Lecture: 4 - Measuring Risk (Return Volatility)
I.
Uncertain Cash Flows - Risk Adjustment
II.
We Want a Measure of Risk With the Following
Features
a. Easy to Calculate
b. Ranks Assets According to Compensated Risk
c. Can be Translated into a Discount Rate, k
III.
Economy-Wide or Systemic Risk -> Beta
Works best for a portfolio of assets.
IV.
Non-Systematic or Company Specific Risk ->
Variance
Works best for a single asset.
Lecture: 4 - Measuring Risk (Return Volatility)
TO MEASURE RISK WE NEED A GOAL VARIABLE.
INVESTORS’ GOAL VARIABLE IS RETURNS
% Return =
=
[ P1  P0  D1 ]
P0
( P1  P0 )
P0
+
D1
P0
= % capital gain (loss) + % Dividend
Yield
%R
=
(12  10  2)
10
= .40 or 40%
QUESTION: What is risk , or, what does risk-free mean?
ANSWER: If exante expected returns always equal expost
returns for an investment then we say it is risk- free. If actual
returns are sometimes larger and sometimes smaller than
expected, the investment carries risk. (we are happy with
large ones but unhappy with small ones).
A measure of risk should tell us the likelihood that we will not
get what we expect and the magnitude of how different our
returns will be from the expected.
Lecture: 4 - Measuring Risk (Return Volatility)
HOW TO MEASURE WHAT TO EXPECT
•
Enumerate outcomes i.e., the different risk
scenarios.
•
Generate a probability distribution - attach
probabilities to each scenario that sum to 1
(remember statistics course)
EXAMPLE
Economic Scenarios
High growth ( 5%)
Low growth (3%)
Recession (-3%)
Prob
.30
.40
.30
Sum = 1
IBM Return
.25
.15
.05
Get mean return - expected return - best guess (Note: Book uses k instead of R)
n
E(R) =

i 1
_
P i Ri =
R
E(R) = (.3 * .25) + (.4 * .15) + (.3 * .05) = .15
Lecture: 4 - Measuring Risk (Return Volatility)
USE VARIANCE TO MEASURE TOTAL RISK
2
n
=

i 1
_
(Ri - R )2 Pi
or standard deviation;
 = [ 2].5
For IBM
 2IBM = (.25 - .15)2(.3) + (.15 - .15)2(.4) + (.05 - .15)2(.3)
= .006
QUESTION: What is the variance of a stock which has a
mean of .15 and returns of .15 in all states of the
economy?
- Zero!
VARIANCE FOR A SINGLE ASSET CONTAINS
a. Diversifiable Risk (Firm Specific) easily by diversification at little or no cost.
b. Undiversified (System) Risk cannot be eliminated through diversification.
Variance Measures the Dispersion of a
Distribution Around Its Mean
Lecture: 4 - Measuring Risk (Return Volatility)
1
2
These two distributions have the same mean but 1’s variance is smaller than 2’s.
If these represent stock returns, a risk averse investor should choose stock 1.
A Standarized Risk Measure
Coefficient of Variation = Standard
Deviation/Mean
Lecture: 4 - Measuring Risk (Return Volatility)
1
2
When two stock return distributions have different means and variances, a risk
averse investor choosing between them needs a method that compares mean return
relative to risk, such as coefficient of variation or the capital asset pricing model.
Portfolio Mean Return and Variance
Lecture: 4 - Measuring Risk (Return Volatility)
TO GET THE VARIANCE OF A PORTFOLIO WE NEED TO
CALCULATE THE PORTFOLIO MEAN RETURN.
Portfolio mean return is a linear, weighted average of
individual mean returns of the assets in the portfolio.
GETTING THE WEIGHTS
_
INVESTMENT
Wi
Ri
100/500 = .2
200/500 = .4
200/500 = .4
.10
.05
.15
$ INVESTED
1
2
3
100
200
200
E(Rp)
= W1 R 1 + W2 R 2 + W3 R 3
_
_
_
= .2(.1) + .4(.05) + .4(.15) = .10
n
GENERAL =>
E(Rp)
=

i 1
_
_
Wi R i = R p
Portfolio Variance is More Complex A Nonlinear Function
Lecture: 4 - Measuring Risk (Return
Volatility)
For a two asset portfolio:
 p2 = W12  12 + W22  22 + 2W1W2Cov12
where: Cov12 = covariance = Corr12  1  2
and Corr12 = correlation
QUESTION: Diversification reduces variance of portfolio
even when corr=0. WHY?- Some asset-specific risk offset
one another.
Portfolio
Risk
Diversifiable Risk
Nondiversifiable Risk
Number of securities in the portfolio
Diversifiable risk drops as more securities
are added to a portfolio.
It’s usually best to diversify, except in this
case.
Lecture: 4 - Measuring Risk (Return Volatility)
Correlation
•
Statistical Measure of the Degree of Linear
Relationship Between Two Random Variables
•
Range: + 1.0 to -1.0
•
+ 1.0 - Move Up and Down Together - Exactly
the Same Rate
•
0.0 - No Relationship Between the Returns
•
- 1.0 - Move Exactly Opposite Each Other
Stock 1
Return

Stock 1
Return

   












Stock 1
Return

Negative
Correlation






 



 


