6-1
Risk and Return:
Modern Portfolio Theory
6-2
Returns and Return Distribution
 Dollar return
 dividend income + capital gains
 Percentage return
 Rt+1 = [Dt+1 + (Pt+1 – Pt)] / Pt
 Holding period return
 RT=3 = (1 + R1) × (1 + R2) × (1 + R3)
 Average return (Arithmetic)
 R = (R1 + ··· + RT) / T
 Variance and Standard Deviation
 SD = (Var) = [(1 / T –1) × ((R1 –R)2 + ··· + (RT –R)2)]
6-3
The Historical Record
 What is the average year-to-year return on
the following financial investments:
 Large-company stocks
 Small-company stocks
 Long-term corporate bonds
 Long-term government bonds
 U.S. Treasury bills
 Consumer Price Index (CPI)
A $1 Investment in Different Types of
Portfolios: 1926-1996
Index ($)
10000
Small Company
Stocks
$4,495.99
$1,370.95
1000
Large
Company
$33.73
Stocks
100
10
Long-Term
Government
Bonds
$13.54
$8.85
1
Treasury Bills
Inflation
0.1
1925 1935 1945 1955 1965 1975 1985 1995
Year-End
6-4
6-5
What does capital market history tell
us about risk and return?
Historical Average Returns and Standard Deviations, 1926 - 1996
Series
Common Stocks
Average
Annual
Return
Standard
Deviation
Risk
Premium
12.7%
20.3%
8.9%
17.7
34.1
13.9
Long-term Corporate Bonds
6.0
8.7
2.2
Long-term Gov' t Bonds
5.4
9.2
1.6
U.S Treasury Bills
3.8
3.3
0.0
Inflation
3.2
4.5
NA
Small Stocks
Source: Stocks, Bonds, Bills, and Inflation 1997 Yearbook, Ibbotson Associates, Inc.
6-6
Risk Premium
Risk Premium: the excess return
required from an investment in a
risky asset over that required
from a risk-free asset.
Treasury bill rate is used as the
risk-free rate.
Historical Returns, Standard Deviations, and
Frequency Distributions: 1926-1996
Series
Average
Annual Return
Standard
Deviation
Large Company Stocks
12.7%
20.3%
Small Company Stocks
17.7
34.1
Long-Term Corporate Bonds
6.0
8.7
Long-Term Government Bonds
5.4
9.2
U.S. Treasury Bills
3.8
3.3
Inflation
3.2
4.5
– 90%
Distribution
0%
+ 90%
Source: © Stocks, Bonds, Bills, and Inflation 1997 Yearbook™, Ibbotson Associates, Inc., Chicago (annually updates work by
Roger G. Ibbotson and Rex A. Sinquefield). All rights reserved.
Frequency Distribution of Returns on
Common Stocks, 1926-1996
18
16
16
14
13
12
11
Number 10
of Years
10
8
8
6
6
3
4
2
2
1
1
0
Return (%)
-55 -45
-35 -25 -15
-5
5
15
25
35
45
55
The Normal Distribution
Probability
68%
95%
> 99%
–3
– 48.2%
–2
– 27.9%
–1
– 7.6%
0
12.7%
+1
33.0%
+2
53.3%
+3
73.6%
Return on
large company
stocks
6-10
What is investment risk?
Investment risk pertains to the probability of
realized returns being less than expected return.
The greater the chance of low or negative
returns, the riskier the investment.
Normal distribution is assumed
States of Nature  Future scenarios
6-11
Expected Return and Variance
 Expected Return: Return on a risky asset expected in the
future
S
E( Ri ) =  p(s)* R(s )i
s=1
 Variance: Measures the dispersion of an asset's returns
around its expected return.
