Download Market Returns without Downside Risk Or The Difference Between

Market Returns without Downside Risk
Or
The Difference Between Beta and the
Equity Premium
Max R. Arai, Ph.D., CFA
Head of Portfolio Construction and Trading
Acadian Asset Management, Inc.
Q3 2007
Abstract
Reproducing results from Haugen and Clarke et al, I find that minimum variance equity portfolios in
the US and globally exhibit returns comparable to the returns of capitalization weighted portfolios.
Following the standard textbook description of CAPM but using empirical data, one finds that the
frontier corresponding to market clearing prices has been essentially flat; capitalization weighted
portfolios (CWP) have added risk with little or no incremental return. In contrast, historical manager
alpha forecasts lead to an efficient frontier dominating the market frontier. This demonstrates the
inefficiency of the capitalization weighted portfolios.
To gain insight into the inefficiency, I analyze the implied alpha of the CWP using a performance
attribution system, and find that the style and stock bets in CWP have no efficacy. Using quintile
portfolios to test these implied alphas confirms the lack of efficacy. I interpret these findings to
demonstrate that “cheap beta” does not efficiently realize the equity premium. Asset allocation
frameworks frequently assume that capitalization weighted portfolios are effective proxies for the
equity premium. The use of minimum variance portfolios as a more efficient proxy can markedly
increase the target allocation to equities, and improve the returns of the overall plan.
Minimum variance portfolios represent a passive investment. Active alternatives further strengthen
the case for portfolios managed relative to total risk. Various applications of leverage further
improve on the expected returns.
I conclude with a discussion of the possible limitations of minimum variance portfolios, and
specifically discuss trading and capacity limitations for these strategies. Due to the presence of
investors with dramatically varying levels of assets, these effects are likely to be persistent.
Outline
• Exploring a client request
– Return characteristics of systematic risk factors
– CAPM and Minimum Variance Portfolios:
theory, history and simulations
•
•
•
•
•
Asset Allocation
Active management relative to a cash benchmark
Capacity limitations, persistence
Why this is timely
Concluding remarks
Exploring a client request
• A client requests a portfolio seeking equity marketlike returns with limited downside risk
• Possible approaches:
– Call options (downside risk is the premium)
– Market Neutral (low absolute risk, high returns with good
manager forecasts)
– Long only equity but control (negative) fat tails
• Consider long only fully invested equity
– Client seeks to market the product globally
Some initial ideas to limit downside risk
• Possible ways to improve portfolio skewness
– Take advantage of factor return skewness, esp. country and
risk factors
– Large emerging market exposure
• Portfolio total risk scales the width of the probability
distribution of returns
– Low total volatility
• Incorporate alpha
– To pay transaction costs
– To compensate for downside
Controlling Fat Tails
• Moments of a probability distribution
(
µ n = E ( X − µ )n
)
• Dimensionless moments
⎛⎛ X − µ ⎞ n ⎞
µˆ n = E ⎜⎜⎜
⎟ ⎟⎟
⎝⎝ σ ⎠ ⎠
– Standard deviation σ sets scale of moments
– Skewness proportional to σ3
– Kurtosis proportional to σ4
• To control fat tails, can we control σ?
