II: Portfolio Theory II
5: Modern Portfolio Theory
Theory vs Practice
 Theory: Efficient portfolios
 Practice: Calculate
correlation coefficients
for all possible pairs of over
10,000 stocks? (?!)
 Perhaps measure the
portfolio directly.
Chapter 5: Modern Portfolio Theory
© Oltheten & Waspi 2012
Limits of Diversification
 Unsystematic Risk
Standard Deviation
 Industry or firm specific – can be diversified away
 Systematic Risk
 Economy wide - cannot be diversified away
Unsystematic Risk
market portfolio
Systematic Risk
0
Chapter 5: Modern Portfolio Theory
20
Number of Stocks in the portfolio 40
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Modern Portfolio Theory
 Calculate the correlation with the basic
underlying value that all stocks have in
common: the market.
Chapter 5: Modern Portfolio Theory
© Oltheten & Waspi 2012
Modern Portfolio Theory
Proctor & Gamble
Boeing
Exxon
Mobile
Citigroup
Hypothetical
Resources
Tardis
Intertemporal
US Steel
Microsoft
Ford
Caterpillar
Chapter 5: Modern Portfolio Theory
© Oltheten & Waspi 2012
Modern Portfolio Theory
Proctor & Gamble
Boeing
Exxon
Mobil
Citigroup
Hypothetical
Resources
Tardis
Intertemporal
Market
US Steel
Microsoft
Ford
Caterpillar
Chapter 5: Modern Portfolio Theory
© Oltheten & Waspi 2012
Market Model
 RStock =  + β RMarket
Return for taking
market risk
Return for taking
undiversifiable, firmspecific risk
Chapter 5: Modern Portfolio Theory
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Market Model
 RStock =  + β RMarket
 β =  (Rs,Rm) *. Rs .
Rm
 Captures the correlation between Rs and Rm.
 Reflects market risk exposure
Chapter 5: Modern Portfolio Theory
© Oltheten & Waspi 2012
Market Model
R
Rt, RMt
et
R = α + β RM
Intercept α
slope=β
RM
Chapter 5: Modern Portfolio Theory
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Market Model
Stock
Prices
Index
Values
R
Rates of Return
RM
Return on the
Market
Market Model
R =  + b RM + e

Alpha
b
e
Regression
Errors
Beta
Chapter 5: Modern Portfolio Theory
© Oltheten & Waspi 2012
Capital Asset Pricing Model
 E[R] = rf + β( E[RM] – rf)
 E[R] is the normal return for an investment
with a risk exposure = β
Chapter 5: Modern Portfolio Theory
© Oltheten & Waspi 2012
Capital Asset Pricing Model
T-Bill
Yields
Stock
Prices
Index
Values
Rf
Risk Free Rates
R
Rates of Return
RM
Return on the
Market
Market Model
R =  + b RM + e
Rf
Expected Risk
Free Rate
E[R]
Expected
Return on
Equity
E[RM]
Expected
Return on the
Market

b
Alpha
e
Regression
Errors
Beta
CAPM
E[R] - { Rf + b (E[RM] - Rf) } = e
Buy
e
Abnormal Return
Hold
e>0
Sell
Chapter 5: Modern Portfolio Theory
© Oltheten & Waspi 2012
CAPM - Example
 You have $1,000,000 to invest and can
invest in:
Completely
Diversified
 T-Bills (E[R]=1.0%, β=0)
 Equity Index Fund (E[R]=6.3%, β=1)
 The beta of a portfolio equals the weighted
average of the betas of the components
Chapter 5: Modern Portfolio Theory
© Oltheten & Waspi 2012
CAPM
 β=0
 $1,000,000 in T-Bills
$1,000,000 @
1.0% =
$0 @
6.3% =
$1,000,000
=> __ __ . __%
CAPM: 1.0% + __.__ (6.3% - 1.0%) => __ __ . __%
Beta
Chapter 5: Modern Portfolio Theory
E[R]
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CAPM
 β=1
 $1,000,000 in the Equity Fund
$0 @
1.0% =
$1,000,000 @
6.3% =
$1,000,000
=> __ __ . __%
CAPM: 1.0% + __.__ (6.3% - 1.0%) => __ __ . __%
Beta
Chapter 5: Modern Portfolio Theory
E[R]
© Oltheten & Waspi 2012
CAPM
 β = 0.5
 $500,000 in the Equity Fund
 $500,000 in T-Bills
$500,000 @
1.0% =
$500,000 @
6.3% =
$1,000,000
=> __ __ . __%
CAPM: 1.0% + __.__ (6.3% - 1.0%) => __ __ . __%
Beta
Chapter 5: Modern Portfolio Theory
E[R]
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CAPM
 β = 2.0


in the Equity Fund
in T-Bills
@
1.0% =
@
6.3% =
$1,000,000
=> __ __ . __%
CAPM: 1.0% + __.__ (6.3% - 1.0%) => __ __ . __%
Beta
Chapter 5: Modern Portfolio Theory
E[R]
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CAPM – Example

1.0% + 2.0 (6.3% - 1.0%)
Spread: Borrow at 1.0% to
invest at 6.3%
The first million you borrow
from yourself
Chapter 5: Modern Portfolio Theory
© Oltheten & Waspi 2012
Security Market Line
 For any Beta
 we can generate a portfolio composed of TBills (or borrowing) and Equity Index Funds
with that Beta
 The portfolio has a normal return of E[R]
where E[R] = rf + β (E[RM] – rf)
Chapter 5: Modern Portfolio Theory
© Oltheten & Waspi 2012
Security Market Line
E[R]
12%
10%
SML:
Normal Return
8%
6%
Slope:
Spread on risky asset
4%
2%
0%
0
Chapter 5: Modern Portfolio Theory
0.5
1
1.5
Beta
2
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CAPM: Investment by Investment
 For any investment with market risk
exposure β,
 we can see if the investment generated any
abnormal return
Chapter 5: Modern Portfolio Theory
© Oltheten & Waspi 2012
CAPM – Investment by Investment
 Hypothetical Resources
 Market Model:
 E[R] = 9.56%
 β = 1.20
Chapter 5: Modern Portfolio Theory
Expectations of actual
return formed from
past data
© Oltheten & Waspi 2012
CAPM – Investment by Investment
 Hypothetical Resources
 Market Model:
 E[R] = 9.56%
 β = 1.20
 CAPM:
 E[R] = 7.36%
 β =1.20
Expectations of actual
return formed from
past data
Expectations of normal
return formed from the
CAPM
 Abnormal return =
Chapter 5: Modern Portfolio Theory
© Oltheten & Waspi 2012
CAPM – Example
12%
10%
8%
E[R]
6%
4%
2%
0%
0
Chapter 5: Modern Portfolio Theory
0.5
1
1.5
Beta 2
© Oltheten & Waspi 2012
Risk Adjusted Measures
 CV:
 Sharpe Ratio:
 Treynor Ratio:
Chapter 5: Modern Portfolio Theory
Rp
1

CV  p
R Sharpe 
R Treynor 
R p  rf
p
R p  rf
bp
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Practice Questions
Chapter 5: Modern Portfolio Theory
© Oltheten & Waspi 2012
Q&P 5-2:
 Derive the CAPM Equation
 Graph the normal and abnormal return on
Discovery Café in this market
 Calculate the risk-adjusted returns
Investment
Annual
Return
Standard
Deviation
3.3%
0.0%
Market Index Fund
12.3%
15.0%
Discovery Café
14.8%
27.3%
T-Bills
Chapter 5: Modern Portfolio Theory
Beta
0.8
© Oltheten & Waspi 2012
Portfolio Theory II