Empirical Financial Economics
5. Current Approaches to
Performance Measurement
Stephen Brown NYU Stern School of
Business
UNSW PhD Seminar, June 19-21 2006
Overview of lecture
Standard approaches
Theoretical foundation
Practical implementation
Relation to style analysis
Gaming performance metrics
Performance measurement
Leeson
Market
Short-term
Investment
(S&P 500)
Government
Managemen Benchmark Benchmark
t
Average .0065
Return
.0050
.0036
Std. .0106
Deviation
.0359
.0015
1.0
.0
Beta .0640
Alpha .0025
.0
.0
(1.92)
100% in cash.0
at close of
Sharpe Style:
Ratio Index
.2484 Arbitrage,.0318
trading
-1
.0
-0 0 %
.5
0
0. %
00
0. %
50
%
1.0
0%
1.5
0%
2.
00
2. %
50
3. %
00
3. %
50
4. %
00
4. %
50
5. %
00
5. %
50
6. %
0
0
6. %
50
%
Frequency distribution of
monthly returns
35
30
25
20
15
10
5
0
Universe Comparisons
40%
Brownian Management
35%
S&P 500
30%
25%
20%
15%
10%
5%
One Quarter
1 Year
3 Years
Periods ending Dec 31 2002
5 Years
Total Return comparison
Average Return
A
B
C
D
Total Return comparison
Average Return
RS&P = 13.68%
rf = 1.08%
A
S&P 500
B
C
D
Treasury Bills
Manager A best
Manager D worst
Total Return comparison
Average Return
A
B
C
D
Sharpe ratio comparison
Average Return
A
B
D
C
Standard
Deviation
Sharpe ratio comparison
Average Return
A
S&P 500
RS&P = 13.68%
B
D
rf = 1.08%
C
Treasury Bills
^
σS&P = 20.0%Standard
Deviation
Sharpe ratio comparison
Average Return
A
S&P 500
RS&P = 13.68%
B
Sharpe ratio =
Average return – rf
D
C
Manager C worst
Manager D best
Standard Deviation
rf = 1.08%
Treasury Bills
^
σS&P = 20.0%Standard
Deviation
Sharpe ratio comparison
Average Return
A
S&P 500
RS&P = 13.68%
B
D
rf = 1.08%
C
Treasury Bills
^
σS&P = 20.0%Standard
Deviation
Jensen’s Alpha comparison
Average Return
A
S&P 500
RS&P = 13.68%
Jensen’s alpha =
Average return –
C
D
B
Manager B worst
Manager C best
{rf + β (RS&P - rf )}
rf = 1.08%
Treasury Bills
βS&P = 1.0 Beta
Intertemporal equilibrium model
 j

Max Et   U (ct  j ) 
Multiperiod problem:
 j 0

First order conditions:
U (ct )   j Et  (1  ri ,t  j )U (ct  j ) 
Stochastic discount factor interpretation:
1  Et  (1  ri ,t  j ) mt , j  ,
U (ct  j )
mt , j  
U (ct )
j
 mt , j “stochastic discount factor”, “pricing
Value of Private Information
I1  I 0
Investor has access to information
R0
I1  I 0 is givenEby
Rwhere
Value of
t [( R1  R0 )mt ]
1
I1
I0
and
are returns on optimal
portfolios given and
Et [( R1  CAPM
R0 )mt ]  (Chen
1Knez
rft )  1
Under
1t  rft &
t (  mt 1996)
Jensen’s alpha measures value of private
information
The geometry of mean variance
a  2bE  cE 2
 
