factor model risk lecture handout

8/6/2019 Factor Model Risk Lecture Handout

1/22

Factor Model Risk Analysis

Eric ZivotUniversity of Washington

BlackRock Alternative Advisors

March 11, 2011


2/22

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.


3/22

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.


4/22

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


5/22

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


6/22


7/22

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


8/22

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


9/22

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()


10/22

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


11/22

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


12/22

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


13/22

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)


14/22

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


15/22

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


16/22

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


17/22

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


18/22

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


19/22

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.


20/22

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.


21/22

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.


22/22

[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.

factor model risk lecture handout

Documents