6-2
Allocation to Risky Assets
• Investors will avoid risk unless there
is a reward.
Risk Aversion and Capital
Allocation to Risky Assets
• 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
6-6
Table 6.1 Available Risky Portfolios (Riskfree Rate = 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?
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
U  E (r ) 
6-8
Table 6.2 Utility Scores of Alternative Portfolios for
Investors with Varying Degree of Risk Aversion
1
As 2
2
6-9
Mean-Variance (M-V) Criterion
Estimating Risk Aversion
• Portfolio A dominates portfolio B if:
• Use questionnaires
E rA   E rB 
• And
6-10
• Observe individuals’ decisions when
confronted with risk
sA sB
• Observe how much people are willing to
pay to avoid risk
6-11
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.
6-12
Basic Asset Allocation
Total Market Value
$300,000
Risk-free money market
fund
$90,000
Equities
Bonds (long-term)
$113,400
$96,600
Controlling Risk:
• Simplest way:
Manipulate the
fraction of the
portfolio invested in
risk-free assets
versus the portion
invested in the risky
assets
$113,400
WTotal
E  risk assets  0.54
$210,000
W 
$96,600
 0.46
$210,00
B
$210,000
6-13
The Risk-Free Asset
Basic Asset Allocation
• Only the government can issue
default-free bonds.
• Let y = weight of the risky portfolio, P,
in the complete portfolio; (1-y) = weight
of risk-free assets:
y
E:
$210,000
 0.7
$300,000
$113,400
 .378
$300,000
– Risk-free in real terms only if price
indexed and maturity equal to investor’s
holding period.
$90,000
1 y 
 0.3
$300,000
B:
6-14
• T-bills viewed as “the” risk-free asset
• Money market funds also considered
risk-free in practice
$96,600
 .322
$300,000
6-15
Figure 6.3 Spread Between 3-Month
CD and T-bill Rates
6-16
Portfolios of One Risky Asset and a Risk-Free
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.
6-17
6-18
Example Using Chapter 6.4 Numbers
rf = 7%
srf = 0%
E(rp) = 15%
sp = 22%
y = % in p
(1-y) = % in rf
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
6-19
6-20
Example (Ctd.)
Example (Ctd.)
• Rearrange and substitute y=sC/sP:
• The risk of the complete portfolio is
the weight of P times the risk of P:
s C  ys P  22 y
E rC   rf 
Slope 
sC
ErP   rf   7  228 s C
sP
E rP   rf
sP

8
22
6-21
Figure 6.4 The Investment
Opportunity Set
6-22
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
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:
E (rc )  rf  y  E (rP )  rf 
– Variance:
s C2  y 2s P2
6-25
6-26
Figure 6.6 Utility as a Function of
Allocation to the Risky Asset, y
Table 6.4 Utility Levels for Various Positions in Risky
Assets (y) for an Investor with Risk Aversion A = 4
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.
7-36
The Investment Decision
• Top-down process with 3 steps:
Optimal Risky Portfolios
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
7-38
Figure 7.1 Portfolio Risk as a Function of the
Number of Stocks in the Portfolio
Diversification and Portfolio Risk
• Market risk
– Systematic or nondiversifiable
• Firm-specific risk
– Diversifiable or nonsystematic
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
rp

rP
 Portfolio Return
wr
D
D
Two-Security Portfolio: Risk
 wEr E
s p2  wD2 s D2  wE2s E2  2wD wE CovrD , rE 
wD  Bond Weight
rD
s D2 = Variance of Security D
 Bond Return
wE  Equity Weight
rE
7-42
s E2 = Variance of Security E
 Equity Return
CovrD , rE  = Covariance of returns for
E (rp )  wD E (rD )  wE E (rE )
Security D and Security E
7-43
Two-Security Portfolio: Risk
7-44
Covariance
Cov(rD,rE) = DEsDsE
• Another way to express variance of the
portfolio:
s P2  wD wDCov(rD , rD )  wE wE Cov(rE , rE )  2wD wE Cov(rD , rE )
D,E = Correlation coefficient of
returns
sD = Standard deviation of
returns for Security D
sE = Standard deviation of
returns for Security E
7-45
Correlation Coefficients
Correlation Coefficients: Possible Values
• When ρDE = 1, there is no diversification
Range of values for 1,2
+ 1.0 >
7-46
s P  wEs E  wDs D
 > -1.0
If  = 1.0, the securities are perfectly
positively correlated
• When ρDE = -1, a perfect hedge is possible
wE 
If  = - 1.0, the securities are perfectly
negatively correlated
sD
 1  wD
s D s E
7-47
Table 7.2 Computation of Portfolio
Variance From the Covariance Matrix
7-48
Three-Asset Portfolio
E (rp )  w1E (r1 )  w2 E (r2 )  w3 E (r3 )
s p2  w12s 12  w22s 22  w32s 32
 2w1w2s 1, 2  2w1w3s 1,3  2w2 w3s 2,3
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
7-54
Figure 7.6 The Opportunity Set of the Debt and Equity
Funds and Two Feasible CALs
Correlation Effects
• 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.
7-55
The Sharpe Ratio
• Maximize the slope of the CAL for any
possible portfolio, P.
• The objective function is the slope:
SP 
E (rP )  rf
sP
• The slope is also the Sharpe ratio.
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 riskreturn opportunities available.
– All portfolios that lie on the minimumvariance 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 Tbills 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
The Power of Diversification
n
• Remember: s 2 

P
i 1
n
 w w Cov(r , r )
j 1
i
j
i
• We can then express portfolio variance as:
j
• If we define the average variance and average
covariance of the securities as:
1
n
s P2  s 2 
1 n
s   s i2
n i 1
2
Cov 
n
1

n(n  1) j 1
j i
7-66
n 1
Cov
n
n
 Cov(r , r )
i 1
i
j
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 Sharing
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.
• 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.
• The insurance principle: risk increases less than
proportionally to the number of policies insured
when the policies are uncorrelated
• True diversification means spreading a portfolio
of fixed size across many assets, not merely
adding more risky bets to an ever-growing risky
portfolio.
– Sharpe ratio increases
7-71
Investment for the Long Run
Long Term Strategy
Short Term Strategy
• Invest in the risky
portfolio for 2 years.
• Invest in the risky
portfolio for 1 year and
in the risk-free asset for
the second year.
– 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.
7-70