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8/6/2019 Factor Model Risk Lecture Handout
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Factor Model Risk Analysis
Eric ZivotUniversity of Washington
BlackRock Alternative Advisors
March 11, 2011
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Outline
Risk measures
Factor Risk Budgeting
Portfolio Risk Budgeting
Factor Model Monte Carlo
Risk Measures
Let Rt be an iid random variable, representing the return on an asset at time
t, with pdf f, cdf F, E[Rt] = and var(Rt) = 2.
The most common risk measures associated with Rt are
1. Return standard deviation: = SD(Rt) = qvar(Rt)2. Value-at-Risk: V aR = q = F1(), (0.01, 0.10)
3. Expected tail loss: ET L = E[Rt|Rt V aR], (0.01, 0.10)
Note: V aR and ET L are tail-risk measures.
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Risk Measures: Normal Distribution
Rt iid N(, 2)
Rt = + Z, Z iid N(0, 1)
= FZ, = fZ
Value-at-Risk
V aRN = + z, z = 1()
Expected tail loss
ET LN = 1
(z)
Estimation
=1
T
TXt=1
Rt, 2 =
1
T 1
TXt=1
(Rt )2, =
q2
dV aR
N = + z
dET LN = 1
(z
)
Note: Standard errors are rarely reported for dV aRN and dET LN , but are easyto compute using delta method or bootstrap.
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Risk Measures: Factor Model and Normal Distribution
Rt = + 0ft + t
ft iid N(f,f), t iid N(0, 2), cov(fk,t, s) = 0 for all k ,t ,s
Then
E[Rt] = F M = + 0f
var(Rt) = 2F M =
0f +
2
F M =q0f +
2
V aRN,FM = F M + F M z
ET LN,FM = F M F M1
(z)
Note: In practice, = 0 is typically imposed so that F M = 0f.
Long-Dated and Short-Dated Estimated Risk Measures
Given estimates F M = + 0f and
2F M =
0f +
2,
dV aRN,FM = F M + F M zdET LN,FM = F M F M1(z)Long-dated estimates
f, 2 based on equally weighted full sample
Short-dated estimates
f, 2 based on exponentially weighted sample
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EWMA Covariance Matrix Estimate
RiskMetricsTM pioneered the exponentially weighted moving average (EWMA)
covariance matrix estimate
f,t = (1 )X
s=0
sfts+1
= (1 )ft1f0t1 + f,t1
ft = ft f, f = T1
TXt=1
ft
0 < < 1
Given , the half-life h is the time lag at which the exponential decay is cut in
half:
h = 0.5 h = ln(0.5)/ ln()
Tail Risk Measures: Non-Normal Distributions
Stylized fact: The empirical distribution of many asset returns exhibit asym-metry and fat tails
Some commonly used non-normal distributions for
Skewed Students t (fat-tailed and asymmetric)
Asymmetric Tempered Stable
Generalized hyperbolic
Cornish-Fisher Approximations
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Tail Risk Measures: Non-parametric estimates
Assume Rt is iid but make no distributional assumptions:
{R1
, . . . , RT} = observed iid sample
Estimate risk measures using sample statistics (aka historical simulation)
dV aRHS = q = empirical quantiledET LHS = 1
[T ]
TXt=1
Rt 1 {Rt q}
1 {Rt q} = 1 if Rt q; 0 otherwise
Factor Risk Budgeting
Additively decompose (slice and dice) individual asset or portfolio return
risk measures into factor contributions
Allow portfolio manager to know sources of factor risk for allocation and
hedging purposes
Allow risk manager to evaluate portfolio from factor risk perspective
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Factor Risk Decompositions
Assume asset or portfolio return Rt can be explained by a factor model
Rt
= + 0ft
+ t
ft iid (f,f), t iid (0, 2), cov(fk,t, s) = 0 for all k ,t ,s
Re-write the factor model as
Rt = + 0ft + t = +
0ft + zt
= + 0ft
= (0, )0, ft = (ft, zt)
0, zt =t
iid (0, 1)
Then
2F M = 0f, f = f 00 1 !
