VaR
ES
Density forecast
Coherence
Backtesting
Financial Econometrics – E892
Risk measures
Mannheim University
28th April 2015
Risk measures
E892 - Financial Econometrics
VaR
ES
Density forecast
Coherence
Backtesting
Contents
1
Value-at-Risk
2
Expected Shortfall
3
Density forecasting
4
Coherent risk measures
5
Backtesting financial risk forecasts
Risk measures
E892 - Financial Econometrics
VaR
ES
Density forecast
Coherence
Backtesting
Contents
1
Value-at-Risk
2
Expected Shortfall
3
Density forecasting
4
Coherent risk measures
5
Backtesting financial risk forecasts
Risk measures
E892 - Financial Econometrics
VaR
ES
Density forecast
Coherence
Backtesting
Value-at-Risk
Denote Pt the value of a portfolio at time t. The α-Value-at-Risk
(α-VaR) is defined as the largest number, such that
P Pt − Pt−1 < −VaR = α .
This means α-VaR is just the (negative) α-quantile.
Example: For Rt ∼ N(µ, σ 2 ) we have α-VaR= Pt−1 (−µ − σΦ−1 (α)),
Φ cdf of N(0, 1).
The α-percentage-Value-at-Risk (α-%VaR) is defined as the largest
number, such that
Pt − Pt−1
P Rt =
< −%VaR = α .
Pt−1
Risk measures
E892 - Financial Econometrics
VaR
ES
Density forecast
Coherence
Backtesting
Conditional VaR
The conditional α-Value-at-Risk is defined by
P(Rt+1 < −VaRt+1|t |Ft ) = α .
Prominent approaches:
1
2
RiskMetrics: σt+1
= (1 − λ)Rt2 + λσt2 and
VaRt+1|t = −σt+1 Φ−1 (α).
2
Parametric ARCH models, e.g.
2
Rt+1 = µ + σt+1 t+1 , σt+1
= ω + γ1 σt2 2t + β1 σt2 ,
(PM)
iid
3
Risk measures
with E[t ] = 0, E[2t ] = 1, t ∼ F , F known with variance 1 such
that VaRt+1|t = −ˆ
µ−σ
ˆt+1 F −1 (α).
Pt
Weighted historical simulation: Fˆ (x) = i=1 wi 1(Ri ≤ x) and
solve VaRt+1|t = maxx {Fˆ (x) ≤ α}.
E892 - Financial Econometrics
VaR
ES
Density forecast
Coherence
Backtesting
Example: Conditional VaR for S&P 500 returns
Adopted from Sheppard.
Risk measures
E892 - Financial Econometrics
VaR
ES
Density forecast
Coherence
Backtesting
Unconditional VaR
Parametric: Rt ∼ Fθ , θ ∈ Θ, such that %VaR = −Fθ−1 (α). Rely
on parametric estimation of θ, usually Maximum-Likelihood.
Nonparametric (historical simulation): %VaR = −Fˆ −1 (α) with
PT
Fˆ (x) = T −1 t=1 1(Rt ≤ x) the empirical distribution function.
Beware that in line with the literature we do not use different symbols
for theoretical known VaR and estimated VaR.
The example for historical S&P 500 data is contained in our sample
data analysis from Topic 1.
Risk measures
E892 - Financial Econometrics
VaR
ES
Density forecast
Coherence
Backtesting
A VaR “paradox”
Consider one portfolio P1 which consists of long option positions that
have a maximum downside of $100, where the worst 1% cases over
a week all result in maximum loss. A second portfolio P2 , which has
the same face value as P1 , consists of short futures positions that
allow for an unbounded maximum loss. We can choose P2 such that
its 1%-VaR is $100 over a week.
In summary, this means
For portfolio P1 , the 1% worst case losses are all equal $100.
For portfolio P2 , the 1% worst case losses range from $100 to
some unknown higher value.
According to 1%-VaR, however, both portfolios bear the same risk!
This illustrates the narrow view of VaR on the riskiness of portfolios.
Example adopted from Danielsson, which is a rich source of
information about risk measures and forecasts.
Risk measures
E892 - Financial Econometrics
VaR
ES
Density forecast
Coherence
Backtesting
Contents
1
Value-at-Risk
2
Expected Shortfall
3
Density forecasting
4
Coherent risk measures
5
Backtesting financial risk forecasts
Risk measures
E892 - Financial Econometrics
VaR
ES
Density forecast
Coherence
Backtesting
Expected Shortfall aka tail VaR
The α-Expected Shortfall (ES) is defined as the expected value of the
portfolio loss, given an α-VaR exceedance has occurred:
h
i
Pt − Pt−1 ES = −E Rt =
Rt < −VaR .