ac  b2
2
E
1  1/ b 
1
x

1/
b



2  0 
a
b
  a  1
2


a
b
Note: returns are in excess of the risk freerf
Informed portfolio strategy
R1  rf  R0  rf  
Excess return
on informed
 is the return on an optimal
strategy where
orthogonal portfolio (MacKinlay 1995)
Sharpe ratio squared of informed strategy
12  (0  rf )1 (0  rf )   1  02  2  02
Assumes well diversified portfolios
Informed portfolio strategy
R1  rf  R0  rf  
Excess return
on informed
 is the return on an optimal
strategy where
orthogonal portfolio (MacKinlay 1995)
Sharpe ratio squared of informed strategy
12  (0  rf )1 (0  rf )   1  02  2  02
Assumes well diversified portfolios
Used in tests of mean variance efficiency of
benchmark
Practical issues
Sharpe ratio sensitive to
diversification, but invariant to
leverage
 Risk
premium and standard deviation
proportionate to fraction of investment
financed by borrowing
Jensen’s alpha invariant
2  0 to  
diversification, but sensitive to
leverage
 In
a complete market
implies
Changes in Information Set
1t  1t  rft  1t ( mt  rft )
How do we measure alpha
I1t
when information set
is not constant?
 Rolling
regression, use subperiods to estimate
1  1  rf  1 ( m  rft )
(no t subscript) – Sharpe (1992)
macroeconomic variable controls – Ferson
and Schadt(1996)
 Use
GSC procedure – Brown and Goetzmann
(1997)
 Use
Style management is crucial …
Economist, July 16, 1995
But who determines styles?
Characteristics-based Styles
rjt   Jt   Jt I t   jt
jJ
 Traditional approach …
  Jt are changing characteristics (PER,
Price/Book)
It
 are returns to characteristics
 Jt
 Style benchmarks are given by
rjt   Jt   jt
jJ
Returns-based Styles
rjt   Jt   Jt I t   jt
jJ
 Sharpe (1992) approach …
  Jt are a dynamic portfolio strategy
 I t are benchmark portfolio returns
 Jt by
 Style benchmarks are given
rjt   Jt   jt
jJ
Returns-based Styles
rjt   Jt   Jt I t   jt
jJ
 GSC (1997) approach …
  jT ,  Jt vary through time but are fixed forJ
style
 Jt
 Allocate funds to styles directly
using
 Jt
 Style benchmarks are given by
rjt   Jt   jt
jJ
Eight style decomposition
10 0 %
80%
60%
40 %
20 %
0%
GSC1 GSC2 GSC3 GSC4 GSC5 GSC6
Ot her
Pure Emerging Market
Global Macro
Event Driven
US Equit y Hedge
GSC7 GSC8
Pure Propert y
Pure Leveraged Currency
Non Direct ional/Relat ive Value
Non-US Equit y Hedge
Five style decomposition
10 0 %
80%
60%
40 %
20 %
0%
GSC1
GSC2
Ot her
Pure Emerging Market
Global Macro
Event Driven
US Equit y Hedge
GSC3
GSC4
GSC5
Pure Propert y
Pure Leveraged Currency
Non Direct ional/Relat ive Value
Non-US Equit y Hedge
Style classifications
GSC1
GSC2
GSC3
GSC4
GSC5
GSC6
GSC7
GSC8
Event driven international
Property/Fixed Income
US Equity focus
Non-directional/relative value
Event driven domestic
International focus
Emerging markets
Global macro
Regressing returns on
2
classifications: Adjusted R
Year N
1992 149
1993 212
1994 288
1995 405
1996 524
1997 616
1998 668
Average
GSC8
classifications
0.3827
0.2224
0.1662
0.0576
0.1554
0.3066
0.2813
0.2246
GSC5
classifications
0.1713
0.1320
0.1040
0.0548
0.0769
0.1886
0.2019
0.1328
TASS 17
classifications
0.4441
0.1186
0.0986
0.0446
0.1523
0.2538
0.1998
0.1874
Variance explained by prior
returns-based classifications
Year N
1992 198
1993 276
1994 348
1995 455
1996 557
1997 649
1998 687
Average
8 GSC
Classifications
0.3622
0.1779
0.1590
0.0611
0.1543
0.2969
0.2824
0.2134
8 Principal
Components
0.0572
0.0351
0.0761
0.0799
0.0286
0.0211
0.2862
0.0835
8 Benchmarks
(predetermined)
0.1769
0.1748
0.0481
0.0862
0.0691
0.0642
0.2030
