portfolio advice for a multifactor world

8/11/2019 Portfolio Advice for a Multifactor World

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59Federal Reserve Bank of Chicago

Introduction and summarIntroduction and summarIntroduction and summarIntroduction and summarIntroduction and summar y y y y y


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6 0 Economic Perspectives

The traditional view


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FIGURE 1

Mean-variancefrontier

R f

average return E(R)

volatility (R)

Risky-assetfrontier

Investors want

Market portfolio

Original assets

Optimal portfolios


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New portfolio theory

FIGURE 2

E(R)

A. No risk-free rate

B. Risk-free rate

E(R)

Notes: Panel A shows an indifference surface and optimal portfolio in the case with no risk-free rate.The dot marks the optimal portfolio where the indifference surface touches the multifactor efficientfrontier. Panel B shows the set of multifactor efficient portfolios with a risk-free rate. The two cone-shaped surfaces intersect on the black line with two dots. The two dots are the market portfolio and anadditional multifactor-efficient portfolio; all multifactor-efficient portfolios on the outer cone can bereached by combinations of the risk-free rate, the market, and the extra multifactor-efficient portfolio.


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Predictable returns


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1

2

1 1 1

1 1

)

) ,

R R a bx

x c x

t t TB

t t

t t t

FIGURE 3

percent allocation to stocks

Notes: The investor maximizes the utility of terminal wealth viaa buy-and-hold investment in stocks versus bonds. The investorhas constant relative risk aversion utility with a risk aversioncoefficient of 10. The top calculation (black) includes predicta blereturns modeled by a regression on d/p ratios ( equation 1). Thesecond calculation (color) includes predictable ret urns and theeffects of parameter uncer tainty. The third calculation (black dash)assumes unpredictable returns, and no parameter uncertainty. Thebottom calculation (color dash) assumes unpredictable returns, butadds parameter uncer tainty. All distributions are conditional on adividend/price ratio equal to its historical mean.Source: Barberis (1999).

100

0

80

60

40

202 4 6 8 10

horizon, years

Predictable

Predictable,uncertain parameters

Unpredictable

Unpredictable, uncertain parameters


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BOX 1

f x x xt t (R 1 1 2, , , ..., ),

f R x x xt t ( , ... ).1 1 2

f x x xt ( , .... ), 1 2

f R x x x f R f x x x d t t t t ( , ... ) ( ) ( , ... ) . 1 1 2 1 1 2

f x x xt ( , .... ), 1 2 f ( )

f x x xt ( )1 2, ...

f x x x f x x x f f x x x

f x x x f x x x f d

t t

t

t t

( ) ( ) ( )( )

( ) ( ) ( )

1 21 2

1 2

1 2 1 2

, ... , ..., ...

, ... , ... .

f R x x xt t ( , ... ),1 1 2


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TABLE 1

AnnualizedHorizon (years) R 2 Sharpe ratio

Buy & hold 0.501 0.17 0.712 0.26 0.723 0.38 0.785 0.59 0.95

Notes: Maximum unconditional Sharpe ratios availablefrom market-timing based on regressions of value-weightedNYSE index returns on the dividend/price ratio. The tablereports annualized Sharpe ratios corresponding to each R 2.

The formula isS

k

R

k R

*. / 05 12

22

and is derived in

the appendix.

FIGURE 4

2.5

2.0

1.5

1.0

0.5

0.048

0

1224

26

2.83.3

3.94.4

4.95.5

f r a c

t i o n w e a

l t h i n e q u

i t i e s

h o r i z o n ( m o n t h s ) d i v i d e

n d / p r i c e r a

t i o

Notes: Optimal allocation to stoc ks as a function of horizon and dividend yield.Source: Brandt (1999).


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FIGURE 5

allocation to stocks, percent

Notes: Optimal allocation to stocks as a function of the expectedreturn implied by a regression that forecasts stock returns fromdividend/price ratios. The line extends from a d/p ratio twostandard deviations above its mean (low expected returns) toone standard deviation below its mean (high expected returns).Risk aversion is 4.0.Source: Campbell and Vicera (1999).

5100

log expected gross excess return, percent4 2 0 2 4 6 8 10 12

60

20

20

60

100

140

180

220

FIGURE 6

Notes: Risk aversion = 4.00 (black line) and = 20.00 (colored dashed line).Source: Campbell and Vicera (1999).


0

80

160

240

320

400

1940 1950 1960 1970 1980 1990 2000


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FIGURE 7


A. Risk aversion coefficient 10


B. Risk aversion coefficient 20

2.06 3.75

100

80

60

40

0

20

5.43d/p, percent

Notes: The colored line ignores parameter un certainty, as in Campbell and Vicera (1999).The black line includes parameter uncertainty, as in Barberis (1999). Data sample is in months (523).

100

80

60

40

0

20

2.06 3.75 5.43d/p, percent

No uncertainty

Parameter uncertainty

No uncertainty

Parameter uncertainty


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12/20


Notes of caution


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14/20



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Conclusion


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Multifactor portfolio mathematics

w

w

w

w

R

R

R

R N N

F

F

N F

1

2

1

2

1

21

1

1

1

; ; ; .

,

,

,

R R p w ;

1 1 w.

E R E w w E R w E p( ) ( ) ( ) R .

p w .

var( ) R w Vw p ,

min s. t. Ew

pw Vw w w w12

1 1 ; ; .

12

1 10 1 2 w Vw w w w p

( ) ( ) . E

w V E V A 1 0 1 211


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A E

p

1

1

0 1 2

.

1

1 1

1 1 1

,

( )

( ) .

A w

AV A

A V A

w V A A V A

=

=

=

=

var .( ) ( ) R A V A p w Vw 1 1

Var( ) ( ) R A V A p p

p

1 11 1 .

var ,( ) ( ) ( ) R A V A A V V A A V A p 1 1 1 1 1 1

Finding the benefits of a market timing strategy without computing the strategy

s E st *2 , 2

s E R R R R f f * / max ( ) ( )

s E R R R Rt t f

t

f

max ( ) ( ) /

E Z

Z

m

E m( )( )

( )( )

,

max E

( *)

( *),m

E m

m m

E m Z E m R

m

t t t t t t f

* argmin ( )

( ) ; ( ) / .

{ }

s.t.

1 1 10 1

2

2

2 2

2

2

2

( )( )

[ ( ) ] [ ( )]( )

( )( )

.m

E m

E m E m

E m E

m

E mt t t

t

ss R

R

* ,

02 2

21

s E R R R R f f 0 ( ) / ( )


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t

t

t

t

t m

E m

E Z

Z

EZ b x Ex( )( )

( )( )

( ),

2 2 2

s s R R* / . 02 2 21

Z EZ b x Ext t t 1 1( ) ,

22

2

2 2 2

2

2 2 2

2 2 2 2

2 2

2 2 2

2

22

( ) ( )( )

( ( )) ( )

( ( )) ( )(1 ) ( ) (1 ) ( )

1 ( )(1 ) ( ) (1 )

1 ( ).

(1 ) ( )

t t

t

m EZ b x Ex E E

E m

E Z b x

E Z b x R Z R Z

EZ R R Z R

E Z R

R Z

+ =

+=

= +

= +

= +


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3 01 1) [( ) ] , E c Z t t

E R R Z f t m

t [( ( ) ) ] . 1 01 1

E u ct t t ( )


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portfolio advice for a multifactor world

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