n
s=
S(
i=1
Ri
-
R
)2(
Pi
)
Standard Deviation（标准差）, s, is a statistical measure of the variability of a
distribution around its mean.
It is the square root of variance（方差）.
Long-term Government Bonds
U.S. Treasury Bills
6.1% 5.7% 3.9%
Inflation
4-8
3.1%
Risk Premiums（风险溢价）
The “extra” return earned for taking on risk Treasury bills are considered to be risk-free The risk premium is the return over and above the risk-free rate
State
Probability
C
T
Boom
0.3
0.15
0.25
Normal
0.5
0.10
0.20
Recession
???
0.02
0.01
RC = RT =
4-14
How to Determine the Expected Return and Standard Deviation
Stock BW
4-22
Risk Attitude Example
You have the choice between (1) a guaranteed dollar reward or (2) a coin-flip gamble of
$100,000 (50% chance) or $0 (50% chance). The expected value of the gamble is $50,000.
4-10

Expected Returns
Expected returns are based on the probabilities of possible outcomes In this context, “expected” means average if the process is repeated many times The “expected” return does not even have to be a possible return
4-23
Risk Attitude Example
What are the Risk Attitude tendencies of each?
Mary shows “risk aversion” because her “certainty equivalent” < the expected value of the gamble. Raleigh exhibits “risk indifference” because her “certainty equivalent” equals the expected value of the gamble. Shannon reveals a “risk preference” because her “certainty equivalent” > the expected value of the gamble.
4-3
4-4
Defining Return
Income received on an investment plus any change in market price, usually expressed as a percent of the beginning market price of the investment.
4-21
Determining Standard Deviation (Risk Measure)
n
s=
S(
i=1
Ri
-
R
)2
(n)
Note, this is for a continuous distribution where the distribution is
for a population. R represents the population mean in this example.
4-9
Historical Risk Premiums
Large stocks: 12.7 – 3.9 = 8.8% Small stocks: 17.3 – 3.9 = 13.4% Long-term corporate bonds: 6.1 – 3.9 =2.2% Long-term government bonds: 5.7 – 3.9 = 1.8%
-5% 4% 13% 22% 31% 40% 49% 58% 67%
Determining Expected Return (Discrete Dist.离散型分布)
n
R
=S i=1
(
Ri
)(
Pi
)
R is the expected return （期望报酬）for the asset,
Ri is the return for the ith possibility,
Sum 1.00
4-17
(Ri)(Pi) -.015 -.006 .036 .042 .033 .090
(Ri - R )2(Pi) .00576 .00288 .00000 .00288 .00576 .01728
Determining Standard Deviation (Risk Measure)
s2 = s= Stock T s2 = s=
4-19
Coefficient of Variation
（变化系数）
The ratio of the standard deviation of a distribution to the mean of that distribution. It is a measure of RELATIVE risk.
R = Dt + (Pt - Pt-1 )
Pt-1
4-5
Return Example
The stock price for Stock A was $10 per share 1 year ago. The stock is currently trading at $9.50 per share and shareholders just received a $1 dividend. What return
CV = s / R
CV of BW = .1315 / .09 = 1.46
4-20
Determining Expected Return (Continuous Dist.连续型分布)
n
R
=
S
i=1
(
Ri
)
/
(
n
)
R is the expected return for the asset,
Ri is the return for the ith observation, n is the total number of observations.
Mary requires a guaranteed $25,000, or more, to call off the gamble. Raleigh is just as happy to take $50,000 or take the risky gamble. Shannon requires at least $52,000 to call off the gamble.
Chapter 4
Risk and Return
风险与报酬
4-1
Dollar Returns
Total dollar return = income from investment + capital gain (loss) due to change in price
Example: You bought a bond for $950 1 year ago. You have received two coupons of $30 each. You can sell the bond for $975 today. What is your total dollar return? Income = Capital gain = Total dollar return =
4-11
Discrete vs. Continuous Distributions
Discrete
Continuous
0.4
0.035
0.35
0.03
0.3
0.025
0.25
0.02
0.2
0.015
0.15
0.01
0.1
0.005
0.05
0
0
-15% -3% 9% 21% 33%
4-12
-50% -41% -32% -23% -14%
n
s=
S
i=1
(
Ri
-
R
)2(
Pi
)
s = .01728
s = .1315 or 13.15%
4-18
Example: Variance and Standard Deviation
Consider the previous example. What are the variance and standard deviation for each stock? Stock C