Chapter 6
Risk Aversion and Capital Allocation to Risky
Assets
1
Risk vs. Return
⚫First we have to make some assumptions about
human behaviors.
⚫A fundamental decision of the asset allocation
problem: percentage invested in riskless asset
and risky asset, which depends on the risk
aversion.
⚫Investors will avoid risk unless there is a reward,
i.e., there is always a price for taking risk.
⚫The utility model gives the optimal allocation
between a risky portfolio and a risk-free asset.
2
The von Neumann-Morgenstern
Theory： Risk & Risk Aversion
The first important use of the expected theory was that of John
von Neumann and Oskar Morgenstern who used the assumption of
expected utility maximization in their formulation of game theory.
3
Risk Aversion
4
The von Neumann-Morgenstern
Theory： Risk & Risk Aversion
⚫Utility of Wealth: marginal utility is positive but
decreasing
⚫Expected Utility
⚫Certainty Equivalent Wealth (Amount) or
Certainty Rate
⚫Risk Premiums
⚫Insurance
5
Example
⚫Suppose U(W) = ln(W)
⚫Start with W0 = $100,000
Face gamble: W1 = 50,000 w/ p = .5
W1 = 150,000 w/ p = .5
E(W) = .5x50,000 + .5x150,000 = 100,000
===> Fair Game
(Note: a fair game is a risky investment with a risk
premium of zero)
6
Example
⚫Compare utility of 100,000 for sure with
expected utility of gamble:
ln(100,000) = 11.51
E(U) = .5xln(50000) + .5xln(150000) = 11.37
Since utility of 100,000 for sure > expected utility,
person is said to be risk averse.
A risk averse person is unwilling to accept a fair
gamble.
7
Fair games and expected utility
U(w)
U[E(w)]
E[U(w)]
W1
W0
W1
W
A person is risk averse at wealth level W if that person is
unwilling to accept a fair gamble at that wealth level.
U[E(W)] > E[U(W)]
Function lies above chord
8
Risk Aversion - Intuition
⚫Utility gained is less than utility lost
⚫Example:
ln(100,000) = 11.51 (utility for sure)
ln(150,000) = 11.92
11.92 - 11.51 = .41 utility gained
ln(50,000) = 10.82
11.51 - 10.82 = .68 utility lost
A risk averser will reject a fair game.
9
Implications
⚫Risk aversion explains the inclination to
agree to a situation with a lower average
payoff that is more predictable rather than
another situation with a less predictable
payoff that is higher on average.
⚫For example, a risk-averse investor might
choose to put their money into
a bank account with a low but guaranteed
interest rate, rather than into a stock that
may have high expected returns, but also
involves a chance of losing value.
10
Three Questions
⚫What amount of wealth for sure gives same utility
of gamble? => Certainty Equivalent Wealth
LN(X) = 11.37 ==> X = e11.37
X = $86,682
⚫What would an investor pay to avoid the gamble?
==> Insurance =100,000-86,682=13,318
⚫What expected value would induce gamble?
==> Risk premium
This risk premium is the amount a risk averse individual would
accept to play an actuarially fair game (or the amount a risk averse
individual would pay to avoid playing an actuarially fair game) 11
Example
⚫Suppose probabilities remain same. What
upside is necessary to induce investment?
We want:
.5ln(50000) + .5ln(X) = ln(100000) = 11.51
.5ln(X) = 11.51 - 5.4099
ln(X) = 12.2002
X = e12.2002 = 198,829
E(W) = .5 x 50,000 + .5 x 198,829 = 124,414
E(R) = 24.414%
Risk Premium=24,414 ➔ the expected value
would induce gamble
12
Summary
⚫Marginal utility of wealth is positive
⚫Decreasing marginal utility of wealth
⚫Individual is risk averse
⚫Will pay to avoid risk (insurance) and will reject
a fair game.
⚫Demands risk premium to accept risk
⚫All the assumptions that will be used later in
the book
https://youtu.be/tCreeXzCNRc
13
Table 6.1 Available Risky Portfolios (Riskfree Rate = 5%)
How to rank alternative investment choice?