Pt−1
For a return density (pdf) f this yields
Z qα
xf (x)
ES = −
dx , qα = −α-VaR .
α
−∞
Example: For Rt ∼ N(µ, σ 2 ), we obtain ES = µ + α−1 σϕ(−Φ−1 (α)),
with ϕ the pdf of N(0, 1). For N(0, 1):
α
VaR
ES
Risk measures
0.5
0
0.798
0.1
1.282
1.755
0.05
1.645
2.063
0.025
1.960
2.338
0.01
2.326
2.665
0.001
3.090
3.367
E892 - Financial Econometrics
VaR
ES
Density forecast
Coherence
Backtesting
Conditional ES & Implementation
R-implementation of above example:
p <−c ( 0 . 5 , 0 . 1 , 0 . 0 5 , 0 . 0 2 5 , 0 . 0 1 , 0 . 0 0 1 )
VaR <− qnorm ( p )
ES <− dnorm ( qnorm ( p ) ) / p
The conditional α-Expected Shortfall (ES) is defined by
h
i
ESt+1|t = −Et Rt+1 Rt+1 < −VaRt+1|t .
ES is a conditional expectation or exceedance mean.
Risk measures
E892 - Financial Econometrics
VaR
ES
Density forecast
Coherence
Backtesting
Contents
1
Value-at-Risk
2
Expected Shortfall
3
Density forecasting
4
Coherent risk measures
5
Backtesting financial risk forecasts
Risk measures
E892 - Financial Econometrics
VaR
ES
Density forecast
Coherence
Backtesting
Multi-step ahead forecast
Consider the model (PM). The 1-step ahead forecast is
d
2
ˆft+1|t =
,
f µ
ˆ, σ
ˆt+1|t
iid
and quite clear in ARCH models. If t ∼ N(0, 1):
2
).
Rt+1 |Ft ∼ N(ˆ
µ, σ
ˆt+1|t
2
The naive 2-step ahead forecast Rt+2 |Ft ∼ N(ˆ
µ, σ
ˆt+2|t
) is incorrect!
2
2
Observe that σt+2|t unlike σt+1|t is random. The correct 2-step ahead
R∞
2
forecast is Rt+2 |Ft ∼ −∞ ϕ(µ, σt+2|t+1
)ϕ(x) dx.
Multi-step density forecasts are usually difficult (often impossible) to
compute.
Risk measures
E892 - Financial Econometrics
VaR
ES
Density forecast
Coherence
Backtesting
Fan plots
Fan plots are a graphical device to illustrate future changes in
uncertainty. The plots have been introduced by the Bank of England
for inflation outlook (as example below).
They can be used to depict forecasts with confidence or predition
error intervals.
Risk measures
E892 - Financial Econometrics
VaR
ES
Density forecast
Coherence
Backtesting
Fan plots and a historical backtest
Taken from “The Norges Bank’s key rate projections and the news
element of monetary policy: a wavelet based jump detection
approach” by Lars Winkelmann.
Risk measures
E892 - Financial Econometrics
VaR
ES
Density forecast
Coherence
Backtesting
Contents
1
Value-at-Risk
2
Expected Shortfall
3
Density forecasting
4
Coherent risk measures
5
Backtesting financial risk forecasts
Risk measures
E892 - Financial Econometrics
VaR
ES
Density forecast
Coherence
Backtesting
Coherence
Let ρ be a generic risk measure and P, P1 , P2 portfolios. The
following properties are desired for risk measures to apply:
Translation invariance: ρ(P + c) = ρ(P) − c.
Positive homogeneity: ρ(λP) = λρ(P) for any λ > 0.
Monotonicity: If P1 first-order stochstically dominates P2 :
ρ(P1 ) ≤ ρ(P2 ).
Subadditivity: ρ(P1 + P2 ) ≤ ρ(P1 ) + ρ(P2 ) as a manifestation of
the diversification principle.
A risk measure satisfying the above axioms is called coherent. ES is
coherent. Positive homogeneity could be restricted in practice by
liquidity risk.
Risk measures
E892 - Financial Econometrics
VaR
ES
Density forecast
Coherence
Backtesting
VaR’s problem
VaR is coherent for Gaussian losses. In general, however, VaR can
fail the subadditivity and be superadditive.
