Fina2802: Investments and Portfolio Analysis
Spring, 2010
Dragon Tang
Lecture 9
Capital Allocation
February 9, 2010
Readings: Chapter 6
Practice Problem Sets: 1-5,12-14,26-28
Fin 2802, Spring 10 - Tang
Chapter 6: Asset Allocation
1
Asset Allocation
Objectives:
•
Characterize the risk and return of portfolios
containing risky and risk-free assets.
1. Evaluate the performance of a passive strategy.
Fin 2802, Spring 10 - Tang
Chapter 6: Asset Allocation
2
Portfolio Selection
1. Asset allocation
2. Security selection
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Chapter 6: Asset Allocation
3
Asset Allocation
• John Bogle: “Asset allocation accounts for
94% of the differences in pension fund
performance”
• Identify investment opportunities (risk-return
combinations)
• Choose the optimal combination according to
investor’s risk attitude
Fin 2802, Spring 10
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Chapter 6: Asset Allocation
4
Portfolios: Basic Asset Allocation
The complete portfolio is composed of:
• The risk-free asset: Risk can be reduced by allocating
more to the risk-free asset
• The risky portfolio: Composition of risky portfolio
does not change (market portfolio)
This is called Two-Fund Separation Theorem.
The proportions depend on your risk aversion.
Fin 2802, Spring 10 - Tang
Chapter 6: Asset Allocation
5
Risk-free Investment
1.00
0.80
0.60
0.40
0.20
0.00
-5%
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0%
5%
Chapter 6: Asset Allocation
10% 15%
6
Stock Returns Are Uncertain
Example: Risky investment with ten possible rates of return
1.00
0.80
0.60
0.40
0.20
0.00
-40% -20% 0%
Fin 2802, Spring 10
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Chapter 6: Asset Allocation
20% 40%
7
Risk and Risk Premium
• Risk-free rate: determined by demand/supply and
intermediaries (such as Fed)
• Risk premium (=Risky return –Risk-free return)
Example
The expected return on the S&P500 is 9%
The return on a 1-month T-bill is 3%
The risk premium is 6% (9%-3%)
• Risk aversion
E(rP) – rf = ½ A σp2
Fin 2802, Spring 10
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Chapter 6: Asset Allocation
8
Complete Portfolio Expected Return
Example: Let the expected return on the risky portfolio, E(rP),
be 15%, the return on the risk-free asset, rf, be 7%. What is the
return on the complete portfolio if all of the funds are invested
in the risk-free asset? What is the risk premium?
7%
0
What is the return on the portfolio if all of the funds are invested
in the risky portfolio?
15%
8%
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Chapter 6: Asset Allocation
9
Complete Portfolio Expected Return
Example: Let the expected return on the risky portfolio, E(rP),
be 15%, the return on the risk-free asset, rf, be 7%. What is the
return on the complete portfolio if 50% of the funds are
invested in the risky portfolio and 50% in the risk-free asset?
What is the risk premium?
0.5*15%+0.5*7%=11%
4%
Fin 2802, Spring 10 - Tang
Chapter 6: Asset Allocation
10
Complete Portfolio Risk Premium
In general:
Equal to 0



E rC   rf  y E rP   rf  (1  y ) rf  rf

y  fraction of funds invested in the risky asset
E rC   rf  risk premium on the complete portfolio
E rP   rf  risk premium on the risky asset
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Chapter 6: Asset Allocation
11
Portfolio Standard Deviation
 C2  y 2 P2  1  y 2  r2  2 y1  y  Pr  P r
f
f
f
 C  y P if  r  0
f
where
c - standard deviation of the complete portfolio
P - standard deviation of the risky portfolio
rf - standard deviation of the risk-free rate
y - weight of the complete portfolio invested in the
risky asset
Fin 2802, Spring 10 - Tang
Chapter 6: Asset Allocation
12
Portfolio Standard Deviation
Example: Let the standard deviation on the risky portfolio,
P, be 22%. What is the standard deviation of the complete
portfolio if 50% of the funds are invested in the risky
portfolio and 50% in the risk-free asset?
22%*0.5=11%
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Chapter 6: Asset Allocation
13
Capital Allocation Line
We know that given a risky asset (p) and a riskfree asset, the expected return and standard
deviation of any complete portfolio (c) satisfy
the following relationship:
E (rc )  rf  y ( E (rp )  rf )
 c  y p
Where y is the fraction of the portfolio invested in the risky asset