• Recall Chebyshev: P(|X-µ|>kσ) < k-2
Non-parametric
Factor returns - US model
• Measure using performance attribution for unit
exposure assets over entire model history
(198611-200706)
• Factor skewness of both signs
>0: Growth, Earnings Yield, Volatility…
<0: Value, Size, Momentum,…
• All factors have excess kurtosis
Factor returns – US model
Descriptive Statistics for UB model risk factors
8.00
7.00
6.00
5.00
stdev
4.00
mean
skewness
3.00
kurtosis
2.00
1.00
0.00
Source: underlying data from MSCI Barra
UB_YLD
UB_VOL
UB_VAL
UB_TRA
UB_SNL
UB_SIZ
UB_NEU
UB_MOM
UB_LEV
UB_GRO
UB_EYL
UB_EVR
UB_CUR
-1.00
Factor returns – US model
QQ plots for Ub model risk factor payoffs
6.00
UB_CUR
4.00
UB_EVR
UB_EYL
2.00
UB_GRO
UB_LEV
-4.00
-3.00
-2.00
-1.00
0.00
0.00
-2.00
UB_MOM
1.00
2.00
3.00
4.00
UB_NEU
UB_SIZ
UB_SNL
-4.00
UB_TRA
UB_VAL
-6.00
UB_VOL
UB_YLD
-8.00
Source: underlying data from MSCI Barra
Factor returns - Global model
• Measure using performance attribution for unit
exposure assets over entire model history
(198802-200705)
• Factor skewness of both signs
>0: Growth, Earnings Yield, Volatility…
<0: Value, Size, Momentum,…
• All factors have excess kurtosis
MS__SIZ
MS__VAL
MS__AEROSP
MS__AIRTRN
MS__APPLNC
MS__AUTO
MS__BANKS
MS__BEVTOB
MS__BLDG
MS__BRDCST
MS__BUSSVC
MS__CHEM
MS__CONSTR
MS__DATAPR
MS__ELCTEQ
MS__ELECMP
MS__ENERGY
MS__ENRGEQ
MS__FINANC
MS__FOODHS
MS__GOLD
MS__HEALTH
MS__INSURE
MS__LEISUR
MS__MACHIN
MS__MATER
MS__MRCHND
MS__MULTI
MS__NFMET
MS__PAPER
MS__REALES
MS__RECCON
MS__RRTRAN
MS__SCS
MS__SHPTRN
MS__SPR
MS__STEEL
MS__TELECM
0.00
-5.00
MS__SIZ
MS__VAL
MS__AEROSP
MS__AIRTRN
MS__APPLNC
MS__AUTO
MS__BANKS
MS__BEVTOB
MS__BLDG
MS__BRDCST
MS__BUSSVC
MS__CHEM
MS__CONSTR
MS__DATAPR
MS__ELCTEQ
MS__ELECMP
MS__ENERGY
MS__ENRGEQ
MS__FINANC
MS__FOODHS
MS__GOLD
MS__HEALTH
MS__INSURE
MS__LEISUR
MS__MACHIN
MS__MATER
MS__MRCHND
MS__MULTI
MS__NFMET
MS__PAPER
MS__REALES
MS__RECCON
MS__RRTRAN
MS__SCS
MS__SHPTRN
MS__SPR
MS__STEEL
MS__TELECM
MS__TEXTL
MS__UTIL
MS__AUSMKT
MS__AUTMKT
MS__BELMKT
MS__CANMKT
MS__DENMKT
MS__FINMKT
MS__FRAMKT
MS__GERMKT
MS__GREMKT
MS__HKGMKT
MS__IREMKT
MS__ITAMKT
MS__JPNMKT
MS__NETMKT
MS__NORMKT
MS__NZEMKT
MS__PORMKT
MS__SINMKT
MS__SPAMKT
MS__SWEMKT
MS__SWIMKT
MS__UKIMKT
MS__USAMKT
MS__ARGMKT
MS__BHRMKT
MS__BRAMKT
MS__CHIMKT
MS__CHNMKT
MS__COLMKT
MS__CZEMKT
MS__EGYMKT
MS__HUNMKT
MS__IDNMKT
MS__INDCMP
MS__ISRMKT
MS__JORMKT
MS__KORMKT
MS__MALMKT
MS__MEXMKT
MS__MORMKT
MS__OMNMKT
MS__PAKMKT
MS__PERMKT
MS__PHIMKT
MS__POLMKT
MS__RUSMKT
MS__SAFMKT
MS__SLVMKT
MS__SRIMKT
MS__TAIMKT
MS__THAMKT
MS__TURMKT
Factor Returns – Global Model
25.00
250
20.00
200
15.00
150
mean
10.00
100
5.00
9.00
8.00
7.00
6.00
5.00
4.00
3.00
2.00
1.00
0.00
-1.00
-2.00
100
50
0
Source: underlying data from MSCI Barra
stdev
skewness
excess kurtosis
50
count
0
250
200
mean
150
stdev
skewness
excess kurtosis
count
Hard to Predict Higher Moments
• Stationarity of factor returns suspect
– Are any country returns actually thin tailed?
