[PDF] - Overview and Compa risons of Long-T erm Finanial Risk Mo PDF Document

SLIDE 1 Overview and Compa risons

f

Long-T erm Finan ial Risk Mo dels Roger Kaufmann, Pierre P atie RiskLab ETH Z uri h O tob er 20, 2000 http:/ /www.risklab. h/Proje ts.html#SL TFR http:/ /www.math.ethz. h/k aufmann http:/ /www.math.ethz. h/patie Overview and Compa risons

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Long-T erm Finan ial Risk Mo dels R. Kaufmann, P . P atie I Intro du tion I I Mo del Des ription I I I Ba ktesting Idea IV Exp e ted Sho rtfall Estimate V Ba ktesting Results VI Con lusions 2000 (R. Kaufmann and P . P atie, RiskLab) 1 I Intro du tion

Aim
f

the p roje t

Key

questions

Risk

measures 2000 (R. Kaufmann and P . P atie, RiskLab) 2 Aim

f

the p roje t

Measurement
f

long-term nan ial risk

f

investment p

rtfolios.

First steps:

Mo

delling the sto hasti evolution

f

risk fa to rs asso iated to p

rtfolio

p

sitions.
T

est the go

dness
f

su h mo dels fo r a long time ho rizon (e.g. 1 y ea r). 2000 (R. Kaufmann and P . P atie, RiskLab) 3

SLIDE 2 Risk measure denition W e

nsider

as risk measure the exp e ted sho rtfall. Denition 1 The exp e ted sho rtfall ES

at

a level

is

dened b y ES

(R

) = E[R jR < V aR

(R

)℄; where R 2 L 1 (): Denition 2 Given

2

℄0; 1[, the value-at-risk V aR

at

level

f

the returns R with distribution P, is V aR

(R

) =

inf

fx 2 R j P[R

x℄
g;

i.e. V aR is the negative

f

the

quantile
f

R . The exp e ted sho rtfall is a

herent

risk measure in the sense

f

Artzner, Delbaen, Eb er and Heath. In general, value-at-risk is not ! 2000 (R. Kaufmann and P . P atie, RiskLab) 4

yearly returns in % P&L distribution

60
40
20

20 40 60 VaR(5%) ES(5%)

2000 (R. Kaufmann and P . P atie, RiskLab) 5 Key questions 1. Whi h frequen y do w e use to t mo dels?

Are

long datasets stationa ry?

What

a re the statisti al restri tions? (la k

f

y ea rly returns)

Ho

w an w e k eep as mu h info rmation as p

ssible?

2. Do the p rop erties

f

nan ial data hange when w e ho

se

another time ho rizon? 3. What is the reliabilit y

f

the time aggregation rule

f

ea h mo del if there is any? 4. Ho w an w e

mpa

re dierent time ho rizons and mo dels? 2000 (R. Kaufmann and P . P atie, RiskLab) 6 I I Mo del Des ription

Random

W alk

GARCH(1,1)
Heavy-tailed

distribution 2000 (R. Kaufmann and P . P atie, RiskLab) 7

SLIDE 3 Random W alk Assumption 1: exp e ted log returns a re equal to zero E t [r t+1 ℄ = 0: Assumption 2: No rmally distributed, inde- p endent log returns with standa rd deviation

in

ea h p erio d [t; t + 1℄. Ass. 1 & 2 ) r t+1 iid s N (0;

2

): ! The loga rithmi asset p ri e follo ws a random w alk with zero drift. Time aggregation: r h;t := h1 X i=0 r ti iid s N (0; h 2 ): 2000 (R. Kaufmann and P . P atie, RiskLab) 8 GARCH(1,1) Let (X t ; t 2 N) b e a stri tly stationa ry time series rep resenting

bservations
f

entered log returns

n

a nan ial asset p ri e. A GARCH(1,1) mo del fo r X is dened b y X t =

t
t

fo r t 2 N;

2

t =

+
1

X 2 t1 +

1
2

t1 ;

t

iid; E[ t ℄ = 0; E[ 2 t ℄ = 1: Stationa rit y

nditions:

<

<

1,

1
0,
1
and
1

+

1

< 1: Fit the GARCH(1,1) p ro ess b y pseudo-maximum- lik eliho

d

estimation to

btain

the value

f

the pa rameters

f

the

nditional

volatilit y . 2000 (R. Kaufmann and P . P atie, RiskLab) 9 Time aggregation: GARCH

eÆ ients

fun - tion Assume: Centered 1-da y log returns X t follo w a GARCH(1,1) p ro ess with a no rmally distributed innovation. X t =

t
t

;