E (rc )  rf
c
Fin 2802, Spring 10 - Tang

E (rp )  rf
p
for every complete portfolio c
Chapter 6: Asset Allocation
14
Capital Allocation Line
Risk Tolerance and Asset Allocation:
•More risk averse - closer to point F
•Less risk averse - closer to P
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Chapter 6: Asset Allocation
15
Slope of the CAL
S
E  rP   r f
P
S is the increase in expected return per
unit of additional standard deviation
S is the reward-to-variability ratio or
Sharpe Ratio
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Chapter 6: Asset Allocation
16
Slope of the CAL
Example: Let the expected return on the risky portfolio, E(rP),
be 15%, the return on the risk-free asset, rf, be 7% and the
standard deviation on the risky portfolio, P, be 22%. What is
the slope of the CAL for the complete portfolio?
S = (15%-7%)/22% = 8/22
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Chapter 6: Asset Allocation
17
Historical Risk-Return Trade-off
Asset Class
Sm Stk
Lg Stk
LT Gov
T-Bills
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Risk
Prem.(%)
13.9
9.3
1.9
---
Real
Retn(%)
14.6
8.9
2.6
0.7
Chapter 6: Asset Allocation
Sharp
Ratio
0.35
0.45
0.23
---
18
Measuring Risk-Return Trade-off
• Mean-Variance Plot
20
18
Expected Return (E(r))
16
14
12
10
8
6
4
2
0
0
5
10
15
20
25
30
35
40
45
Standard Deviation ()
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Chapter 6: Asset Allocation
19
Borrowing
y>1
S
E  rP   rB
P
where rB is the borrowing rate
Usually borrowing rate > lending rate
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Chapter 6: Asset Allocation
20
Borrowing Example
Example: Let the expected return on the risky portfolio, E(rP),
be 15%, the return on the risk-free asset, rf, be 7%, the borrowing
rate, rB, be 9% and the standard deviation on the risky portfolio,
P, be 22%. What is the slope of the CAL for the complete
portfolio for points where y > 1?
S=(15%-9%)/22%=6/22
Note: For y  1, the slope is as indicated above if the lending
rate is rf.
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Chapter 6: Asset Allocation
21
Investment Opportunity Set with
Differential Borrowing and Lending Rates
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Chapter 6: Asset Allocation
22
Passive Strategies
•Assumes securities are fairly priced
•Avoids cost of security analysis
•Indexing - value-weighted portfolio
•Assume that the search for mispriced securities
(performed by active strategies) keeps prices fair
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Chapter 6: Asset Allocation
23
Capital Market Line (CML)
SPECIAL CASE OF CAL (I.e., P=MKT)
The line provided by one-month T-bills and a
broad index of common stocks (e.g. S&P500)
Consequence of a passive investment strategy
based on stocks and T-bills
Fin 2802, Spring 10 - Tang
Chapter 6: Asset Allocation
24
Which Portfolio to Choose?
E(r)
P3?
E(Rm) = 12%
M
P2?
P1?
S=0.45
rf = 3%
F
0
Fin 2802, Spring 10
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20%
Chapter 6: Asset Allocation

