Class Business
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Debate #1
Upcoming Groupwork
Hedge Funds
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A private investment pool, open to wealthy or
institutional investors.
– Minimum investment at least $1million (by law)
Not registered as mutual funds and not subject to
SEC regulation.
Pursues more speculative policies.
Name comes from the fact that hedge funds want to
create market-neutral strategies by going long in
some assets and going short in related assets.
Hedge Funds vs. Mutual Funds
Mutual Funds
Hedge Funds
Investment methods
Buy publicly traded securities. Little
use of leverage or short-sales.
Buy also non-public securities, currencies
and commodities. Wide use of leverage and
short-sales.
Diversification
Hold broad mix of assets.
Holdings are often concentrated.
Fees
Relatively low fees that do not
depend on performance
Relatively high fees that depend on
performance.
Share buybacks
Usually daily after close.
Often limited to a few times a year.
Regulation
Heavy Regulation
Light Regulation
Initial investments
Relatively low
Very high investments necessary.
Other Investment Companies
Real estate investment trusts (REITS)
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closed-end fund that holds real
estate assets
some hold properties directly usually have 70% debt
•
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some hold mortgages on properties
exempt from taxes as long as 95%
of taxable income is distributed
Chapter 17: Investors and the
Investment Process
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Specify objectives
Identify constraints
Formulate an investment policy
Monitor performance
Reevaluate and modify portfolio as
determined from monitoring
Portfolios
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Suppose we have (1-w) of our wealth in a
risk-free asset and w of our wealth in some
portfolio of stocks.
Suppose we know the rate of return on the
risk-free asset, rf (e.g., 3%)
We expect the return on S&P 500 to be E[rS]
(e.g., 8%)
Portfolios
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Intuitively:
– The more we invest in the risk-free asset, and the
less in the stock portfolio, the lower will be our
expected return, and the lower the variance (or the
risk) of the portfolio
– . . . and vice versa
– If portfolio standard deviation = risk
expected return
= reward
what is the reward-risk tradeoff?
Portfolio of Risk-Free Asset
and One Risky Asset
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Return:
rP  wrS  (1  w)rf
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Expected Return:
E(rp )  wE(rS )  1  w rf
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Variance:
 w
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Standard Deviation:
 p  w S
2
p
2
s
S
Capital Allocation Line:
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Note: since  p  w  S
it follows that w   p / S
We also know E (r )  wE(r )  1  w r
P
S
f
Substituting for w, gives the Capital Allocation Line
(CAL):

E (rp )  rf 
E (rS )  rf
S

P
Capital Allocation Line
E (rp )  rf 
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E (rS )  rf
S
This is just the equation for a line!
y=b+mx
where
y=E(rp)
b=rf
m=[E(rS)-rf]/s
x=p
P
Capital Allocation Line
0.12
CAL
Expected Return
0.1
125% Stocks
-25% T-Bills
0.08
100% Stocks
0.06
50% Stocks
50% T-Bills
0.04
100% T-Bills
0.02
0
0
0.05
0.1
0.15
Standard Deviation
0.2
0.25
0.3
Capital Allocation Line
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The Capital Allocation Line shows the risk-return
combinations available by changing the proportion invested
in a risk-free asset and a risky asset.
The slope of the CAL is the reward-to-variability ratio
The choice is determined by the risk aversion of investors.
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Risk-averse investors will invest more in the
risk-free asset.
Risk-tolerant investors will invest more in the
risky asset.
Example
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Expected return on risky portfolio: 12%
Stdev of risky portfolio is .32
Risk free rate is 7%
What is formula for CAL line?
.12  .07 P
E (r )  .07 

.32
P
E (r P )  .07  (0.156) P
Passive Investing
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Select a broad diversified portfolio
Invest a fraction of your wealth in the
portfolio according to your level of risk
aversion, and the rest in a risk free asset.
Benefits:
– No need to spend time researching
stocks
– No need to pay someone else to
research
Performance vs. Active strategy?
Another CAL Example
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E[rs]=8%
s=.12
rf=4%
E[rp]=wE[rS]+(1-w)rf
p=ws
Risky
A: 0%
B: 100%
C: 50%
D: 150%
Risk-Free
100%
0%
50%
-50%
E[rp]
E[rp ]  rf 
E[rs ]  rf
s
D
10%
8%
6%
A
4%
p
B
C
p
.06
.12
.18
CAL Example Continued
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Suppose you want expected return of 9%, what are the
weights?
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w=1.25
E[rp]=w(.08)+(1-w)(.04) = .09
p=1.25(.12)=.15
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Risk
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We don’t like uncertainty (variance)
We don’t like assets that “lose” when bad
things happen
We like assets that “win” when bad things
happen: insurance
To incorporate these ideas into a concrete
theory, we need to understand covariance.
Covariance
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Covariance is a measure of “how much two
variables move with each other”.
When one variable is abnormally high, is the
other variable abnormally high or low?
It is measured as the “expected product of
the deviations from the mean.”
Cov[r1,r2] =E[(r1-E[r1]) (r2 -E[r2])]
Covariance
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Cov[r1,r2] =E[(r1-E[r1]) (r2 -E[r2])]
Positive covariance:
x
y
Covariance
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Cov[r1,r2] =E[(r1-E[r1]) (r2 -E[r2])]
Negative covariance:
x
y
Covariance: From Probability
model
Probability
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r1
State 1
0.80
10%
5%
State 2
0.20
-5%
0%
Cov[r1,r2] =E[(r1-E[r1]) (r2 -E[r2])]
Steps:
1)
2)
3)
4)
r2
Find expected return for each asset
Find deviations from mean for each asset in each state.
Take product of deviations
Find expected product of deviations
Covariance: Probability Model
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Probability
r1
r2
State 1
0.80
10%
5%
State 2
0.20
-5%
0%
Step 1: Find expected return for each asset
E[r1]=.80(.10) + .20(-.05) = 0.07
E[r2]=.80(.05) + .20(0.0) =0.04
Step 2: Find Deviations
Dev1
Dev2
State 1
.10-.07= .03
.05-.04= .01
State 2
-0.05-0.07= -.12
0-.04= -.04
Covariance: Probability Model
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Step 3: Find product of deviations in each state
– Product1=.03(.01)=.0003
– Product2= (-.12)(-.04)= .0048
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Step 4: Find expected product of deviations
– Covariance = .8(.0003) +.2(.0048)=.0012
Covariance: Probability Model
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Even if we don’t know correct probability model, we can still estimate
the covariance from past data.
r1
r2
0.005 0.014
-0.036 -0.063
0.016 -0.004
0.074 0.110
Average 0.015
0.014
r1  r1 r2  r2 product
-0.010 0.000
-0.051 -0.077
0.001 -0.018
0.059 0.096
0.000
0.004
0.000
0.006
0.003
Covariance is average product of deviations.