Asset return
Every investment has some required rate of
return, based on its level of perceived risk.
Of course, your realized return from an investment may
differ substantially from what you had originally
expected or required – uncertainty concerning the final
outcome is unavoidable.
People tend to be risk-adverse, therefore they
demand higher rates of return (additional
compensation) for investments of increasing risk.
Now we relax the assumption of certainty!!
1
Asset return
The return on a security comes in two forms: dividends
and capital gains
P0= price of the stock at the beginning of the year
P1= price of the stock at the end of the year
Div1= dividend paid on the stock during the year
X1= total amount of return received (dividends+capital gains)
X0= total amount invested
Return of a risky security purchased at t=0 and sold at
t=1:
Dividend yield: Div1 / P0
Capital gain: (P1 - P0)/ P0
Rate of return: r= (X1 - X0)/ X0
Total return on the investment:
R=X1 / X0
R=dividend yield + capital gain
R=1+r
2
Asset return
Ex. Sarah purchased 100 shares of stock at the beginning of the
year at p=37$ per share. Over the year the stock paid a dividend of
1.85$ per share. At the end of the year the market price of the
stock is 40.33$ per share. Which is the total return on the
investment?
Total investment=100*37$=3,700$ (X0)
Total dividends=100*1.85$=185$ (Div1)
Total gain=(40.33$-37$)*100=333$
Total return received=Div+gain=185$+333$=518$ (X1)
Total cash received after selling the shares at the end of the
year: 3,700$+518$=4,218$
Dividend yield=1.85$/37$=5%
Capital gain=(40.33$-37$)/37$=9%
r=(518$-3,700$)/3,700$=-0.86
R=1+r=1-0.86=0.14=14% or R=5%+9%=14% or
R=518$/3,700$=14%
3
Short selling
(receive X0)
(pay X1)
You bet that the asset price falls. If x0>X1, you
realize a profit.
Total return from short-selling:
R=X1 / X0
Short squeeze
4
Mean-variance portfolio theory
Two risky securities A and B and there are 4 equally likely states of
the economy:
Depression
Recession
Normal
Boom
RA
RB
Depression
-0.20
0.05
Recession
0.10
0.20
Normal
0.30
-0.12
Boom
0.50
0.09
Expected return:
it is the return that an individual expects a stock to earn over the next
period
It can be calculated as the average return that the security has earned in
the past

R A  17.5%
R B  5 .5 %
5
Mean-variance portfolio theory
Variance and Standard Deviation are used to assess
the volatility of a security’s return
2
2
VARIANCE (RA)= A = expected value of ( RA  RA )

1. Calculate the deviations from expected returns
for each state
2. Calculate the squared deviations for each state
3. Calculate the average squared deviation for the
security’s return
6
Mean-variance portfolio theory
RA
RA  RA
( RA  RA ) 2
RB
RB  RB
Depression
-0.20
-0.375
0.140625
0.05
-0.005
0.000025
Recession
0.10
-0.075
0.005625
0.20
0.145
0.021025
Normal
0.30
0.125
0.015625
-0.12
-0.175
0.030625
Boom
0.50
0.325
0.105625
0.09
0.035
0.001225
0.2675
( RB  RB ) 2
0.0529
RA  RA (dep)  0.20  0.175  0.375
RA  RA (rec )  0.10  0.175  0.075
RA  RA (norm)  0.30  0.175  0.125
RA  RA (boom)  0.50  0.175  0.325
0.2675
 