Stock 2 Return



 
Stock 2 Return
Positive
Correlation
Stock 2 Return
Zero
Correlation
Covariance is a Measure of Risk and Beta is a
Standardized Covariance
Lecture: 4 - Measuring Risk (Return
Volatility)
Covariance Measures How Closely Returns For Two
AssetsTrack Each Other Other (Closeness to the
Regression Line)
All else equal, covariance is large when the data points fall
along the regression line instead of away from it because,
on the line, the deviations from the means of each variable
are equal – the products are squares - larger than
otherwise.
Beta Is a Standardized Covariance Measure
Lecture: 4 - Measuring Risk (Return Volatility)
We Need Beta (Standardized Covariance Measure) in Order
to Make Comparisons of Risk Between Assets or Portfolios.
•
Measured Relative to the Market Portfolio (the most
diversified portfolio is the standard)
•
Slope of the Regression Line
•
Slopes Measured Relative to Market Return
General Formula
Betai
=
=
Covim
 m2
Corrim i  m
 m2
Beta and the Market (Illustration)
•
Beta = 1 - Same as Market Risk
•
Beta > 1 - Riskier than Market
•
Beta < 1 - Less Risky than Market
•
Beta = 2 - Twice as Risky as Market
Positive Beta
Lecture: 4 - Measuring Risk (Return Volatility)
Annual return pairs for the S&P 500 and Homestake Mining's stock
H
o
m
e
s
t
a
k
e
'
s
Year
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1
0.8
0.6
Slope is 0.54
0.4
0.2
R
0
e
t
-0.2
u
r
n -0.4
-0.1
0
0.1
0.2
0.3
0.4
S&P 500 Return
The correlation between Homestake and the S&P 500 is 0.18 and its beta is 0.54
S&P Homestake
0.23
0.01
0.06
-0.26
0.32
0.1
0.18
0.09
0.05
0.39
0.17
-0.27
0.31
0.55
-0.03
-0.09
0.3
-0.16
0.08
-0.25
0.1
0.83
0.01
-0.19
Negative Beta
Lecture: 4 - Measuring Risk (Return Volatility)
Year
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
Annual return pairs for the S&P 500 and gasoline
0.8
G
a
s
o
l
i
n
e
R
e
t
u
r
n
0.7
0.6
0.5
Slope is -2.11
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.1
0
0.1
0.2
0.3
0.4
S&P 500 Return
Gasoline's correlation with the S&P 500 is -0.47 and its beta is -2.11.
S&P
0.23
0.06
0.32
0.18
0.05
0.17
0.31
-0.03
0.3
0.08
0.1
0.01
Gas
0.08
-0.1
0.09
-0.45
0.19
-0.04
-0.08
0.73
-0.33
-0.07
-0.29
0.2
Zero Beta
Lecture: 4 - Measuring Risk (Return Volatility)
Annual return pairs for the S&P 500 and Gold
Year
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
0.4
G
o
l
d
0.2
R
e
t
u
r
n
Slope is zero
0
-0.2
-0.1
0
0.1
0.2
0.3
0.4
S&P 500 Return
The correlation between gold and the S&P 500 and its beta is approximately zero.
S&P
0.23
0.06
0.32
0.18
0.05
0.17
0.31
-0.03
0.3
0.08
0.1
0.01
Gold
-0.1
-0.16
0
0.25
0.2
-0.17
0
-0.05
-0.05
-0.06
0.12
0.02
Positive Beta
Stock
Return
Negative Beta
Positive and negative beta stock returns
move opposite one another.
High Beta
Stock
Return
Market
Low Beta
During this time period the market rises, falls, and then rises again. A high (low)
beta stock varies more (less) than the market.
Port folio Beta
Lecture: 4 - Measuring Risk (Return Volatility)
GENERAL FORMULA
n
Bp =  W iBi
i 1
Example: Beta for a portfolio containing three stocks.
INVESTMENT
$ INVESTED
Wi
1
2
3
100
400
500
100/1000 = .1
400/1000 = .4
500/1000 = .5
Bp
= W1B1 + W2B2 + W3B3
Bi
2.0
1.5
0.5
= .1(2) + .4(1.5) + .5(.5) = 1.05
CAPM
“Beta is Useful Because it Can Be Precisely Translated into
a Required Return, k, Using the Capital Asset Pricing Model”
Lecture 4 - Measuring Risk (Return Volatility)
CAPM (Capital Asset Pricing Model)
General Formula
Ri
=
ki
= Rf + Bi(Rm - Rf)
= time value + (units of risk) x (price per unit)
= time value + risk premium
where, Rf
= Risk-Free Rate -> T-Bill
Rm = Expected Market Return -> S&P 500
Bi
= Beta of Stock i
Example:
Suppose that a firm has only equity, is twice as
risky as the market and the risk free rate is 10%
and expected market return is 15%. What is the
firm’s required rate?
Ri
= ki
= Rf + Bi(Rm - Rf)
= 10% + 2(15% - 10%)
= 10% + 10%
= 20%
Lecture: 4 - Measuring Risk (Return
Volatility)
QUESTION: If an asset has a B = 0, what is its return?
-> Rf
QUESTION: If an asset has a B = 1, what is its return?
-> Rm
QUESTION: Suppose E(R1) > E(R2) AND B1 < B2, which
asset do you choose? -> 1
QUESTION: How about if E(R1) > E(R2) and B1 > B2 ?
Now we need to know B1 and B2 and use the CAPM
Lecture: 4 - Measuring Risk (Return
Volatility)
CONSIDER STOCK PRICE AND CAPM
P =
D1
kg
Ri =
ki = Rf + Bi(Rm - Rf)
QUESTIONS:
What happens to price as growth increases? P increases!
What happens to price if k increases? P Decreases!
What happens to price if Beta decreases? P increases!
What happens to price if Rf increases? B >1->P increase
B<1 -> P decrease
What happens to price if Rm decreases? P increases!
QUESTION: As financial managers, what variables should
we try to change and in what directions?
1. Increase cash flows - or growth in CF’s make superior investment decisions, use the
lowest cost financing or manipulate debt/equity
ratio
2. Bring cash flows in closer to the present
3. Decrease Beta - Manipulate assets (LaborCapital ratio).