– Time dependent coverage
• Stock returns often have non–stationary
distributions - true even at the index level
Higher Moments for Some Sample
Indexes
Skewness -
3 year rolling window monthly returns
1.5
1
0.5
-1
-1.5
Source: underlying data from MSCI Barra, JP Morgan, Acadian simulated portfolios
6/30/07
4/30/07
2/28/07
12/30/06
8/30/06
10/30/06
6/30/06
4/30/06
2/28/06
12/30/05
8/30/05
10/30/05
6/30/05
4/30/05
2/28/05
12/30/04
8/30/04
-0.5
10/30/04
6/30/04
4/30/04
2/29/04
12/30/03
8/30/03
10/30/03
6/30/03
4/30/03
2/28/03
12/30/02
8/30/02
10/30/02
6/30/02
4/30/02
2/28/02
12/30/01
8/30/01
10/30/01
6/30/01
4/30/01
2/28/01
12/30/00
8/30/00
10/30/00
6/30/00
0
Active Absolute
JPM-1-3Yr Bond Index
JPM-10yr Bond Index
Passive Absolute
MSCI Australia
MSCI Europe
MSCI Japan
MSCI World
MSCI US
MSCI EAFE Small
MSCI EMF
MSCI GCC
Higher Moments for Some Sample
Indexes
Kurtosis -
3 year rolling window monthly returns
3
2
1
-1
-2
-3
-4
Source: underlying data from MSCI Barra, JP Morgan, Acadian simulated portfolios
6/30/07
4/30/07
2/28/07
12/30/06
8/30/06
10/30/06
6/30/06
4/30/06
2/28/06
12/30/05
8/30/05
10/30/05
6/30/05
4/30/05
2/28/05
12/30/04
8/30/04
10/30/04
6/30/04
4/30/04
2/29/04
12/30/03
8/30/03
10/30/03
6/30/03
4/30/03
2/28/03
12/30/02
8/30/02
10/30/02
6/30/02
4/30/02
2/28/02
12/30/01
8/30/01
10/30/01
6/30/01
4/30/01
2/28/01
12/30/00
8/30/00
10/30/00
6/30/00
0
Active Absolute
JPM-1-3Yr Bond Index
JPM-10yr Bond Index
Passive Absolute
MSCI Australia
MSCI Europe
MSCI Japan
MSCI World
MSCI US
MSCI EAFE Small
MSCI EMF
MSCI GCC
Review of CAPM
Return
Capital market line
Risk Free Rate
Maximum Sharpe Ratio portfolio
Minimum variance portfolio
Risk
Minimum Variance Portfolios in the
context of CAPM
• CAPM makes heroic assumptions (equilibrium, fully
informed investors who agree, no taxes, borrowing
and lending at the same risk-free rate, …)
• Under CAPM, the Maximum Sharpe ratio portfolio is
the market portfolio.
• CAPM has an efficient frontier (relative to the risk
free asset)
• Assuming CAPM, minimum variance portfolio has
lower return and Sharpe ratio than the market
portfolio
Empirical cash relative frontier
Return
Capital market line
Maximum Sharpe Ratio portfolio
Cap weighted portfolio
Risk Free Rate
Minimum variance portfolio
Risk
Source: illustration of Acadian simulations
Market consensus
efficient frontier
Minimum Variance Portfolios (MVP)
• Clarke et al.
“Minimum-Variance Portfolios in the U.S. Equity Market” , Journal
of Portfolio Management, Fall 2006
• Haugen and Baker
“The Efficient Market Inefficiency Of Capitalization- Weighted
Stock Portfolios”, Journal of Portfolio Management Spring 1991
• Acadian simulations confirm Clarke's findings for US, Australian
and global portfolios. Global results illustrate the benefits of
currency hedging for long term results.
