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Chapter 10 Capital Market Theory: An Overview10.1 Returns
10.2 Holding-Period Returns
10.3 Return Statistics
10.4 Average Stock Returns and Risk-Free Returns
10.5 Risk Statistics
10.6 Summary and Conclusions
10.1 Returns
10.2 Holding-Period Returns
10.3 Return Statistics
10.4 Average Stock Returns and Risk-Free Returns
10.5 Risk Statistics
10.6 Summary and Conclusions
10.1 Returns: Example
• % return = (cash in - cash out)/(cash out) = (cash in / cash out) - 1e.g. Bought a share of XYZ at $100 a year ago. Today you received $4
dividend and just sold the share at $106.Cash in = $4 + $106 Cash out = $100 % return = ((4+106)/100) – 1 = 10% p.a.% return= (4/100) + (106/100) – 1
= (4/100) + (106 – 100)/100= dividend yield + capital gains yield= 4% + 6% = 10% p.a.
Qu: What are two parts of (total %) return?• Dollar returns vs percentage returns
• % return = (cash in - cash out)/(cash out) = (cash in / cash out) - 1e.g. Bought a share of XYZ at $100 a year ago. Today you received $4
dividend and just sold the share at $106.Cash in = $4 + $106 Cash out = $100 % return = ((4+106)/100) – 1 = 10% p.a.% return= (4/100) + (106/100) – 1
= (4/100) + (106 – 100)/100= dividend yield + capital gains yield= 4% + 6% = 10% p.a.
Qu: What are two parts of (total %) return?• Dollar returns vs percentage returns
10.2 Holding Period Return: Example
• Suppose your investment provides the following returns over a four-year period:
• Suppose your investment provides the following returns over a four-year period:
Year Return1 10%2 -5%3 20%4 15%
Your holding period return =
= (1.10)*(0.95)*(1.2)*(1.15)
= (1 + r1)* (1 + r2)* (1 + r3)* (1 + r4) – 1
= 44.21%
0.1
10
1000
1957 1962 1967 1972 1977 1982 1987 1992 1997 2002
The Future Value of an Investment of $1 in 1957
$42.91
$20.69
17.86$)1()1()1(1$ 200319581957 rrr
Common StocksLong BondsT-Bills
Holding Period Return: Example
• What would be the annual return on her investment?Suppose our investor made r% p.a. on her money for four years. At the end of four years, she has $1.4421: 1.4421 = (1+r%)4
Solving for r yields the geometric return: 9.58% p.a.
• What would be the annual return on her investment?Suppose our investor made r% p.a. on her money for four years. At the end of four years, she has $1.4421: 1.4421 = (1+r%)4
Solving for r yields the geometric return: 9.58% p.a.
Year Return1 10%2 -5%3 20%4 15% %58.9095844.
1)15.1()20.1()95(.)10.1(
)1()1()1()1()1(
return average Geometric
4
43214
g
g
r
rrrrr
Holding Period Return: Example
• Note that the geometric average is not same as the arithmetic average:
• Note that the geometric average is not same as the arithmetic average:
Year Return1 10%2 -5%3 20%4 15%
%104
%15%20%5%104
return average Arithmetic 4321
rrrr
10.2 Holding-Period Returns
• The holding period return is the return that an investor would get when holding an investment over a period of n periods.
• The holding period return is given as, where ri represents the return during period i:
• The holding period return is the return that an investor would get when holding an investment over a period of n periods.
• The holding period return is given as, where ri represents the return during period i:
1)1()1()1(
returnperiod holding
21 nrrr
• Table 10.1 shows annual (holding period) returns on several portfolios.
•Qu on market history.
10.3 Return Statistics• One can summarize the data with the average
(mean), the standard deviation, and the frequency distribution of the returns.
• Average return: What is the return that an investor could have realized during a year over the 1948-2000 period?
• Example: Annual returns for 1997-2000 are 14.98%, -1.58%, 31.71%, and 7.41%. Average return is (14.98% -1.58% + 31.71% + 7.41%)/4 =13.13%.
• One can summarize the data with the average (mean), the standard deviation, and the frequency distribution of the returns.
• Average return: What is the return that an investor could have realized during a year over the 1948-2000 period?
• Example: Annual returns for 1997-2000 are 14.98%, -1.58%, 31.71%, and 7.41%. Average return is (14.98% -1.58% + 31.71% + 7.41%)/4 =13.13%.
T
RRR T )( 1
Average Standard Investment Annual Return Deviation Distribution
Canadian common stocks 10.64% 16.41%
Long Bonds 8.96 10.36
Treasury Bills 6.80 4.11
Inflation 4.29 3.63
Historical Returns, 1957-2003
– 60% + 60%0%
10.4 Average Stock Returns and Risk-Free Returns
• The Risk Premium is the difference between risky and risk-free return.
• For 1957-2003, risk premium for • large cap common stocks: 3.84% = 10.64% – 6.8%
• long-term (corporate+gov’t) bonds: 2.16%= 8.96% – 6.8%
• One of the most significant observations of stock market data is this long-run risk premium.
