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AP ECON 4410 Week 1, September 14 and 16, 2015 Instructor: Dr. David K. LeeDepartment of Economics York University
Topic: Risk and Return I
Readings: Chapters 10 and 11
Chapter 10: Risk and Return: Lessons from Market History
Returns
Dollar Return = Dividend + Change in Market Value
Returns: Example
• Suppose you bought 100 shares of XYC Inc. 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?
• Quite well. You invested $25 × 100 = $2,500. At the end of the year, you have shares
worth $3,000 and cash dividends of $20. Your dollar gain was $520 = $20 + ($3,000 –
$2,500).
• Your percentage gain for the year is
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 r i:
Holding Period Return: Example
yieldgainscapitalyielddividend
uemarket val beginning
uemarket valinchangedividend
uemarket val beginning
returndollarreturn percentage
500,2$520$%8.20
1)1()1()1(
return periodholding
21
nr r r
Year Return
1 10%
2 -5%
3 20%
4 15% %21.444421.
1)15.1()20.1()95(.)10.1(
1)1()1()1()1(
return periodholdingYour
4321
r r r r
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So, our investor made 9.58% on his money for four years, realizing a holding period
return of 44.21%
• The geometric average is not the same thing as the arithmetic average.
Return Statistics
• The history of capital market returns can be summarized by describing the
– average return
– the standard deviation of those returns
– the frequency distribution of the returns.
Average Stock Returns and Risk-Free Returns
• The Risk Premium is the additional return (over and above the risk-free rate) resulting
from bearing risk.
• One of the most significant observations of stock and bond market data is this long-run
excess of security return over the risk-free return.
•
The average return on T-bills was 5.97%.
–
The average excess return from Canadian common shares for the period 1957
through 2013 was:
Year Return
1 10%
2 -5%
3 20%
4 15% %58.9095844.
1)15.1()20.1()95(.)10.1(
)1()1()1()1()1(
returnaverageGeometric
4
4321
4
g
g
r
r r r r r
4)095844.1(4421.1
%104
%15%20%5%10
4return averageArithmetic 4321
r r r r
T
R R R
T )( 1
1
)()()( 2222
1
T
R R R R R RVARSD
T
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4.46% = 10.43% – 5.97%
– The average excess return from Canadian long-term bonds for the period 1957
through 2013 was:
2.41% = 8.38% – 5.97%
Risk Premia
• Suppose that the current rate for one-year Treasury bills is 2%.
• What is the expected return on the market of Canadian stocks?
•
Recall that the average excess return from Canadian common stocks for the period
1957 through 2013 was 4.46%.
• Given a risk-free rate of 2%, we have an expected return on the market of Canadian
common shares of: 6.46% = 4.46% + 2%
Risk Statistics
•
The measures of risk that we discuss are variance and standard deviation.
– The variance, and its square root, standard deviation measure variability.
– Its interpretation is facilitated by a discussion of the normal distribution.
Normal Distribution – Figure 10.6
The probability that a yearly return will fall within 16.64-percent of the mean of 10.43-
percent will be approximately 2/3.
The probability that a yearly return will fall within 34.10-percent (2×17.05) of the mean
of 10.43-percent will be 0.9544.
The probability that a yearly return will fall within 51.15-percent (3×17.05) of the mean
of 10.43-percent will be 0.9974.
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More on Average Returns
• Arithmetic average – return earned in an average period over multiple periods
– Represents the return earned in a typical year
• Geometric average – average compound return per period over multiple periods
– Represents actual return earned each year, compounded annually
Forecasting Return
• To address the time relation in forecasting returns, use Blume’s formula:
where, T is the forecast horizon and N is the number of years of historical data we are
working with. T must be less than N.
2008: A Year of Financial Crisis
•
2008: one of the worst years for stock investment in history.
•
Many of the world’s major markets declined more than those in the US and Canada.
• On the other hand, government bond values increased.
• What lessons should investors take away?
– Stocks have significant risk.
– Losses on a diversified portfolio of stocks and bonds would have been much
smaller.
Average Arithmetic N
T N AverageGeometric
N
T T R
1
1
1)(
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Chapter 11: Risk and Return: The Capital Asset Pricing Model
Expected Return, Variance, and Covariance
Consider the following two risky asset worlds. There is a 1/3 chance of each state of
the economy and the only assets are a stock fund and a bond fund.