Performance attribution for cap weighted
portfolios
• Use minimum variance portfolio as the benchmark
• Neither style nor asset selection added value
• The benchmark to assess the cap weighted portfolio
(CWP) and active manager portfolios should be the
minimum variance portfolio.
• Manager frontier and market consensus frontier can
differ, and can reveal manager skill (using CWP can
obscure manager skill).
Example performance attribution
ATTRIBUTION REPORT
Annualized Contributions To Total Return
Managed
vs minimum variance
MSCI World
vs minimum variance
Managed
vs MSCI World
Source
Contribution
Risk
Contribution
Risk
Contribution
Risk
of Return
(% Return)
(% Std Dev)
(% Return)
(% Std Dev)
(% Return)
(% Std Dev)
1 Risk Free
3.59
N/A
3.59
N/A
3.59
N/A
2 Total Benchmark
8.83
8.80
8.83
8.80
7.07
14.41
3 Country Selection
-0.52
1.94
-0.90
1.60
0.40
2.48
4 Currency Selection
0.60
1.21
0.69
1.39
-0.10
1.21
5 Cash-Equity Policy
-0.07
0.06
-0.09
0.11
0.00
0.00
6 Asset Allocation [3+4+5]
0.01
2.12
-0.30
2.06
0.31
2.71
7 Local Market Timing
0.53
1.84
1.70
8.14
-2.02
7.70
8 Risk Indices
1.42
1.21
-1.29
2.10
2.60
2.74
9 Industries
0.25
1.20
-0.09
1.70
0.54
2.01
10 Asset Selection
2.69
2.26
-1.72
1.87
5.15
1.93
11 Within Market [7+8+9+10]
4.88
3.30
-1.40
9.35
6.27
9.53
12 Trading
0.00
0.40
N/A
N/A
0.00
0.40
-1.24
N/A
N/A
N/A
-1.23
N/A
3.62
3.42
-1.76
9.41
5.38
9.45
12.45
10.15
7.07
14.41
12.45
10.15
13 Transaction Cost
14 Total Active [6+11+12+13]
15 Total Managed [2+14]
Source: Acadian simulated performance
Efficacy of implied alpha
• Extract implied alpha for a cap weighted
benchmark
• Use implied alphas to construct quintile
portfolios to measure spreads
• The market consensus offers negligible
forecast value (no free lunch).
Beta ≠ Equity Premium
• Minimum variance portfolio is fully invested
and captures the equity premium
• Beta relative to the cap weighted portfolio
adds uncompensated risk
• Better market proxy (missing assets)?
– Capitalization would have to be enormous to
change the result
Simplified Asset Allocation Framework
• Select some investible proxies for asset classes
of interest
– JPMorgan 1-3 year bond index
– JPMorgan 10 year bond index
– MSCI indexes (World, AU, JP, US, Europe, EAFE
Small cap, EMF)
– Passive and active absolute return portfolios
Simplified Asset Allocation Framework (2)
• Assume long only allocations
• Use historical covariance
• Optimize allocations using historical returns
and forecast returns
– Historical data provides a reference to place the
covariance data in context
– Forecast data seeks to avoid the rear view mirror
Asset Allocation Methods
• Compute covariance from monthly returns
(in Excel)
• Use Solver to optimize mean variance returns at
varying levels of risk aversion
–
–
–
–
U = ∑ R w − λ∑ w Γ w
U is utility
wk is weight of kth asset
Rk is return of kth asset
Γkk’ is covariance of kth and k’th assets
k
k
k
k kk'
k,k'
k'
JPM-1-3Yr Bond Index
JPM-10yr Bond Index
MSCI EMF
MSCI Japan
MSCI Australia
MSCI US
MSCI Europe
MSCI EAFE Small
MSCI World
Passive Absolute
Active Absolute
100%
71%
-32%
-16%
-18%
-30%
-30%
-24%
-32%
-16%
-19%
71%
100%
-23%
-13%
-12%
-22%
-21%
-7%
-23%
-6%
-6%
Historical Returns
Historical Risk
Forecast Returns
4.5%
1.7%
5%
