Value-at-Risk and Expected Tail Loss: A Comprehensive Guide to Tail Risk Measures, Lecture notes of Investment Management and Portfolio Theory

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Factor Model Risk Analysis

Eric Zivot

University of Washington

BlackRock Alternative Advisors

March 11, 2011

Outline

• Risk measures
• Factor Risk Budgeting
• Portfolio Risk Budgeting
• Factor Model Monte Carlo

Risk Measures

Let R t

be an iid random variable, representing the return on an asset at time

t, with pdf f , cdf F, E[R t

] = μ and var(R t

) = σ

2

.

The most common risk measures associated with R t

are

1. Return standard deviation: σ = SD(R t

q

var(R t

2. Value-at-Risk: V aRα = qα = F

− 1

(α), α ∈ (0. 01 , 0 .10)

3. Expected tail loss: ET L α

= E[R

t

|R

t

≤ V aR α

], α ∈ (0. 01 , 0 .10)

Note: V aR α

and ET L α

are tail-risk measures.

Risk Measures: Factor Model and Normal Distribution

R

t

= α + β

0

f t

• ε t

f t

∼ iid N (μ f

f

), ε t

∼ iid N (0, σ

2

ε

), cov(f k,t

, ε s

) = 0 for all k, t, s

Then

E[R

t

] = μ F M

= α + β

0

μ f

var(R t

) = σ

2

F M

= β

0

Ω f

β + σ

2

ε

σ F M

q

β

0

Ω f

β + σ

2

ε

V aR

N,F M

α

= μ F M

• σ F M

× z α

ET L

N,F M

α

= μ F M

− σ F M

α

φ(z α

Note: In practice, α = 0 is typically imposed so that μ F M

= β

0

μ f

Long-Dated and Short-Dated Estimated Risk Measures

Given estimates μˆ F M

= ˆα +

β

0

μˆ f

and ˆσ

2

F M

β

0

ˆ Ω f

β + ˆσ

2

ε

d V aR

N,F M

α

= ˆμ F M

• ˆσ F M

× zα

d ET L

N,F M

α

= ˆμ F M

− σˆ F M

α

φ(zα)

Long-dated estimates

f

, σˆ

2

ε

based on equally weighted full sample

Short-dated estimates

f

, σˆ

2

ε

based on exponentially weighted sample

EWMA Covariance Matrix Estimate

RiskMetrics

TM pioneered the exponentially weighted moving average (EWMA)

covariance matrix estimate

f,t

= (1 − λ)

∞ X

s=

λ

sˇ

f t−s+

= (1 − λ)

f t− 1

f

0

t− 1

• λ

f,t− 1

f t

= f t

f,

f = T

− 1

T X

t=

f t

0 < λ < 1

Given λ, the half-life h is the time lag at which the exponential decay is cut in

half:

λ

h

= 0. 5 ⇒ h = ln(0.5)/ ln(λ)

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 Student’s t (fat-tailed and asymmetric)
• Asymmetric Tempered Stable
• Generalized hyperbolic
• Cornish-Fisher Approximations

Tail Risk Measures: Non-parametric estimates

Assume R t

is iid but make no distributional assumptions:

{R

1

,... , R

T

} = observed iid sample

Estimate risk measures using sample statistics (aka historical simulation)

d V aR

HS

α

= qˆ α

= empirical α − quantile

d ET L

HS

α

[T α]

T X

t=

R

t

· 1 {R

t

≤ qˆ α

1 {R

t

≤ qˆ α

} = 1 if R t

≤ qˆ α

; 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

Factor Risk Decompositions

Assume asset or portfolio return R t

can be explained by a factor model

R

t

= α + β

0

f t

• ε t

f t

∼ iid (μ f

f

), ε t

∼ iid (0, σ

2

ε

), cov(f k,t

, εs) = 0 for all k, t, s

Re-write the factor model as

R

t

= α + β

0

f t

• ε t

= α + β

0

f t

• σ (^) ε × z t

= α +

β

0

˜ f t

β = (β

0

, σ (^) ε)

0

,

f t

= (f t

, z t

0

, z t

ε t

σ (^) ε

∼ iid (0, 1)

Then

σ

2

F M

β

0

˜ f

β, Ω ˜ f

Ã

f

!

Linearly Homogenous Risk Functions

Let RM(

β) denote the risk measures σ F M

, V aR

F M

α

and ET L

F M

α

as func-

tions of

β

Result 1: RM(

β) is a linearly homogenous function of

β for RM = σ F M

V aR

F M

α

and ET L

F M

α

. That is, RM(c ·

β) =c · RM(

β) for any constant

c ≥ 0

Example: Consider RM(

β) = σ F M

β). Then

σ F M

(c ·

β) =

³

c ·

β

0

˜ f

c ·

β

´ 1 / 2

= c ·

³

β

0

˜ f

β

´ 1 / 2

Analytic Results for RM(

β) = σ F M

β)

σ F M

β) =

³

β

0

˜ f

β

´ 1 / 2

∂σ F M

β)

β

σ F M

β)

˜ f

β

Factor j = 1,... , K percent contribution to σ F M

β)

β 1

β j

cov(f 1 t

, f jt

) + · · · + β

2

j

var(f jt

) + · · · + β K

β j

cov(f Kt

, f jt

σ

2

F M

β)