Rate of Return Scenario Probab i l i ty Stock fund Bond fund
Recession 33.3% -7% 17%
Normal 33.3% 12% 7%
Boom 33.3% 28% -3%
Stock fund Bond Fund
Rate o f Squared Rate o f Squared
Scenario Retu rn Dev iati on Retu rn Devi ati on
Recession -7% 3.24% 17% 1.00%
Normal 12% 0.01% 7% 0.00%
Boom 28% 2.89% -3% 1.00%
Expected return 11.00% 7.00%
Variance 0.0205 0.0067
Standard Deviation 14.3% 8.2%
%11)(
%)28(3
1%)12(3
1%)7(3
1)(
S
S
r E
r E
%7)(
%)3(3
1%)7(3
1%)17(3
1)(
B
B
r E
r E
%24.3%)7%11( 2 %01.%)12%11( 2 %89.2%)28%11( 2
%)89.2%01.0%24.3(3
1%05.2
0205.0%3.14
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Note that stocks have a higher expected return than bonds and higher risk. Let us turn
now to the risk-return tradeoff of a portfolio that is 50% invested in bonds and 50%
invested in stocks.
The rate of return on the portfolio is a weighted average of the returns on the stocks
and bonds in the portfolio:
Note that stocks have a higher expected return than bonds and higher risk. Let us turn
now to the risk-return tradeoff of a portfolio that is 50% invested in bonds and 50%
invested in stocks.
The rate of return on the portfolio is a weighted average of the returns on the stocks
and bonds in the portfolio:
The expected rate of return on the portfolio is a weighted average of the expected
returns on the securities in the portfolio.
Rate of Return
Scenario Stock fund Bond fund Portfol io squared deviation
Recession -7% 17% 5.0% 0.160%
Normal 12% 7% 9.5% 0.003%
Boom 28% -3% 12.5% 0.123%
Expected return 11.00% 7.00% 9.0%
Variance 0.0205 0.0067 0.0010
Standard Deviation 14.31% 8.16% 3.08%
0205.0%3.14
S S B B P
r wr wr
%)17(%50%)7(%50%5
%)7(%50%)12(%50%5.9
%)3(%50%)28(%50%5.12
)()()(S S B B P
r E wr E wr E
%)7(%50%)11(%50%9
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The variance of the rate of return on the two risky assets portfolio is
where BS is the correlation coefficient between the returns on the stock and bond
funds.
The Efficient Set for Two Assets
We can consider other portfolio weights besides 50% in stocks and 50% in bonds …
BS S S B B
2
S S
2
B B
2
P ) ρσ )(wσ 2(w )σ (w )σ (wσ
% in stocks Risk Return
0% 8.2% 7.0%
5% 7.0% 7.2%
10% 5.9% 7.4%
15% 4.8% 7.6%
20% 3.7% 7.8%
25% 2.6% 8.0%
30% 1.4% 8.2%
35% 0.4% 8.4%
40% 0.9% 8.6%
45% 2.0% 8.8%
50.00% 3.08% 9.00%
55% 4.2% 9.2%
60% 5.3% 9.4%
65% 6.4% 9.6%
70% 7.6% 9.8%
75% 8.7% 10.0%
80% 9.8% 10.2%
85% 10.9% 10.4%
90% 12.1% 10.6%
95% 13.2% 10.8%
100% 14.3% 11.0%
5.0%
6.0%
7.0%
8.0%
9.0%
10.0%
11.0%
12.0%
0.0% 5.0% 10.0% 15.0% 20.0%
P o r t f o l i o R e t u r n
Portfolio Risk (standard deviation)
Portfolo Risk and Return Combinations
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Two-Security Portfolios with Various Correlations
Portfolo Risk and Return Combinations
5.0%
6.0%
7.0%
8.0%
9.0%
10.0%
11.0%
12.0%
0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%
Portfolio Risk (standard deviation)
PortfolioR
eturn
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The Efficient Set for Many Securities
Consider a world with many risky assets; we can still identify the opportunity set of risk-
return combinations of various portfolios.
Given the opportunity set we can identify the minimum variance portfolio
The section of the opportunity set above the minimum variance portfolio is the
efficient frontier.
Diversification
• Diversification can substantially reduce the variability of returns without an equivalent
reduction in expected returns.
• This reduction in risk arises because worse than expected returns from one asset are
offset by better than expected returns from another.
• However, there is a minimum level of risk that cannot be diversified away, and that is
the systematic portion.
•
The variance (risk) of a single security’s return can be broken down into:
• Systematic (Market) Risk
Economy-wide random events that affect almost all assets to a certain degree
•
Unsystematic (diversifiable) Risk
Random events that affect single security or small groups of securities
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• The Effect of Diversification:
unsystematic risk will significantly diminish in large portfolios
systematic risk is not affected by diversification since it affects all securities in any large
portfolio
Riskless Borrowing and Lending
In addition to stocks and bonds, consider a world that also has risk-free securities like T-
bills.
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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.
Relationship between Risk and Expected Return (CAPM)
• Expected Return on the Market:
• Expected return on an individual security:
̅ − =
)(
)(2
,
M
M i
i
R
R RCov
PremiumRiskMarket F
M R R
)(β F
M i F
i R R R R
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Expected Return on an Individual Security
• the Capital Asset Pricing Model (CAPM)
Assume = 0, then the expected return is RF.
Assume = 1, then
)(β F
M i F
i R R R R
M i R R