2

t =

+
1

X 2 t1 +

1
2

t1 ;

t

iid s N (0; 1): Drost-Nijman: X h;t : = h1 X i=0 X ti is w eak GARCH(1,1): X h;t =

h;t
h;t

;

2

h;t =

h;0

+

h;1

X 2 h;t1 +

h;1
2

h;t1 ;

h;t

iid s N (0; 1);

h;1

! 0;

h;1

! as h ! 1: 2000 (R. Kaufmann and P . P atie, RiskLab) 10 Heavy-tailed distributions W e

nsider

(r t ; t 2 N) to b e indep endent and identi ally distributed (i.i.d.), rep resenting

b-

servations

f

the log returns

n

a nan ial asset p ri e. W e assume P[r 1 < x℄ = C x

L(x)

as x ! 1; (1) where C ;

2

R + and L is a slo wly va rying fun tion, i.e. 8t > : lim x!1 L(tx) L(x) = 1: Distributions satisfying (1) a re alled heavy- tailed distributions sin e the k th moment is innite fo r k > . (1) is also a ha ra terisation

f

the maximum domain

f

attra tion

f

the F r

e het

distribution. 2000 (R. Kaufmann and P . P atie, RiskLab) 11

SLIDE 4 Time aggregation F eller's theo rem (1971) Theo rem: Assuming that (r t ; t 2 N) have heavy- tailed distributions leads to P 2 4 h X t=1 r t < x 3 5 = hC L(x)x

[1

+

(1)℄;

fo r x ! 1; where the s ale fa to r C is as in (1) . P h t=1 r t

rresp
nds

to the h-da y log returns. When appli able, this theo rem supplements the entral limit theo rem b y p roviding info rmation

n erning

the tails. Da o rogna, M uller, Pi tet and de V ries sho w the follo wing theo rem: Supp

se

r has a nite va rian e (i.e.

>

2). A t a

nstant

risk level p, in reasing the time ho rizon h in reases the V aR and the exp e ted sho rtfall numb ers fo r the heavy tailed mo del b y a fa to r h 1

.

2000 (R. Kaufmann and P . P atie, RiskLab) 12 I I I Ba ktesting Idea

Mo

del

mpa

rison

Ba ktesting

des ription 2000 (R. Kaufmann and P . P atie, RiskLab) 13 Mo del Compa rison No

ne
f

the p rop

sed

mo dels

bviously
ut-

p erfo rms the

thers.

Ea h

f

them has its de ien ies. All mo dels

nly

p erfo rm w ell fo r relatively sho rt time ho rizons. W e have to x a ho rizon h < 1 y ea r, fo r whi h w e an use

ur

mo dels. F

r

the gap b et w een h and 1 y ea r, w e will have to use a s aling rule.

suitable model scaling rule today h 1 year

2000 (R. Kaufmann and P . P atie, RiskLab) 14 Ba ktesting: T est Des ription Idea:

mpa

re fo re asted exp e ted sho rtfall d ES t; with empiri al estimation

f

exp e ted sho rtfall. Measure 1: Evaluate values b elo w the negative

f

the estimated value-at-risk d V aR t; . W e build the dieren e b et w een the real (i.e.

b-

served)

ne-y

ea r returns R t+1 and the negative

f

the estimation d ES t; . W e al ulate the

nditional

average

f

these dieren es,

nditioned
n

fR t+1 <

d

V aR t; g, V ES 1 = P t 1 t=t

R

t+1

(

d ES t; )

1

fR t+1 < d V aR t; g P t 1 t=t 1 fR t+1 < d V aR t; g : A go

d

estimation fo r exp e ted sho rtfall will lead to a lo w absolute value

f

V ES 1 . 2000 (R. Kaufmann and P . P atie, RiskLab) 15

SLIDE 5 Measure 2: Evaluate values b elo w the \1 in 1 /

event"

(fo r

=

1%:

ne

in hundred event). W e build the dieren e b et w een the real (i.e.

