Chapter 6
Risk and Rates of Return
Chapter 6 Objectives
Inflation and rates of return
How to measure risk
(variance, standard deviation, beta)
How to reduce risk
(diversification)
How to price risk
(security market line, CAPM)
2
Historical Risk and Return
Annual From 1926 to 1999
Avg. Return
Small Stocks
17.6%
Large Co. Stocks
13.3%
L-T Corp Bonds
5.9%
L-T Govt. Bonds
5.5%
T-Bills
3.8%
Inflation
3.2%
Std Dev.
33.6%
20.1%
8.7%
9.3%
3.2%
4.6%
3
Why are these rates different?
90-day Treasury Bill
90-day Commercial Paper
2-year US Treasury Note
10-year US Treasury Note
10-year Corporate Bond
1.7%
1.8%
3.0%
5.0%
6.9%
4
Inflation, Rates of Return, and the
Fisher Effect
Interest
Rates
Conceptually:
Interest Rates
Conceptually:
Nominal
risk-free
Interest
Rate
krf
=
Interest Rates
Interest Rates
Conceptually:
Nominal
risk-free
Interest
Rate
krf
=
Real
risk-free
Interest
Rate
k*
Interest Rates
Conceptually:
Nominal
risk-free
Interest
Rate
krf
=
Real
risk-free
Interest
Rate
k*
+
Inflationrisk
premium
IRP
Interest Rates
Conceptually:
Nominal
risk-free
Interest
Rate
=
Real
risk-free
Interest
Rate
krf
k*
+
Inflationrisk
premium
IRP
Mathematically:
(1 + krf) = (1 + k*) (1 + IRP)
This is known as the “Fisher Effect”
Interest Rates
Suppose the real rate is 3%, and
the nominal rate is 8%. What is
the inflation rate premium?
(1 + krf) = (1 + k*) (1 + IRP)
(1.08) = (1.03) (1 + IRP)
(1 + IRP) = (1.0485), so
IRP = 4.85%
11
Term Structure of Interest Rates
The pattern of rates of return for
debt securities that differ only in the
length of time to maturity.
Term Structure of Interest Rates
The pattern of rates of return for
debt securities that differ only in the
length of time to maturity.
yield
to
maturity
time to maturity (years)
Term Structure of Interest Rates
The pattern of rates of return for
debt securities that differ only in the
length of time to maturity.
yield
to
maturity
time to maturity (years)
Term Structure of Interest Rates
The yield curve may be downward
sloping or “inverted” if rates are
expected to fall.
yield
to
maturity
time to maturity (years)
Term Structure of Interest Rates
The yield curve may be downward
sloping or “inverted” if rates are
expected to fall.
yield
to
maturity
time to maturity (years)
Recent US Treasury Yield Curve
6.00%
5.00%
Yield
4.00%
3.00%
2.00%
1.00%
0.00%
0
5
10
15
20
25
30
35
Time to Maturity
Last Semester
Last Week
17
For a Treasury security, what is
the required rate of return?
Required
rate of
return
=
Risk-free
rate of
return
Since Treasuries are essentially free of
default risk, the rate of return on a
Treasury security is considered the
“risk-free” rate of return.
For a corporate stock or bond, what is
the required rate of return?
Required
rate of
return
=
For a corporate stock or bond, what is
the required rate of return?
Required
rate of
return
=
Risk-free
rate of
return
For a corporate stock or bond, what is
the required rate of return?
Required
rate of
return
=
Risk-free
rate of
return
+
Risk
premium
How large of a risk premium should we
require to buy a corporate security?
Returns
Expected Return - the return that
an investor expects to earn on an
asset, given its price, growth
potential, etc.
Required Return - the return that
an investor requires on an asset
given its risk and market interest
rates.
Holding Period (Actual) Returns
The realized return over a period of
time (HPR).
HPR=(Ending Price - Beginning Price +
Distributions Received)/Beginning Price
Example: What is your HPR if you buy
a stock for $20, receive $1 in dividends,
and then sell it for $25.
