends (= 20 cents per share × 100
shares). At the end of the year, the stock sells for $30.
How did you do?
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Holding-Period Returns
• The holding period return is the return
that an investor would get when
holding an investment over a period of
n years, when the return during year i
is given as ri:
holding period return 
 (1  r1 )  (1  r2 )   (1  rn )  1
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Example
• Suppose your investment provides the
following returns over a four-year period:
Year Return
1
10%
2
-5%
3
20%
4
15%
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Holding Period Returns
• Year-by-year historical rates of return starting in
1926 for the following five important types of
financial instruments in the United States:
–
–
–
–
–
Large-Company Common Stocks
Small-company Common Stocks
Long-Term Corporate Bonds
Long-Term U.S. Government Bonds
U.S. Treasury Bills
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Future Value of an Investment of $1 in 1925
$1,775.34
1000
$59.70
$17.48
10
Common Stocks
Long T-Bonds
T-Bills
0.1
1930
1940
1950
1960
1970
1980
1990
2000
Source: © Stocks, Bonds, Bills, and Inflation 2003 Yearbook™, Ibbotson Associates, Inc., Chicago (annually updates work by
Roger G. Ibbotson and Rex A. Sinquefield). All rights reserved.
Chapters 9 & 10 – MBA504
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Return Statistics
• The history of capital market returns can be
summarized by describing the
– average return
( R1    RT )
R
T
– the standard deviation of those returns
( R1  R) 2  ( R2  R) 2   ( RT  R) 2
SD  VAR 
T 1
– the frequency distribution of the returns.
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Historical Returns, 1926-2002
Average
Annual Return
Series
Standard
Deviation
Large Company Stocks
12.2%
20.5%
Small Company Stocks
16.9
33.2
Long-Term Corporate Bonds
6.2
8.7
Long-Term Government Bonds
5.8
9.4
U.S. Treasury Bills
3.8
3.2
Inflation
3.1
4.4
– 90%
Distribution
0%
+ 90%
Source: © Stocks, Bonds, Bills, and Inflation 2003 Yearbook™, Ibbotson Associates, Inc., Chicago (annually updates work by
Roger G. Ibbotson and Rex A. Sinquefield). All rights reserved.
Chapters 9 & 10 – MBA504
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Average Stock Returns and Risk-Free Returns
• Risk Premium is the additional return (in excess of riskfree rate) resulting from bearing risk.
• One of the most significant observations of stock market
data is this long-run excess of stock return over the riskfree return.
– The average excess return from large company common stocks
for the period 1926 through 1999 was 8.4% = 12.2% – 3.8%
– The average excess return from small company common stocks
for the period 1926 through 1999 was 13.2% = 16.9% – 3.8%
– The average excess return from long-term corporate bonds for
the period 1926 through 1999 was 2.4% = 6.2% – 3.8%
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Risk Premia
• Suppose that The Wall Street Journal announced
that the current rate for 1-year Treasury bills is
5%.
• What is the expected return on the market of
small-company stocks?
• Recall that the average excess return from small
company common stocks for the period 1926
through 1999 was 13.2%
• Given a risk-free rate of 5%, we have an expected
return on the market of small-company stocks of
18.2% = 13.2% + 5%
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The Risk-Return Tradeoff
18%
Small-Company Stocks
Annual Return Average
16%
14%
Large-Company Stocks
12%
10%
8%
6%
T-Bonds
4%
T-Bills
2%
0%
5%
10%
15%
20%
25%
30%
35%
Annual Return Standard Deviation
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Risk Premiums
• Rate of return on T-bills is essentially risk-free.
• Investing in stocks is risky, but there are
compensations.
• The difference between the return on T-bills and
stocks is the risk premium for investing in stocks.
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You can either sleep well or eat well.
-- An old saying on Wall Street
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Risk Statistics
• There is no universally agreed-upon
definition of risk.
• The measures of risk that we discuss are
variance and standard deviation.
– The standard deviation is the standard statistical
measure of the spread of a sample
– In terms of normal distribution
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Normal Distribution
• A large enough sample drawn from a normal
distribution looks like a bell-shaped curve.
Probability
The probability that a yearly return
will fall within 20.1 percent of the
mean of 13.3 percent will be
approximately 2/3.
– 3s
– 49.3%
– 2s
– 28.8%
– 1s
– 8.3%
0
12.2%
+ 1s
32.7%
+ 2s
53.2%
68.26%
+ 3s
73.7%
Return on
large company common
stocks
95.44%
99.74%
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Risk and Return: Individual Securities
• The characteristics of individual securities
that are of interest are the:
– Expected Return
– Variance and Standard Deviation
– Covariance and Correlation
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Expected Return and Variance
Rate of Return
Scenario Probability Stock fund Bond fund
Recession
33.3%
-7%
17%
Normal
33.3%
12%
7%
Boom
33.3%
28%
-3%
Consider the following two risky asset world. There
is a 1/3 chance of each state of the economy and the
only assets are a stock fund and a bond fund.
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Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock fund
Rate of
Squared
Return Deviation
-7%
3.24%
12%
0.01%
28%
2.89%
11.00%
0.0205
14.3%
Chapters 9 & 10 – MBA504
Bond Fund
Rate of
Squared
Return Deviation
17%
1.00%
7%
0.00%
-3%
1.00%
7.00%
0.0067
8.2%
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Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock fund
Rate of
Squared
Return Deviation
-7%
3.24%
12%
0.01%
28%
2.89%
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
1.00%
7%
0.00%
-3%
1.00%
7.00%
0.0067
8.2%
Consider a portfolio that is 50% invested in bonds and
50% invested in stocks.
