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Risk Aversion and Capital Allocation to Risky Assets
6-2
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.
6-3
Risk and Risk Aversion
• Speculation
– Taking considerable risk for a
commensurate gain
– Parties have heterogeneous
expectations
6-4
Risk and Risk Aversion
• Gamble
– Bet or wager on an uncertain outcome
for enjoyment
– Parties assign the same probabilities to
the possible outcomes
6-5
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?
6-6
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
6-7
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( )
2U E r As
6-8
Table 6.2 Utility Scores of Alternative Portfolios for Investors with Varying Degree of Risk Aversion
6-9
Mean-Variance (M-V) Criterion
• Portfolio A dominates portfolio B if:
• And
BA rErE
BA ss
6-10
Estimating Risk Aversion
• Use questionnaires
• Observe individuals’ decisions when
confronted with risk
• Observe how much people are willing to
pay to avoid risk
6-11
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
6-12
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$BW
6-13
Basic Asset Allocation
• Let y = weight of the risky portfolio, P,
in the complete portfolio; (1-y) = weight
of risk-free assets:
7.0000,300$
000,210$y 3.0
000,300$
000,90$1 y
378.000,300$
400,113$: E 322.
000,300$
600,96$: B
6-14
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
6-15
Figure 6.3 Spread Between 3-Month CD and T-bill Rates
6-16
• 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
6-17
rf = 7% srf = 0%
E(rp) = 15% sp = 22%
y = % in p (1-y) = % in rf
Example Using Chapter 6.4 Numbers
6-18
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
6-19
Example (Ctd.)
• The risk of the complete portfolio is
the weight of P times the risk of P:
yy PC 22 ss
6-20
Example (Ctd.)
• Rearrange and substitute y=sC/sP:
CfP
P
CfC rrErrE s
s
s
22
87
22
8
P
fP rrESlope
s
6-21
Figure 6.4 The Investment Opportunity Set
6-22
• 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
6-23
Figure 6.5 The Opportunity Set with Differential Borrowing and Lending Rates
6-24
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
6-25
Table 6.4 Utility Levels for Various Positions in Risky Assets (y) for an Investor with Risk Aversion A = 4
6-26
Figure 6.6 Utility as a Function of Allocation to the Risky Asset, y
6-27
Table 6.5 Spreadsheet Calculations of Indifference Curves
6-28
Figure 6.7 Indifference Curves for U = .05 and U = .09 with A = 2 and A = 4
6-29
Figure 6.8 Finding the Optimal Complete Portfolio Using Indifference Curves
6-30
Table 6.6 Expected Returns on Four Indifference Curves and the CAL
6-31
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
6-32
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).
6-33
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.
6-34
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.
Optimal Risky Portfolios
7-36
The Investment Decision
• Top-down process with 3 steps:
1.Capital allocation between the risky portfolio
and risk-free asset
2.Asset allocation across broad asset classes
3.Security selection of individual assets within
each asset class
7-37
Diversification and Portfolio Risk
• Market risk
– Systematic or nondiversifiable
• Firm-specific risk
– Diversifiable or nonsystematic
7-38
Figure 7.1 Portfolio Risk as a Function of the Number of Stocks in the Portfolio
7-39
Figure 7.2 Portfolio Diversification
7-40
Covariance and Correlation
• Portfolio risk depends on the
correlation between the returns of the
assets in the portfolio
• Covariance and the correlation
coefficient provide a measure of the
way returns of two assets vary
7-41
Two-Security Portfolio: Return
Portfolio Return
Bond Weight
Bond Return
Equity Weight
Equity Return
p D ED E
P
D
D
E
E
r
r
w
r
w
r
w wr r
( ) ( ) ( )p D D E EE r w E r w E r
7-42
= Variance of Security D
= Variance of Security E
= Covariance of returns for
Security D and Security E
Two-Security Portfolio: Risk
EDEDEEDD rrCovwwww ,222222
p sss
2
Es
2
Ds
ED rrCov ,
7-43
Two-Security Portfolio: Risk
• Another way to express variance of the
portfolio:
2 ( , ) ( , ) 2 ( , )P D D D D E E E E D E D Ew w Cov r r w w Cov r r w w Cov r rs
7-44
D,E = Correlation coefficient of
returns
Cov(rD,rE) = DEsDsE
sD = Standard deviation of
returns for Security D
sE = Standard deviation of
returns for Security E
Covariance
7-45
Range of values for 1,2
+ 1.0 > > -1.0
If = 1.0, the securities are perfectly
positively correlated
If = - 1.0, the securities are perfectly
negatively correlated
Correlation Coefficients: Possible Values
7-46
Correlation Coefficients
• When ρDE = 1, there is no diversification
• When ρDE = -1, a perfect hedge is possible
DDEEP ww sss
D
ED
DE ww
1
ss
s
7-47
Table 7.2 Computation of Portfolio Variance From the Covariance Matrix
7-48
Three-Asset Portfolio
1 1 2 2 3 3( ) ( ) ( ) ( )pE r w E r w E r w E r
2
3
2
3
2
2
2
2
2
1
2
1
2 ssss wwwp
3,2323,1312,121 222 sss wwwwww
7-49
Figure 7.3 Portfolio Expected Return as a Function of Investment Proportions
7-50
Figure 7.4 Portfolio Standard Deviation as a Function of Investment Proportions
7-51
The Minimum Variance Portfolio
• The minimum variance portfolio is the portfolio composed of the risky assets that has the smallest standard deviation, the portfolio with least risk.