Each portfolio receives a utility score to assess the
investor’s risk/return trade off
6-14
Quadratic Utility Functions
When we use quadratic utility functions, we can nicely express our
utility functions; one example is the following:
U = E ( R) − 1
A 2
2
Hence, the investor just considers risk and return.
In other words, you require a higher expected return, the
higher the risk of the investment.
U = utility
E ( R ) = expected return on the asset or portfolio
A = coefficient of risk aversion
 = variance of returns
½ = a scaling factor
15
Table 6.2 Utility Scores of Alternative Portfolios for
Investors with Varying Degree of Risk Aversion
6-16
⚫ If A>0, then the investor is risk averse.
⚫ If A<0, the investor is risk-loving.
⚫ If A=0, the investor is risk-neutral.
17
Risk and Risk Aversion
⚫ 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
6-18
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?
6-19
Mean-Variance (M-V) Criterion
⚫Portfolio A dominates portfolio B if:
E (rA )  E (rB )
⚫And
A B
6-20
Return
I
(preferred direction)
II
III
IV
P
Risk
Consider moving from point P to other quadrants
21
Dominance Principle
Expected Return
4
2
3
1
How about 1 and 3?
Variance or Standard Deviation
• 2 dominates 1; has a higher return
• 2 dominates 3; has a lower risk
• 4 dominates 3; has a higher return
22
Return
Indifference Curve
Risk
23
Indifference Curves
Expected Return
Based on previous PPT
Increasing Utility
Standard Deviation
24
U(w)
W
Utility Function
25
Discrete R.V.’s - Summary Measures
Expected Value
–
–
Mean of Probability Distribution
Weighted Average of All Possible Values
X = E(X) = Xi P(Xi)
= p(X1) X1+ p(X2) X2+ … + p(Xn) Xn
Variance (𝝈𝟐 , a risk measure)
–
Weighted Average Squared Deviation about Mean
X2 = E[ (Xi-X)  =  (Xi-X) P(Xi)
= p(X1)(X1- X)2 + p(X2 )(X2- X)2 + … + p(Xn) )(Xn- X)2
Standard deviation (𝝈)
–
Square root of the variance. Measure of risk shows dispersion of
possible outcomes around expected level of outcomes.
26
Asset Expected Return & Risk
⚫ Calculate the Expected Returns of alternative assets.
– Rki= Return to Asset k if Event i occurs
– P(Rki) = Probability Event i occurs.
Economic
Conditions
Recession
Stable Economy
Moderate Growth
Boom
Asset A
Asset B
Probability % Returns % Returns
.10
.40
.30
.20
3.0
7.0
10.0
15.0
5.0
3.0
25.0
40.0
27
Calculating ER and Risk
⚫ Can use the previous table to calculate ER and variance (risk)
for each asset.
– Show calculations for Asset A below.
Economic
Conditions P(Ri) Ri for A P()Ri for A Ri - ER
Recession
Stable
Moderate
Boom
.10
.40
.30
.20
ERA = 9.1%
3.0
7.0
10.0
15.0
.3
.28
.30
.30
ER =
9.1
A = 3.56%
-6.1
-2.1
0.9
5.9
(Ri - ER)2
P(Ri) (Ri - ER)2
37.21
4.41
.81
34.81
3.721
1.764
.243
6.962
2 =
12.69
CV=ERA/A = 2.55
28
Covariance
⚫ Covariance between two random variables X and Y is
defined as
Cov( X , Y ) =  ( X , Y ) =  XY = E( X − EX )(Y − EY )
– A negative covariance between X and Y means that when X is above its
mean its is likely that Y is below its mean value.
– If the covariance of the two random variables is zero then on average
the values of the two variables are unrelated.
– A positive covariance between X and Y means that when X is above its
mean its is likely that Y is above its mean value.
– Covariance of a random variable with itself, its own covariance, is equal
to its variance.
29
Correlation
⚫ Correlation between two random variables X and Y
measured as:
Corr( X , Y ) =  XY =  XY
= Cov( X , Y )
 XY
 XY
– Correlation takes on values between –1 and +1. Correlation is a
standardized measure of how two random variables move together, i.e.
correlation has no units associated with it.
