Download Hedge Fund Risk and Return Modeling

Modeling Risk and Return for
Hedge Funds and Fund of
Hedge Funds
Seattle Economic Council,
December 2, 2015
Eric Zivot
Robert Richards Chaired Professor of Economics
Adjunct Professor, Departments of Applied Mathematics
Finance and Statistics
Co-Director of CF&RM
University of Washington
and
BlackRock Alternative Advisors
Outline
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Motivation
Fund of hedge funds environment
Characteristics of hedge fund data
Linear factor model for hedge fund returns
Risk Measures
Factor model Monte Carlo methodology
Risk decompositions
Examples
© Eric Zivot 2011
Motivation
• Teach quantitative risk management in UW
CF&RM program.
• Risk management consultant to BlackRock
Alternative Advisors, a large fund-of-hedge
fund.
• General problem: model risk and return for
portfolios of hedge fund investments.
• Hedge fund returns have unique properties that
present interesting challenges for modeling.
© Eric Zivot 2011
Fund of Hedge Funds Environment
• FoHFs are hedge funds that invest in other hedge
funds
– 20 to 30 portfolios of hedge funds
– Typical portfolio size is 30 funds
• Hedge fund universe is large: 5000 live funds
– Segmented into 10-15 distinct strategy types
• Hedge funds voluntarily report monthly performance
to commercial databases
– Altvest, Barclay, CISDM, Eureka Hedge,
HedgeFund.net, HFR, Lipper TASS, CS/Tremont
• Limited transparency is typical
© Eric Zivot 2011
Hedge Fund Universe
Live funds
3% 1%
6%
6%
26%
10%
5%
5%
6%
4%
28%
© Eric Zivot 2011
Convertible Arbitrage
Dedicated Short Bias
Emerging Markets
Equity Market Neutral
Event Driven
Fixed Income Arbitrage
Global Macro
Long/Short Equity Hedge
Managed Futures
Multi-Strategy
Fund of Funds
Risk Measurement and Management
• Quantify exposures to risk drivers
– Equity, rates, credit, volatility, currency,
commodity, etc.
• Quantify fund and portfolio risk
– Return standard deviation, value-at-risk (VaR),
expected tail loss (ETL)
• Perform risk decomposition
– Contribution of risk factors, contribution of
constituent funds to portfolio risk
• Stress testing and scenario analysis
© Eric Zivot 2011
Challenges of Hedge Fund Return Data
• Reporting biases
– Survivorship, backfill
• Non-normal behavior
– Asymmetry (skewness) and fat tails (excess
kurtosis)
• Serial correlation
– Performance smoothing, illiquid positions
• Small sample sizes
– Mostly monthly returns
• Unequal histories
© Eric Zivot 2011
Characteristics of Hedge Fund Data
fund1
fund2
fund3
fund4
fund5
Observations
122.0000 107.0000 135.0000 135.0000 135.0000
NAs
13.0000 28.0000
0.0000
0.0000
0.0000
Minimum
-0.0842 -0.3649 -0.0519 -0.1556 -0.2900
Quartile 1
-0.0016 -0.0051
0.0020 -0.0017 -0.0021
Median
0.0058
0.0046
0.0060
0.0073
0.0049
Arithmetic Mean
0.0038 -0.0017
0.0063
0.0059
0.0021
Geometric Mean
0.0037 -0.0029
0.0062
0.0055
0.0014
Quartile 3
0.0158
0.0129
0.0127
0.0157
0.0127
Maximum
0.0311
0.0861
0.0502
0.0762
0.0877
Variance
0.0003
0.0020
0.0002
0.0008
0.0013
Stdev
0.0176
0.0443
0.0152
0.0275
0.0357
Skewness
-1.7753 -5.6202 -0.8810 -2.4839 -4.9948
Kurtosis
5.2887 40.9681
3.7960 13.8201 35.8623
Rho1
0.6060
0.3820
0.3590
0.4400
0.383
Sample: January 1998 – March 2009
© Eric Zivot 2011
Characteristics of Hedge Fund Data
X153684
-0.08
-0.04
0.00
-0.15 -0.10 -0.05 0.00 0.05
X104314
-0.04
-0.1
-0.3 -0.2
-0.02
-0.06
X156145
0.02
0.00 0.02 0.04
