Chapter 3 Risk and Return: Part II
Topics
 Portfolio Theory
 Efficient Frontier
 Capital Market Line (CML)
 Capital Asset Pricing Model (CAPM)
 Security Market Line (SML)
 Beta calculation
 Arbitrage pricing theory
 Fama-French 3-factor model
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Two axioms regarding individuals' preferences:
1. Individuals prefer more wealth to less.
2. Individuals are averse to taking risk.
The first axiom needs little explanation or justification. Let’s motivate
the second a little further.
Ex: Would you rather have: a) $30K for sure or b) a 50-50 chance of
receiving $100K or paying $30K
Q. How does your choice demonstrate your aversion to risk?
IV. The Mean and Standard Deviation Rule (M-S rule)
Utilizing the two axioms above (and that security returns are normality
distributed) the rule for choosing between risky assets.
Asset A is preferred to asset B if either:
E(rA) > E(rB) and σ (rA) < = σ (rA)
σ (rA) < σ (rA) and E(rA) >= E(rB)
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Ex: Consider the expected return and risk of four securities
Stock
E(r)
σ(r)
A
.20
.15
B
.20
.20
C
.15
.15
D
.15
.20
Q. According to the M-S rule, which risky asset would we prefer;
(A or B), (A or C), (A or D)? What about (B or C)?
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Conclusions of the M-S Rule:
Assets which lie the furthest to the NW are mean-variance efficient
because they offer the highest expected return for a given level of risk.
Goal of portfolio theory: combine assets in such a way that the
portfolio lies as far to the NW as possible.
Note: For pts B vs. C, the M -S rule is inconclusive. One cannot make
general statements regarding which of these assets is preferred because it
depends on the risk aversion of the individual.
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If we have 2 risky assets, the risk-return combinations are as follows
(with ρ=1, 0,-1, respectively).
FIGURE 3.1
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If we have 2 risky assets, the risk-return combinations are as follows
(with ρ=1, 0,-1, respectively).
End points: 100% in A
r̂A = 5, σA = 4
100% in B
r̂B = 8, σB = 10
Middle points
Case I, ρA,B= +1
-no decline in σ (risk) from combining assets A &B
-no benefits from diversification (perfectly correlated)
 risk/return graph a straight line
Case II, ρA,B= 0 -there is a dip in σ when combining assets A &B
-benefits of diversification of uncorrelated assets
 risk/return graph backward bending
From pt A → you can ↑ return while ↓ing risk
Case III, ρA,B= -1
-sharp decline in risk by combining negatively
correlated assets
-greatest benefit of diversification possible
- notice there is a combo that eliminates all risk
 sharp backward bending risk/return graph
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Putting the graphs in the last column together (covering difft ρ), for the 2
asset portfolio:
Conclusion: There are gains to diversification (combining assets to make
up portfolio) as long as ρ < 1.
What are the gains?
The elbow- the ability to ↑ r̂p while ↓’ing σp .
Notice: You will never choose a point at the bottom of the elbow,
’cause there is always a pt on the top of elbow that has the same σp,
but has a higher r̂p
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Feasible and Efficient Portfolios
The feasible set of portfolios - represents all portfolios that can be
constructed from a given set of stocks.
An efficient portfolio offers:
1. the most return for a given amount of risk, or
2. the least risk for a given amount of return.
 The collection of efficient portfolios is called the efficient set or
efficient frontier.
-the set of all possible optimal portfolios (curve B to E)
Remember: Goal is NW corner
Note: Column c of 3.1 illustrates the feasible sets when combining 2 assets (you get a line or a
curve).
What is the feasible set if the portfolio consists of more than 2 assets?
Pts A, H, G & E → portfolio with 100% weight in 1 asset (0% in others)
Each point in the feasible set represents a portfolio w/ a r̂p & σp
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Aside: The Theory of Indifference Curves
Indifference curve- the set of points in which the consumer is indifferent
between each point on the curve.
Given 2 products, each individual faces a family of indifferences
curves that represent that individual’s preferences.
Ex: The consumer likes 1 product (good) and dislikes the other (bad).
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Consumer is indifferent between A, B, C and D on I1.
Goal: To be on the highest indifference curve possible  highest
satisfaction, given your preferences.
In Portfolio Theory: Indifference curves reflect investor’s attitude
toward risk. They differ among investors because of differences in risk
aversion.
the good  r̂p, the bad  σp
Individual A: risk averse, steep indifference curves
Individual B: not risk averse, relatively flat indifference curves
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Bringing together the efficient frontier and indifference curves
Optimal Portfolios
 Recall: investor’s goal to get on highest indifference curve that
represents her preferences.
 An investor’s optimal portfolio is defined by the tangency point
between the efficient set and the investor’s indifference curve.
 Slope of indif curve = slope of efficient frontier
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Now let’s combine our optimal risky portfolio with a risk-free asset.
First we need to define the Capital Market Line (CML):
 Define: pt rrf → 100% invested in risk-free asset
pt M →100% in optimal risky portfolio
 The Capital Market Line (CML) is all linear combinations of the
risk-free asset and Portfolio M.
 Note: M→Z indicates short sales
Efficient Set with a Risk-Free Asset
The CML Equation
 Portfolios below the CML are inferior.
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- The CML defines our efficient frontier when combining
M and a risk-free asset.
- All investors will choose a portfolio on the CML.
 The CML tells us the expected rate of return on any efficient
portfolio is equal to the risk-free rate plus a risk premium.
Let’s add some indifference curves to our graph
FIGURE 3.5
The optimal portfolio for any investor is the point of tangency between
the CML and the investor’s indifference curves (pt R).
 Investors will move from N (the optimal risky portfolio) to R
(optimal portfolio which includes a risk free asset.
 Notice R is on a higher indifference curve than N.
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