 0.066875
4
0.0529
 B2 
 0.013225
4
2
A
7
Mean-variance portfolio theory
SD is the square root of the variance
SDRA   A   A2
SDRA   A  0.066875  0.2586  25.86%
SDRB   B  0.013225  0.1150  11.50%
Co-variance and correlation measure the
interrelationship between two securities
Cov( RA, RB )   AB  Expected value of
[( RA  RA )( RB  RB )]
Cov( RA, RB )   AB   AB A B
Corr ( RA, RB )   AB 
Cov( RA , RB )
 A B
8
Mean-variance portfolio theory
1. Multiply together the deviations from expected returns
of A and B
2. Calculate the average value of the four states
Cov( RA, RB )   AB 
 0.0195
 0.004875
4
Corr ( RA, RB )   AB 
 0.004875
 0.1639
0.2586 * 0.1150
RA
RA  RA
( RA  RA ) 2
RB
RB  RB
( RB  RB ) 2 ( RA  RA )( RB  RB )
Depression
-0.20
-0.375
0.140625
0.05
-0.005
0.000025
0.001875
Recession
0.10
-0.075
0.005625
0.20
0.145
0.021025
-0.010875
Normal
0.30
0.125
0.015625
-0.12
-0.175
0.030625
-0.021875
Boom
0.50
0.325
0.105625
0.09
0.035
0.001225
0.011375
0.0529
-0.0195
0.2675
9
Return correlations
Perfect positive correlation: both the returns on
security A and B are higher (lower) than average at
the same time
Perfect negative correlation: security A has an
higher than average return when security B has a
lower than average return (viceversa)
Zero correlation: the return of security A is
completely unrelated to that of security B
10
Return correlations
R
R
Corr ( RA, RB )  1
t
Corr ( RA, RB )  1
t
R
Corr ( RA, RB )  0
t
11
Mean-variance portfolio theory
Let’s consider a portfolio of risky securities A
and B.
wA,B are the proportions of the assets A and B in the
portfolio
Expected return of the portfolio:
Rp  wA RA  wB RB
The expected return on a portfolio is the weighted average of the
expected returns on the individual assets of the portfolio.
From previous example, suppose the investor has 100$ and he invests 60$
in A and 40$ in B.
Rp  0.6(17.5%)  0.4(5.5%)  12.7%
12
Mean-variance portfolio theory
Variance of the portfolio:
 p2  wA 2 A2  2wA  wB   AB  wB 2 B2
The variance of a portfolio depends on both the variances of individual
securities and the co-variance between them.
Since  AB   A, B A B it can be also written as:
For given variances of individual securities, a positive co-variance
increases the variance of the portfolio.
Hedging effect if the variance of one security increases and the
variance of the other security decreases.
SDp   p   p2
13
Mean-variance portfolio theory
 p2  wA 2 A2  2wA  wB   AB  wB 2 B2
 p2  (0.6)2 (0.066875)  2[0.6 * 0.4 * (0.004875)]  (0.4)2 (0.013225)  0.023851
 p2  (0.6)2 (0.066875)  2[0.6 * 0.4 * (0.1639) * 0.2586 * 0.115]  (0.4)2 (0.013225)  0.023851
SDp   p  0.023851  0.1544  15.44%
14
Mean-variance portfolio theory
Extention to the case of n risky securities:
n
R p   wi Ri
i 1
n
n





2
 p  E   wi ( Ri Ri )   w j ( R j R j )  
 j 1

 i 1
n
 n

 E   wi w j ( Ri Ri )( R j  R j )    wi w j ij
 i , j 1
 i , j 1
The variance of a portfolio of n assets can be calculated from
the co-variances of the pairs of securities and the proportions
of the securities in the portfolio.
15
Mean-variance portfolio theory
The diversification of the risk:
1 n
R p   ri
n i 1
1
2
p  2
n
n
2
i 1
n
2

 
Portfolio standard deviation
The variance of a portfolio can be reduced by adding
more securities in the portfolio.
Let’s suppose to have n securities, with returns not
correlated. Their expected return is r with variance 2.
All securities are equally weighted in the portfolio:
0
5
10
15
Number of Securities
16
Mean-variance portfolio theory
In the previous example, the choice of 60% A and 40% B is just one of an
infinite number of portfolios
If we change the proportions of A and B (keeping the same correlation), we
have a set of infinite portfolios.
Rp
17.5%
5.5%
Y
X
12.7%
L’
Z
SD p  15.44% 
X
R p  12.7% 
 A, B  0.1639
L
SD p  11.5% 
Z
R p  5.5% 
SD p  25.86% 
Y
R p  17.5% 
p
Diversification occurs when the correlation between A and B is lower than 1
No diversification occurs if the correlation between A and B is =1 (straight line)
11.50%
15.44%
25.86%