-0.04
X113169
X29554
0.0
0.1-0.08
X165939
0.00
fundData.z[, QFundIdxLessCash[c(1, 2, 3, 4, 31, 32)]]
1998
2000
2002
2004
2006
2008
1998
Index
2000
2002
2004
Index
© Eric Zivot 2011
2006
2008
Fund Level Linear Factor Model
Rit   i  i1 F1t   ik Fkt   it ,
  i  βi Ft   it
i  1,
, n; t  ti ,
,T
Ft ~ (μ F , Σ F )
 it ~ (0,  )
2
 ,i
cov( Fjt ,  is )  0 for all j, i, s and t
cov( it ,  js )  0 for i  j, s and t
© Eric Zivot 2011
Performance Attribution
E[ Rit ]   i   i1 E[ F1t ]     ik E[ Fkt ]
Expected return due to systematic “beta” exposure
i1E[ F1t ]   ik E[ Fkt ]
Expected return due to manager specific “alpha”
 i  E[ Rit ]  (i1E[ F1t ]    ik E[ Fkt ])
© Eric Zivot 2011
Covariance and Correlation
R t  α  B Ft  ε t
n1
n1
nk k 1
n1
var(R t )  Σ FM  BΣ F B  D
D  diag ( 2,1 ,
cor(R t )  D
1
FM
,  2,n )
Σ FM D
1
FM
D FM  diag ( Σ FM )
Note:
cov( Rit , R jt )  βi var(Ft )β j  βi ΣF β j
© Eric Zivot 2011
Portfolio Linear Factor Model
w  ( w1 , , wn )  portfolio weights
n
 w  1, w  0 for i  1,
i
i 1
i
,n
R p ,t  w R t  w α  w BFt  w ε t
n
n
n
n
i 1
i 1
i 1
i 1
  wi Rit  wi i   wi βi Ft   wi it
  p  βp Ft   p ,t
© Eric Zivot 2011
Factor Structure
• Primary Factors
– Tradable indices that measure broad asset class
performance of equity, rates, credit, commodities
and currency. Factor is excess return on index.
• Secondary Factors
– Tradable market factors that measure intra-asset
class biases
• Geography, Sector, Style, Capitalization, Credit Quality,
Capital Structure, Yield Curve Shape
– Tradable factors that capture performance of
passive investment strategies utilized by hedge
funds
© Eric Zivot 2011
Example Factor Descriptions
Factor Name
Type
Definition
Equity
Primary
G3 equity return less LIBOR
US Regional Equity
Secondary
US equity return less global equity return
Equity Volatility
Secondary
Change in 2-yr implied equity volatility
Rates
Primary
G3 government bond rate less LIBOR
Yield Curve Slope
Secondary
2-yr less 10-yr interest rate swaps
Credit – Corporate
Secondary
Investment grade yield less gov’t bond yield
Commodity – Energy
Primary
Broad energy commodity index return
Energy Type
Secondary
Natural gas index return less oil index return
Currency
Primary
US dollar performance versus basket currency
Trend following
Secondary
Passive trend following strategy index return
Note: All factors are returns on tradable index-type
securities
© Eric Zivot 2011
Practical Considerations
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Many potential risk factors ( > 50)
High collinearity among some factors
Risk factors vary across discipline/strategy
Nonlinear effects
Dynamic (lagged) effects
Factor sensitivities change over time
Common histories for factors; unequal
histories for fund performance
© Eric Zivot 2011
Estimation Methodology
• Use prior information to specify small factor
set for specific hedge funds
– Pure data-mining techniques to select factors often
produce nonsensical results
• Estimate linear factor model by least squares
– Exponentially weight data to account for time
varying coefficients
• Use proxy factor models for hedge funds with
insufficient history
© Eric Zivot 2011
Factor Model Fit for Example Fund
-0.02
-0.04
-0.06
-0.08
Monthly performance
0.00
0.02
AG Super Fund, L.P. Class A
Actual
Fitted
2005
2006
2007
Index
© Eric Zivot 2011
2008
2009
Risk Measures
Return Standard Deviation (SD, aka active risk)
  SD( Rt )   βΣ F β  

