mafinrisk market risk course value at risk models: simulation approaches session 8 andrea sironi

MafinriskMarket Risk Course

Value at Risk Models: simulation approaches

Session 8

Andrea Sironi

Mafinrisk - Simulation Approaches 2

Agenda

Common features of simulation approaches

Historical simulations The hybrid approach Monte Carlo simulations Stress testing


Simulation Approaches

Problems of the parametric approach Non-normal distribution of market factors’

returns: higher kurtosis (fat tails) + skewness

Serial correlation of market factors’ returns Non linear positions (bonds, options, etc.)

Simulation approaches Historical & Monte Carlo simulations



Full valuation approaches Every position is repriced for each scenario No use of sensitivity coefficients (delta,

duration, beta, etc.) No normal distribution assumption

Historical simulations: every position is revalued at the historical conditions (returns)

Monte Carlo simulation: random generation of a large number of scenarios

Logic of the distribution percentile


Figure 1 – Main Features of the Simulation Approaches

2. Portfolio: 3. Risk measures:

stocks

rates

commodities

fx

1. Risk factors:A high number of scenarios is generated for changes in market variables, based either on their past changes (historical simulation) or on some chosen (e.g. normal) distribution (Montecarlo simulation).

ConfidentialReportfor theCompany’sC.E.O.

ConfidentialReportfor theCompany’sC.E.O.

Each scenario is translated into a simulated value change for the bank’s portfolio, usually based on the full valuation logic and some appropriate pricing formulae

VaR (or other risk measures) is derived from the distribution of simulated portfolio value changes, e.g. by computing the appropriate percentile.

0%

2%

4%

6%

8%

10%

12%

14%

16%

-60

7

-543

-479

-41

5

-35

1

-28

8

-224

-160 -9

6

-32 32 96

160

224

288

351

415

479

543

607

Variazioni di valore del portafoglio (euro, valore centrale)

% d

i cas

i


Table 1 – Problems and Solutions in VaR Simulation Models

Features of the simulation approach

c) Simulation approach (normal and other distributions)

Monte Carlo simulation

a) Full valuation

b) Percentile approach Historical

simulation With non-

normal distributions

With normal distributions

Non-linear payoffs

Problems Non-normal

market returns

Legend: = solves the problem; = does not solve the problem



Historical simulations

Four phases Selection of an historical sample of market

factors’ returns (e.g. 100 days) Revaluation of the portfolio for each of the

historical values of the market factor Reconstruction of the empirical frequency

distribution of the portofolio market values Identification of the desired distribution

percentile, corresponding to the desired confidence level



1. Revalue the position/ portfolio based on historical conditions

2. Rank P&L 3. Cut the distribution

at the desidered percentile level

Ex. 99% VaR for a long USD position 5.42%

Ex. 95% VaR for a short USD position 5.91%

Monthly returns ITL/USD exchange rate 100 data (June 1987-Sept. 1995) Ordered from lowest to highest

Month ReturnMay 93 -6,81%

November 87 -5,42%November 88 -5,10%

January 88 -5,09%August 88 -4,71%

July 92 -4,71%… …… …

January 93 5,91%September 89 6,54%February 88 7,46%

November 92 7,87%apr-91 8,56%

October 92 17,47%


Table 3 – Example of Historical Simulation for a call option position Data in chronological order Data ranked based on daily log returns

Date S&P500

Daily log returns of the S&P

500 Rank

Daily log returns of the S&P

500

Simulated value of the

S&P500

Simulated value of the call

Change in the

value of the call

03/01/2003 908.6 0.0% 1 -3.6% 1170.8 0.18 -2.11 06/01/2003 929.0 2.2% 2 -3.0% 1178.1 0.30 -2.00 07/01/2003 922.9 -0.7% 3 -2.6% 1182.2 0.39 -1.90 08/01/2003 909.9 -1.4% 4 -2.5% 1183.3 0.42 -1.88 09/01/2003 927.6 1.9% 5 -2.3% 1185.8 0.49 -1.80 10/01/2003 927.6 0.0% 6 -1.9% 1190.4 0.65 -1.65 13/01/2003 926.3 -0.1% 7 -1.9% 1191.2 0.68 -1.61 14/01/2003 931.7 0.6% 8 -1.8% 1192.0 0.72 -1.58 15/01/2003 918.2 -1.5% 9 -1.8% 1192.1 0.72 -1.58 16/01/2003 914.6 -0.4% 10 -1.6% 1193.7 0.79 -1.50 17/01/2003 901.8 -1.4% 11 -1.6% 1193.9 0.80 -1.50 21/01/2003 887.6 -1.6% 12 -1.6% 1194.5 0.83 -1.47 22/01/2003 878.4 -1.0% 13 -1.6% 1194.7 0.84 -1.46 23/01/2003 887.3 1.0% 14 -1.6% 1194.8 0.84 -1.46 24/01/2003 861.4 -3.0% 15 -1.5% 1194.9 0.85 -1.45 27/01/2003 847.5 -1.6% 16 -1.5% 1195.1 0.85 -1.44 28/01/2003 858.5 1.3% 17 -1.5% 1195.1 0.86 -1.44 29/01/2003 864.4 0.7% 18 -1.5% 1195.4 0.87 -1.43 30/01/2003 844.6 -2.3% 19 -1.5% 1195.8 0.89 -1.41 31/01/2003 855.7 1.3% 20 -1.5% 1195.8 0.89 -1.40