17
Mean-variance portfolio theory
MV=minimum variance portfolio. It is the portfolio with
the minimum standard deviation
The choice of the portfolio depends on the propensity to
risk of the investor
Y
RY
X
RZ
MV
Opportunity set
Z
Z
Y
The curve from MV to Y is the EFFICIENT FRONTIER
18
Mean-variance portfolio theory
For different cases of correlation between the two
assets:
 AB  1
RB
 AB  0
RA
The lower the correlation,
the more bend there is in
the curve
 AB  1
A
B
19
Mean-variance portfolio theory
The opportunity set of a portfolio of n securities :
If the portfolio has at least 3 securities (with different expected
returns and not perfectly correlated), the opportunity set is a
continous two-dimensional region.
The opportunity set is convex on the left. Each combination of 2
securities is on the left (or on the same line) of the straight line
that joins them.
R
11
4
MV
2
3

20
Mean-variance portfolio theory
R
R
Efficient frontier


Efficient frontier: the set of portfolios that risk adverse investors
would choose.
21
Mean-variance portfolio theory
We have a risk-free security available for investment
(risk-free return rf and   0) and a risky security ( r ,  2)
The co-variance between them is 0.
Consider a portfolio with a invested in the risk-free
asset and 1-a invested in the risky asset:
R p  arf  (1  a)r
 p  (1  a) 2  2  (1  a)
22
Mean-variance portfolio theory
Expected return and standard deviation vary linearly with a
R
rf
R p  arf  (1  a)r
 p  (1  a) 2  2  (1  a)

23
Mean-variance portfolio theory
Now consider portfolios consisting of some combination of n risky
assets and the risk-free asset
R
When you include a risk-free
asset, the possible region is
an infinite triangular shape!
rf

24
Mean-variance portfolio theory
ONE-FUND
THEOREM
R
F
rf

25
Mean-variance portfolio theory
Q is a portfolio of securities within the feasible set of risky securities. If an
investor wants to combine Q with an investment in the risky-less asset moves
along line I (choosing point 1 or 2 or 3).
Line II is tangent in F to the efficient set of risky securities and it is the
efficient set of all assets (risky and risk-less)
The portfolio of risky assets held by an investor will always be point F,
combined with the risk-less asset.
R
Line II
5
F
4
rf
1
2
Q
3
Line I