• The Risk Premium is the difference between risky and risk-free return.
• For 1957-2003, risk premium for • large cap common stocks: 3.84% = 10.64% – 6.8%
• long-term (corporate+gov’t) bonds: 2.16%= 8.96% – 6.8%
• One of the most significant observations of stock market data is this long-run risk premium.
Risk Premia• Suppose the current one-year Treasury bills rate is
5%. What is the expected return on the market of small-company stocks?
• The average risk premium for small cap (capitalization) stocks for the period 1976-2000 was 5.68%
• Given a risk-free rate of 5%, expected return on small-cap stocks is 14.8% = 5.68% + 5%.
• Why risk premium?
• Suppose the current one-year Treasury bills rate is 5%. What is the expected return on the market of small-company stocks?
• The average risk premium for small cap (capitalization) stocks for the period 1976-2000 was 5.68%
• Given a risk-free rate of 5%, expected return on small-cap stocks is 14.8% = 5.68% + 5%.
• Why risk premium?
Rates of Return 1948-2000
-30
-20
-10
0
10
20
30
40
50
60
1945 1955 1965 1975 1985 1995
Common Stocks
Long Bonds
T-Bills
Why Risk Premiums?
• Rate of return on T-bills is essentially risk-free.
• Investing in stocks is risky, but there are compensations.
• The risk premium is the reward to bearing risk in stocks.
• Rate of return on T-bills is essentially risk-free.
• Investing in stocks is risky, but there are compensations.
• The risk premium is the reward to bearing risk in stocks.
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
11.00
12.00
0.00 5.00 10.00 15.00 20.00 25.00
Annual Return Standard Deviation
An
nu
al
Ret
urn
Aver
age
The Risk-Return Tradeoff (1957-2003)
Long Bonds
T-Bills
Large-Company Stocks
Stock Market Volatility
Source: © Stocks, Bonds, Bills, and Inflation 2000 Yearbook™, Ibbotson Associates, Inc., Chicago (annually updates work by Roger G. Ibbotson and Rex A. Sinquefield). All rights reserved.
The volatility of stocks is not constant from year to year.
-30
-20
-10
0
10
20
30
40
50
1955 1965 1975 1985 1995 2005
Common StocksLong BondsT-bills
10.5 Risk Statistics• There is no universally agreed-upon definition of risk.
• We mainly use variance and standard deviation.
• Example: Annual returns for 1997-2000 are 14.98%, -1.58%, 31.71%, and 7.41%, with an average return of 13.1%.
var = [(14.98%-13.1%)2 + (-1.58%-13.1%)2 +(31.71%-13.1%)2 + (7.41%-13.1%)2 ]/3 = 0.019925
stdev = sqrt (var) = sqrt (0.019925) = 0.1222
• There is no universally agreed-upon definition of risk.
• We mainly use variance and standard deviation.
• Example: Annual returns for 1997-2000 are 14.98%, -1.58%, 31.71%, and 7.41%, with an average return of 13.1%.
var = [(14.98%-13.1%)2 + (-1.58%-13.1%)2 +(31.71%-13.1%)2 + (7.41%-13.1%)2 ]/3 = 0.019925
stdev = sqrt (var) = sqrt (0.019925) = 0.1222
1
)()()( 222
21
T
RRRRRRVARSD T
Normal Distribution
• A large sample from a normal distribution looks like a bell-shaped curve.
• A large sample from a normal distribution looks like a bell-shaped curve.
Probability
Return onlarge company commonstocks
99.74%
– 3 – 36.35%
– 2 – 19.87%
– 1 – 3.39%
013.09%
+ 1 29.57%
+ 2 46.05%
+ 3 62.53%
The probability that an annual return will fall between –3.39% and 29.57% is approximately 2/3.
68.26%
95.44%
Normal Distribution S&P 500 Return Frequencies
0
2
5
11
16
9
1212
1
2
110
0
2
4
6
8
10
12
14
16
62%52%42%32%22%12%2%-8%-18%-28%-38%-48%-58%
Annual returns
Ret
urn
fre
qu
ency
Normal approximationMean = 12.8%Std. Dev. = 20.4%
Source: © Stocks, Bonds, Bills, and Inflation 2002 Yearbook™, Ibbotson Associates, Inc., Chicago (annually updates work by Roger G. Ibbotson and Rex A. Sinquefield). All rights reserved.
10.6 Summary and Conclusions• Stocks have outperformed bonds over most of the
twentieth century.• Stocks have exhibited more risk than bonds.• The small cap stocks have outperformed large cap
stocks over most of the twentieth century, again with more risk.
• The statistical measures in this chapter are necessary building blocks for the material of the next three chapters.
• Stocks have outperformed bonds over most of the twentieth century.
• Stocks have exhibited more risk than bonds.• The small cap stocks have outperformed large cap
stocks over most of the twentieth century, again with more risk.
• The statistical measures in this chapter are necessary building blocks for the material of the next three chapters.
