Post on 19-Jan-2016
•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
© Eric Zivot 2011
Outline
• 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
3% 1%
6%
6%
10%
5%
4%
28%
6%
5%
26%
Live funds
Convertible ArbitrageDedicated Short BiasEmerging MarketsEquity Market NeutralEvent DrivenFixed Income ArbitrageGlobal MacroLong/Short Equity HedgeManaged FuturesMulti-StrategyFund of Funds
© Eric Zivot 2011
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 fund5Observations 122.0000 107.0000 135.0000 135.0000 135.0000NAs 13.0000 28.0000 0.0000 0.0000 0.0000Minimum -0.0842 -0.3649 -0.0519 -0.1556 -0.2900Quartile 1 -0.0016 -0.0051 0.0020 -0.0017 -0.0021Median 0.0058 0.0046 0.0060 0.0073 0.0049Arithmetic Mean 0.0038 -0.0017 0.0063 0.0059 0.0021Geometric Mean 0.0037 -0.0029 0.0062 0.0055 0.0014Quartile 3 0.0158 0.0129 0.0127 0.0157 0.0127Maximum 0.0311 0.0861 0.0502 0.0762 0.0877Variance 0.0003 0.0020 0.0002 0.0008 0.0013Stdev 0.0176 0.0443 0.0152 0.0275 0.0357Skewness -1.7753 -5.6202 -0.8810 -2.4839 -4.9948Kurtosis 5.2887 40.9681 3.7960 13.8201 35.8623Rho1 0.6060 0.3820 0.3590 0.4400 0.383
Sample: January 1998 – March 2009
© Eric Zivot 2011
Characteristics of Hedge Fund DataX
165939
-0.0
8-0
.04
0.0
0
X29554
-0.3
-0.2
-0.1
0.0
0.1
-0.0
40.0
00.0
20.0
4
X113169
1998 2000 2002 2004 2006 2008
Index
X104314
-0.1
5-0.1
0-0.0
50.0
00.0
5
X153684
-0.0
8-0
.04
0.0
0-0
.06
-0.0
20.0
2
X156145
1998 2000 2002 2004 2006 2008
Index
fundData.z[, QFundIdxLessCash[c(1, 2, 3, 4, 31, 32)]]
© Eric Zivot 2011
Fund Level Linear Factor Model
1 1
2,
,
1, , ; , ,
~ ( , )
~ (0, )
cov( , ) 0 for all , , and
cov( , ) 0 for , and
it i i t ik kt it
i i t it
i
t F F
it i
jt is
it js
R F F
i n t t T
F j i s t
i j s t
β F
F μ Σ
© Eric Zivot 2011
Performance Attribution
][][][ 11 ktiktiiit FEFERE
][][ 11 ktikti FEFE
])[][(][ 11 ktiktiiti FEFERE
Expected return due to systematic “beta” exposure
Expected return due to manager specific “alpha”
© Eric Zivot 2011
Covariance and Correlation
2 2,1 ,
1 1
var( )
( , , )
cor( )
( )
t FM
n
t FM FM FM
FM FM
diag
diag
FR Σ BΣ B D
D
R D Σ D
D Σ
11 1 1t t
n knn k n R α B F εt
cov( , ) var( )it jt i t j i jR R Fβ F β β Σ βNote:
© Eric Zivot 2011
Portfolio Linear Factor Model
,
1 1 1 1
,
p t t t t
n n n n
i it i i i i t i iti i i i
p p t p t
R
w R w w w
w R w α w BF w ε
β F
β F
1
1
( , , ) portfolio weights
1, 0 for 1, ,
n
n
i ii
w w
w w i n
w
© 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 DescriptionsFactor 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
• 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
.08
-0.0
6-0
.04
-0.0
20
.00
0.0
2
Index
Mo
nth
ly p
erf
orm
an
ce
2005 2006 2007 2008 2009
AG Super Fund, L.P. Class A
ActualFitted
© Eric Zivot 2009
Risk Measures
1( ), 0.01 0.10
of return t
VaR q F
F CDF R
Value-at-Risk (VaR)
Expected Tail Loss (ETL)
[ | ]t tETL E R R VaR
Return Standard Deviation (SD, aka active risk)
1/22( )t FSD R β Σ β
© Eric Zivot 2011
Risk Measures
Returns
De
nsity
-0.15 -0.10 -0.05 0.00 0.05
05
10
15
20
25
± SD
5% VaR5% ETL
© Eric Zivot 2009
Tail Risk Measures: Normal Distribution
2
2
1
1
2
~ ( , ),
, ( ), 0.01 0.10
1 1( ), ( )
2
t
N
zN
R N
VaR z z
ETL z z e
Note: Not realistic assumption for hedge fund returns
© Eric Zivot 2011
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, Gram-
Charlier, a-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 non-parametrically from simulated return data
© Eric Zivot 2011
Simulation Algorithm• Simulate B values of the risk factors by re-sampling
from empirical distribution:
• Simulate B values of the factor model residuals from fitted non-normal distribution:
• Create factor model returns from fitted factor model parameters, simulated factor variables and simulated residuals:
1 , , B* *F F
* *1ˆ ˆ, , , 1, ,i iB i n
* * *ˆˆ ˆ , 1, , ; 1, ,it i i t itR t B i n β F
© Eric Zivot 2009
What to do with ?