26
Market equilibrium
We now use the general results of the meanvariance portfolio theory to identify the market
equilibrium relations between price and returns
of asset portfolios.
We will then turn to the CAPM developed by
Sharpe, Lintner and Mossin
We focus on all investors which we assume base
their expectations from the same publicly
available information on past price movements
27
Market equilibrium
28
Market equilibrium
A broad-based index such as S&Poor’s 500 is a proxy
for the market portfolio
29
Market equilibrium
An asset’s weight (wi) in a market portfolio is
equal to the proportion of that asset’s total capital
value to the total market capitalization
The capitalization weights are proportional to the
total market capitalization, not to the number of
shares.
Stock
N. shares
Price
capitalization
weight
A
10,000
6
60,000
0.15
B
30,000
4
120,000
0.3
C
40,000
5.5
220,000
0.55
capitalization A/total
capitalization=60,000/400,000
400,000
30
Market equilibrium
Changes in prices affect the estimates of assets returns
and investors will recalculate their optimal portfolios.
This process continues until demand=supply
(equilibrium)
Prices adjust to drive the market to efficiency.
In this context, the optimal portfolio (or the efficient
fund of risky assets) is the market portfolio.
31
Assumptions-CAPM
CAPM is based on the following assumptions:
1. All investors have identical expectations about expected
returns, standard deviations, and correlation coefficients for
all securities.
2. All investors have the same one-period investment time
horizon.
3. All investors can borrow or lend money at the risk-free rate
of return.
4. There are no transaction costs.
5. There are no personal income taxes so that investors are
indifferent between capital gains and dividends.
6. There are many investors, and no single investor can affect
the price of a stock through his or her buying and selling
decisions. Therefore, investors are price-takers.
7. Capital markets are in equilibrium.
32
CAPITAL ASSET PRICING MODEL
( )
r
The expected return (and SD) of an arbitrary efficient portfolio is r ( )
rM  rf  (rM  rf )
Risk premium
33
CAPITAL ASSET PRICING MODEL
The CML states that as
risk increases, the
corresponding expected
rate of return must also
increase!
34
CAPITAL ASSET PRICING MODEL
It is impossible for an individual to assemble the market portfolio.
There are some index funds (mutual funds) that try to duplicate the
portfolio of a major stock market index.
Hp: every investor has the same info about the uncertain returns
of stocks
CAPM can be used to evaluate the performance of an investment
portfolio (mutual funds/pension funds)
35
CAPITAL ASSET PRICING MODEL
ri  rf   i (rM  rf )
The expected return on a security is linearly (and positively)
related to to its beta.
rM  r f
is usually positive because the average return on
the market is usually higher than the average risk-free rate.
36
CAPITAL ASSET PRICING MODEL
Beta measures the risk of a security in a large portfolio. More
precisely, it measures the responsiveness (correlation) of a
security to movements in the market portfolio.
Virtually no stock has a negative beta. Stocks with negative
beta are hedges and reduce the risk of a portfolio.
The average beta across all securities, when weighted by the
proportion of each security’s market value (Xi) to that of the
market portfolio, is 1
n
 xi  i  1
i 1
37
CAPITAL ASSET PRICING MODEL
ri  rf   i (rM  rf )
38
CAPITAL ASSET PRICING MODEL
An individual who holds 1 security should use as a measure of
the security’s risk the variance (SD) of his security’s returns.
For individual who holds a diversified portfolio, the contribution
of a security to the risk of the portfolio is best measured by
beta.
39
CAPITAL ASSET PRICING MODEL
The asset is
completely
uncorrelated
with the
market
 i  1  ri  rM
 i  1  ri  rM
 i  1  ri  rM
40
CAPITAL ASSET PRICING MODEL
The estimation of beta:
Betas are usually
estimated by financial
services companies
 They use records of
past stock values
Highly leveraged
companies have usually
higher betas
41
CAPITAL ASSET PRICING MODEL
A security’s
market risk is
measured by
beta, its
expected
sensitivity to
the market.
42
CAPITAL ASSET PRICING MODEL
If you cumulate data on the security return and market return,
you obtain a set of points, which are interpolated by a straight
line where beta represents the inclination.
43
CAPITAL ASSET PRICING MODEL
The beta of a portfolio is the weighted average of the
betas of the assets, with the weights being identical to
those that define the portfolio:
n
 P   wi  i
i 1
44
CAPITAL ASSET PRICING MODEL
The security market line
(SML)
The expected rate of return
increases linearly as Beta
increases
r
rM
M
The slope is upward sloping
because r  r
M
f
1

45
SML and CML
SML
It relates expected
returns to beta
It holds both for
individual securities
than for possible
portfolios
CML
It traces the efficient set
of portfolios formed from
both risky and risk-less
assets
It holds only for efficient
portfolios
46
Systematic and idiosyncratic risk
ri  rf  (rM  rf )  i   i
     var(  i )
2
i
2
i
2
M
The first part is called SYSTEMATIC RISK (associated with
the market). It cannot be diversified.
The second term is IDIOSYNCRATIC or NON- SYSTEMATIC
RISK (firm-specific, uncorrelated with the market). It can
be diversified.
47
Systematic and idiosyncratic risk
Assets on the CML have only systematic risk.
Assets carrying non-systematic risk do not fall on the CML.
As the non-systematic risk increases, the points drift to
the right
The horizontal distance of a point from the CML is a
measure of the non-systematic risk
r
Asset with
systematic risk
CML
rf

Assets with nonsystematic risk
48
CAPM as a pricing model
The CAPM is a pricing model
Suppose that an asset is purchased at price P and sold at price Q.
P is known, Q is random
If all the assumptions of CAPM are satisfied it follows:
Q P
r
 r f  (rM  r f ) 
P
Q
P
1  r f  (rM  rf ) 
49