S

var( Ri ) =  p(s)* R(s )i - E( Ri )
2

s=1
 Standard deviation: The square root of the variance.
std( Ri ) = var( Ri )
1
2
9-5
Investment Alternatives:
2 risky investments
Estimated Rate of Return
State of
the Weather
Very Cold
Cold
Average
Hot
Probability
 100%
0.1
0.3
0.4
0.2
Amusement Park Ski Resort
-15.0%
-5.0
10.0
30.0
Adds up to 100%
35.0%
15.0
5.0
-5.0
6-13
Calculate the expected rate of
return on each investment alternative.
Mean or expected value:
E(X) = p(1)  X1 + p(2)  X2 + … + p(n)  Xn
where i
= possible outcome
p(i)
= probability of outcome i
Xi
= return if outcome I happens
n
= total number of possible outcomes
E(RA) = (-15%)  0.1 + (-5%)  0.3 + (10%)  0.4 + (30%)  0.2 = 7.0%
E(RS) = (35%)  0.1 + (15%)  0.3 + (5%)  0.4 + (-5%)  0.2 = 9.0%
6-14
What is the standard deviation of
returns for each alternative?
 =
n
Variance =  p(i) × [Xi - E(X) ]2
i=
1
• measure of total or “stand-alone” risk
• the larger the  the lower the probability
that actual returns will be close to the
expected returns.
=
n
 p(i) × [Xi - E(X)]2
i=
1
For the amusement park and ski resort we
have:
 A  0.1(.15  .07) 2  0.3(.05  .07) 2  0.4(.1  .07) 2  0.2(.3  .07) 2
 0.0201  0.14177
 S  0.1(.35  .09) 2  0.3(.15  .09) 2  0.4(.05  .09) 2  0.2(.05  .09) 2
 0.0124  0.11136
6-15
6-16
What is stand-alone risk?
Stand-alone risk consists of:
 diversifiable risk
 company specific, unique, or unsystematic
 non-diversifiable risk
 market or systematic
 It is measured by dispersion of returns
about the mean and is relevant only for
assets held in isolation.
6-17
Probability Distributions
Probability
Firm X
Firm Y
-70
0
15
Expected Rate of Return
Rate of
100 Return%
6-18
A = B but A is riskier
Probability
because it has a larger
probability of losses.
A
0
B
Expected Return
6-19
What is diversifiable risk?
 Caused by company or industry specific
events like lawsuits, strikes, winning or
losing major contracts, etc.
 Effects of such events on a portfolio can
be eliminated by diversification and
should therefore not be rewarded.
6-20
What is market risk?
 Stems from such external events as war,
inflation, recession, and interest rates.
 Because all firms are affected simultaneously
by these factors, market risk cannot be eliminated
by diversification.
 Market risk is also known as systematic risk
since it shows the degree to which a stock moves
systematically with other stocks.
Define coefficient of variation (CV)
Standardized measure of dispersion
about the expected value:
Std. Dev.
CV =
=
Mean

E(x)
Shows risk per unit of return.
6-21
6-22
What is the covariance of
returns for each alternative?
n
Covariance =  p(i) × (Xi - E(X) ) × (Yi - E(Y))
i=
1
• measure of how 2 (X and Y) investments move
together, or how an investment moves with the entire
market
• For our amusement park and ski resort:
AS = 0.1(-.15 - .07)(.35 - .09) + 0.3(-.05 - .07)(.15 - .09)
+ 0.4(.10 - .07)(.05 - .09) + 0.2(.30 - .07)(-.05 - .09)
= - 0.0148
6-23
What is the correlation coefficient?
CORRXY = COVXY / X Y = XY
 The correlation coefficient is standardized to
always be between -1.0 and +1.0
 For the amusement park and ski resort we
have:
AS = AS /
A S
= – 0.0148 / (0.1418)(0.1114)
= – 0.9375
Returns for Two Perfectly
Negatively Correlated ( = -1.0)
Stocks and Portfolio WM

Stock W
Portfolio
WM



Stock M
6-24
Returns for Two Perfectly
Positively Correlated ( = +1.0)
Stocks and Portfolio MM’:




Stocks M and M ’
(Identical returns)
Portfolio MM’
6-25
6-26
Portfolio Theory
 Portfolio: A group of securities, such as stocks
and bonds, held by an investor.