b-

served)

ne-y

ea r return R t+1 and the negative

f

the estimation d ES t; : D t := R t+1

(

d ES t; ): W e al ulate the

nditional

average

f

these dieren es,

nditioned
n

fD t < D

g,

V ES 2 = P t 1 t=t D t 1 fD t <D

g

P t 1 t=t 1 fD t <D

g

; where D

denotes

the empiri al

quantile
f

fD t g t tt 1 . A go

d

estimation fo r exp e ted sho rtfall will lead to a lo w absolute value

f

V ES 2 . 2000 (R. Kaufmann and P . P atie, RiskLab) 16 W e evaluate t w

additional

measures that p ro- vide info rmation ab

ut

the go

dness
f
ur

estimato rs:

F

requen y

f

ex eedan e Cal ulate the p er entage

f

times the

b-

served returns fall b elo w the negative V aR estimate: V freq = 1 t 1

t

+ 1 t 1 X t=t 1 fR t+1 <

d

V aR t; g :

Relative

amount

f

ex eedan e Relative size

f

values ex eeding V aR: V size = P t 1 t=t R t+1

(

d V aR t; )

d

V aR t; 1 fR t+1 <

d

V aR t; g P t 1 t=t 1 fR t+1 <

d

V aR t; g : 2000 (R. Kaufmann and P . P atie, RiskLab) 17 Overview and Compa risons

f

Long-T erm Finan ial Risk Mo dels R. Kaufmann, P . P atie I Intro du tion I I Mo del Des ription I I I Ba ktesting Idea IV Exp e ted Sho rtfall Estimate V Ba ktesting Results VI Con lusions 2000 (R. Kaufmann and P . P atie, RiskLab) 18 No rmal Distribution W e assume fo r the h-da y log returns: r h iid s N (0;

2

h ): W e estimate the

ne

y ea r exp e ted sho rtfall at a level p b y: d ES p t (r ) = s 261 h ^

t;h

'(x p ) p ; where: x p : pquantile

f

a standa rd no rmal random va riable, ' : densit y

f

a standa rd no rmal random va riable, ^

2

t;h = 1 N 1 P N 1 i=0 r 2 ti;h ; the h-da y sample va rian e. 2000 (R. Kaufmann and P . P atie, RiskLab) 19

SLIDE 6 GARCH(1,1) A GARCH(1,1) mo del with no rmally distributed innovation p ro ess fo r the h-da y log returns r h is dened b y r t;h =

t;h
t

fo r t 2 N;

2

t;h =

0;h

+

1;h

r 2 t1;h +

1;h
2

t1;h ; where

t

iid s N (0; 1): W e estimate the 1 y ea r exp e ted sho rtfall at a level p in 4 steps: 1. w e t the GARCH(1,1) p ro ess with the h-da y log returns, 2. w e apply the Drost-Nijman rule (s aling rule) to get the pa rameters

f

the y ea rly

nditional

volatilit y , 3. w e fo re ast the y ea rly volatilit y using the follo wing re ursive relation: ^

2

t+1;y = ^

0;y

+ ^

1;y

r 2 t;y + ^

1;y

^

2

t;y ; sta rting with ^

2

1;y = 261 N 1 P N i=1 (r i

r

) 2 , 4. d ES p t (r ) = ^

t+1;y

'(x p ) p : 2000 (R. Kaufmann and P . P atie, RiskLab) 20 Heavy T ailed Distribution W e assume the h-da y log returns to b e inde- p endent, further P[r 1 < x℄ = C x

L(x)

as x ! 1; (2) where C ;

2

R + and L is a slo wly va rying fun tion. By inverting (2) and using the s aling rule fo r heavy tailed distributions (F eller's theo rem) w e an easily derive estimates fo r the

ne

y ea r exp e ted sho rtfall at a level p: d E S p t =

261

h

1

^

k

;N ^

k

;N ^

k

;N

1

! ^ x p k ;N ; ^

k

;N > 1; where b x p k ;N = r k ;N

k

p N

1

^

k

;N b

1

k ;N = 1 k P k i=1 log

r

i;N r k ;N

(Hill

estimato r), N is the sample size, r k ;N is the k th

rder

statisti s. 2000 (R. Kaufmann and P . P atie, RiskLab) 21 Ba ktesting in A tion Problem: Not enough y ea rly data fo r estimat- ing mo del pa rameters and p ro eeding the ba k- testing! Solution: W e use 22 sto k samples (German sto ks), ea h