2 1/2

Value-at-Risk (VaR)
VaR  q  F ( ), 0.01    0.10
1
F  CDF of return Rt
Expected Tail Loss (ETL)
ETL  E[ Rt | Rt  VaR ]
© Eric Zivot 2009
5% ETL
5% VaR
10
Density
15
20
25
Risk Measures
0
5
± SD
-0.15
-0.10
-0.05
Returns
© Eric Zivot 2011
0.00
0.05
Tail Risk Measures: Normal Distribution
Rt ~ N (  ,  2 ),
VaRN      z , z   1 ( ), 0.01    0.10
ETL    
N
1

 ( z ),  ( z ) 
1
e
2
1
 z2
2
Note: Not realistic assumption for hedge fund returns
© Eric Zivot 2009
Tail Risk Measures: Non-Normal
Distributions
• Hedge fund returns are often heavily skewed
with fat tails
• Many possible univariate non-normal
distributions
– Student-t, skewed-t, generalized hyperbolic, GramCharlier, -stable, generalized Pareto, etc.
• Straightforward numerical problem to compute
VaR and ETL for individual funds. However,
not straightforward to compute for portfolio.
© Eric Zivot 2011
Factor Model Monte Carlo
• Use fitted factor model to simulate pseudo
hedge fund return data preserving empirical
characteristics
– Do not assume full parametric distributions for
hedge fund returns and risk factor returns
• Estimate tail risk and related measures nonparametrically from simulated return data
© Eric Zivot 2011
Simulation Algorithm
• Simulate B values of the risk factors by re-sampling
from empirical distribution:
F
*
1
,
, FB* 
• Simulate B values of the factor model residuals from
fitted non-normal distribution:
*
*
ˆ
ˆ

,
,

,n
 i1
iB  , i  1,
• Create factor model returns from fitted factor model
parameters, simulated factor variables and simulated
residuals:
*
*
*
ˆ
ˆ
ˆ

Rit  i  βi Ft   it , t  1, , B; i  1, , n
© Eric Zivot 2011
What to do with R  ?
B
*
it t 1
• Backfill missing fund performance
• Compute fund and portfolio performance
measures (e.g., Sharpe ratios)
• Compute non-parametric Estimates of fund
and portfolio tail risk measures
• Standard errors can be computed using a
bootstrap procedure
© Eric Zivot 2009
Risk Decomposition 1
Given linear factor model for fund or portfolio returns,
Rt    βFt   t    βFt     zt    βFt
β  (β ,   ), Ft  (Ft, zt ), zt ~ (0,1)
SD, VaR and ETL are linearly homogenous functions
of factor sensitivities β . Euler’s theorem gives
additive decomposition
RM (β)
RM (β)    j
, RM  SD, VaR , ETL
 j
j 1
k 1
© Eric Zivot 2011
Risk Decomposition 1
Marginal Contribution to
Risk of factor j:
Contribution to Risk
of factor j:
RM (β)
 j
RM (β)
j
 j
Percent Contribution to Risk  RM (β) RM (β)
j
 j
of factor j:
© Eric Zivot 2011
Risk Decomposition 1
For RM = SD, analytic results are available
   βΣ F β 
1/2
ΣF
 1
 Σ F β, Σ F  
β 
 0
0
 2 
Percent contribution of factor j to SD
1

2
 
1
j cov( F1t , F jt ) 
  j2 var( Fjt ) 
© Eric Zivot 2011
  k  j cov( Fkt , Fjt ) 
Risk Decomposition 1
For RM = VaR, ETL it can be shown that
VaR (β)
 E[ Fjt | Rt  VaR ], j  1, , k  1
 j
ETL (β)
 E[ Fjt | Rt  VaR ], j  1, , k  1
 j
Notes:
1. Intuitive interpretations as stress loss scenarios
2. Analytic results are available under normality
© Eric Zivot 2011
Risk Decomposition 1
Factor Model Monte Carlo semi-parametric
estimates
B
1
Eˆ [ Fjt | Rt  VaR ]   Fjt* 1VaR    Rt*  VaR   
m t 1
B
1
*
*
Eˆ [ Fjt | Rt  VaR ] 
F