7.7% 16.9% 2.9% 11.3% 6.2% 9.2% 13.7% 6.3%
8.7% 21.9% 20.2% 18.4% 15.8% 16.7% 15.8% 15.0%
6% 14%
6% 10%
8%
9% 14%
9%
-32%
-23%
100%
55%
73%
70%
70%
77%
77%
58%
55%
-16%
-13%
55%
100%
60%
47%
45%
73%
61%
65%
61%
Source: underlying data from MSCI Barra, JP Morgan, Acadian simulated portfolios
-18%
-12%
73%
60%
100%
64%
67%
75%
74%
71%
66%
-30%
-22%
70%
47%
64%
100%
81%
58%
96%
70%
70%
-30%
-21%
70%
45%
67%
81%
100%
68%
92%
75%
70%
-24%
-7%
77%
73%
75%
58%
68%
100%
71%
77%
72%
-32%
-23%
77%
61%
74%
96%
92%
71%
100%
80%
77%
-16%
-6%
58%
65%
71%
70%
75%
77%
80%
100%
95%
Active Absolute
Passive Absolute
MSCI World
MSCI EAFE Small
MSCI Europe
MSCI US
MSCI Australia
MSCI Japan
MSCI EMF
JPM-10yr Bond Index
JPM-1-3Yr Bond Index
Historical Correlations
-19%
-6%
55%
61%
66%
70%
70%
72%
77%
95%
100%
6.4% 11.4%
9.4% 11.0%
9% 11%
Allocation Results using Forecast
Returns
Three Asset Control Study
Four Asset Allocation: Two Bonds, MSCI World, Passive Absolute Strategy
Dominance of Active Absolute Strategy
100%
100%
100%
80%
80%
80%
60%
60%
60%
MSCI World
JPM-10yr Bond Index
JPM-1-3Yr Bond Index
40%
20%
MSCI World
Passive Absolute
JPM-10yr Bond Index
JPM-1-3Yr Bond Index
40%
20%
0%
20%
0%
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
0%
1
-20%
MSCI World
Passive Absolute
Active Absolute
JPM-10yr Bond Index
JPM-1-3Yr Bond Index
40%
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
1
-20%
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
-20%
Three Asset Control Study
Four Asset Allocation: Two Bonds, MSCI World, Passive Absolute Strategy
Dominance of Active Absolute Strategy
12.00%
120
12.00%
120
12.00%
120
10.00%
100
10.00%
100
10.00%
100
8.00%
8.00%
80
6.00%
60
4.00%
2.00%
0.00%
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
expected return
expected risk
utility
lambda
80
expected return
expected risk
utility
lambda
8.00%
80
6.00%
60
6.00%
60
40
4.00%
40
4.00%
40
20
2.00%
20
2.00%
20
0
0.00%
0
0.00%
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Source: underlying data from MSCI Barra, JP Morgan, Acadian simulated portfolios
25
26
27
28
29
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
expected return
expected risk
utility
lambda
Allocation Results using Historical
Returns
Three Asset Control Study
Four Asset Allocation: Two Bonds, MSCI World, Passive Absolute Strategy
Five Asset Allocation: Two Bonds, MSCI World, Active and Passive Absolute Strategy
100%
100%
100%
80%
80%
80%
60%
60%
MSCI World
JPM-10yr Bond Index
JPM-1-3Yr Bond Index
40%
20%
60%
MSCI World
Passive Absolute
JPM-10yr Bond Index
JPM-1-3Yr Bond Index
40%
20%
0%
20%
0%
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
0%
1
-20%
MSCI World
Passive Absolute
Active Absolute
JPM-10yr Bond Index
JPM-1-3Yr Bond Index
40%
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
1
-20%
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
-20%
Three Asset Control Study
Four Asset Allocation: Two Bonds, MSCI World, Passive Absolute Strategy
Five Asset Allocation: Two Bonds, MSCI World, Active and Passive Absolute Strategy
12.00%
120
12.00%
120
12.00%
120
10.00%
100
10.00%
100
10.00%
100
8.00%
80
6.00%
60
4.00%
2.00%
0.00%
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
8.00%
80
6.00%
60
40
4.00%
20
2.00%
0
0.00%
expected return
expected risk