ntaining

23.5 y ea rs

f

data, w e a rry

ut

the ba ktesting

n

ea h sample indep endently , then w e aggregate the results. F

r

ea h sample w e p ro eed as follo ws: 1. fo r ea h mo del w e estimate the y ea rly fo re- asted exp e ted sho rtfall d ES p t

n

a windo w

f

size N (e.g. N=2000 daily data). W e also use non-overlapping data fo r lo w er frequen y , 2. w e

mpa

re the estimates with the follo w- ing returns R t+1 using dierent measures, 3. w e move the windo w b y

ne,

then w e re- p eat steps 1 and 2 up to the end

f

the whole dataset. 2000 (R. Kaufmann and P . P atie, RiskLab) 22 Ba ktesting Measures Measure 1: Evaluate values b elo w the negative

f

the estimated value-at-risk d V aR p t : V ES 1 = P t 1 t=t

R

t+1 ( ES p t )

1

fR t+1 < d V aR p t g P t 1 t=t 1 fR t+1 < d V aR p t g : Measure 2: Evaluate values b elo w the \1 in 1 /

event"

(fo r

=

1%:

ne

in hundred event): D t := R t+1

(

d ES p t ); V ES 2 = P t 1 t=t D t 1 fD t <D

g

P t 1 t=t 1 fD t <D

g

; where D

is

the

quantile
f

fD t g t tt 1 . F requen y

f

ex eedan e: V freq =

1

t 1 t +1 P t 1 t=t 1 fR t+1 <

d

V aR p t g

:

Relative amount

f

ex eedan e: V size = P t 1 t=t R t+1

(

d V aR p t )

d

V aR p t 1 fR t+1 <

d

V aR p t g P t 1 t=t 1 fR t+1 <

d

V aR p t g : 2000 (R. Kaufmann and P . P atie, RiskLab) 23

SLIDE 7 The 5% One Y ea r Exp e ted Sho rtfall Mo del F req ES V aR V ES 1 V ES 2 V freq V size Exp e ted 0% 0% 5% No rmal 1 6 :4 10 :9 :5 1:1 5 4 :3 7 :2 3 :2 9:1 22 3 :7 6 :8 4 :1 11 : 65 1 :9 2 :2 5 :4 30 :7 GARCH 1 2:7 7:4 8 :6 42:9 5 2:5 6:8 8 :2 40:9 22 2:7 6:5 7 :7 41:2 HT 1 1 :6 5 :3 7 :8 40 :5 Ba ktesting Results 22 ba ktesting samples

f

length 3884 ea h. F req N N GARCH 1 2000 3070 5 400 614 22 90 139 65 30 47 261 7 11 Numb er

f

data used with resp e t to the frequen y (in da ys). 2000 (R. Kaufmann and P . P atie, RiskLab) 24 Con lusions

The

No rmal app roa h gives go

d

results but

nly

fo r lo w frequen y (i.e. mo re than 3 months). This ma y

me

from the fa t that lo w frequen y data get loser to the no rmal distribution. Also the squa re ro

t
f

time rule do es not suit fo r high frequen y data. It leads to

verestimation
f

the risk.

Heavy

tailed distributions p erfo rm w ell fo r high frequen y data (daily data).

GARCH

underestimates the risk: p roblem

f

stationa rit y ,

r

{ as fo r the No r- mal app roa h { the s aling rule do es not p erfo rm w ell fo r high frequen y data (in the

pp
site

w a y). Next steps: { investigate the No rmal inverse Gaussian distribution, {

nsider

the p roblem

f

risk aggregation. 2000 (R. Kaufmann and P . P atie, RiskLab) 25 Referen es Artzner P ., Delbaen F., Eb er J.M. and Heath D. (1999) Coherent Measures

f

Risk, Mathemat- i al Finan e 9, no. 3, 203-228. Emb re hts P ., Kl upp elb erg C. and Mik

s h

T. (1997), Mo delling Extremal Events fo r Insur- an e and Finan e, Sp ringer, New Y

rk.

Da o rogna M.M., M uller U.A., Pi tet O.V. and de V ries C.G. (1999), Extremal F

rex

Returns in Extremely La rge Data Sets. Drost F.C. and Nijman T.E. (1993) T emp

ral

Aggregation

f

GARCH Pro esses, E onomet- ri a, 61, 909-927. http:/ /www.risklab. h/Proje ts.html#SL TFR 2000 (R. Kaufmann and P . P atie, RiskLab) 26