… … … … … … … … 30/11/2004 1173.8 -0.4% 481 1.6% 1233.1 5.44 3.15 01/12/2004 1191.4 1.5% 482 1.6% 1233.1 5.46 3.16 02/12/2004 1190.3 -0.1% 483 1.6% 1233.2 5.48 3.18 03/12/2004 1191.2 0.1% 484 1.6% 1233.4 5.51 3.22 06/12/2004 1190.3 -0.1% 485 1.7% 1234.7 5.80 3.50 07/12/2004 1177.1 -1.1% 486 1.8% 1235.2 5.92 3.63 08/12/2004 1182.8 0.5% 487 1.9% 1236.6 6.26 3.96 09/12/2004 1189.2 0.5% 488 1.9% 1237.1 6.38 4.08 10/12/2004 1188.0 -0.1% 489 1.9% 1237.2 6.41 4.11 13/12/2004 1198.7 0.9% 490 1.9% 1237.2 6.41 4.12 14/12/2004 1203.4 0.4% 491 1.9% 1237.3 6.43 4.14 15/12/2004 1205.7 0.2% 492 2.1% 1239.6 7.02 4.72 16/12/2004 1203.2 -0.2% 493 2.1% 1239.9 7.11 4.81 17/12/2004 1194.2 -0.8% 494 2.2% 1240.7 7.32 5.02 20/12/2004 1194.7 0.0% 495 2.2% 1240.7 7.33 5.03 21/12/2004 1205.5 0.9% 496 2.2% 1240.8 7.36 5.06 22/12/2004 1209.6 0.3% 497 2.3% 1241.4 7.53 5.23 23/12/2004 1210.1 0.0% 498 2.6% 1245.2 8.66 6.36 27/12/2004 1204.9 -0.4% 499 3.4% 1255.4 12.23 9.93 28/12/2004 1213.5 0.7% 500 3.5% 1256.5 12.70 10.40


Figure 2 – The 500 Simulated Values

0

2

4

6

8

10

12

14

1160 1180 1200 1220 1240 1260 1280

Simulated values for S&P500 ($)

Sim

ulat

ed v

alue

s fo

r th

e ca

ll ($

)

Figure 3 – Frequency Distribution of Simulated Changes in the Value of the Call

0%

5%

10%

15%

20%

25%

Change in the value of the call ($)

Per

cen

tag

e o

f ca

ses

-1.65


Table 4 – Example of a Stock Portfolio Historical Simulation

Daily log returns in chronological order Data ranked based on daily log returns

Date FTSE100 DAX S&P500 Average Rank FTSE100 DAX S&P500 Average

22/07/2004 -1.6% -2.0% 0.3% -1.1% 1 -1.7% -2.7% -1.6% -2.0% 23/07/2004 0.5% -0.1% -1.0% -0.2% 2 -1.6% -2.0% 0.3% -1.1% 26/07/2004 -0.9% -1.2% -0.2% -0.8% 3 -1.1% -2.1% -0.1% -1.1% 27/07/2004 0.9% 1.6% 1.0% 1.2% 4 -0.9% -1.1% -1.1% -1.0% 28/07/2004 0.7% -0.2% 0.1% 0.2% 5 -0.3% -1.2% -1.4% -1.0% 29/07/2004 1.4% 2.1% 0.5% 1.3% 6 -0.8% -1.5% -0.2% -0.8% 30/07/2004 -0.1% 0.2% 0.1% 0.0% 7 -0.8% -0.9% -0.6% -0.8% 02/08/2004 0.1% -0.8% 0.4% -0.1% 8 -0.9% -1.1% -0.3% -0.8% 03/08/2004 0.3% 0.4% -0.6% 0.0% 9 -0.9% -1.2% -0.2% -0.8% 04/08/2004 -0.5% -1.4% -0.1% -0.7% 10 -0.8% -1.3% 0.0% -0.7%

