Post on 19-Apr-2022
VaR ES Density forecast Coherence Backtesting
Financial Econometrics – E892Risk 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, σ2) 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 largestnumber, such that
P(
Rt =Pt − Pt−1
Pt−1< −%VaR
)= α .
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 RiskMetrics: σ2t+1 = (1− λ)R2
t + λσ2t and
VaRt+1|t = −σt+1Φ−1(α).
2 Parametric ARCH models, e.g.
Rt+1 = µ+ σt+1εt+1 , σ2t+1 = ω + γ1σ
2t ε
2t + β1σ
2t , (PM)
with E[εt ] = 0, E[ε2t ] = 1, εtiid∼ F , F known with variance 1 such
that VaRt+1|t = −µ̂− σ̂t+1F−1(α).
3 Weighted historical simulation: F̂ (x) =∑t
i=1 wi1(Ri ≤ x) andsolve VaRt+1|t = maxx{F̂ (x) ≤ α}.
Risk measures 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 = −F−1θ (α). Rely
on parametric estimation of θ, usually Maximum-Likelihood.
Nonparametric (historical simulation): %VaR = −F̂−1(α) withF̂ (x) = T−1∑T
t=1 1(Rt ≤ x) the empirical distribution function.
Beware that in line with the literature we do not use different symbolsfor theoretical known VaR and estimated VaR.
The example for historical S&P 500 data is contained in our sampledata 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 thathave a maximum downside of $100, where the worst 1% cases overa week all result in maximum loss. A second portfolio P2, which hasthe same face value as P1, consists of short futures positions thatallow for an unbounded maximum loss. We can choose P2 such thatits 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 tosome 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 ofinformation 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 theportfolio loss, given an α-VaR exceedance has occurred:
ES = −E[Rt =
Pt − Pt−1
Pt−1
∣∣Rt < −VaR].
For a return density (pdf) f this yields
ES = −∫ qα
−∞
xf (x)
αdx , qα = −α-VaR .
Example: For Rt ∼ N(µ, σ2), we obtain ES = µ+ α−1σϕ(−Φ−1(α)),with ϕ the pdf of N(0,1). For N(0,1):
α 0.5 0.1 0.05 0.025 0.01 0.001VaR 0 1.282 1.645 1.960 2.326 3.090ES 0.798 1.755 2.063 2.338 2.665 3.367
Risk measures E892 - Financial Econometrics
VaR ES Density forecast Coherence Backtesting
Conditional ES & Implementation
R-implementation of above example:
p <−c (0 .5 ,0 .1 ,0 .05 ,0 .025 ,0 .01 ,0 .001 )VaR <− qnorm ( p )ES <− dnorm ( qnorm ( p ) ) / p
The conditional α-Expected Shortfall (ES) is defined by
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
f̂t+1|td= f(µ̂, σ̂2
t+1|t),
and quite clear in ARCH models. If εtiid∼ N(0,1):
Rt+1|Ft ∼ N(µ̂, σ̂2t+1|t ).
The naive 2-step ahead forecast Rt+2|Ft ∼ N(µ̂, σ̂2t+2|t ) is incorrect!
Observe that σ2t+2|t unlike σ2
t+1|t is random. The correct 2-step aheadforecast is Rt+2|Ft ∼
∫∞−∞ ϕ(µ, σ2
t+2|t+1)ϕ(x) dx .
Multi-step density forecasts are usually difficult (often impossible) tocompute.
Risk measures E892 - Financial Econometrics
VaR ES Density forecast Coherence Backtesting
Fan plotsFan plots are a graphical device to illustrate future changes inuncertainty. The plots have been introduced by the Bank of Englandfor inflation outlook (as example below).
They can be used to depict forecasts with confidence or preditionerror 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 newselement of monetary policy: a wavelet based jump detectionapproach” 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. Thefollowing 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 ofthe diversification principle.
A risk measure satisfying the above axioms is called coherent. ES iscoherent. Positive homogeneity could be restricted in practice byliquidity risk.
Risk measures E892 - Financial Econometrics
VaR ES Density forecast Coherence Backtesting
VaR’s problemVaR is coherent for Gaussian losses. In general, however, VaR canfail the subadditivity and be superadditive.Counterexample: Let L,L1,L2 be continuously distributed lossrandom 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
L1 =
{L if L ≤ 1000 if L > 100
, L2 =
{0 if L ≤ 100L 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 mostprominent 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 isforecast (testing window).
Example:
Estimation windowStart End VaR forecast for
01/01/2000 12/31/2000 VaR(01/01/2001)01/02/2000 01/01/2001 VaR(01/02/2001)
......
...04/29/2014 04/28/2015 VaR(04/29/2015)
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 =∑WT
t=1 ηt , with
ηt =
{1 if Rt ≤ −VaRt
0 if Rt > −VaRt.
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
The hypothesis η = (ηt )t=1,...,WT
iid∼ B(α) is a sequence ofi.i.d. Bernoulli trials can be tested with α̂ = v/WT using√
WT(α̂− α)√α(1− α)
weakly−→ N(0,1) .
More prominent is the likelihood ratio test by Kupiec exploiting
LR = 2 logα̂v (1− α̂)WT−v
αv (1− α)WT−vweakly−→ χ2
1 .
Backtesting ES considers NSt = Rt/ESt for respective times; testH0 : 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 dramaticallyunderforecast risk. VRs for 01/30/1998–11/01/2006: EWMA 1.4, MA1.6, 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 andPractice of Forecasting Market Risk with Implementation in R andMatlab.Wiley, ISBN: 9780470669433
Dowd, K., 2002. An Introduction to Market Risk Measurement.Wiley, ISBN: 9780470847480
Kupiec, P. 1995. Techniques for Verifying the Accuracy of RiskManagement Models.Journal of Derivatives, 3, 73-84.
Sheppard, K., 2013. Financial Econometrics Notes. Lecture Notes
Risk measures E892 - Financial Econometrics