Asset specific factor contribution to risk

σ

2

ε

σ

2

F M

β)

, j = K + 1

Results for RM(

β) = V aR

F M

α

β), ET L

F M

α

β)

Based on arguments in Scaillet (2002), Meucci (2007) showed that

∂V aR

F M

α

β)

β j

= E[ ˜f jt

|R

t

= V aR

F M

α

β)], j = 1,... , K + 1

∂ET L

F M

α

β)

β j

= E[ ˜f jt

|R

t

≤ V aR

F M

α

β)], j = 1,... , K + 1

Remarks

• Intuitive interpretation as stress loss scenario
• Analytic results are available under normality

Marginal Contributions to Tail Risk: Non-Parametric Estimates

Assume R t

and

f t

are iid but make no distributional assumptions:

{(R

1

f 1

),... , (R

T

f T

)} = observed iid sample

Estimate marginal contributions to risk using historical simulation

E

HS

[ ˜f jt

|R

t

= V aR α

] =

m

T X

t=

f jt

½

d V aR

HS

α

− ε ≤ R t

d V aR

HS

α

• ε

¾

E

HS

[ ˜f jt

|R

t

≤ V aRα] =

[T α]

T X

t=

f jt

½

d V aR

HS

α

≤ R

t

¾

Problem: 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

Terminology

Asset i marginal contribution to risk

∂RM(w)

∂w i

Asset i contribution to risk

w i

∂RM(w)

∂w i

Asset i percent contribution to risk

w i

∂RM(w)

∂w i

RM(w)

Analytic Results for RM(w) = σ(w)

R

p,t

= w

0

R t

, var(R t

σ(w) =

³

w

0

Ωw

´ 1 / 2

∂σ(w)

∂w

σ(w)

Ωw

Note

Ωw =

⎛

⎜

⎝

cov(R 1 t

, R

p,t

cov(R Nt

, R

p,t

⎞

⎟

⎠ = σ (w)

⎛

⎜

⎝

β 1 ,p

β N,p

⎞

⎟

⎠

β i,p

= cov(R it

, R

p,t

)/σ

2

(w)

Results for RM(w) = V aR α

(w), ET L α

(w)

Gourieroux (2000) et al and Scalliet (2002) showed that

∂V aRα(w)

∂w i

= E[R

it

|R

p,t

= V aR α

(w)], i = 1,... , N

∂ET Lα(w)

∂w i

= E[R

it

|R

p,t

≤ V aR α

(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 R t

is iid but make no distributional as-

sumptions:

{R

t

,... , R

T

} = observed iid sample

R

p,t

= w

0

R t

Estimate marginal contributions to risk using historical simulation

E

HS

[R it

|R

p,t

= V aR α

] =

m

T X

t=

R

it

½

d V aR

HS

α

− ε ≤ R p,t

d V aR

HS

α

• ε

¾

E

HS

[R it

|R

p,t

≤ V aR α

] =

[T α]

T X

t=

R

it

½

d V aR

HS

α

≤ R

p,t

¾

Problem: Very few observations used for estimates

• Estimate tail risk and related measures non-parametrically from simulated

return data

Unequal History

f 1 T

· · · f KT

R

iT

f 1 ,T −T i

· · · f 1 ,T −T i

R

i,T −T i

f 11

· · · f 1 K

• Observe full history for factors {f 1

,... , f T

• Observe partial history for assets (monotone missing data)

{R

i,T −T i

,... , R

iT

i = 1 ,... , n; t = T − T i

+ 1,... , T

Simulation Algorithm

• Estimate factor models for each asset using partial history for assets and

risk factors

R

it

= ˆα i

β

0

i

f t

• ˆε it

, t = T − T i

+ 1,... , T

• Simulate B values of the risk factors by re-sampling with replacement from

full history of risk factors {f 1

,... , f T

{f

∗

1

,... , f

∗

B

• Simulate B values of the factor model residuals from empirical distribution

or fitted non-normal distribution:

{ˆε

∗

i 1

,... , ˆε

∗

iB

• Create pseudo factor model returns from fitted factor model parameters,

simulated factor variables and simulated residuals:

{R

∗

1

,... , R

∗

B

R

∗

it

β

0

i

f

∗

t

• ˆε

∗

it

, t = 1,... , B

Simulating Residuals: Distribution choices

• Empirical
• Normal
• Skewed Student’s 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 (typically

subject 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.

Optimized Portfolios

Suppose that the objective is to form a portfolio by maximizing a generalized

expected return-to-risk (Sharpe) ratio:

max

w

μ p

(w)

RM(w)

μ p

(w) = w

0

μ

RM(w) = linearly homogenous risk measure

The F.O.C.’s of the optimization are (i = 1,... , n)

∂w i

Ã

μ p

(w)

RM(w)

!

RM(w)

∂μ p

(w)

∂w i

μ p

(w)

RM(w)

2

∂RM(w)

∂w i

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

μ

implied

i

(w) =

μ p

(w)

RM(w)

×

∂RM(w)

∂w i

Result: fund i’s implied return is proportional to its marginal contribution to

risk, with the constant of proportionality being the generalized Sharpe ratio of

the portfolio.