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…