CHAPTER 6
Risk Aversion
and Capital
Allocation to
Risky Assets
Three Steps in Investment Decisions –
Top-down Approach
I.
Capital Allocation Decision
 Allocate funds between risky and risk-free assets
 Made at higher organization levels
II. Asset Allocation Decision
 Distribute risk investments across asset classes – smallcap stocks, large-cap stocks, bonds, & foreign assets
III. Security Selection Decision
 Select particular securities within each asset class
 Made at lower organization levels
6-2
Risk and Risk Aversion
• Speculation
– Considerable risk
• Sufficient to affect the decision
– Commensurate gain
• Gamble
– Bet or wager on an uncertain outcome
6-3
Risk Aversion and Utility Values
• Risk averse investors reject investment
portfolios that are fair games or worse
• These investors are willing to consider only
risk-free or speculative prospects with
positive risk premiums
• Intuitively one would rank those portfolios as
more attractive with higher expected returns
6-4
6-5
Table 6.1 Available Risky Portfolios
(Risk-free Rate = 5%)
6-6
Utility Function
1
2
U  E (r )  As
2
Where
U = utility
E ( r ) = expected return on the asset or
portfolio
A = coefficient of risk aversion
s2 = variance of returns
6-7
Table 6.2 Utility Scores of Alternative
Portfolios for Investors with Varying
Degree of Risk Aversion
6-8
Estimating Risk Aversion
• Observe individuals’ decisions when
confronted with risk
• Observe how much people are willing to pay
to avoid risk
– Insurance against large losses
6-9
Figure 6.2 The Indifference Curve
6-10
Table 6.3 Utility Values of Possible
Portfolios for an Investor with Risk
Aversion, A = 4
6-11
Capital Allocation Across Risky and
Risk-Free Portfolios
• Control risk
– Asset allocation choice
• Fraction of the portfolio invested in
Treasury bills or other safe money
market securities
6-12
The Risky Asset Example
Total portfolio value
= $300,000
Risk-free value
= 90,000
Risky (Vanguard & Fidelity) = 210,000
Vanguard (V) = 54%
Fidelity (F) = 46%
6-13
The Risky Asset Example Continued
Vanguard
113,400/300,000 = 0.378
Fidelity
96,600/300,000 = 0.322
Portfolio P
Risk-Free Assets F
Portfolio C
210,000/300,000 = 0.700
90,000/300,000 = 0.300
300,000/300,000 = 1.000
6-14
The Risk-Free Asset
• Only the government can issue default-free
bonds
– Guaranteed real rate only if the duration of
the bond is identical to the investor’s desire
holding period
• T-bills viewed as the risk-free asset
– Less sensitive to interest rate fluctuations
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 split investment funds
between safe and risky assets.
• Risk free asset: proxy; T-bills
• Risky asset: stock (or a portfolio)
6-17
Example Using Chapter 6.4 Numbers
rf = 7%
srf = 0%
E(rp) = 15%
sp = 22%
y = % in p
(1-y) = % in rf
6-18
Expected Returns for Combinations
E (rc )  yE (rp )  (1  y )rf
rc = complete or combined portfolio
For example, y = .75
E(rc) = .75(.15) + .25(.07)
= .13 or 13%
6-19
Combinations Without Leverage
If y = .75, then
sc
= .75(.22) = .165 or 16.5%
If y = 1
s
c
= 1(.22) = .22 or 22%
If y = 0
s c = (.22) = .00 or 0%
6-20
Capital Allocation Line (CAL)
E(rc) = yE(rp) + (1 – y)rf
= rf +[(E(rp) – rf)]y
(1)
σc = yσp → y = σc/σp
(2)
From (1) and (2)
E(rc) = rf +[(E(rp) - rf)/σp]σc (CAL)
6-21
Figure 6.4 The Investment Opportunity Set with
a Risky Asset and a Risk-free Asset in the
Expected Return-Standard Deviation Plane
6-22
Capital Allocation Line with Leverage
Borrow at the Risk-Free Rate and invest in
stock.
Using 50% Leverage,
rc = (-.5) (.07) + (1.5) (.15) = .19
sc = (1.5) (.22) = .33
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
– Trade-off between risk and return
– Expected return of the complete portfolio is
given by:
E (rc )  rf  y  E (rP )  rf 
– Variance is:
s ys
2
C
2
2
P
6-25
Table 6.5 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
Analytical Solution
where
U = E(rc) – (1/2)Aσc2
(1)
E(rc) = yE(rp) + (1-y)rf
σc = yσp
(2)
(3)
Substituting (2) and (3) into (1), we obtain
U = yE(rp) + (1-y)rf – (1/2)A(yσp)2
From
dU/dy = E(rp) – rf – Ayσp2 = 0,
y* = (E(rp) – rf)/Aσp2
6-28
y* = (E(rp) – rf)/Aσp2
= (0.15 – 0.07)/4*(0.22)2
= 0.413
6-29
Indifference curve
We can trace combinations of E(rc) and σc
for given values of U and A.
From
U = E(rc) – (1/2)Aσc2
E(rc) = U + (1/2)Aσc2
Example: E(rc) = 0.05 + (1/2)(2)σc2
6-30
Table 6.6 Spreadsheet Calculations of
Indifference Curves
6-31
Figure 6.7 Indifference Curves for
U = .05 and U = .09 with A = 2 and A = 4
6-32
Figure 6.8 Finding the Optimal Complete
Portfolio Using Indifference Curves
6-33
Passive Strategies: The Capital Market
Line
• E(rc) = rf +[(E(rM) - rf)/σM]σc
• Passive strategy involves a decision that avoids
any direct or indirect security analysis
• Supply and demand forces may make such a
strategy a reasonable choice for many investors
6-34
Passive Strategies:
The Capital Market Line Continued
• A natural candidate for a passively held risky
asset would be a well-diversified portfolio of
common stocks
• Because a passive strategy requires devoting
no resources to acquiring information on any
individual stock or group we must follow a
“neutral” diversification strategy
6-35