Prof. Dr. Roland Kirstein
Economics of Business and Law
Faculty of Economics and Management
http://econbizlaw.de
The slope of the indifference curve
is the marginal rate of substitution between risk and (expected) return
=> how many units of additional return a (risk-averse) decision-maker
demands in order to accept one additional unit of risk.
U(µ, σ)= µ-cσ2 => dµ/dσ = - [∂U/∂σ] / [∂U/∂µ] = -(-c)/1=c
The parameter c denotes a (risk-averse) decision-makers degree of riskaversion (his propensity towards risk): the higher the absolute value of c,
the more risk-averse the decision-maker, the steeper his/her
indifference curves (lower absolute value of c => flatter IC).
118
Prof. Dr. Roland Kirstein
Economics of Business and Law
Faculty of Economics and Management
http://econbizlaw.de
Beta analysis
Beta: contribution of an individual asset to the portfolio risk.
Market Portfolio - Portfolio of all assets in the economy.
In practice a broad stock market index, such as the
S&P Composite, is used to represent the market.
Market beta - Sensitivity of an individual stock’s return to the return on
the market portfolio.
• β>0: stock price increases with index
• β>1: stock price increases faster than index
What’s the beta of an average (representative) stock/portfolio?
119
Prof. Dr. Roland Kirstein
Economics of Business and
an Law
Faculty of Economics and Management
http://econbizlaw.de
Beta and Unique Risk
1. Total risk = diversifiable
risk + market risk
Expected
stock return
2. Market risk is measured
by beta, the sensitivity to
market changes
120
Prof. Dr. Roland Kirstein
Economics of Business and Law
Faculty of Economics and Management
http://econbizlaw.de
Formal Definition of Beta
σ im
βi = 2
σm
Covariance of stock i with
the market (=index)
Variance of the market
βi measures the contribution of stock i to the portfolio risk.
An asset can have a beta with regard to a portfolio
or to the market (just plug in the adequate co-/variances).
121
Prof. Dr. Roland Kirstein
Economics of Business and Law
Faculty of Economics and Management
http://econbizlaw.de
Portfolio Risk and Beta
Coca Cola vs. Reebok example
CC: portfolio share x1= .65, σ1 = 31.5
Reebok: share x2= .35, σ2 = 58.5
Assume a correlation coefficient ρ = .2
=> Portfolio var σ2=1,006.1
=> Portfolio σ=31.7
122
Prof. Dr. Roland Kirstein
Economics of Business and Law
Faculty of Economics and Management
http://econbizlaw.de
Coca - Cola
Coca - Cola
Reebok
x12 σ12 = .652 × 31.52
x1x 2ρ12 σ1σ 2 =
.65 × .35 × .2 × 31.5 × 58.5
Reebok
x1x 2ρ12 σ1σ 2 =
.65 × .35 × .2 × 31.5 × 58.5
x 22 σ 22 = .352 × 58.52
CC’s contribution to the portfolio risk is given by the sum of the left
column (=sum of the upper line) in this table, and depends on its share
(x2=1-x1) and its (average) covariance with the stock in the portfolio.
Covariance between CC and R: σ12 = ρ x σ1 x σ2 = .2x31.5x58.5=368.55
123
Prof. Dr. Roland Kirstein
Economics of Business and Law
Faculty of Economics and Management
http://econbizlaw.de
CC
x1 ( x1 σ12 + x 2 σ12 ) = .65 × (.65 × 31.52 + .35 × 368.55) = .65 × 774
R
x 2 ( x 2σ 22 + x1σ12 ) = .35 × (.35 × 58.52 + .65 × 368.55) = .35 ×1437.3
In both rows, the outcome equals 503.07
=> If the portfolio shares are x1=.65 and x2=.35, then both stocks
contribute the same to the portfolio variance of 1,006.14 (σ=31.71)
774 is CC’s (weighted or) average covariance with the stocks in the
portfolio, i.e., with CC (itself) and with R
1437.3 is R’s average covariance).