 Portfolio weights: Percentages of the portfolio's
total value invested in each security.
 Example: Your portfolio consists of IBM stock
and GM stock. You have $2,500 invested in IBM
and $7,500 invested in GM. What are the portfolio
weights?
6-27
Portfolio Return and Variance
 Expected Return on a portfolio: Weighted average
of the expected returns on the individual securities in
the portfolio.
E( R p ) =  wn E( R n )
N
n=1
 Portfolio Variance: Unlike the expected return, the
variance of a portfolio is not a simple weighted
average of the individual security variances.
Var( R p ) = w 2a var( R A ) + w 2B var( R B ) + 2 w A w B cov( R A , R B )
6-28
Expected Portfolio Return
 In our earlier example, there were two stocks: the
amusement park and the ski resort.
 E(RA) = 7%
 E(RS) = 9%
A = 14.177%
S = 11.136%
Cov(A,S) = – 0.0148
 Assume we have $100 and invest $50 in A and
$50 in S. What is our expected portfolio return?
 E(RP) = 0.5 (7%) + 0.5 (9%) = 8%
 wA = 50/100 = 0.5
 wS = 50/100 = 0.5
6-29
Expected Portfolio Risk
 To measure the risk of the portfolio, we must
account for the risk of the individual stocks and
how they move together.
 Var(Rp)
= (0.5)2 (0.1418)2 + (0.5)2 (0.1114)2
+ 2 (0.5)(0.5)(-0.0148)
=0.000725
 Standard Deviation (Rp) = (0.000725)1/2
= 0.0269
= 2.69%
Investment Alternatives
9-22
Estimated Rate of Return
State of
the Weather
Very Cold
Cold
Average
Hot
Prob
0.1
0.3
0.4
0.2
1.0
Amusement
Park
Ski
Resort
50-50
Portfolio
-15.0%
-5.0
10.0
30.0
35.0%
15.0
5.0
-5.0
10.0%(a)
5.0%
7.5%
12.5%
8.0% (b)
(a) RP(very cold) = 0.5 (-15%) + 0.5 (35%) = 10%
(b) RP = 0.1(10) + 0.3(5) + 0.4(7.5) + 0.2(12.5) = 8.0%
Var(RP) = 0.1(10 - 8)2 + 0.3(5 - 8)2 + 0.4(7.5 - 8)2 + 0.2(12.5 - 8)2 = 7.25
Std. Dev. (RP) = (7.25)0.5 = 2.69%
6-31
If we alter the weights on the two stocks:
WA (%)
WS (%) SD(RP) (%)
E(RP) (%)
100.00
0.00
14.18
7.00
90.00
10.00
11.72
7.20
80.00
20.00
9.29
7.40
70.00
30.00
6.89
7.60
60.00
40.00
4.60
7.80
50.00
50.00
2.69
8.00
40.00
60.00
2.40
8.20
30.00
70.00
4.09
8.40
20.00
80.00
6.33
8.60
10.00
90.00
8.71
8.80
0.00
100.00
11.14
9.00
6-32
Plotting the risk-return combinations gives:
100% S
100% A
6-33
Expected
Portfolio
Return, RP
Efficient Set
B
C
E
A
D
Feasible Set
Risk, P
6-34
Risk Free Assets
What happens if one of the assets in
our portfolio is risk free, i.e. Y = 0?