25
Key Determinant of Asset Allocation:
Attitude towards Risk
• Risk Preference
– Risk averse
» Require compensation for taking risk
– Risk neutral
» No requirement of risk premium
– Risk loving
» Pay to take risk
• Utility Values: A is risk aversion parameter
U  E (r )  0.5 A
Fin 2802, Spring 10
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2
Chapter 6: Asset Allocation
26
Utility Function
U = E ( r ) – 1/2 A 2
Where
U = utility
E ( r ) = expected return on the asset or portfolio
A = coefficient of risk aversion
2 = variance of returns
Fin 2802, Spring 10
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Chapter 6: Asset Allocation
27
Utility Scores of Alternative Portfolios
for Investors with Varying Risk Aversion
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Chapter 6: Asset Allocation
28
The Trade-off Between Risk and Returns
of a Potential Investment Portfolio
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Chapter 6: Asset Allocation
29
Utility Values of Possible Portfolios for an Investor
with Risk Aversion, A = 4
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Chapter 6: Asset Allocation
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Figure 6.2 The Indifference Curve
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Chapter 6: Asset Allocation
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Utility Indifference Curves
Utility Indifference Curves
20
18
A=5
16
Expected Return (E(r))
14
A=2
12
10
8
U1
6
Increasing utility
4
U2
2
0
0
5
10
15
20
25
30
35
40
45
Standard Deviation ()
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Chapter 6: Asset Allocation
32
Risk Aversion and Asset 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 would result in leveraged
combinations.
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Chapter 6: Asset Allocation
33
Investor’s Willingness to Pay for Catastrophe Insurance
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Chapter 6: Asset Allocation
34
Spread Between 3-Month CD and T-bill Rates
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Chapter 6: Asset Allocation
35
Find the Optimal Allocation
•
Solve the maximization problem:
Max U  E (r )  0.5 A 2
 rf  w1  (rm  rf )  0.5 A( w1 m ) 2
•
Two approaches:
1. Try different w1
2. Use calculus: dU  r  r  Aw  2  0
m
f
1 m
dw
1
•
Solution:
Fin 2802, Spring 10
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w1 
Rm  rf
A
2
m
Chapter 6: Asset Allocation
36
Asset Allocation Rules:
•
•
•
•
When rm-rf increases, w1 increases
When A increases, w1 decreases
When m increases, w1 decreases
W1 is constant when all three are fixed
Fin 2802, Spring 10
08 - Tang
Chapter 6: Asset Allocation
37
Illustration of Solution
If A=4 then w1=0.56
E(r)
Utility indifference curves
(A=4)
P3
E(Rm) = 12%
M
P2
8%
P1!
S=0.45
rf = 3%
F
0
Fin 2802, Spring 10
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11.2%
Chapter 6: Asset Allocation
20%

38
Which Portfolio to Choose?
• For Jack, risk aversion A = 2, the optimal choice is
112.5% (of total capital) in the market, financed by selling
short 12.5% (of total capital) in T-bills
• For Jill, risk aversion A = 5, the optimal choice is 45% in
the market and 55% in T-bills.
• The weight in the market:
w1 
Fin 2802, Spring 10
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Rm  rf
Chapter 6: Asset Allocation
A
2
m
39
Utility Comparison
10
Utility Value
5
0
-5
-10
Jack
Jill
-15
-20
0%
45%
50%
113%
100%
150%
200%
250%
Weight in Market
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Chapter 6: Asset Allocation
40
Utility Levels for Positions in Risky Assets for an Investor
with Risk Aversion A = 4
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Chapter 6: Asset Allocation
41
Utility as a Function of Allocation to the Risky Asset, y
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Chapter 6: Asset Allocation
42
Finding the Optimal Complete Portfolio Using Indifference Curves
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Chapter 6: Asset Allocation
43
Expected Returns on Four Indifference Curves and the CAL
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Chapter 6: Asset Allocation
44
Average Annual Return on Stocks and 1-Month T-bills;
S. Dev. and Reward to Variability of Stocks Over Time
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Chapter 6: Asset Allocation
45
Summary
•Capital allocation line (CAL)
All combinations of the risky and risk-free asset
Slope is the reward-to-variability ratio
•Capital market line (CML)
Passive strategy
Market index portfolio as the risky asset
•Risk aversion determines position on the capital allocation line
•Next: Market Efficiency
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Chapter 6: Asset Allocation
46