… … … … … … … … … … 25/11/2004 0.7% 0.8% 0.0% 0.5% 91 1.1% 1.3% 0.0% 0.8% 26/11/2004 -0.3% -0.1% 0.1% -0.1% 92 0.5% 1.6% 0.6% 0.9% 29/11/2004 0.2% -0.2% -0.3% -0.1% 93 0.9% 1.0% 0.9% 0.9% 30/11/2004 -1.0% -0.5% -0.4% -0.6% 94 0.8% 0.8% 1.3% 1.0% 01/12/2004 0.7% 1.4% 1.5% 1.2% 95 0.9% 1.6% 1.0% 1.2% 02/12/2004 0.3% 0.7% -0.1% 0.3% 96 0.7% 1.4% 1.5% 1.2% 03/12/2004 -0.1% -0.2% 0.1% -0.1% 97 1.1% 1.4% 1.4% 1.3% 06/12/2004 -0.5% -0.4% -0.1% -0.3% 98 1.0% 1.7% 1.3% 1.3% 07/12/2004 0.1% 0.4% -1.1% -0.2% 99 1.4% 2.1% 0.5% 1.3%

08/12/2004 -0.5% -0.3% 0.5% -0.1% 100 1.9% 2.6% 1.5% 2.0%


Table 5 – Compared Approaches

Variances / Covariances

Historical simulation

VaR at 95% - long position 1.03% 0.85% VaR at 99% - long position 1.46% 1.12% VaR at 95% - short position 1.03% 1.2% VaR at 99% - short position 1.46% 1.3% Mean 0.00% 0.08% Standard Deviation 0.63% 0.63% Skewness 0.000 -0.013

(Excess) kurtosis 0.000 0.868

Historical simulations vs parametric approach


Figure 4 – Historical Distribution and Normal Distribution

0%

2%

4%

6%

8%

10%

12%

14%

16%

Change in value of the stock portfolio ($)

Per

cen

tag

e o

f ca

ses

Historical simulations vs parametric approach



Advantages Easy to understand and communicate No explicit underlying assumption

concerning the functional form of the returns distribution

No need to estimate the variance-covariance matrix

Allows to capture the risk profile of portfolios with non linear and non monotonic sensitivity to market factors returns



Disadvantages Assumption of stability of the distribution of

market factors’ returns Computationally hard because of full valuation Size of the historical sample, particularly when

time horizon > 1 day Bad definition of the distributions tails Risk of overweighting or underweighting the extreme

events in the historical sample Increasing the size of the historical sample there is

the risk of deviating from the distribution stationarity assumption


Hybrid approach

Boudoukh, Richardson e Whitelaw (1998) Attempt to combine the advantages of the

parametric approach (decreasing weights through exponentially weighted moving averages) e those of historical simulations (no normal distribution assumption)

Long historical series but more weight to recent data Weight attributed to each individual historical

return:

n

i

i

i

itP

1

10


Table 6 – Example of a Simulation Based Upon the Hybrid Method

Daily log returns in chronological order Data ranked based on daily log returns

Date t - i Simulated portfolio returns

Weights Wi (i/i)

Date t - i Simulated portfolio returns

Weights Wi (i/i)

Cumulated weights (Wi)

22/07/2004 t - 100 -1.12% 0.01% 06/08/2004 t - 89 -1.99% 0.03% 0.03% 23/07/2004 t - 99 -0.20% 0.01% 22/07/2004 t - 100 -1.12% 0.01% 0.04% 26/07/2004 t - 98 -0.76% 0.01% 25/10/2004 t - 33 -1.09% 0.83% 0.87% 27/07/2004 t - 97 1.16% 0.02% 19/11/2004 t - 14 -1.04% 2.69% 3.56% 28/07/2004 t - 96 0.20% 0.02% 22/09/2004 t - 56 -0.99% 0.20% 3.76% 29/07/2004 t - 95 1.34% 0.02% 12/10/2004 t - 42 -0.85% 0.48% 4.23% 30/07/2004 t - 94 0.05% 0.02% 27/09/2004 t - 53 -0.78% 0.24% 4.48% 02/08/2004 t - 93 -0.12% 0.02% 11/08/2004 t - 86 -0.77% 0.03% 4.51% 03/08/2004 t - 92 0.02% 0.02% 26/07/2004 t - 98 -0.76% 0.01% 4.52% 04/08/2004 t - 91 -0.66% 0.02% 20/10/2004 t - 36 -0.70% 0.69% 5.21% 05/08/2004 t - 90 -0.46% 0.02% 04/08/2004 t - 91 -0.66% 0.02% 5.23%