E( R p ) = W X E( R X ) +W Y E( RY )
SD( R p ) = W X2 X2 + WY2 Y2 + 2  W X WY Cov XY
or
SD( R p ) = W X2 X2 + WY2 Y2 + 2  W X WY  X  Y CorrXY
 WX  X
6-35
Risk Free Assets
 Assume there is a risky asset (X) and a risk free asset (F):
• Risky Asset: E(RX) = 0.16, X = 8%
• Risk Free Asset: E(RF) = 0.06, F = 0%
 You have $100, you put $50 in X and $50 in F (i.e., lending
$50 at the risk free rate). The weights are:
amount in X
50
=
= 0.50
WX=
my initial wealth 100
amount in F
50
=
= 0.50
WF=
my initial wealth 100
E(Rp) = WXE(RX) + WYE(RY) = 0.5 (16) + 0.5 (6) = 11%
SD(RP) = WXX = 0.5 (8) = 4%
6-36
Risk Free Assets
 What happens if you have $100 and you borrow $50 from F
and put $150 in X?
amount in X
150
=
= 1.50
WX=
my initial wealth 100
amount in F
- 50
=
= - 0.50
WF=
my initial wealth 100
Note: The weights always add up to one!
E(Rp)=WXE(RX) + WYE(RY)=1.5 (16%) + (-0.5) (6%)=21%
SD(RP) = WXX = 1.5 (8) = 12%
6-37
If we compute the expected return and standard
deviation for a variety of weights, we can build a
table as we did before:
WF (%)
WX (%)
SD(RP) (%)
E(RP) (%)
100.00
0.00
0.00
6.00
80.00
20.00
1.60
8.00
50.00
50.00
4.00
11.00
20.00
80.00
6.40
14.00
0.00
100.00
8.00
16.00
-50.00
150.00
12.00
21.00
6-38
6-39
Expected
Portfolio
Return, RP
Borrowing
M
Lending
.
.
C
Q
D
Rf
E
Risk, P
6-40
General statements about risk
 Most stocks are positively correlated.
 average correlationx,y = 0.65
 Average i = 49.24%
 Average p = 20%
 Combining stocks in a portfolio usually
lowers the risk.
This is referred to as diversification.
 Exception: Correlation = +1
6-41
What would happen to the
risk of a 1 stock portfolio
as more randomly selected
stocks were added?
The standard deviation of the portfolio
would decrease because the added
stocks would not be perfectly correlated.
6-42
Standard Deviations of Annual Portfolio Returns (Table 13.7)
Number of Stocks
in Portfolio
1
10
50
100
300
500
1,000
Ratio of Portfolio
Average Standard Standard Deviation to
Deviation of Annual
Standard Deviation
Portfolio Returns
of a Single Stock
49.24%
23.93
20.20
19.69
19.34
19.27
19.21
1.00
0.49
0.41
0.40
0.39
0.39
0.39
These figures are from Table 1 in Meir Statman, “How Many Stocks Make a Diversified
Portfolio?” Journal of Financial and Quantitative Analysis 22 (September 1987), pp. 353–64.
They were derived from E. J. Elton and M. J. Gruber, “Risk Reduction and Portfolio Size: An
Analytic Solution,” Journal of Business 50 (October 1977), pp. 415–37.
6-43
Portfolio Diversification (Figure 13.1)
Average annual
standard deviation (%)
49.2
Diversifiable (firm-specific)risk
23.9
19.2
Nondiversifiable
(Market) risk
1
10
20
30
40
1000
Number of stocks
in portfolio
6-44
If you hold a one-stock
portfolio and thus are
exposed to more risk than
diversified investors, would
you be compensated for all
the risk that you bear?
6-45
 No! If you hold only one stock, you will not
be compensated for the additional risk you
bear.
 Stand-alone risk is not as important to a welldiversified investor, and most of it can be
eliminated at virtually no cost through
diversification.
 Rational risk averse investors are concerned
with P, which is based on market risk.
6-46
The Systematic Risk Principal
 The reward for bearing risk depends only upon
systematic risk since unsystematic risk can be
diversified away.
 This implies that the expected return on any asset
depends only on that asset's systematic risk
 Hence, the discount rate will depend only on the
systematic risk of the project
6-47
Measuring Systematic Risk
 Beta () measures a stock’s market
(or systematic) risk. It shows the
relative volatility of a given stock
compared to the average stock. An
average stock (or the market
portfolio) has a beta = 1.0.