Average covariance => portfolio beta
124
Prof. Dr. Roland Kirstein
Economics of Business and Law
Faculty of Economics and Management
http://econbizlaw.de
Portfolio Beta
The proportion of the risk of the Coca Cola component in the example
portfolio is computed by its share times its portfolio beta, i.e., the ratio
of average covariance over portfolio variance:
CovCC
774
share x
= .65 x
= .65 x.769 = .5
portfolioVar
1,006.14
x1*β1 = 1’s contribution to the portolio risk
The same computation for R yields .35 x 1.43 = .5
125
Prof. Dr. Roland Kirstein
Economics of Business and Law
Faculty of Economics and Management
http://econbizlaw.de
The portfolio beta is a measure of a stock’s contribution to the portfolio
risk.
The “portfolio beta of the portfolio” is the weighted average of the betas
of its components, which yields 1 (of course): x1*β1 + (1-x1)*β2=1
If the basis for asset i’s β is not this portfolio, but the market, then this
computation does not (necessarily) yield 1, but the portfolio’s market β.
In the example portfolio, the two components have an equal impact on
the portfolio risk:
• R is riskier (beta=1.43 against CC’s beta of .77),
• but the portfolio contains more of CC (.65 vs. .35).
126
Prof. Dr. Roland Kirstein
Economics of Business and Law
Faculty of Economics and Management
http://econbizlaw.de
Section 8: Portfolio Selection and CAPM
Topics covered
• Markowitz’ Portfolio Selection Theory
• Risk and Return Relationship
• The CAPM
• CAPM Alternatives
127
Prof. Dr. Roland Kirstein
Economics of Business and Law
Faculty of Economics and Management
http://econbizlaw.de
Markowitz Portfolio Theory
Combining stocks to portfolios (“diversification”) can reduce risk
below the level obtained from a simple weighted average calculation,
and even below the lowest SD of all stocks involved (depending on
correlation coefficients and the composition of the portfolio).
Even more striking: If you own stock with a low SD, you can reduce your
risk by mixing in a high-risk stock.
The smaller the correlation coefficients, the lower the remaining risk.
All this is different for “zero investment strategies.”
A stock’s contribution to the portfolio risk depends on its SD and its
correlation coefficient with the other components (=> its portfolio beta)
The combinations of stocks that minimize portfolio SD for each level of
expected portfolio return constitute the set of efficient portfolios.
128
Prof. Dr. Roland Kirstein
Economics of Business and
an Law
Faculty of Economics and Management
http://econbizlaw.de
Efficient Portfolios
Expected Returns and Standard Deviations vary when the composition of
the portfolio changes.
The blue line is called (set of) “efficient
portfolios”, as these portfolios minimize
the portfolio risk for each attainable
a
level
of expected portfolio return.
If the efficiency criterion is to maximize
return for each attainable risk level, then
only the upper part of the blue line is
efficient (from the bliss point on)
129
Prof. Dr. Roland Kirstein
Economics of Business and Law
Faculty of Economics and Management
http://econbizlaw.de
Markowitz Portfolio Theory
The efficient portfolio line is “bulkier”, the lower the correlation
coefficient - remember the figure
E(r)
TWO-SECURITY PORTFOLIOS WITH
DIFFERENT CORRELATIONS
13%
ρ = -1
ρ=0
ρ = .3
ρ=1
8%
12%
20%
St. Dev
130
Prof. Dr. Roland Kirstein
Economics of Business and
an Law
Faculty of Economics and Management
http://econbizlaw.de
Even with two stocks of identical SD and return which, however, are not
perfectly correlated, diversification would decrease risk.
The
he portfolio betas of the tw
two
o stocks are identical; the minimum
portfolio depends on the correlation coefficient
coefficient.
131
Prof. Dr. Roland Kirstein
Economics of Business and
an Law
Faculty of Economics and Management
http://econbizlaw.de
Now consider a situation in which not just two, but many risky stocks are
available (with different risk/expected return combinations)
132
Prof. Dr. Roland Kirstein
Economics of Business and
an Law
Faculty of Economics and Management
http://econbizlaw.de
Efficient Frontier
Each “half egg shell”” represents the possible weighted combinations of
two stocks (individual stocks left out from the figure)
figure).
Risk-return-combinations
Risk
ON
these egg shells can easily be
reached, but how can we obtain
points “between” these “egg
133
Prof. Dr. Roland Kirstein
Economics of Business and Law
Faculty of Economics and Management
http://econbizlaw.de
Example: more than two assets
Stocks
σ
A Corp
28
60%
15%
B Inc.