0.1175
Variance explained by prior
factor loadings
Year N
1992 198
1993 276
1994 348
1995 455
1996 557
1997 649
1998 687
Average
8 GSC
Classifications
0.2742
0.2170
0.1760
0.0670
0.1444
0.3135
0.2752
0.2096
8 Principal
Components
0.1607
0.0928
0.1577
0.0783
0.0888
0.3069
0.3744
0.1799
8 Benchmarks
(predetermined)
0.2552
0.0932
0.0700
0.0829
0.0349
0.0899
0.3765
0.1432
Percentage in cash (monthly)
120 %
10 0 %
80%
60%
40 %
20 %
0%
31-Dec-198 9
15-May-1991
26 -Sep-1992
8 -Feb-1994
Examples of riskless index
arbitrage …
Percentage in cash (daily)
20 0 %
10 0 %
0%
-10 0 %
-20 0 %
-30 0 %
-40 0 %
-50 0 %
-6 0 0 %
31-Dec-198 9
15-May-1991
26 -Sep-1992
8 -Feb-1994
“Informationless” investing
Concave payout strategies
Zero net investment overlay strategy (Weisman
2002)
Uses only public information
Designed to yield Sharpe ratio greater than benchmark
Using strategies that are concave to benchmark
Concave payout strategies
Zero net investment overlay strategy (Weisman
2002)
Uses only public information
Designed to yield Sharpe ratio greater than benchmark
Using strategies that are concave to benchmark
Why should we care?
Sharpe ratio obviously inappropriate here
But is metric of choice of hedge funds and derivatives
traders
We should care!
Delegated fund management
Fund flow, compensation based on
historical performance
Limited incentive to monitor high
Sharpe ratios
Behavioral issues
Prospect theory: lock in gains, gamble
on loss
Are there incentives to control this
behavior?
Sharpe Ratio of Benchmark
10 0 %
50 %
0%
-50 %
Benchmark
-10 0 %
-150 %
-20 0 %
-50 %
0%
50 %
Sharpe ratio = .631
10 0 %
Maximum Sharpe Ratio
10 0 %
50 %
Benchmark
0%
-50 %
Maximum
Sharpe Rat io
St rat egy
-10 0 %
-150 %
-20 0 %
-50 %
0%
50 %
Sharpe ratio = .748
10 0 %
Concave trading strategies
10 0 %
50 %
Benchmark
0%
Loss Averse
Trading
(Median)
Maximum
Sharpe Rat io
St rat egy
-50 %
-10 0 %
-150 %
-20 0 %
-50 %
0%
50 %
10 0 %
Examples of concave payout
strategies
Long-term asset mix guidelines
Examples of concave payout
strategies
Unhedged short volatility
Writing out of the money calls and
puts
Examples of concave payout
strategies
Loss averse trading
a.k.a. “Doubling”
Examples of concave payout
strategies
Long-term asset mix guidelines
Unhedged short volatility
Writing out of the money calls and
puts
Loss averse trading
a.k.a. “Doubling”
Forensic Finance
Implications of concave payoff
strategies
Patterns of returns
Forensic Finance
Implications of Informationless
investing
Patterns of returns
 are
returns concave to benchmark?
Forensic Finance
Implications of concave payoff
strategies
Patterns of returns
 are
returns concave to benchmark?
Patterns of security holdings
Forensic Finance
Implications of concave payoff
strategies
Patterns of returns
 are
returns concave to benchmark?
Patterns of security holdings
 do
security holdings produce concave
payouts?
Forensic Finance
Implications of concave payoff
strategies
Patterns of returns
 are
returns concave to benchmark?
Patterns of security holdings
 do
security holdings produce concave
payouts?
Patterns of trading
Forensic Finance
Implications of concave payoff
strategies
Patterns of returns
 are
returns concave to benchmark?
Patterns of security holdings
 do
security holdings produce concave
payouts?
Patterns of trading
 does
pattern of trading lead to concave
Conclusion
Value of information interpretation of
standard performance measures
New procedures for style analysis
Return based performance measures
only tell part of the story