Linearly Homogenous Risk Functions
Let RM() denote the risk measures F M, V aRF M and ET L
F M as func-
tions of
Result 1: RM() is a linearly homogenous function of for RM = F M,
V aRF M and ET LF M . That is, RM(c ) =c RM() for any constant
c 0
Example: Consider RM() = F M(). Then
F M(c ) =
c 0fc
1/2= c
0f
1/2
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Eulers Theorem and Additive Risk Decompositions
Result 2: Because RM() is a linearly homogenous function of, by Eulers
Theorem
RM() =k+1Xj=1
jRM()
j
= 1RM()
1+ + k+1
RM()
k+1
= 1RM()
1+ + k
RM()
k+
RM()
Terminology
Factor j marginal contribution to risk
RM()
j
Factor j contribution to risk
j RM(
)j
Factor j percent contribution to risk
jRM()
j
RM()
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Analytic Results for RM() = F M()
F M() =0
f1/2
F M()
=
1
F M()
f
Factor j = 1, . . . , K percent contribution to F M()
1jcov(f1t, fjt) + + 2
j var(fjt) + + Kjcov(fKt, fjt)
2F M(),
Asset specific factor contribution to risk
2
2F M(), j = K + 1
Results for RM() = V aRF M (), ETLF M ()
Based on arguments in Scaillet (2002), Meucci (2007) showed that
V aRF M ()
j= E[fjt|Rt = V aR
F M ()], j = 1, . . . , K + 1
ETLF M ()
j= E[fjt|Rt V aR
F M ()], j = 1, . . . , K + 1
Remarks
Intuitive interpretation as stress loss scenario
Analytic results are available under normality
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Marginal Contributions to Tail Risk: Non-Parametric Estimates
Assume Rt and ft are iid but make no distributional assumptions:
{(R1
, f1
), . . . , (RT
, fT
)} = observed iid sample
Estimate marginal contributions to risk using historical simulation
EHS[fjt|Rt = V aR] =
1
m
TXt=1
fjt 1 dV aRHS Rt dV aRHS +
EHS[fjt|Rt V aR] =1
[T ]
TXt=1
fjt 1
dV aRHS RtProblem: Not reliable with small samples or with unequal histories for R
t
Portfolio Risk Budgeting
Additively decompose (slice and dice) portfolio risk measures into asset
contributions
Allow portfolio manager to know sources of asset risk for allocation and
hedging purposes
Allow risk manager to evaluate portfolio from asset risk perspective
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Portfolio Risk Decompositions
Portfolio return:
Rt
= (R1t
, . . . , RNt
), wt
= (w1
, . . . , wn
)0
Rp,t = w0Rt =
NXi=1
wiRit
Let RM(w) denote the risk measures , V aR and ET L as functions of
the portfolio weights w.
Result 3: RM(w) is a linearly homogenous function of w for RM = ,
V aR and ET L. That is, RM(c w) =c RM(w) for any constant c 0
Result 4: Because RM(w) is a linearly homogenous function ofw, by Eulers
Theorem
RM(w) =NX
i=1
wiRM(w)
wi
= w1RM(w)
w1+ + wN
RM(w)
wN
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Terminology
Asset i marginal contribution to risk
RM(w)
wi
Asset i contribution to risk
wiRM(w)
wi
Asset i percent contribution to risk
wiRM(w)
wiRM(w)
Analytic Results for RM(w) = (w)
Rp,t = w0Rt, var(Rt) =
(w) =w
0w
1/2(w)
w=
1
(w)w
Note
w = cov(R1t, Rp,t)...
cov(RNt, Rp,t)
= (w) 1,p...
N,p
i,p = cov(Rit, Rp,t)/2 (w)
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Results for RM(w) = V aR(w), ETL(w)
Gourieroux (2000) et al and Scalliet (2002) showed that
V aR(w)
wi = E[Rit|Rp,t = V aR(w)], i = 1, . . . , N
ETL(w)
wi= E[Rit|Rp,t V aR(w)], i = 1, . . . , N
Remarks
Intuitive interpretation as stress loss scenario
Analytic results are available under normality and Cornish-Fisher expansion
Marginal Contributions to Tail Risk: Non-Parametric Estimates
Assume the N 1 vector of returns Rt is iid but make no distributional as-
sumptions:
{Rt, . . . ,RT} = observed iid sample
Rp,t = w0Rt
Estimate marginal contributions to risk using historical simulation
EHS[Rit|Rp,t = V aR] =
1
m
TXt=1
Rit 1 dV aRHS Rp,t dV aRHS +
EHS[Rit|Rp,t V aR] =1
[T ]
TXt=1
Rit 1 dV aRHS Rp,t
Problem: Very few observations used for estimates
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Marginal Contributions to Tail Risk: Cornish-Fisher Expansion
Boudt, Peterson and Croux (2008) derived analytic expressions for
V aR(w)
wi = E[Rit|Rp,t = V aR(w)], i = 1, . . . , N
ETL(w)
wi= E[Rit|Rp,t V aR(w)], i = 1, . . . , N
based on the Cornish-Fisher quantile expansions for each asset.
Results depend on asset specific variance, skewness, kurtosis as well as all
pairwise covariances, co-skewnesses and co-kurtosises
Factor Model Monte Carlo
Problem: Short history and incomplete data limits applicability of historical
simulation, and risk budgeting calculations are extremely difficult for non-
normal distributions
Solution: Factor Model Monte Carlo (FMMC)
Use fitted factor model to simulate pseudo hedge fund return data preserv-
ing empirical characteristics
Use full history for factors and observed history for asset returns
Do not assume full parametric distributions for hedge fund returns and
risk factor returns
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Estimate tail risk and related measures non-parametrically from simulated
return data
Unequal History
f1T fKT RiT... ... ... ...