42
40%
21%
% of Portfolio
Avg Return
Correlation Coefficient = .4
Return (weighted average) of portfolio = 17.4
Standard Deviation of portfolio = 28.1
This portfolio AB has a higher expected return and about the same risk
as stock A.
Use formulas from previous section to compute the share of stock A in a risk-minimizing
portfolio AB* (80,48%). Note that AB* contains a positive amount of the riskier stock B.
134
Prof. Dr. Roland Kirstein
Economics of Business and
an Law
Faculty of Economics and Management
http://econbizlaw.de
Example: more than two assets
135
Prof. Dr. Roland Kirstein
Economics of Business and Law
Faculty of Economics and Management
http://econbizlaw.de
Now we increase our investment by adding stock of “C Corp” to our
portfolio, with σN =30 and r=19%. Assume a correlation coefficient
between the portfolio AB and C of .3
Stocks
σ
% of Portfolio
Avg Return
Portfolio (AB) 28.1
50%
17.4%
C Corp.
50%
19%
30
Expected portfolio return (weighted avg.) = 18.20
Standard Deviation of new portfolio = 23.43
(Risk min. portfolio would contain 45% of C, and 55% of AB)
136
Prof. Dr. Roland Kirstein
Economics of Business and
an Law
Faculty of Economics and Management
http://econbizlaw.de
Example: more than two assets
By
y adding a third stock (C) to our portfolio we can obtain a risk-return
risk
combination that is not on the „half egg shell“ between A and B => with
enough
nough risky assets, each point can be reached
reached.
137
Prof. Dr. Roland Kirstein
Economics of Business and
an Law
Faculty of Economics and Management
http://econbizlaw.de
Efficient Frontier
The outer convex hull of all possible curves constitutes the efficient
frontier of the market. Efficiency criterion: min σ for all levels of r.
If you look at max r for all levels of σ,, then only the north-west
north
part of
the bold curve would be called efficient
efficient.
138
Prof. Dr. Roland Kirstein
Economics of Business and
an Law
Faculty of Economics and Management
http://econbizlaw.de
If (and only if) two stocks exist with a correlation coefficient of -1, then a
portfolio composition of these two stocks exist with zero risk
In the diagram no combination of stocks is characterized by ρ=-1.
139
Prof. Dr. Roland Kirstein
Economics of Business and
an Law
Faculty of Economics and Management
http://econbizlaw.de
Markowitz Portfolio Theory
Now consider a situation in which many risky stocks are available (with
different risk/expected return combinations), and one risk free asset
exists.
140
Prof. Dr. Roland Kirstein
Economics of Business and
an Law
Faculty of Economics and Management
http://econbizlaw.de
Market portfolio
Draw a line starting in the parameters of the risk
risk-free
free asset (r
( f,0) that is
tangent to the set of efficient portfolios (not above => not feasible;
not intersecting => leaves out more efficient portfolios).
141
Prof. Dr. Roland Kirstein
Economics of Business and
an Law
Faculty of Economics and Management
http://econbizlaw.de
The tangent point is the average market portfolio (of risky assets),
denoted S, with (rS, σS); it can be created by an adequate combination of
two (or more) risky assets S1 and S2.
There is no reason to hold, e.g., the portfolios T or V.
142
Prof. Dr. Roland Kirstein
Economics of Business and
an Law
Faculty of Economics and Management
http://econbizlaw.de
Lending orr borrowing at the risk free rate (rf) allows us to choose
r- σ -combinations outside the efficient frontier by creating linear
combinations of the risk
risk-free
free asset and the market portfolio S
(or: we create a portfolio that also contains the risk
risk-free
free asset with pos.
or neg. weight).
Lending: split
it your budget among
stock and risk free bonds (short
selling excluded).
Borrowing: short sell bonds and
buy more stock (or short buy
stock).
143
Prof. Dr. Roland Kirstein
Economics of Business and
an Law
Faculty of Economics and Management
http://econbizlaw.de
Best Efficient Portfolio
The tangent point is the best (of the) efficient
portfolio(s) as it offers the highest ratio of market risk
premium (rm-rf) over portfolio SD = slope of tangent line
144
Prof. Dr. Roland Kirstein
Economics of Business and
an Law
Faculty of Economics and Management
http://econbizlaw.de
Portfolio Choice
145
Prof. Dr. Roland Kirstein
Economics of Business and
an Law
Faculty of Economics and Management
http://econbizlaw.de
146