Financial Econometrics E892 Risk measures

VaR ES Density forecast Coherence Backtesting

Financial Econometrics – E892Risk measures

Mannheim University

28th April 2015

Risk measures E892 - Financial Econometrics


Contents

1 Value-at-Risk

2 Expected Shortfall

3 Density forecasting

4 Coherent risk measures

5 Backtesting financial risk forecasts



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 .



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}.



Example: Conditional VaR for S&P 500 returns

Adopted from Sheppard.



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.



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.



Contents

1 Value-at-Risk







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



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.



Contents

1 Value-at-Risk







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.



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.



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.



Contents

1 Value-at-Risk







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.



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.



Contents

1 Value-at-Risk







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.



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.



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.



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.




Period of lower volatility.





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 .




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


Financial Econometrics E892 Risk measures

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