1
R


jt
t  VaR 
[ B ] t 1
© Eric Zivot 2011
Risk Decomposition 2
Given portfolio returns,
n
R p ,t  wR t   wi Rit
i 1
SD, VaR and ETL are linearly homogenous functions
of portfolio weights w. Euler’s theorem gives
additive decomposition
RM (w)
RM (w)   wi
, RM  SD, VaR , ETL
wi
i 1
n
© Eric Zivot 2011
Risk Decomposition 2
Marginal Contribution to
Risk of fund i:
RM ( w )
wi
RM (w )
wi
wi
Contribution to Risk
of fund i:
Percent Contribution to Risk
of fund i:
RM ( w )
wi
wi
© Eric Zivot 2011
RM ( w )
Risk Decomposition 2
For RM = SD, analytic results are available
 p   wBΣ F Bw  wD w 
1/ 2
 p
w

1
p
 BΣ F Bw  D w 
© Eric Zivot 2011
Risk Decomposition 2
For RM = VaR, ETL it can be shown that
VaR (w )
 E[ Rit | R p ,t  VaR (w )], i  1, , n
wi
ETL (w )
 E[ Rit | R p ,t  VaR (w )], i  1, , n
wi
Note: Analytic results are available under normality
© Eric Zivot 2011
Risk Decomposition 2
Factor Model Monte Carlo semi-parametric
estimates
B
1
Eˆ [ Rit | R p ,t  VaR (w )]   Rit* 1VaR (w )    R*p ,t  VaR (w )   
m t 1
B
1
*
*
Eˆ [ Rit | Rt  VaR (w )] 
R

1
R


it
p ,t  VaR ( w )
[ B ] t 1
© Eric Zivot 2011
Example FoHF Portfolio Analysis
• Portfolio of 50 funds; largest allocation is 7.7%
• Diverse across strategy and geography
• Relatively larger allocation to relative value,
event driven and credit-based strategies
• No directional trading allocation
• R2 of factor model for portfolio ≈ 90%, Average
R2 of factor models for individual hedge funds ≈
50-60%
© Eric Zivot 2011
Portfolio Factor Risk Budgeting
Factor Risk Budgeting
Factor Group CTR Vol
%CTR Vol CTR ETL
%CTR ETL
Equity
0.4%
19.6%
1.2%
19.0%
Rates
0.0%
-0.4%
0.0%
0.0%
Credit
0.9%
47.1%
2.3%
36.5%
Currency
0.0%
-0.1%
0.0%
-0.3%
Commodity
0.0%
0.1%
0.0%
0.0%
Strategy
0.5%
28.2%
2.7%
42.2%
Residual
0.1%
5.5%
0.2%
2.6%
Portfolio
1.8%
100.0%
6.3%
100.0%
© Eric Zivot 2011
Portfolio Fund Risk Budgeting
Fund
Fund 5
Fund 8
Fund 22
Fund 40
Fund 33
Fund 6
Fund 17
Fund 9
All Others
Portfolio
Fund Risk Budgeting
Allocation CTR Vol
%CTR Vol CTR ETL %CTR ETL
7.7%
0.31%
16.9%
1.18%
18.7%
5.2%
0.13%
7.1%
0.49%
7.7%
3.3%
0.12%
6.4%
0.35%
5.5%
2.8%
0.10%
5.7%
0.30%
4.7%
2.8%
0.09%
4.7%
0.29%
4.6%
3.6%
0.09%
4.7%
0.27%
4.3%
2.7%
0.08%
4.3%
0.25%
3.9%
3.6%
0.07%
3.6%
0.23%
3.7%
68.3%
0.85%
46.6%
2.95%
46.9%
100%
1.8%
100%
6.3%
100%
© Eric Zivot 2011
Summary and Conclusions
• Factor models for asset returns are widely used
in academic research and industry practice
• Risk measurement and management of hedge
funds poses unique challenges
• Factor model Monte Carlo is a simple and
powerful technique for computing individual
fund and portfolio risk measures without
making strong and unrealistic distributional
assumptions.
© Eric Zivot 2011