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Risk and Return for a Portfolio
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Rate of Return
Stock fund Bond fund Portfolio
-7%
17%
5.0%
12%
7%
9.5%
28%
-3%
12.5%
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
squared deviation
0.160%
0.003%
0.123%
9.0%
0.0010
3.08%
The rate of return on the portfolio is a weighted average of
the returns on the stocks and bonds in the portfolio:
rP  wB rB  wS rS
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% in stocks
Risk
Return
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50.00%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
8.2%
7.0%
5.9%
4.8%
3.7%
2.6%
1.4%
0.4%
0.9%
2.0%
3.08%
4.2%
5.3%
6.4%
7.6%
8.7%
9.8%
10.9%
12.1%
13.2%
14.3%
7.0%
7.2%
7.4%
7.6%
7.8%
8.0%
8.2%
8.4%
8.6%
8.8%
9.00%
9.2%
9.4%
9.6%
9.8%
10.0%
10.2%
10.4%
10.6%
10.8%
11.0%
Portfolio Return
The Efficient Set for Two Assets
Portfolo Risk and Return Combinations
12.0%
11.0%
100%
stocks
10.0%
9.0%
8.0%
7.0%
100%
bonds
6.0%
5.0%
0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%
Portfolio Risk (standard deviation)
We can consider other
portfolio weights besides
50% in stocks and 50% in
bonds …
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Risk
Return
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
8.2%
7.0%
5.9%
4.8%
3.7%
2.6%
1.4%
0.4%
0.9%
2.0%
3.1%
4.2%
5.3%
6.4%
7.6%
8.7%
9.8%
10.9%
12.1%
13.2%
14.3%
7.0%
7.2%
7.4%
7.6%
7.8%
8.0%
8.2%
8.4%
8.6%
8.8%
9.0%
9.2%
9.4%
9.6%
9.8%
10.0%
10.2%
10.4%
10.6%
10.8%
11.0%
Portfolio Return
% in stocks
Portfolo Risk and Return Combinations
12.0%
100%
stocks
11.0%
10.0%
9.0%
8.0%
7.0%
6.0%
100%
bonds
5.0%
0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%
Portfolio Risk (standard deviation)
Note that some portfolios are
“better” than others. They have
higher returns for the same level of
risk or less.
These compromise the efficient frontier.
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Portfolio Risk as a Function of the
Number of Stocks in the Portfolio
s
In a large portfolio the variance terms are effectively
diversified away, but the covariance terms are not.
Diversifiable Risk;
Nonsystematic Risk;
Firm Specific Risk;
Unique Risk
Portfolio risk
Nondiversifiable risk;
Systematic Risk;
Market Risk
n
Thus diversification can eliminate some, but not all of the
risk of individual securities.
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return
Efficient Set for Many Securities
Individual Assets
sP
Consider a world with many risky assets; we can still
identify the opportunity set of risk-return
combinations of various portfolios.
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return
minimum
variance
portfolio
Individual Assets
sP
The section of the opportunity set above the
minimum variance portfolio is the efficient
frontier.
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return
Optimal Risky Portfolio with a RiskFree Asset
100%
stocks
rf
100%
bonds
s
In addition to stocks and bonds, consider a world
that also has risk-free securities like T-bills
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return
Riskless Borrowing and Lending
100%
stocks
Balanced
fund
rf
100%
bonds
s
Now investors can allocate their money across the T-bills and
a balanced mutual fund
Investor risk aversion is revealed in their choice of where to
stay along the capital allocation line.
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return
Market Equilibrium
100%
stocks
Optimal
Risky
Portfolio
rf
100%
bonds
s
All investors have the same CML because they all
have the same optimal risky portfolio given the
risk-free rate.
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Definition of Risk When Investors
Hold the Market Portfolio
• Researchers have shown that the best
measure of the risk of a security in a large
portfolio is the beta (b)of the security.
• Beta measures the responsiveness of a
security to movements in the market
portfolio.
Cov( R R )
bi 
i,
M
s ( RM )
2
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Security Returns
Estimating b with regression
Slope = bi
Return on
market %
Ri = a i + biRm + ei
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Relationship between Risk and
Expected Return (CAPM)
• Expected Return on the Market:
R M  RF  Market Risk Premium
• Expected return on an individual security:
Ri  RF  βi  ( R M  RF )
Market Risk Premium
This applies to individual securities held within welldiversified portfolios.
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• This formula is called the Capital Asset Pricing
Model (CAPM)
Expected
return on
a security
RiskBeta of the
=
+
×
free rate
security
Market risk
premium
• Assume bi = 0, then the expected return is RF.
• Assume bi = 1, then Ri  R M
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Expected
return
Relationship Between Risk & Expected Return
13.5%
3%
βi  1.5
RF  3%
1.5
b
R M  10%
R i  3%  1.5  (10%  3%)  13.5%
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Empirical Approaches to Asset Pricing
• Empirical methods are based less on theory
and more on looking for some regularities
in the historical record.
– E(R) = Rf + k1(beta) + k2(B/M) + k3(size)
– k1, k2>0, and k3<0
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Style Portfolios
• Related to empirical methods is the practice
of classifying portfolios by style e.g.
– Value portfolio
– Growth portfolio
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