INVESTMENTS | BODIE, KANE, MARCUS
Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin
CHAPTER 6
Risk Aversion and Capital
Allocation to Risky Assets
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Allocation to Risky Assets
• Investors will avoid risk unless there
is a reward.
• The utility model gives the optimal
allocation between a risky portfolio
and a risk-free asset.
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Risk and Risk Aversion
• Speculation
– Taking considerable risk for a
commensurate gain
– Parties have heterogeneous
expectations
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Risk and Risk Aversion
• Gamble
– Bet or wager on an uncertain outcome
for enjoyment
– Parties assign the same probabilities to
the possible outcomes
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Risk Aversion and Utility Values
• Investors are willing to consider:
– risk-free assets
– speculative positions with positive risk
premiums
• Portfolio attractiveness increases with
expected return and decreases with risk.
• What happens when return increases
with risk?
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Table 6.1 Available Risky Portfolios (Risk-
free Rate = 5%)
Each portfolio receives a utility score to
assess the investor’s risk/return trade off
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Utility Function
U = utility
E ( r ) = expected
return on the asset
or portfolio
A = coefficient of risk
aversion
s2 = variance of
returns
½ = a scaling factor
21( )
2
U E r As 
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Table 6.2 Utility Scores of Alternative Portfolios for
Investors with Varying Degree of Risk Aversion
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Mean-Variance (M-V) Criterion
• Portfolio A dominates portfolio B if:
• And
   BA rErE 
BA ss 
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Estimating Risk Aversion
• Use questionnaires
• Observe individuals’ decisions when
confronted with risk
• Observe how much people are willing to
pay to avoid risk
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Capital Allocation Across Risky and Risk-
Free Portfolios
Asset Allocation:
• Is a very important
part of portfolio
construction.
• Refers to the choice
among broad asset
classes.
Controlling Risk:
• Simplest way:
Manipulate the
fraction of the
portfolio invested in
risk-free assets
versus the portion
invested in the risky
assets
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Basic Asset Allocation
Total Market Value $300,000
Risk-free money market
fund
$90,000
Equities $113,400
Bonds (long-term) $96,600
Total risk assets $210,000
54.0
000,210$
400,113$
EW 46.0
00,210$
600,96$
BW
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Basic Asset Allocation
• Let y = weight of the risky portfolio, P,
in the complete portfolio; (1-y) = weight
of risk-free assets:
7.0
000,300$
000,210$
y 3.0
000,300$
000,90$
1  y
378.
000,300$
400,113$
: E 322.000,300$
600,96$
: B
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The Risk-Free Asset
• Only the government can issue
default-free bonds.
– Risk-free in real terms only if price
indexed and maturity equal to investor’s
holding period.
• T-bills viewed as “the” risk-free asset
• Money market funds also considered
risk-free in practice
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Figure 6.3 Spread Between 3-Month
CD and T-bill Rates
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• It’s possible to create a complete portfolio
by splitting investment funds between safe
and risky assets.
– Let y=portion allocated to the risky portfolio, P
– (1-y)=portion to be invested in risk-free asset,
F.
Portfolios of One Risky Asset and a Risk-Free
Asset
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rf = 7% srf = 0%
E(rp) = 15% sp = 22%
y = % in p (1-y) = % in rf
Example Using Chapter 6.4 Numbers
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Example (Ctd.)
The expected
return on the
complete
portfolio is the
risk-free rate
plus the weight
of P times the
risk premium of
P
( ) ( )c f P fE r r y E r r    
   7157  yrE c
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Example (Ctd.)
• The risk of the complete portfolio is
the weight of P times the risk of P:
yy PC 22 ss
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Example (Ctd.)
• Rearrange and substitute y=sC/sP:
     CfP
P
C
fC rrErrE s
s
s
22
8
7
 
22
8



P
fP rrE
Slope
s
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Figure 6.4 The Investment
Opportunity Set
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• Lend at rf=7% and borrow at rf=9%
– Lending range slope = 8/22 = 0.36
– Borrowing range slope = 6/22 = 0.27
• CAL kinks at P
Capital Allocation Line with Leverage
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Figure 6.5 The Opportunity Set with
Differential Borrowing and Lending Rates
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Risk Tolerance and Asset Allocation
• The investor must choose one optimal
portfolio, C, from the set of feasible
choices
– Expected return of the complete
portfolio:
– Variance:
( ) ( )c f P fE r r y E r r    
2 2 2
C Pys s
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Table 6.4 Utility Levels for Various Positions in Risky
Assets (y) for an Investor with Risk Aversion A = 4
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Figure 6.6 Utility as a Function of
Allocation to the Risky Asset, y
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Table 6.5 Spreadsheet Calculations of
Indifference Curves
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Figure 6.7 Indifference Curves for
U = .05 and U = .09 with A = 2 and A = 4
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Figure 6.8 Finding the Optimal Complete
Portfolio Using Indifference Curves
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Table 6.6 Expected Returns on Four
Indifference Curves and the CAL
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Passive Strategies:
The Capital Market Line
• The passive strategy avoids any direct or
indirect security analysis
• Supply and demand forces may make such
a strategy a reasonable choice for many
investors
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Passive Strategies:
The Capital Market Line
• A natural candidate for a passively held
risky asset would be a well-diversified
portfolio of common stocks such as the
S&P 500.
• The capital market line (CML) is the capital
allocation line formed from 1-month T-bills
and a broad index of common stocks (e.g.
the S&P 500).
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Passive Strategies:
The Capital Market Line
• The CML is given by a strategy that
involves investment in two passive
portfolios:
1. virtually risk-free short-term T-bills (or
a money market fund)
2. a fund of common stocks that mimics
a broad market index.
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Passive Strategies:
The Capital Market Line
• From 1926 to 2009, the passive risky
portfolio offered an average risk premium
of 7.9% with a standard deviation of
20.8%, resulting in a reward-to-volatility
ratio of .38.