Beta shows how risky a stock is if
the stock is held in a well-diversified
portfolio.
6-48
Beta Coefficients for Selected Companies (Table 13.8)
Company
Exxon
Beta
Coefficient (i)
0.65
AT&T
0.90
IBM
0.95
Wal-Mart
1.10
General Motors 1.15
Microsoft
1.30
Harley-Davidson 1.65
America Online 2.40
Source: From Value Line Investment Survey, April 19, 1996.
6-49
Portfolio Betas
 Portfolio Betas: While portfolio variance is not
equal to a simple weighed sum of individual
security variances, portfolio betas are equal to
the weighed sum of individual security betas.
N
 P =  wi  i
i=1
 You have $6,000 invested in IBM, $4,000 in GM.
The beta of IBM and GM is 0.95 and 1.15
respectively. What is the beta of the portfolio?
6-50
How are betas calculated?
Run a regression line of past returns
on Stock i versus returns on the
market.
The regression line is called the
characteristic line.
The slope coefficient of the
characteristic line is defined as the
beta coefficient.
6-51
Illustration of beta calculation.
_
Ri
20
.
15
.
10
Regression line
Year
1
2
3
RM
15%
-5
12
Ri
18%
-10
16
5
-5
0
-5
.
-10
5
10
15
20
^
^
Ri = -2.59 + 1.44 RM
_
RM
6-52
If beta = 1.0, stock is average risk.
If beta > 1.0, stock is riskier than
average.
If beta < 1.0, stock is less risky than
average.
Most stocks have betas in the range
of 0.5 to 1.5.
6-53
Can a beta be negative?
Answer: Yes, if the correlation
between the market and the stock is
negative. Then, in a “beta graph”, the
characteristic (regression) line will
slope downward.
But, in the real world, we have never
seen a negative beta stock.
6-54
Beta and Risk Premium
 A risk free asset has a beta of zero.
 When a risky asset is combined with a risk free
asset, the resulting portfolio expected return is a
weighted average of their expected returns and
the portfolio beta is the weighted average of their
betas.
 Consider various portfolios comprised of an
investment in stock A with a beta () of 1.2 and
expected return of 18%, and a Treasury bill with a
7% return. Compute the expected return and beta
for different portfolios of stock A and a Treasury
bill.
6-55
p
wA
wrf
E(Rp)
0.0
1.00
0*18% + 1*7% = 7%
0.25
0.75
.25*18% + .75*7% = 9.75
0.25*1.2 + 0.75*0 = 0.3
0.50
0.50
.50*18% + .50*7% = 12.5
0.50*1.2 + 0.50*0 = 0.6
0.75
0.25
.75*18% + .25*7% = 15.25
0.75*1.2 + 0.25*0 = 0.9
1.00
0.00
1.50
-0.50
1*18% + 0*7% = 18%
1.5*18% + (-.5)*7% = 23.5%
0*1.2 + 1*0 = 0
1*1.2 + 0*0 = 1.2
1.5*1.2 + (-.5)*0 = 1.8
6-56
Expected Return - Beta Plot
25
18%
E(R)
20
15
10
5
7%
0
0
0.3
0.6
0.9
Beta
1.2
1.5
1.8
6-57
Reward to Risk Ratio
 We can vary the amount invested in each
type of asset and get an idea of the
relation between portfolio expected return
and portfolio beta.
Reward - to - Risk Ratio =
E( R P ) - R f
P
6-58
Consider asset B, with a beta of 1.6 and an expected return of 20%.