HPR = ($25-$20+$1)/$20 = 0.3 = 30%
23
Calculation of Expected Returns
Expected Rate of Return (Expected Value) given a
probability distribution of possible returns(ki): E(k) or
k
_
n
E(k)=k =  ki P(ki)
i=1
Realized or Average Return on Historical Data:
-
n
k = 1/n k i
i=1
24
Expected Return and Standard
Deviation Example
St a t e of
Con t ra ry
Econ om y Prob a b ilit y M AD I n c. Co. ( CON )
Boom
0.25
80%
5%
Norm al
0.60
30%
10%
Recession
0.15
- 30%
15%
MAD E(r) = .25(80%) + .60(30%) + .15(30%) = 33.5%
CON E(r) = .25(5%) + .60(10%) +
.15(15%) = 9.5%
25
Definition of Risk
Risk is an uncertain outcome or chance of an
adverse outcome.
Concerned with the riskiness of cash flows
from financial assets.
Namely, the chance that actual cash flows will
be different from forecasted cash flows.
Standard Deviation can measure this type
of risk.
26
How do we Measure Risk?
A more scientific approach is to
examine the stock’s standard
deviation of returns.
Standard deviation is a measure of
the dispersion of possible
outcomes.
The greater the standard deviation,
the greater the uncertainty, and
therefore , the greater the risk.
Standard Deviation
s
n
=

i=1
(ki - k)2 P(ki)
Expected Return and Standard
Deviation Example
St a t e of
Con t ra ry
Econ om y Prob a b ilit y M AD I n c. Co. ( CON )
Boom
0.25
80%
5%
Norm al
0.60
30%
10%
Recession
0.15
- 30%
15%
MAD E(r) = .25(80%) + .60(30%) + .15(30%) = 33.5%
CON E(r) = .25(5%) + .60(10%) +
.15(15%) = 9.5%
29
s=
n
 (ki -
2
k)
P(ki)
i=1
MAD, Inc.
( 80% - 33.5%)2 (.25) = 540.56
(30% - 33.5%)2 (.6) =
7.35
(-30% - 33.5%)2 (.15) = 604.84
Variance
=
1152.75%
Stand. dev. =
1152.75 = 34.0%
Expected Return and Standard
Deviation Example
St a t e of
Cont ra ry
Econom y Proba bilit y M AD I nc. Co. ( CON )
Boom
0.25
80%
5%
Norm al
0.60
30%
10%
Recession
0.15
- 30%
15%
MAD E(r) = .25(80%) + .60(30%) + .15(30%) = 33.5%
CON E(r) = .25(5%) + .60(10%) + .15(15%)
= 9.5%
31
s=
n
 (ki -
2
k)
P(ki)
i=1
Contrary Co.
(5% - 9.5%)2 (.25) =
5.06
(10% - 9.5%)2 (.6) = 0.15
(15% - 9.5%)2 (.15) = 4.54
Variance
=
9.75%
Stand. dev. =
9.75
= 3.1%
Which stock would you prefer?
How would you decide?
Which stock would you prefer?
How would you decide?
It depends on your tolerance
for risk!
Return
Risk
Remember, there’s a tradeoff
between risk and return.
Coefficient of Variation
A relative measure of risk. Whereas, s is an
absolute measure of risk.
Relates risk to expected return.
CV = s/E(k)
MAD’s CV = 34%/33.5% = 1.01
CON’s CV = 3.1%/9.5% = 0.33
CONtrary is the less risky of the two
investments. Would choose CON if risk
averse.
36
Portfolios
Expected Portfolio Return is weighted
average of the expected returns of the
individual stocks = Σwjkj.
However, portfolio risk (standard
deviation) is NOT the weighted average
of the standard deviations of the
individual stocks.
Combining several securities in a
portfolio can actually reduce overall
risk.
How does this work?
Suppose we have stock A and stock B.
The returns on these stocks do not tend
to move together over time (they are
not perfectly correlated).
kA
rate
of
return
kB
time
What has happened to the
variability of returns for the
portfolio?
kA
rate
of
return
kp
kB
time
Diversification
Investing in more than one security to
reduce risk.
If two stocks are perfectly positively
correlated, diversification has no
effect on risk.
If two stocks are perfectly negatively
correlated, the portfolio is perfectly
diversified.
If you owned a share of every
stock traded on the NYSE and
NASDAQ, would you be
diversified?
YES!
Would you have eliminated all
of your risk?
NO! Common stock portfolios
still have risk.
Some risk can be diversified
away and some cannot.