• When correlation is less than +1, the portfolio standard deviation may be smaller than that of either of the individual component assets.
• When correlation is -1, the standard deviation of the minimum variance portfolio is zero.
7-52
Figure 7.5 Portfolio Expected Return as a Function of Standard Deviation
7-53
• The amount of possible risk reduction
through diversification depends on the
correlation.
• The risk reduction potential increases as
the correlation approaches -1.
– If = +1.0, no risk reduction is possible.
– If = 0, σP may be less than the standard
deviation of either component asset.
– If = -1.0, a riskless hedge is possible.
Correlation Effects
7-54
Figure 7.6 The Opportunity Set of the Debt and Equity Funds and Two Feasible CALs
7-55
The Sharpe Ratio
• Maximize the slope of the CAL for any
possible portfolio, P.
• The objective function is the slope:
• The slope is also the Sharpe ratio.
( )P f
P
P
E r rS
s
7-56
Figure 7.7 The Opportunity Set of the Debt and Equity Funds with the Optimal CAL and the Optimal Risky Portfolio
7-57
Figure 7.8 Determination of the Optimal Overall Portfolio
7-58
Markowitz Portfolio Selection Model
• Security Selection
– The first step is to determine the risk-
return opportunities available.
– All portfolios that lie on the minimum-
variance frontier from the global
minimum-variance portfolio and upward
provide the best risk-return
combinations
7-59
Figure 7.10 The Minimum-Variance Frontier of Risky Assets
7-60
Markowitz Portfolio Selection Model
• We now search for the CAL with the
highest reward-to-variability ratio
7-61
Figure 7.11 The Efficient Frontier of Risky Assets with the Optimal CAL
7-62
Markowitz Portfolio Selection Model
• Everyone invests in P, regardless of their
degree of risk aversion.
– More risk averse investors put more in the
risk-free asset.
– Less risk averse investors put more in P.
7-63
Capital Allocation and the Separation Property
• The separation property tells us that the
portfolio choice problem may be
separated into two independent tasks
– Determination of the optimal risky
portfolio is purely technical.
– Allocation of the complete portfolio to T-
bills versus the risky portfolio depends
on personal preference.
7-64
Figure 7.13 Capital Allocation Lines with Various Portfolios from the Efficient Set
7-65
The Power of Diversification
• Remember:
• If we define the average variance and average
covariance of the securities as:
2
1 1
( , )n n
P i j i j
i j
w w Cov r rs
2 2
1
1 1
1
1( , )
( 1)
n
i
i
n n
i j
j ij i
n
Cov Cov r rn n
s s
7-66
The Power of Diversification
• We can then express portfolio variance as:
2 21 1P
nCov
n ns s
7-67
Table 7.4 Risk Reduction of Equally Weighted Portfolios in Correlated and Uncorrelated Universes
7-68
Optimal Portfolios and Nonnormal Returns
• Fat-tailed distributions can result in extreme
values of VaR and ES and encourage smaller
allocations to the risky portfolio.
• If other portfolios provide sufficiently better VaR
and ES values than the mean-variance efficient
portfolio, we may prefer these when faced with
fat-tailed distributions.
7-69
Risk Pooling and the Insurance Principle
• Risk pooling: merging uncorrelated, risky
projects as a means to reduce risk.
– increases the scale of the risky investment by
adding additional uncorrelated assets.
• The insurance principle: risk increases less than
proportionally to the number of policies insured
when the policies are uncorrelated
– Sharpe ratio increases
7-70
Risk Sharing
• As risky assets are added to the portfolio, a
portion of the pool is sold to maintain a risky
portfolio of fixed size.
• Risk sharing combined with risk pooling is the
key to the insurance industry.
• True diversification means spreading a portfolio
of fixed size across many assets, not merely
adding more risky bets to an ever-growing risky
portfolio.
7-71
Investment for the Long Run
Long Term Strategy
• Invest in the risky portfolio for 2 years.
– Long-term strategy is riskier.
– Risk can be reduced by selling some of the risky assets in year 2.
– “Time diversification” is not true diversification.
Short Term Strategy
• Invest in the risky
portfolio for 1 year and
in the risk-free asset for
the second year.
