Risk and Rates of Return
Stand-alone risk (statistics review)
Portfolio risk (investor view) --
Risk & return: CAPM/SML (market
Risk is viewed primarily from the 2
Management cares about risk because
stockholders care about risk.
If stockholders like or dislike something about a
company (like risk) it affects the stock price.
Risk affects the discount rate for future returns --
directly affecting the multiple (P/E ratio)
Thus, the concern is still about the stock price.
Stockholders have portfolios of investments –
they have stock in more than just one company
and a great deal of flexibility in which stocks they
What is investment risk?
Investment risk pertains to the
uncertainty regarding the rate of return.
Especially when it is less than the
expected (mean) return.
The greater the chance of low or
negative returns, the riskier the
Return = dividend + capital gain or loss
Dividends are relatively stable
Stock price changes (capital gains/losses)
are the major uncertain component
There is a range of possible outcomes and
a likelihood of each -- a probability
Expected Rate of Return
The mean value of the probability
distribution of possible returns
It is a weighted average of the
outcomes, where the weights are the
Expected Rate of Return
k p1k1 p2 k 2 ... pn k n
pi k i
Economy Prob. T-bill HT Coll USR MP
Recession 0.1 8.0% -22.0% 28.0% 10.0% -13.0%
Below avg 0.2 8.0 -2.0 14.7 -10.0 1.0
Average 0.4 8.0 20.0 0.0 7.0 15.0
Above avg 0.2 8.0 35.0 -10.0 45.0 29.0
Boom 0.1 8.0 50.0 -20.0 30.0 43.0
Why is the T-bill
of the economy?
Will return be 8%
regardless of the economy?
Do T-bills really promise a
completely risk-free return?
No, T-bills are still exposed to
the risk of inflation.
However, not much unexpected
inflation is likely to occur over a
relatively short period.
Do the returns of HT and Coll.
move with or counter to the
High Tech: With. Positive correlation.
Negative correlation. Unusual.
Calculate the expected rate of
return for each alternative:
^ = expected rate of return
k = k i pi
^ = (-22%)0.1 + (-2%)0.20
+ (20%)0.40 +
+ (50%)0.1 = 17.4%
Calculate others on your own
HT appears to be the best, but is it really?
What‟s the standard deviation
of returns for each alternative?
= standard deviation
= Variance =
= (ki k i
ˆ )2 p
In a sample of observations
One often assumes that data are from
an approximately normally distributed
about 68.26% of the values are at
within 1 standard deviation away from
95.46% of the values are within two
standard deviations and
99.73% lie within 3 standard
(k i k) Pi
HT = [- 22 - 17.4 2 0.1 + - 2 - 17.4 2 0.2 20 - 17.4 2 0.4 + 35 - 17.4 2 0.20
50 17.4 2 0.1]0.5 0.5 20.0748599
T-bills = 0.0%.
HT = 20.0%. Coll = 13.4%.
USR = 18.8%.
M = 15.3%.
Standard deviation (i)
measures total, or stand-
The larger the i , the lower
the probability that actual
returns will be close to the
Expected Returns vs. Risk:
Security Expected Risk,
High Tech 17.4% 20.0
Market 15.0 15.3
US Rubber 13.8* 18.8*
T-bills 8.0 0.0
Collections 1.7* 13.4*
*Return looks low relative to
Coefficient of variation (CV):
Standardized measure of dispersion
about the expected value:
CV = =
Shows risk per unit of return.
Portfolio Risk & Return
Assume a two-stock portfolio
with $50,000 in HighTech and
$50,000 in Collections.
Calculate kp and p.
kp is a weighted average:
kp = w i k i .
kp = 0.5(17.4%) + 0.5(1.7%) = 9.6%
^ ^ ^
kp is between kHT and kCOLL.
Economy Prob. HT Coll. Port.
Recession 0.10 -22.0% 28.0% 3.0%
Below avg. 0.20 -2.0 14.7 6.4
Average 0.40 20.0 0.0 10.0
Above avg. 0.20 35.0 -10.0 12.5
Boom 0.10 50.0 -20.0 15.0
kp = (3.0%)0.1 + (6.4%)0.20 + (10.0%)0.4
+ (12.5%)0.20 + (15.0%)0.1 = 9.6%
3.0 - 9.6 0.1 6.4 9.6 0.20
P = 10.0 9.6 0.4 12.5 9.6 0.20
15.0 9.6 0.1
CVP = = 0.34.
p = 3.3% is much lower than that of
either stock (20% and 13.4%).
p = 3.3% is also lower than avg. of HT
and Coll, which is 16.7%.
Portfolio provides avg. return but
Negative correlation is present
between HT and Coll but is not
required to have a diversification
General Statements about risk:
Most stocks are positively
correlated. rk,m 0.65.
You still get a lot of
diversification effect at .65
35% for an average stock.
Combining stocks generally
What would happen to the
riskiness of a 1-stock
portfolio as more randomly
selected stocks were added?
p would decrease because the
added stocks would not be
35 Company Specific risk
Total Risk, P
10 20 30 40 ...... 1500+
# stocks in portfolio
As more stocks are added, each
new stock has a smaller risk-
p falls very slowly after about 40
stocks are included. The lower
limit for p is about 20% = M .
Decomposing Risk—Systematic (Market) and
Unsystematic (Business-Specific) Risk
Fundamental truth of the investment world
– The returns on securities tend to move up and
• Not exactly together or proportionately
Events and Conditions Causing Movement in
– Some things influence all stocks (market risk)
• Political news, inflation, interest rates, war, etc.