utility
lambda
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
Source: underlying data from MSCI Barra, JP Morgan, Acadian simulated portfolios
22
23
24
25
26
27
28
29
8.00%
80
6.00%
60
40
4.00%
40
20
2.00%
20
0
0.00%
expected return
expected risk
utility
lambda
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
expected return
expected risk
utility
lambda
Impact of Minimum Variance Portfolios on
Asset Allocation
• The addition of minimum variance portfolios to the
standard asset allocation framework increases the
equity contribution to both risk and return
• Domination of the cap weighted equity proxy by
absolute return portfolios highlights the difference
between beta and the equity premium
• Material improvements in overall plan returns
possible
Active Management Relative to a Cash
Benchmark
• Describe capacity limitations
• Describe turnover
• Describe the noise trader argument for likely
persistence of the effects
• Describe how the persistence of beta versus
equity premium relates to long term market
disequilibrium
Limitations of minimum variance
portfolios
• Capacity limitations
As assets become large, cap weights are the
only investment available
• High tracking error when viewed from a
conventional viewpoint
• Relatively high turnover
(typically over 10% per month)
Capacity considerations
• Underlying experiments need to be completed
Persistence of capitalization weighted
benchmark portfolios
• Could be due to noise trading
– Advances in Behavioral Finance, Thaler Ch.2
– Cap weighting easy to implement
– Huge percentage of market assets managed to cap
weighted benchmark
– Material risk for arbitrageurs
Role of shorting in active management
relative to a cash benchmark
Annualized Contributions To
Total Return
Source of Return
US strategy
Russell 1000
Russell 3000
Minimum
Variance
long only
60bp limit
long only
100bp limit
130/30
100bp limit
1 Risk Free
3.53
3.53
3.53
3.53
3.53
3.53
2 Total Benchmark
7.14
7.14
7.14
7.14
7.14
7.14
3 Expected Active
-0.02
0.00
-3.47
-2.94
-3.10
-3.85
4 Market Timing
-0.05
0.00
2.22
2.29
2.37
2.44
5 Risk Indices
-0.32
0.00
3.55
4.01
4.00
5.31
6 Industries
0.00
0.00
0.21
-0.29
-0.31
-0.35
7 Asset Selection
0.30
0.00
0.83
2.01
2.67
3.77
8 Trading
N/A
N/A
N/A
N/A
N/A
N/A
9 Transaction Cost
N/A
N/A
N/A
N/A
N/A
N/A
10 Total Exceptional
Active[4+...+9]
-0.07
0.00
6.81
8.02
8.73
11.17
11 Total Active [3+10]
-0.09
0.00
3.34
5.09
5.63
7.32
7.05
7.14
10.48
12.23
12.77
14.45
12 Total Managed [2+11]
Source: underlying data from MSCI Barra, JP Morgan, Acadian simulated portfolios
Why is this timely
• 20 years ago, even US transactions costs were high,
~100bp
• Commercial risk models available since 1980’s, but
supporting tools for portfolio construction and trading
both expensive and weak
• Today, direct market access available in most
developed markets
• The bloom is off the rose for cap weighted
benchmarks; active investing well established
Outlook and conclusions
• Absolute return portfolios may be exiting the
pioneering phase and moving into rapid
growth
• Increased attention to liability driven
investment (LDI) and stability of pension plan
funding status may lead to demand for more
efficient investments
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