Download Ch.5 part2

5.5 Asset Allocation Across
Risky and Risk Free
Portfolios
5-1
Allocating Capital Between Risky &
Risk-Free Assets
 Possible to split investment funds between safe and
risky assets
or money market fund
 Risk free asset rf : proxy; T-bills
________________________
risky portfolio
 Risky asset or portfolio rp: _______________________
 Example. Your total wealth is $10,000. You put $2,500
in risk free T-Bills and $7,500 in a stock portfolio
invested as follows:
– Stock A you put $2,500
______
– Stock B you put $3,000
______
– Stock C you put $2,000
______
$7,500
5-2
Allocating Capital Between Risky &
Risk-Free Assets
Stock A $2,500
Weights in rp
– WA = $2,500 / $7,500 = 33.33%
– WB = $3,000 / $7,500 = 40.00%
– WC = $2,000 / $7,500 = 26.67%
100.00%
The complete portfolio includes the riskless
investment and rp.
Stock B $3,000
Stock C $2,000
Wrf = 25% ; Wrp = 75%
In the complete portfolio
WA = 0.75 x 33.33% = 25%; WB = 0.75 x 40.00% = 30%
WC = 0.75 x 26.67% = 20%; Wrf = 25%
5-3
Example
rf = 5%
srf = 0%
E(rp) = 14%
srp = 22%
y = % in rp
(1-y) = % in rf
5-4
Expected Returns for Combinations
E(rC) = yE(rp) + (1 - y)rf
sc = ysrp + (1-y)srf
E(rC) = Return for complete or combined portfolio
For example, let y = 0.75
____
rf = 5%
srf = 0%
E(rp) = 14%
srp = 22%
E(rC) = (.75 x .14) + (.25 x .05)
y = % in rp
(1-y) = % in rf
E(rC) = .1175 or 11.75%
sC = ysrp + (1-y)srf
sC = (0.75 x 0.22) + (0.25 x 0) = 0.165 or 16.5%
5-5
Complete portfolio
E(rc) = yE(rp) + (1 - y)rf
sc = ysrp + (1-y)srf
linear
Varying y results in E[rC] and sC that are ______
combinations
___________ of E[rp] and rf and srp and srf
respectively.
This is NOT generally the case for
the s of combinations of two or
more risky assets.
5-6
E(r)
Possible Combinations
CAL
(Capital
Allocation
Line)
E(rp) = 14%
P
E(rp) = 11.75%
y=1
y =.75
rf = 5%
F
y=0
0
16.5%
22%
s
5-7
Combinations Without Leverage
rf = 5%
srf = 0%
E(rp) = 14%
srp = 22%
y = % in rp
(1-y) = % in rf
Since σrf = 0
σ c= y σ p
E(rc) = yE(rp) + (1 - y)rf
If y = .75, then
y = .75
σc= 75(.22) = 16.5% E(rc) = (.75)(.14) + (.25)(.05) = 11.75%
If y = 1
σc= 1(.22) = 22%
y=1
E(rc) = (1)(.14) + (0)(.05) = 14.00%
If y = 0
σc= 0(.22) = 0%
y=0
E(rc) = (0)(.14) + (1)(.05) = 5.00%
5-8
Using Leverage with Capital
Allocation Line
Borrow at the Risk-Free Rate and invest in stock
Using 50% Leverage
y = 1.5
E(rc) = (1.5) (.14) + (-.5) (.05) = 0.185 = 18.5%
(1.5) (.22) = 0.33 or 33%
rf = 5%
sc =
E(r ) = 14%
p
y = % in rp
E(rC) =18.5%
srf = 0%
srp = 22%
(1-y) = % in rf
y = 1.5
y=0
33%
5-9
Risk Premium & Risk Aversion
• The risk free rate is the rate of return that can be
earned with certainty.
• The risk premium is the difference between the
expected return of a risky asset and the risk-free rate.
E[rasset] – rf
Excess Return or Risk Premiumasset =
Risk aversion is an investor’s reluctance to accept
risk.
How is the aversion to accept risk overcome?
By offering investors a higher risk premium.
5-10
Risk Aversion and Allocation
 Greater levels of risk aversion lead investors to
choose larger proportions of the risk free rate
Lower levels of risk aversion lead investors to
 choose larger proportions of the portfolio of risky
assets
 Willingness to accept high levels of risk for high
levels of returns would result in
leveraged combinations
E(rC) =18.5%
y = 1.5
y=0
33%
5-11
E(r)
P or combinations of
P & Rf offer a return
per unit of risk of
9/22.
CAL
(Capital
Allocation
Line)
P
E(rp) = 14%
E(rp) - rf = 9%
) Slope = 9/22
rf = 5%
0
F
srp = 22%
s
5-12
Quantifying Risk Aversion
E rp   rf  0.5  A  s p
2
E(rp) = Expected return on portfolio p
rf = the risk free rate
0.5 = Scale factor
A x sp2 = Proportional risk premium
The larger A is, the larger will be the
_________________________________________
investor’s added return required to bear risk
5-13
Quantifying Risk Aversion
Rearranging the equation and solving
for A
E ( rp )  rf
A 
0.5  σ 2
p
Many studies have concluded that
investors’ average risk aversion is
between _______
2 and 4
5-14
Using A
E ( rp )  rf
A 
0.5  σ 2
p
What is the maximum
A that an investor
could have and still
choose to invest in the
risky portfolio P?
A
0.14  0.05
0.5  0.22
2

3.719
Maximum A = 3.719
5-15
Sharpe Ratio
• Risk aversion implies that investors will accept a
lower reward (portfolio expected return) in
exchange for a sufficient reduction in risk (std
dev of portfolio return)
• A statistic commonly used to rank portfolios in
terms of the risk-return trade-off is the Sharpe
measure (also reward-to-volatility measure)
• The higher the Sharpe ratio the better
• Also the slope of the CAL
Sharpe ratio
E rp   rf
portfolio risk premium
S

std dev of portfolio excess return
sp
• Example: You manage an equity fund with
an expected return of 16% and an expected
std dev of 14%. The rate on treasury bills is
6%.
Risk premium
16  6
S

 0.71
Standard deviation
14
5.6 Passive Strategies and
the Capital Market Line
5-18
A Passive Strategy
•
Investing in a broad stock index and a risk
free investment is an example of a passive
strategy.
– The investor makes no attempt to actively find
undervalued strategies nor actively switch
their asset allocations.
– The CAL that employs the market (or an index
that mimics overall market performance) is
called the Capital Market Line or CML.
5-19
Active versus Passive Strategies
• Active strategies entail more trading costs than
passive strategies.
• Passive investor “free-rides” in a competitive
investment environment.
• Passive involves investment in two passive
portfolios
– Short-term T-bills
– Fund of common stocks that mimics a broad
market index
– Vary combinations according to investor’s
risk aversion.
5-20