– Some things influence only particular firms
• Earnings reports, unexpected death of key
– Some things affect all companies within an
• A labor dispute, shortage of a raw material
Total = Market + Firm specific
risk risk risk
Market risk is that part of a
security‟s risk that cannot be
eliminated by diversification.
Firm-specific risk is that part
of a security‟s risk which can
be eliminated with
By forming portfolios, we
can eliminate nearly half
the riskiness of individual
stocks (35% vs. 20%).
(actually35% vs. 20% is a
CAPM -- Capital Asset
If you chose to hold a one-
stock portfolio and thus are
exposed to more risk than
diversified investors, would
you be compensated for all
the risk you bear?
Stand-alone risk as
measured by a stock‟s
or CV is not important to
Rational, risk averse
investors are concerned
with p , which is based
on market risk.
Beta measures a stock‟s 34
market risk. It shows a
stock‟s volatility relative
to the market.
Beta shows how risky a
stock is when the stock is
held in a well-diversified
The higher beta, the
higher the expected rate
How are betas calculated?
Run a regression of past
returns on Stock i versus
returns on the market.
The slope coefficient is
the beta coefficient.
Illustration of beta = slope:
ki 20 .
15 . Year
2 -5 -10
10 3 12 16
-5 0 5 10 15 20 kM
ki = -2.59 + 1.44 kM
“By Eye.” Plot points, draw in
„regression‟ line, get slope as
b = Rise/Run. The “rise” is the
difference in ki , the “run” is the
difference in kM . For example,
how much does ki increase or
decrease when kM increases
from 0% to 10%?
Computer: statistics program or
spreadsheet regression program
Calculator. Enter data points, and
calculator does least squares
regression: ki = a + bkM = -2.59 +
1.44kM . r = corr. coefficient = 0.997.
Find someone‟s estimate of beta
for a given stock on the web
In the real world, we would use
weekly or monthly returns, with at
least a year of data, and would
always use a computer.
If beta = 1.0, average risk stock. (The
„market‟ portfolio has a beta of 1.0.)
If beta > 1.0, stock riskier than
If beta < 1.0, stock less risky than
Most stocks have betas in the range
of 0.5 to 1.5.
Some ag. related companies have
betas less than 0.5
=1, get the market expected
<1, earn less than the market
>1, get an expected return
greater than the market
Can a beta be negative?
Answer: Yes, if the
correlation between ki and kM
Then in a “beta graph” the
regression line will slope
Negative beta -- rare
b = 1.29
-20 0 20 40
-20 b = -0.86
Security Return (Beta)
HighTech 17.4% 1.29
“Market” 15.0 1.00
US Rubber 13.8 0.68
T-bills 8.0 0.00
Collections 1.7 -0.86
Riskier securities have higher
returns, so the rank order is O.K.
Given the beta of a stock, a theoretical
required rate of return can be
The Security Market Line (SML) is used.
SML: ki = kRF + (kM - kRF)bi
MRP= market risk premium
ki = kRF + (kM - kRF)bi
For Term Projects
Use KRF = 2.5%; this is the 10 year treasury
rate. Often it is argued to use a shorter
term rate, but we are going to use 2.5%.
Use MRP = 5%
The historical average MRP is about 5%.
Find your own beta from the web
On Yahoo Finance look up your company
and then the “key statistics” tab on the left
will give you their beta
Use the SML to calculate
the required returns (for the example)
SML: ki = kRF + (kM - kRF)bi .
Assume kRF = 8%.
Note that kM = kM is 15%.
MRP = kM - kRF = 15% - 8% = 7%
Required rates of return:
kHT = 8.0% + (15.0% - 8.0%) 1.29
= 8.0% + (7%)1.29
= 8.0% + 9.0% = 17.0%
kM = 8.0% + (7%)1.00 = 15.0%
kUSR = 8.0% + (7%)0.68 = 12.8%
kTbill = 8.0% + (7%)0.00 = 8.0%
kColl = 8.0% + (7%)(-0.86) = 2.0%
Calculate beta for a portfolio with
50% HT and 50% Collections:
bP = weighted average of the
betas of the stocks in the portfolio
= 0.5(bHT) + 0.5(bColl)
= 0.5(1.29) + 0.5(-0.86)
= 0.22 .
Weights are the proportions invested
in each stock.
The required return on the HT/Coll.
kP = Weighted average k
= 0.5(17%) + 0.5(2%) = 9.5% .
Or use SML for the portfolio:
kP = kRF + (kM - kRF) bP
= 8.0% + (15.0% - 8.0%) (0.22)
= 8.0% + 7%(0.22) = 9.5% .
Using Beta—The Capital Asset Pricing
The CAPM helps us determine how stock prices
are set in the market
Developed in 1950s and 1960s by Harry
Markowitz and William Sharpe
The CAPM's Approach
People won't invest unless a stock's expected
return is at least equal to their required return
The CAPM attempts to explain how investors'
required returns are determined
Has the CAPM been verified through
Not completely. Because
statistical tests have
problems which make
Investors seem to be concerned with
both market risk and total risk.
Therefore, the SML may not produce
a correct estimate of ki:
ki = kRF + (kM - kRF)b + ?
Also, CAPM/SML concepts are
based on expectations, yet
betas are calculated using
historical data. A company‟s
historical data may not reflect
investors‟ expectations about