• 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
*
1
B
it tR
© Eric Zivot 2011
Risk Decomposition 1
Given linear factor model for fund or portfolio returns,
SD, VaR and ETL are linearly homogenous functions of factor sensitivities . Euler’s theorem gives additive decomposition
( , ), ( , ) , ~ (0,1)
t t t t t t
t t t t
R z
z z
β F β F β F
β β F F
1
1
( )( ) , , ,
k
jj j
RMRM RM SD VaR ETL
ββ
β
© Eric Zivot 2011
Risk Decomposition 1
Marginal Contribution to Risk of factor j:
Contribution to Risk of factor j:
Percent Contribution to Risk of factor j:
( )
j
RM
β
( )j
j
RM
β
( )( )j
j
RMRM
β
β
© Eric Zivot 2011
Risk Decomposition 1
For RM = SD, analytic results are available
1/2
2
01,
0
F
F
F F
β Σ β
ΣΣ β Σ
β
Percent contribution of factor j to SD
21 12
1cov( , ) var( ) cov( , )j t jt j jt k j kt jtF F F F F
© Eric Zivot 2011
Risk Decomposition 1
For RM = VaR, ETL it can be shown that
( )[ | ], 1, , 1
( )[ | ], 1, , 1
jt tj
jt tj
VaRE F R VaR j k
ETLE F R VaR j k
β
β
Notes: 1. Intuitive interpretations as stress loss scenarios2. Analytic results are available under normality
© Eric Zivot 2011
Risk Decomposition 1
Factor Model Monte Carlo semi-parametric estimates
* *
1
* *
1
1ˆ[ | ] 1
1ˆ[ | ] 1[ ]
B
jt t jt tt
B
jt t jt tt
E F R VaR F VaR R VaRm
E F R VaR F R VaRB
© Eric Zivot 2011
Risk Decomposition 2
Given portfolio returns,
SD, VaR and ETL are linearly homogenous functions of portfolio weights w. Euler’s theorem gives additive decomposition
,1
n
p t t i iti
R w R
w R
1
( )( ) , , ,
n
ii i
RMRM w RM SD VaR ETL
w
ww
© Eric Zivot 2011
Risk Decomposition 2
Marginal Contribution to Risk of fund i:
Contribution to Risk of fund i:
Percent Contribution to Risk of fund i:
( )
i
RM
w
w
( )i
i
RMw
w
w
( )( )i
i
RMw RM
w
w
w
© Eric Zivot 2011
Risk Decomposition 2
For RM = SD, analytic results are available
1/2
1
p F
pF
p
w BΣ Bw w D w
BΣ Bw D ww
© Eric Zivot 2011
Risk Decomposition 2
For RM = VaR, ETL it can be shown that
,
,
( )[ | ( )], 1, ,
( )[ | ( )], 1, ,
it p ti
it p ti
VaRE R R VaR i n
w
ETLE R R VaR i n
w
ww
ww
Note: Analytic results are available under normality
© Eric Zivot 2011
Risk Decomposition 2
Factor Model Monte Carlo semi-parametric estimates
* *, ,
1
* *,
1
1ˆ[ | ( )] 1 ( ) ( )
1ˆ[ | ( )] 1 ( )[ ]
B
it p t it p tt
B
it t it p tt
E R R VaR R VaR R VaRm
E R R VaR R R VaRB
w w w
w w
© 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 BudgetingFactor Risk Budgeting
Factor Group CTR Vol %CTR Vol CTR ETL %CTR ETLEquity 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 BudgetingFund Risk Budgeting
Fund Allocation CTR Vol %CTR Vol CTR ETL %CTR ETLFund 5 7.7% 0.31% 16.9% 1.18% 18.7%Fund 8 5.2% 0.13% 7.1% 0.49% 7.7%Fund 22 3.3% 0.12% 6.4% 0.35% 5.5%Fund 40 2.8% 0.10% 5.7% 0.30% 4.7%Fund 33 2.8% 0.09% 4.7% 0.29% 4.6%Fund 6 3.6% 0.09% 4.7% 0.27% 4.3%Fund 17 2.7% 0.08% 4.3% 0.25% 3.9%Fund 9 3.6% 0.07% 3.6% 0.23% 3.7%All Others 68.3% 0.85% 46.6% 2.95% 46.9%Portfolio 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.