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.
– i.e. Risk Premium should be positive
• Agents preference (taste) gives the
optimal allocation between a risky
portfolio and a risk-free asset.
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Speculation vs. Gamble
• Speculation
– Taking considerable risk for a commensurate
gain
– Parties have heterogeneous expectations
• Gamble
– Bet or wager on an uncertain outcome for
enjoyment
– Parties assign the same probabilities to the
possible outcomes
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Table 6.1 Available Risky Portfolios (Riskfree 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 of portfolio
with return r
E ( r ) = expected
return portfolio
A = coefficient of risk
aversion
s2 = variance of
returns of portfolio
½ = a scaling factor
1
2
U  E (r )  As
2
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Table 6.2 Utility Scores of Alternative Portfolios for
Investors with Varying Degree of Risk Aversion
IN CLASS EXERCISE. Anwer: How high the
risk aversion coefficient (A) has to be so that L
is preferred over M and H?
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Mean-Variance (M-V) Criterion
• Portfolio A dominates portfolio B if:
• And
ErA   ErB 
sA sB
• As noted before: this does not determine the choice of
one portfolio, but a whole set of efficient portfolios.
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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 RiskFree Portfolios
Asset Allocation:
• Is a very important
part of portfolio
construction.
• Refers to the choice
among broad asset
classes.
– % of total Investment in
risky vs. risk-free assets
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 Example
Total Amount Invested
Risk-free money market
fund
Total risk assets
Equities
$300,000
$90,000
Bonds (long-term)
$96,600
$113,400
WE 
 0.54
$210,000
Proportion of Risk
assets on Equities
$210,000
$113,400
$96,600
WB 
 0.46
$210,00
Proportion of Risk
assets on Bonds
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Basic Asset Allocation
• P is the complete portfolio where we have y
as the weight on the risky portfolio and (1-y)
= weight of risk-free assets:
$210,000
y
 0.7
$300,000
1 y 
$113,400
E:
 .378
$300,000
$90,000
 0.3
$300,000
$96,600
B:
 .322
$300,000
• Complete Portfolio is:
(0.3, 0.378, 0.322)
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The Risk-Free Asset
• Only the government can issue defaultfree 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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Portfolios of One Risky Asset and a RiskFree Asset
• 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.
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Example Using Chapter 6.4 Numbers
rf = 7%
srf = 0%
E(rp) = 15%
sp = 22%
y = % in p
(1-y) = % in rf
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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
E (rc )  rf  y  E (rP )  rf 
Erc   7  y15  7
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Example (Ctd.)
• The risk of the complete portfolio is
the weight of P times the risk of P:
s C  ys P  22y
– This follows straight from the formulas we
saw before and the fact that any constant
random variable has zero variance.
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Feasible (var, mean)
• Taken together this determines the
set of feasible (mean,variance)
portfolio return:
Erc   7  y15  7
s C  ys P  22y
– This determines a straight line, which we
call Capital Allocation Line. Next we derive
it’s equation completely
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Example (Ctd.)
• Rearrange and substitute y=sC/sP:
sC
8

E rC   rf 
E rP   rf   7  s C
sP
22
– The sub-index C is to stand for complete
portfolio
Slope
E rP   rf
sP
8

22
– The slope has a special name: Sharpe ratio.
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Figure 6.4 The Investment
Opportunity Set
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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 )  rf  y  E (rP )  rf 
– Variance:
s ys
2
C
2
2
P
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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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Portfolio problem
• Agent’s problem with one risky and one
risk-free asset is thus:
• Pick portfolio (y, 1-y) to maximize utility U
– U(y,1-y) = E(r_C) -0.5*A*Var(r_C)
• Where r_C is the complete portfolio
– This is the same as
– r_f + y[E(r) – r_f] -0.5*A*y^2*Var(r)
– Solution (take foc) is y (proportion on risky
asset) as (1/A)*SharpeRatio
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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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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 )  rf  y  E (rP )  rf 
– Variance:
s ys
2
C
2
2
P
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One word on Indifference Curves
• If you see the IC curves over (mean,st. dev) you
will note that these are all nice smooth concave
curves.
– This is an assumption.
– Note that agents have preference over random variables
(representing payoff/return). A random variable, in
general, is not completely described by (mean, variance).
• That is, in general, we can have X and Y with mean(X) <
mean (Y) and var(X)=var(Y) BUT X is ranked better than Y
nonetheless.
– IF agents have expected utility, we can solve this
issue in two ways.
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One word on Indifference Curves
• First method is:
• Assumption 1: all random variables are normally
distributed
• Assumption 2: agents have expected utility with
Bernoulli given by u(x)= a*x^2 + bx + c
– BLACKBOARD (Expected utility computation)
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Word on our Portfolio problem
• So far we saw how to solver for
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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.
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