Counterexample: Let L, L1 , L2 be continuously distributed loss
random variables with cdfs FL , FL1 , FL2 . Assume
FL (1) = 0.91, FL (90) = 0.95, FL (100) = 0.96 ,
such that .95-VaR(L) = 90. Now if L = L1 + L2 and
(
(
L if L ≤ 100
0 if L ≤ 100
L1 =
, L2 =
,
0 if L > 100
L if L > 100
we derive FL1 (1) = 0.91/0.96, FL1 (90) = 0.95/0.96, FL1 (100) = 1,
FL2 (0) = 0.96, such that
.95-VaR(L1 ) + .95-VaR(L2 ) = 1 < .95-VaR(L) .
Still VaR fails subadditivity only for very fat tails and remains the most
prominent risk measure.
Risk measures
E892 - Financial Econometrics
VaR
ES
Density forecast
Coherence
Backtesting
Contents
1
Value-at-Risk
2
Expected Shortfall
3
Density forecasting
4
Coherent risk measures
5
Backtesting financial risk forecasts
Risk measures
E892 - Financial Econometrics
VaR
ES
Density forecast
Coherence
Backtesting
Estimation and testing windows
WE denotes the number of observations used for forecast
(estimation window).
WT denotes the size of the data sample over which risk is
forecast (testing window).
Example:
Estimation window
Start
End
01/01/2000 12/31/2000
01/02/2000 01/01/2001
..
..
.
.
VaR forecast for
VaR(01/01/2001)
VaR(01/02/2001)
..
.
04/29/2014
VaR(04/29/2015)
04/28/2015
Compare forecasts to actual outcomes.
Risk measures
E892 - Financial Econometrics
VaR
ES
Density forecast
Coherence
Backtesting
Violation ratio
Record the number of VaR violations: v =
(
1 if Rt ≤ −VaRt
ηt =
.
0 if Rt > −VaRt
PWT
t=1
ηt , with
The violation ratio is VR = v /(αWT ).
If VR > 1, the model underforecasts risk.
If VR < 1, the model overforecasts risk.
Risk measures
E892 - Financial Econometrics
VaR
ES
Density forecast
Coherence
Backtesting
Coverage tests
iid
The hypothesis η = (ηt )t=1,...,WT ∼ B(α) is a sequence of
i.i.d. Bernoulli trials can be tested with α
ˆ = v /WT using
p
(ˆ
α − α) weakly
−→ N(0, 1) .
WT p
α(1 − α)
More prominent is the likelihood ratio test by Kupiec exploiting
LR = 2 log
α
ˆ v (1 − α
ˆ )WT −v weakly 2
−→ χ1 .
αv (1 − α)WT −v
Backtesting ES considers NSt = Rt /ESt for respective times; test
H0 : NS = 1.
Risk measures
E892 - Financial Econometrics
VaR
ES
Density forecast
Coherence
Backtesting
Example: VaR backtest for S&P 500
Danielsson performs a backtest study of S&P 500, 02/1994-12/2009,
using 4000 daily observations, α = 0.01 and WE = 1000.
He considers four approaches (EWMA, MA, HS, GARCH).
Adopted from Danielsson.
Risk measures
E892 - Financial Econometrics
VaR
ES
Density forecast
Coherence
Backtesting
Example: VaR backtest for S&P 500
Period of lower volatility.
Adopted from Danielsson.
Risk measures
E892 - Financial Econometrics
VaR
ES
Density forecast
Coherence
Backtesting
Example: VaR backtest for S&P 500
Crisis period. With the crisis period all approaches dramatically
underforecast risk. VRs for 01/30/1998–11/01/2006: EWMA 1.4, MA
1.6, HS 1.05, GARCH 1.25 .
Adopted from Danielsson.
Risk measures
E892 - Financial Econometrics
VaR
ES
Density forecast
Coherence
Backtesting
Literature
Danielsson, J., 2011. Financial Risk Forecasting: The Theory and
Practice of Forecasting Market Risk with Implementation in R and
Matlab.
Wiley, ISBN: 9780470669433
Dowd, K., 2002. An Introduction to Market Risk Measurement.
Wiley, ISBN: 9780470847480
Kupiec, P. 1995. Techniques for Verifying the Accuracy of Risk
Management Models.
Journal of Derivatives, 3, 73-84.
Sheppard, K., 2013. Financial Econometrics Notes. Lecture Notes
Risk measures
E892 - Financial Econometrics