Chapter Six
Capital Allocation to Risky Assets
INVESTMENTS | BODIE, KANE, MARCUS
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Chapter Overview
• Risk aversion and its estimation
• Two-step process of portfolio construction
• Composition of risky portfolio
• Capital allocation between risky and risk-free
assets
• Passive strategies and the capital market
line (CML)
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Risk and Risk Aversion
Speculation Gamble
• Taking considerable • Bet on an uncertain
risk for a outcome for
commensurate gain enjoyment
• Parties have • Parties assign the
heterogeneous same probabilities to
expectations the possible
outcomes
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Risk and Risk Aversion
• Utility Values
• Investors (risk-averse) 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
Each portfolio receives a utility score to
assess the investor’s risk/return trade off
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Risk Aversion and Utility Values
• Utility Function
• U = Utility
• E(r) = Expected return on the asset or portfolio
• A = Coefficient of risk aversion
• σ2 = Variance of returns
• ½ = A scaling factor
U = E (r ) − 1 Aσ 2
2
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Table 6.2 Utility Scores of Portfolios with
Varying Degrees of Risk Aversion
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Certainty equivalent rate
• We can interpret the utility score of risky
portfolios as a certainty equivalent rate of return
• That is the rate of return that if earned with
certainty (risk-free!) would provide the same
utility score as the risky portfolio
• If the certainty equivalent rate is lower than a
risk-free alternative, the investment is not
desirable
• A=0 risk-neutral
• A<0 risk-lover
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Mean-variance
• Mean-Variance (M-V) Criterion
• Portfolio A dominates portfolio B if:
E (rA )≥ E (rB )
and
σA ≤σB
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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
• Utility functions: relevant to understand investors’ preferences
• Portfolio managers can serve investors without knowing their
preferences. HOW?
• Combining risky assets with risk-free assets!
• Asset Allocation
• The choice among broad asset classes that represents a very
important part of portfolio construction
• The simplest way to control risk is to 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 Example
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
$113,400 $96,600
WE = = 0.54 WB = = 0.46
$210,000 $210,00
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Basic Asset Allocation Example
Let
• y = Weight of the risky portfolio, P, in the
complete portfolio
• (1-y) = Weight of risk-free assets
$210,000 $90,000
y= = 0 .7 1− y = = 0.3
$300,000 $300,000
$113,400 $96,600
E: = .378 B: = .322
$300,000 $300,000
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The Risk-Free Asset
• Only the government can issue default-free
securities
• T-bills viewed as “the” risk-free asset
• Money market funds also considered risk-
free in practice
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Portfolios of One Risky Asset and a Risk-
Free Asset
• It’s possible to create a complete portfolio
(let’s call it C) 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
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One Risky Asset and a Risk-Free Asset:
Example
rf = 7% σrf = 0%
E(rp) = 15% σp = 22%
• The expected return on the complete portfolio:
E (rc ) = 7 + y (15 − 7 )
• The risk of the complete portfolio:
σ C = yσ P = 22 y
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One Risky Asset and a Risk-Free Asset:
Example
• Rearrange and substitute y = σC/σP:
σC 8
E (rC ) = rf + [
σP
]
E (rP )− rf = 7 + σ C
22
E (rP ) − rf 8
Slope = =
σP 22
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Figure 6.4 The Investment Opportunity Set
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The investment opportunity set
• The figure graphs the investment opportunity
set:
• when y=0, the complete portfolio is F
• when y=1 the complete portfolio is P
• when 0<y<1 the portfolios will graph on the
straight line connecting point F and P
• The slope is called reward-to-volatility ratio and
measures extra return for extra risk
• The line is called Capital Allocation Line
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Portfolios of One Risky Asset and a Risk-
Free Asset
• Capital allocation line with leverage
• 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
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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:
[
E (rc ) = r f + y E (rp )− r f ]
• Variance:
2 2 2
σ c = y σ p
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Table 6.4 Utility Levels for
Various Positions in Risky Assets
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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
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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
• A natural candidate for a passively held
risky asset would be a well-diversified
portfolio of common stocks such as the
S&P 500
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Passive Strategies:
The Capital Market Line
• The Capital Market Line (CML)
• Is a capital allocation line formed investment in
two passive portfolios:
1. Virtually risk-free short-term T-bills (or a
money market fund)
2. Fund of common stocks that mimics a
broad market index
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Passive Strategies:
The Capital Market Line
• From 1926 to 2012, the passive risky
portfolio offered an average risk premium
of 8.1% with a standard deviation of
20.48%, resulting in a reward-to-volatility
ratio of .40
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