f1,TTi+1 f1,TTi+1 Ri,TTi+1... ... ...f11 f1K
Observe full history for factors {f
1, . . . ,f
T}
Observe partial history for assets (monotone missing data)
{Ri,TTi+1, . . . , RiT},
i = 1, . . . , n; t = T Ti + 1, . . . , T
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Simulation Algorithm
Estimate factor models for each asset using partial history for assets and
risk factors
Rit = i + 0ift + it, t = T Ti + 1, . . . , T
Simulate B values of the risk factors by re-sampling with replacement from
full history of risk factors {f1, . . . , fT}:
{f1 , . . . , fB}
Simulate B values of the factor model residuals from empirical distribution
orfi
tted non-normal distribution:{i1, . . . ,
iB}
Create pseudo factor model returns from fitted factor model parameters,
simulated factor variables and simulated residuals:
{R1, . . . , RB}
Rit = 0ift +
it, t = 1, . . . , B
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Remarks:
1. Algorithm does not assume normality, but relies on linear factor structure
for distribution of returns given factors.
2. Under normality (for risk factors and residuals), FMMC algorithm reduces
to MLE with monotone missing data.
3. Use of full history of factors is key for improved efficiency over truncated
sample analysis
4. Technical justification is detailed in Jiang (2009).
Simulating Factor Realizations: Choices
Empirical distribution
Filtered historical simulation
use local time-varying factor covariance matrices to standardize factors
prior to re-sampling and then re-transform with covariance matrices
after re-sampling
Multivariate non-normal distributions
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Simulating Residuals: Distribution choices
Empirical
Normal
Skewed Students t
Generalized hyperbolic
Cornish-Fisher
Reverse Optimization, Implied Returns and Tail Risk Budgeting
Standard portfolio optimization begins with a set of expected returns and
risk forecasts.
These inputs are fed into an optimization routine, which then produces
the portfolio weights that maximizes some risk-to-reward ratio (typicallysubject to some constraints).
Reverse optimization, by contrast, begins with a set of portfolio weights
and risk forecasts, and then infers what the implied expected returns must
be to satisfy optimality.
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Optimized Portfolios
Suppose that the objective is to form a portfolio by maximizing a generalized
expected return-to-risk (Sharpe) ratio:
maxw
p(w)
RM(w)
p(w) = w0
RM(w) = linearly homogenous risk measure
The F.O.C.s of the optimization are (i = 1, . . . , n)
0 =
wi
p(w)
RM(w)
!=
1
RM(w)
p(w)
wi
p(w)
RM(w)2RM(w)
wi
Reverse Optimization and Implied Returns
Reverse optimization uses the above optimality condition with fixed portfo-
lio weights to determine the optimal fund expected returns. These optimal
expected returns are called implied returns. The implied returns satisfy
impliedi (w) =
p(w)
RM(w)
RM(w)
wi
Result: fund is implied return is proportional to its marginal contribution torisk, with the constant of proportionality being the generalized Sharpe ratio of
the portfolio.
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How to Use Implied Returns
For a given generalized portfolio Sharpe ratio, impliedi (w) is large if
RM(w)wi
is large.
If the actual or forecast expected return for fund i is less than its implied
return, then one should reduce ones holdings of that asset
If the actual or forecast expected return for fund i is greater than its implied
return, then one should increase ones holdings of that asset
References
[1] Boudt, K., Peterson, B., and Christophe C.. 2008. Estimation and decom-
position of downside risk for portfolios with non-normal returns. 2008. The
Journal of Risk, vol. 11, 79-103.
[2] Epperlein, E. and A. Smillie (2006). Cracking VAR with Kernels, Risk.
[3] Goldberg, L.R., Yayes, M.Y., Menchero, J., and Mitra, I. (2009). Extreme
Risk Analysis, MSCI Barra Research.
[4] Goldberg, L.R., Yayes, M.Y., Menchero, J., and Mitra, I. (2009). Extreme
Risk Management, MSCI Barra Research.
8/6/2019 Factor Model Risk Lecture Handout
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[5] Goodworth, T. and C. Jones (2007). Factor-based, Non-parametric Risk
Measurement Framework for Hedge Funds and Fund-of-Funds, The Eu-
ropean Journal of Finance.
[6] Gourieroux, C., J.P. Laurent, and O. Scaillet (2000). Sensitivity Analysis
of Values at Risk, Journal of Empirical Finance.
[7] Hallerback, J. (2003). Decomposing Portfolio Value-at-Risk: A General
Analysis, The Journal of Risk 5/2.
[8] Jiang, Y. (200). Overcoming Data Challenges in Fund-of-Funds Portfolio
Management. PhD Thesis, Department of Statistics, University of Wash-ington.
[9] Meucci, A. (2007). Risk Contributions from Generic User-Defined Fac-
tors, Risk.
[10] Scaillet, O. Nonparametric estimation and sensitivity analysis of expected
shortfall. Mathematical Finance, 2002, vol. 14, 74-86.
[11] Yamai, Y. and T. Yoshiba (2002). Comparative Analyses of ExpectedShortfall and Value-at-Risk: Their Estimation Error, Decomposition, and
Optimization, Institute for Monetary and Economic Studies, Bank of
Japan.