Capital Allocation Between The
Risky And The Risk-Free Asset
Chapter 7
McGraw-Hill/Irwin
Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.
Allocating Capital: Risky & Risk Free Assets
It’s possible to split investment funds
between safe and risky assets.
Risk free asset: proxy; T-bills
Risky asset: stock (or a portfolio)
7-2
Allocating Capital: Risky & Risk Free Assets
Issues
Examine risk/return tradeoff.
Demonstrate how different degrees
of risk aversion will affect
allocations between risky and risk
free assets.
7-3
Example Using Chapter 7.3 Numbers
p refers to a risky portfolio, rf is the risk free
rf = 7%
rf = 0%
E(rp) = 15%
p = 22%
y = % in p
(1-y) = % in rf
7-4
Expected Returns for Combinations
E(rc) = yE(rp) + (1 - y)rf
(Rule 3)
rc = complete or combined portfolio
For example, y = 0.75
E(rc) = 0.75(0.15) + 0.25(.07)
= 0.13 or 13%
7-5
Portfolio Risk for Combinations
  y   (1 y)   2 y(1 y)Cov(rp rf )
2
c
2
2
p
2
2
rf
However,   0 and Cov(rp rf )  0
2
So,  c  y 
2
rf
2 2
p
takesquare root  c  y p whichis Rule 4
Example: let y=0.75. c=0.75*0.22 = 0.165
Note: Both the expected return formula of the
previous slide, and the risk formula for this slide are
linear; thus, all combinations will also be linear
7-6
Risk and Return by y
y
1-y
E(r)
StDev
0.0
1.0
0.070
0.000
0.2
0.8
0.086
0.044
0.4
0.6
0.102
0.088
0.6
0.4
0.118
0.132
0.8
0.2
0.134
0.176
1.0
0.0
0.150
0.220
1.2
-0.2
0.166
0.264
1.4
-0.4
0.182
0.308
1.6
-0.6
0.198
0.352
7-7
Possible Combinations (the CAL)
Note: This is a plot of the Capital
Allocation Line. Each point on the
line is determined by the value of “y”
E(r)
E(rp) = 15%
P
E(rc) = 13%
C
Slope =
rf = 7%
F
0
(E(rp)- rf)/σp
c
p=22%

7-8
Slope of the CAL
Slope = (E(rp)- rf)/σp
=(0.15 – 0.07)/0.22
= .08/0.22 = 0.3636
This is referred to as the “Reward to Volatility
Ratio”
7-9
Capital Allocation Line with Leverage
Borrow at the Risk-Free Rate and invest
in stock. Using 50% leverage means
that y = 1.5, so (1-y) = -0.5
E(rc ) = (1.5) (0.15) + (-0.5) (0.07) = 0.19
c = (1.5) (0.22) = 0.33
7-10
CAL (Capital Allocation Line)
E(r)
P
E(rp) = 15%
E(rp) - rf = 8%
) S = 8/22
rf = 7%
F
0
p = 22%

7-11
CAL with Higher Borrowing Rate
E(r)
P
) S = .27
9%
7%
) S = .36
p = 22%

7-12
Risk Aversion and Allocation
Greater levels of risk aversion lead to larger
proportions of the risk free rate.
Lower levels of risk aversion lead to larger
proportions of the portfolio of risky assets.
Willingness to accept high levels of risk for
high levels of returns will result in leveraged
combinations (i.e. buying the risky asset on
margin).
7-13
Recall: Utility Function
U = E (r) ― 0.005 A 2
Where
U = utility
E (r) = expected return on the asset or
portfolio
A = coefficient of risk aversion
2 = variance of returns
7-14
CAL with Risk Preferences
The lender has a larger A when
compared to the borrower
E(r)
P
Borrower
7%
Lender
p = 22%

7-15
Utility Maximization
Max U = E(r) – 0.005Aσ2
Can be written as:
Max: rf + y[Erp) – rf] – 0.005Ay2σp2
Take the derivative with respect to y and set to
zero to determine the y value that maximized
utility to derive:
y 
*
E (rp )  rf
0.01A
2
p
Note: y* depends on A
7-16
Passive Investment Strategy
A passive strategy concentrates on
selecting the appropriate asset
allocation, and then investing using a
buy and hold approach
Research shows that most of the return
in a portfolio is due to asset allocation
rather than individual security selection
Passive strategies have very low
investment costs
7-17
Capital Market Line
The CAL which uses the risk free asset and
the portfolio of all traded assets (usual proxy
is the S&P 500) is called the Capital Market
Line
1926-2002 shows an equity risk premium of
8.2%, and a 20.8% standard deviation for a
reward to variability ratio of 0.39
Given that overall investors have 74% of their
wealth in risk assets (Table 1.2) the implied
risk aversion coefficient is A=2.6.
7-18