II: Portfolio Theory II 5: Modern Portfolio Theory

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.


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


Modern Portfolio Theory

Calculate the correlation with the basic underlying value that all stocks have in common: the market.



Tardis Intertemporal

Proctor & Gamble

Caterpillar

Microsoft

Exxon Mobile

US Steel

Citigroup

Ford

Boeing

HypotheticalResources



HypotheticalResources

Tardis Intertemporal

Proctor & Gamble

Caterpillar

Microsoft

Exxon Mobil

US Steel

Citigroup

Ford

Boeing

Market


Market Model

RStock = + β RMarket

Return for taking market risk

Return for taking undiversifiable, firm-

specific risk


Market Model

RStock = + β RMarket

β = (Rs,Rm) *. Rs . Rm Captures the correlation between Rs and Rm. Reflects market risk exposure


Market Model

slope=β

Intercept α

Rt, RMt

R = α + β RM

RM

R

et


Market Model

RRates of Return

RMReturn on the

Market

aAlpha b

Beta

eRegression

Errors

Market ModelR = a + b RM + e

Stock Prices

Index Values


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


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


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


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]


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]


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]


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]


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


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)


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


CAPM: Investment by Investment

For any investment with market risk exposure β,

we can see if the investment generated any abnormal return


CAPM – Investment by Investment Hypothetical Resources

Market Model: E[R] = 9.56% β = 1.20

Expectations of actual return formed from past data


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


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

]


Risk Adjusted Measures

CV:

Sharpe Ratio:

Treynor Ratio:

p

fpSharpe

rR

R

p

fpTreynor

rR

b

R

p

pR

CV

1


Practice Questions


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

Documents

II: Portfolio Theory II 5: Modern Portfolio Theory