– A correlation of 0 means there is no straight-line (linear) relationship
between the two variables.
– Increasingly positive (negative) correlations indicate an increasingly
strong positive (negative) linear relationship between the variables.
– When the correlation equals 1 (-1) there is a perfect positive (negative)
linear relationship between the two variables.
30
Portfolio Probability Calculations
Portfolio consisting of two assets A and B, wA invested in A. Asset A has
expected return rA and variance 2A. Asset B has expected return rB and
variance 2B. The correlation between the two returns is AB.
Portfolio Expected Return:
E(rp) = wA rA + (1-wA )rB
Portfolio Variance:
 r2 = wA2 A2 + (1 − wA )2  B2 + 2wA (1 − wA )Cov(rA , rB )
p
or
 r2 = wA2 A2 + (1 − wA )2  B2 + 2wA (1 − wA ) AB A B
p
31
Capital Allocation:
Risky and Risk-Free Portfolios
Asset Allocation decision: Controlling Risk:
⚫ Is a very important part ⚫ Simplest way:
of portfolio construction.
Manipulate the fraction
of the portfolio invested
⚫ Refers to the choice
in risk-free assets versus
among broad asset
the portion invested in
classes especially in
the risky assets
riskless and risky assets.
6-32
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
6-33
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.
6-34
Example Using Chapter 6.4 Numbers
rf = 7%
rf = 0%
E(rp) = 15%
p = 22%
y = % in p
(1-y) = % in rf
6-35
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 ) = (1 − y )rf + yE (rP ) = rf + y  E (rP ) − rf 
E (rc ) = 7 + y (15 − 7 )
Risk premium
Base riskless rate
6-36
Example (Ctd.)
⚫ The risk of the complete portfolio is the
weight of P times the risk of P:
 C = y P = 22 y
Don’t forget the formula at Ch.4:
 = w  + (1 − wA )  B2 + 2wA (1 − wA ) AB A B
2
rp
2
A
2
A
2
6-37
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
risk premium per unit of risk: reward –to-variability ratio
6-38
Figure 6.4 :The Investment Opportunity Set
CAL: depicts all the
feasible risk-return
combinations available
to investors.
6-39
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
⚫If there is a capital market where you can
borrow to invest in the risky assets, then CAL
kinks at P.
6-40
Figure 6.5 The Opportunity Set with
Differential Borrowing and Lending Rates
6-41
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:
 =y
2
C
2
2
P
6-42
How Investors Choose y?
⚫Maximize the utility by choosing the best
allocation to the risky asset y:
U = E ( R) − 1
2
A 2
⚫If y increases, E(R) increases but so does
volatility, so U can increase or decrease.
⚫A is the degree of risk-aversion.
43
Table 6.4 Utility Levels for Various Positions in Risky
Assets (y) for an Investor with Risk Aversion A = 4
Max
U = E (r ) − 1
A 2 = rf + y[ E ( rp ) − rf ] − 1
y* =
=
2
E ( rP ) − rf
A P2
2
Ay 2 2
0.15 − 0.07
= 0.41
4  0.222
6-44
Figure 6.6 Utility as a Function of
Allocation to the Risky Asset, y
6-45
Table 6.5 Spreadsheet Calculations of Returns
for Indifference Curves
Assume two investors with A= 2 and A=4
6-46
Figure 6.7 Indifference Curves for
U = .05 and U = .09 with A = 2 and A = 4
More risk-averse
6-47
Figure 6.8 Finding the Optimal Complete
Portfolio Using Indifference Curves (A=4)
6-48
Table 6.6 Expected Returns on Four
Indifference Curves and the CAL (A=4)
Tagent to CML
Imply y*=0.41
6-49
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
6-50
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 1month T-bills and a broad index of
common stocks (e.g. the S&P 500).
6-51
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.
6-52
Passive Strategies:
The Capital Market Line
⚫From 1926 to 2012, the passive risky portfolio
offered an average risk premium of 8.1% with a
standard deviation of 20.48%, resulting in a
reward-to-volatility ratio of 0.40.
6-53