Class Business

Debate #1 Upcoming Groupwork

Hedge Funds

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 FundsMutual 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)– closed-end fund that holds real

estate assets– some hold properties directly -

usually have 70% debt• some hold mortgages on properties

– exempt from taxes as long as 95% of taxable income is distributed

Chapter 17: Investors and the Investment Process

Specify objectives Identify constraints Formulate an investment policy Monitor performance Reevaluate and modify portfolio as

determined from monitoring

Portfolios

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

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

Return:

Expected Return:

Variance:

Standard Deviation:

( ) ( ) 1p S fE r wE r w r

(1 )P S fr wr w r

2 2 sp Sw

p Sw

Capital Allocation Line:

Note: since

it follows that We also know Substituting for w, gives the Capital Allocation Line

(CAL):

p Sw / p Sw

( ) ( ) 1P S fE r wE r w r

( )( ) S f

p f P

S

E r rE r r

Capital Allocation Line

This is just the equation for a line! y=b+mx where

y=E(rp)

b=rfm=[E(rS)-rf]/s

x=p

( )( ) S f

p f P

S

E r rE r r

Capital Allocation Line

0

0.02

0.04

0.06

0.08

0.1

0.12

0 0.05 0.1 0.15 0.2 0.25 0.3

Standard Deviation

Exp

ecte

d R

etur

n

100% Stocks

100% T-Bills

50% Stocks50% T-Bills

CAL

125% Stocks-25% T-Bills

Capital Allocation Line

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.

– Risk-averse investors will invest more in the risk-free asset.

– Risk-tolerant investors will invest more in the risky asset.

Example

Expected return on risky portfolio: 12% Stdev of risky portfolio is .32 Risk free rate is 7% What is formula for CAL line?

PPrE 32.

07.12.07.)(

PPrE )156.0(07.)(

Passive Investing

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

E[rs]=8%

s=.12

rf=4%

E[rp]=wE[rS]+(1-w)rf

p=ws

Risky Risk-Free

A: 0% 100%

B: 100% 0%

C: 50% 50%

D: 150% -50%

4%

E[rp]

p

A

8%

.12

B

.06

6%C

.18

10% D

[ ][ ] s f

p f p

s

E r rE r r

CAL Example Continued Suppose you want expected return of 9%, what are the

weights?

w=1.25 E[rp]=w(.08)+(1-w)(.04) = .09

p=1.25(.12)=.15

Risk

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

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

Cov[r1,r2] =E[(r1-E[r1]) (r2 -E[r2])] Positive covariance:

y

x

Covariance

Cov[r1,r2] =E[(r1-E[r1]) (r2 -E[r2])] Negative covariance:

y

x

Covariance: From Probability model

Cov[r1,r2] =E[(r1-E[r1]) (r2 -E[r2])]

Steps:

1) Find expected return for each asset2) Find deviations from mean for each asset in each state.3) Take product of deviations4) Find expected product of deviations

Probability r1 r2

State 1 0.80 10% 5%

State 2 0.20 -5% 0%

Covariance: Probability Model

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

Probability r1 r2

State 1 0.80 10% 5%

State 2 0.20 -5% 0%

  Dev1 Dev2

State 1 .10-.07= .03 .05-.04= .01

State 2 -0.05-0.07= -.12 0-.04= -.04

Covariance: Probability Model

Step 3: Find product of deviations in each state– Product1=.03(.01)=.0003

– Product2= (-.12)(-.04)= .0048

Step 4: Find expected product of deviations – Covariance = .8(.0003) +.2(.0048)=.0012

Covariance: Probability Model

Even if we don’t know correct probability model, we can still estimate the covariance from past data.

0.005 0.014 -0.010 0.000 0.000-0.036 -0.063 -0.051 -0.077 0.0040.016 -0.004 0.001 -0.018 0.0000.074 0.110 0.059 0.096 0.006

0.015 0.014 0.003

1 2 1 1 2 2 productr r r r r r

Covariance is average product of deviations.

Average