Market risk (systematic risk) is
nondiversifiable. This type of risk
cannot be diversified away.
Company-unique risk (unsystematic
risk) is diversifiable. This type of risk
can be reduced through diversification.
42
Market Risk
Unexpected changes in interest
rates.
Unexpected changes in cash
flows due to tax rate changes,
foreign competition, and the
overall business cycle.
Company-unique Risk
A company’s labor force goes
on strike.
A company’s top management
dies in a plane crash.
A huge oil tank bursts and
floods a company’s production
area.
As you add stocks to your
portfolio, company-unique risk is
reduced.
portfolio
risk
Market risk
number of stocks
As you add stocks to your
portfolio, company-unique risk is
reduced.
portfolio
risk
companyunique
risk
Market risk
number of stocks
Do some firms have more market
risk than others?
Yes. For example:
Interest rate changes affect all
firms, but which would be
more affected:
a) Retail food chain
b) Commercial bank
Note:
The market compensates investors for
accepting risk - but only for market
risk. Company-unique risk can and
should be diversified away.
So - we need to be able to measure
market risk.
This is why we have Beta.
Beta: a measure of market risk.
Specifically, beta is a measure of
how an individual stock’s returns
vary with market returns.
It’s a measure of the “sensitivity”
of an individual stock’s returns to
changes in the market.
The Concept of Beta
Beta(b) measures how the return of an
individual asset (or even a portfolio) varies with
the market portfolio.
b = 1.0 : same risk as the market
b < 1.0 : less risky than the market
b > 1.0 : more risky than the market
Beta is the slope of the regression line (y = a +
bx) between a stock’s return(y) and the market
return(x) over time, b from simple linear
regression.
bi = Covariancei,m/Mkt. Var. =rimsism/sm2
50
Relating Market Risk and Required Return:
the CAPM
Here’s the word story: a stock’s
required rate of return = risk-free rate
+ the stock’s risk premium.
The main assumption is investors hold
well diversified portfolios = only
concerned with market risk.
A stock’s risk premium = measure of
systematic risk X market risk premium.
51
CAPM Equation
krp= market risk premium = km - krf
stock risk premium = bj(krp)
kj = krf + bj(km - krf )
= krf + bj (krp)
Example: What is Yahoo’s required return if its b =
1.75, the current 3-mo. T-bill rate is 1.7%, and the
historical market risk premium of 9.5% is
demanded?
Yahoo k = 1.7% + 1.75(9.5%) = 18.3%
52
Question: If Yahoo’s exp. Return
= 15%, what to do?
Return
Required vs. Expected Return
25.00%
20.00%
15.00%
10.00%
5.00%
0.00%
19.10%
15%
0
0.5
1
1.5
2
2.5
Beta
Req. Return
Exp Return
53
Portfolio Beta and CAPM
The b for a portfolio of stocks is the weighted
average of the individual stock bs.
bp = wjbj
Example: The risk-free rate is 6%, the market
return is 16%. What is the required return for a
portfolio consisting of 40% AOL with b = 1.7,
30% Exxon with b = 0.85, and 30% Fox Corp.
with b = 1.15.
Bp = .4(1.7)+.3(0.85)+.3(1.15) = 1.28
kp = 6% + 1.28(16% - 6%) = 18.8%
54
More SML Fun!
According to the CAPM and SML equation
with k = 6% + b(16% - 6%)
How would a change in inflation affect
required returns? (Say inflation increases 2%
points)
How would a change in risk aversion (market
risk premium) affect required returns? (Say
market risk premium decreases 2% points.)
55
Changes to SML
Security Market Line
Return
30%
25%
20%
15%
Original SML
10%
5%
0%
0
1
2
3
Beta
56
Changes to SML
Security Market Line
Return
30%
25%
20%
Original SML
15%
10%
Increased
Inflation
5%
0%
0
1
2
3
Beta
57
Changes to SML
Security Market Line
Return
30%
25%
Original SML
20%
15%
Less Risk
Aversion
10%
5%
0%
0
1
2
3
Beta
58
Limitations of CAPM/SML
Don’t really know what the market portfolio
is, which makes it hard to estimate market
expected or required return.
Beta estimates can be unstable and might not
reflect the future.
Maturity debate over proper risk-free
estimate.
Most investors focus on more than systematic
risk.
59