Compute the expected return and beta for different portfolios of B and
a T-bill with a 7% return.
p
WB
wrf
E(Rp) (%)
0.0
1.00
0*20% + 1*7% = 7
0.25
0.75
0.25*20% + .75*7% = 10.25 0.25*1.6 + 0.75*0 = 0.4
0.50
0.50
0.50*20% + .50*7% = 13.5 0.50*1.6 + 0.50*0 = 0.8
0.75
0.25
0.75*20% + .25*7% = 16.75 0.75*1.6 + 0.25*0 = 1.2
1.00
0.00
1.50
-0.50
1*20% + 0*7% = 20
1.5*20% + (-.5)*7% = 26.5
0*1.6 + 1*0 = 0
1*1.6 + 0*0 = 1.6
1.5*1.6 + (-.5)*0 = 2.4
6-59
Expected Return - Beta Plot
E(R)
20
18%
16.75%
15
10
7%
5
0
0
1.2
Beta
6-60
What happens if two assets have
different reward-to-risk ratios?
 Since systematic risk is all that matters in
determining expected return, the reward-to-risk
ratio must be the same for all assets and
portfolios. If not, investors would only buy the
assets (portfolios) that offer a higher reward-torisk ratio.
 Because the reward-to-risk ratio is the same for
all assets, it must hold for the risk free asset as
well as for the market portfolio.
 Result:
E ( RA )  R f E ( RB )  R f

A
B
6-61
The Security Market Line
 The security market line is the line which gives
the expected return-systematic risk (beta)
combinations of assets in a well functioning,
active financial market.
 In an active, competitive market in which only
systematic risk affects expected return, the reward-torisk ratio must be the same for all assets in the market.
 The slope of the SML is the difference between the
expected return on the market portfolio and the risk-free
rate, or, the market risk premium.
E ( Ri )  R f
i
 E ( Rm )  R f
6-62
The Security Market Line (SML) (Figure 13.4)
Asset expected
return (E (Ri))
= E (RM) – Rf
E (RM)
Rf
 M = 1.0
Asset
beta ( i)
6-63
The Capital Asset Pricing Model
 The Capital Asset Pricing Model (CAPM) - an
equilibrium model of the relationship between risk
and required return on assets in a diversified
portfolio.
 What determines an asset’s expected return?
 The risk-free rate - the pure time value of money
 The market risk premium - the reward for bearing
systematic risk
 The beta coefficient - a measure of the amount of
systematic risk present in a particular asset
The CAPM: E(Ri ) = Rf + [E(RM ) - Rf ] x i
6-64
Example of CAPM
 Suppose a stock has 1.5 times the
systematic risk as the market portfolio
(average asset). If the risk-free rate as
measured by the Treasury bill rate is 5%
and the expected risk premium on the
market portfolio is 8%, what is the stock's
expected return according to the CAPM?
E(R) = 0.05 + 1.5  0.08 = 0.17 = 17%
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Summary of Risk and Return
I. Total risk - the variance (or the standard deviation) of an asset’s return.
II. Total return - the expected return + the unexpected return.
III. Systematic and unsystematic risks
Systematic risks are unanticipated events that affect almost all assets to
some degree.
Unsystematic risks are unanticipated events that affect single assets or
small groups of assets.
IV. The effect of diversification - the elimination of unsystematic risk via the
combination of assets into a portfolio.
V. The systematic risk principle and beta - the reward for bearing risk
depends only on its level of systematic risk.
VI. The reward-to-risk ratio - the ratio of an asset’s risk premium to its beta.
VII. The capital asset pricing model: E(Ri) = Rf + [E(RM) - Rf] i.
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Cost of Capital
Cost of Equity (rE)
CAPM: E(Ri) = Rf + i × [E(Rm) – Rf]
E(Ri) = rE if firm is 100% equity financed
i = I,M / (M)2
Financial Leverage
A = D/A × D + E/A× E
E = (1 + D/E) × A if D = 0
E = (1 + (1 – Tc) × D/E) × A if D = 0
Tc = corporate tax rate
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Weighted Average Cost of Capital
rWACC = D/A × (1–Tc) × rD + E/A × rE
D=market value of debt
E=market value of equity
A=market value of assets
Tc=corporate tax rate
rD=expected return on debt
rE=expected return on equity
Weights based on expected
or target values prevailing
over the life of the project