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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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61Federal Reserve Bank of Chicago
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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6 2 Economic Perspectives
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.
8/11/2019 Portfolio Advice for a Multifactor World
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63Federal Reserve Bank of Chicago
Predictable returns
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6 4 Economic Perspectives
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
8/11/2019 Portfolio Advice for a Multifactor World
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65Federal Reserve Bank of Chicago
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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6 6 Economic Perspectives
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).
8/11/2019 Portfolio Advice for a Multifactor World
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67Federal Reserve Bank of Chicago
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).
allocation to stocks, percent
0
80
160
240
320
400
1940 1950 1960 1970 1980 1990 2000
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6 8 Economic Perspectives
FIGURE 7
allocation to stocks, percent
A. Risk aversion coefficient 10
allocation to stocks, percent
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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69Federal Reserve Bank of Chicago
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7 0 Economic Perspectives
Notes of caution
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7 2 Economic Perspectives
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73Federal Reserve Bank of Chicago
Conclusion
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7 4 Economic Perspectives
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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75Federal Reserve Bank of Chicago
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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7 6 Economic Perspectives
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
+ =
+=
= +
= +
= +
8/11/2019 Portfolio Advice for a Multifactor World
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77Federal Reserve Bank of Chicago
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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