CHAPTER 6
RISK AVERSION AND CAPITAL
ALLOCATION TO RISKY ASSETS

6-1

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.

6-2

Risk and Risk Aversion
• Gamble
– Bet or wager on an uncertain outcome for enjoyment, a sure way to make a profit
– Parties assign the same probabilities to the possible outcomes

6-3

Risk and Risk Aversion
• Sports Betting
– Taking considerable risk for a commensurate gain – Parties have heterogeneous expectations

6-4

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? 6-5

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
6-6

1
2
U

E
(r)2A


Utility Function
U = utility
E ( r ) = expected return on the asset or portfolio
A = coefficient of risk aversion  = variance of returns ½ = a scaling factor

6-7

Table 6.2 Utility Scores of Alternative Portfolios for
Investors with Varying Degree of Risk Aversion

6-8

Mean-Variance (M-V) Criterion
• Portfolio A dominates portfolio B if:

E  rA  E  rB 
• And

 A  B
6-9

Estimating Risk Aversion
• Use questionnaires
• Observe individuals’ decisions when confronted with risk
• Observe how much people are willing to pay to avoid risk
6-10

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

6-11

Basic Asset Allocation
Total Market Value
Risk-free money market fund $300,000
$90,000

Equities
Bonds (long-term)
Total risk assets

$113,400
$96,600
$210,000

$113,400
WE 
0.54
$210,000

$96,600
WB 
0.46
$210,00

6-12

Basic Asset Allocation
• Let y = weight of the risky portfolio, P, in the complete portfolio; (1-y) = weight of risk-free assets:
$210,000
y
0.7
$300,000

$113,400
E:
.378
$300,000

$90,000
1 y 
0.3
$300,000

$96,600
B:
.322
$300,000

6-13

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
6-14

Figure 6.3 Spread Between 3-Month
CD and T-bill Rates

6-15

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.

6-16

Example Using Chapter 6.4 Numbers rf = 7%

 rf = 0%

E(rp) = 15%

 p = 22%

y = % in p

(1-y) = % in rf

6-17

E
(rc)
rfP

y
E
(r)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  7  y 15  7 

6-18

Example (Ctd.)
• The risk of the complete portfolio is the weight of P times the risk of P:

 C  y P 22 y

6-19

Example (Ctd.)
• Rearrange and substitute y=C/P:

C
8

E  rC  rf 
E  rP   rf  7   C
P
22
Slope 

E  rP   rf

P

8

22

6-20

Figure 6.4 The Investment
Opportunity Set

6-21

Capital Allocation Line with