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II: Portfolio Theory II
5: Modern Portfolio Theory
Chapter 5: Modern Portfolio Theory © Oltheten & Waspi 2012
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 Industry or firm specific – can be diversified away
Systematic Risk Economy wide - cannot be diversified away
0 20 40
Systematic Risk
Unsystematic Risk
market portfolio
Number of Stocks in the portfolio
Sta
ndar
d D
evia
tion
Chapter 5: Modern Portfolio Theory © Oltheten & Waspi 2012
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
Tardis Intertemporal
Proctor & Gamble
Caterpillar
Microsoft
Exxon Mobile
US Steel
Citigroup
Ford
Boeing
HypotheticalResources
Chapter 5: Modern Portfolio Theory © Oltheten & Waspi 2012
Modern Portfolio Theory
HypotheticalResources
Tardis Intertemporal
Proctor & Gamble
Caterpillar
Microsoft
Exxon Mobil
US Steel
Citigroup
Ford
Boeing
Market
Chapter 5: Modern Portfolio Theory © Oltheten & Waspi 2012
Market Model
RStock = + β RMarket
Return for taking market risk
Return for taking undiversifiable, firm-
specific risk
Chapter 5: Modern Portfolio Theory © Oltheten & Waspi 2012
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
slope=β
Intercept α
Rt, RMt
R = α + β RM
RM
R
et
Chapter 5: Modern Portfolio Theory © Oltheten & Waspi 2012
Market Model
RRates of Return
RMReturn on the
Market
aAlpha b
Beta
eRegression
Errors
Market ModelR = a + b RM + e
Stock Prices
Index Values
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
RRates of Return
RMReturn on the
Market
aAlpha b
Beta
RfRisk Free Rates
CAPME[R] - { Rf + b (E[RM] - Rf) } = e
E[RM]Expected
Return on the Market
eRegression
Errors
E[R]Expected Return on
Equity
eAbnormal Return
Market ModelR = a + b RM + e
Stock Prices
Index Values
T-Bill Yields
RfExpected Risk
Free Rate
e>0
Buy
Sell
Hold
Chapter 5: Modern Portfolio Theory © Oltheten & Waspi 2012
CAPM - Example
You have $1,000,000 to invest and can invest in: 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
Completely Diversified
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 E[R]
Chapter 5: Modern Portfolio Theory © Oltheten & Waspi 2012
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 E[R]
Chapter 5: Modern Portfolio Theory © 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 E[R]
Chapter 5: Modern Portfolio Theory © Oltheten & Waspi 2012
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 E[R]
Chapter 5: Modern Portfolio Theory © Oltheten & Waspi 2012
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 T-Bills
(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
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2%
0%
2%
4%
6%
8%
10%
12%
Beta
E[R
]
SML:Normal Return
Slope:Spread on risky asset
Chapter 5: Modern Portfolio Theory © Oltheten & Waspi 2012
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
Expectations of actual return formed from past data
Chapter 5: Modern Portfolio Theory © 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
Abnormal return =
Expectations of actual return formed from past data
Expectations of normal return formed from the CAPM
Chapter 5: Modern Portfolio Theory © Oltheten & Waspi 2012
CAPM – Example
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20%
2%
4%
6%
8%
10%
12%
Beta
E[R
]
Chapter 5: Modern Portfolio Theory © Oltheten & Waspi 2012
Risk Adjusted Measures
CV:
Sharpe Ratio:
Treynor Ratio:
p
fpSharpe
rR
R
p
fpTreynor
rR
b
R
p
pR
CV
1
Chapter 5: Modern Portfolio Theory © Oltheten & Waspi 2012
Practice Questions
Chapter 5: Modern Portfolio Theory © Oltheten & Waspi 2012
Derive the CAPM Equation Graph the normal and abnormal return on
Discovery Café in this market Calculate the risk-adjusted returns
Q&P 5-2:
Investment Annual Return
Standard Deviation
Beta
T-Bills 3.3% 0.0%
Market Index Fund 12.3% 15.0%
Discovery Café 14.8% 27.3% 0.8
Portfolio Theory II
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