… … … … … … … … … 25/11/2004 t - 10 0.52% 3.45% 01/11/2004 t - 28 0.80% 1.13% 90.01% 26/11/2004 t - 9 -0.11% 3.66% 17/11/2004 t - 16 0.89% 2.38% 92.38% 29/11/2004 t - 8 -0.12% 3.90% 11/11/2004 t - 20 0.94% 1.86% 94.24% 30/11/2004 t - 7 -0.63% 4.15% 10/08/2004 t - 87 0.98% 0.03% 94.27% 01/12/2004 t - 6 1.21% 4.41% 27/07/2004 t - 97 1.16% 0.02% 94.28% 02/12/2004 t - 5 0.32% 4.69% 01/12/2004 t - 6 1.21% 4.41% 98.70% 03/12/2004 t - 4 -0.06% 4.99% 16/08/2004 t - 83 1.30% 0.04% 98.73% 06/12/2004 t - 3 -0.32% 5.31% 27/10/2004 t - 31 1.34% 0.94% 99.67% 07/12/2004 t - 2 -0.18% 5.65% 29/07/2004 t - 95 1.34% 0.02% 99.69%

08/12/2004 t - 1 -0.10% 6.01% 01/10/2004 t - 49 2.01% 0.31% 100.00%


VaR (99%) “prudent”: 1.09% VaR (99%) realistic: linear

interpolation

Hybrid approach

%07.1%)87.0%56.3(

%)87.0%1(%04.1

%)87.0%56.3(

%)1%56.3(%09.1%99

VaR

Table 7 –VaR Measures: Historical Simulations and Hybrid Approach Compared

Historical simulation

Hybrid simulation

VaR at 95% - long position 0.85% 0.72% VaR at 99% - long position 1.12% 1.07% VaR at 95% - short position 1.16% 1.17%

VaR at 99% - short position 1.34% 1.32%


Monte Carlo Simulations Problem lack of data: generate new data

Monte Carlo Originally used for pricing complex

derivatives (i.e. exotic options) for which no closed analytic solution was possible expected value of the payoff present value

Simulate the market factor path n times (respecting arbitrage constraints) and compute the payoff in each simulated scenario average of these values = expected value


Monte Carlo Simulations Risk Management: 5 steps

Identify the distribution – f(x) – that best proxy the actual market factor returns distribution

Simulate the market factor evolution n times

Calculate the position market value in each scenario

Build the empirical probability distribution of the changes of the position’s market value

Cut the empirical distribution at the desired confidence level


Monte Carlo Simulations Pricing

The stochastic process that governs the evolution of the market factor is generally known

The problem concerns the valuation

Risk Management The problem concerns the choice of the

distribution from which to extract the market factor returns


Monte Carlo Simulations

The third step is based on the use of a random generator and the uses a uniform distribution. It can be decomposed into 4 sub-steps Extract a value U from a uniform distribution

[0,1] Calculate the value x of this function f(x)

corresponding to the extracted U value Determine the inverse of the cumulative

function of the original sample distribution Repeat the previous steps a large number of

times


Generation of Normally Distributed Values and Revaluation of the Position

uniform distribution

p

10 p

)(1 pv

1 tx

t SeS)(1 pv

)(1 pFx

tC

1tC

(I) A value for p is drawn from

a uniform distribution

(II) p (shaded area) is convertedinto a standard normally-

distributed value, v

(III)v = -1(p) is converted into a value for x, reflecting the and of the real probability distribution f

(IV) Based on x, the market factorin t+1 is simulated and the change

in value for the call is computed



Example: a bank has bought an at the money call option on the MIB 30 stock index with a maturity of 1 year and a market value of 9.413 euro.

Hp. 1: Rf=3%, Volatility MIB 30 = 20%

Hp. 2: normal distribution with mean 0.15% and standard deviation standard 1.5%


Random Value

Inverse Standardized

NormalActual Normal

Value Index MIB30

Value Call MIB 30

Delta MV Call MIB 30

1 0,19 -0,88 -1,17% 98,84 8,73 -0,682 0,57 0,18 0,42% 100,42 9,67 0,253 0,72 0,58 1,02% 101,02 10,04 0,624 0,94 1,52 2,43% 102,46 10,94 1,535 0,85 1,04 1,71% 101,73 10,48 1,066 0,60 0,26 0,54% 100,54 9,74 0,337 0,74 0,64 1,11% 101,11 10,09 0,688 0,11 -1,25 -1,72% 98,29 8,42 -0,999 0,26 -0,65 -0,83% 99,17 8,93 -0,49

10 0,16 -0,99 -1,34% 98,67 8,63 -0,78




What about a portfolio which is sentitive to more than just one market factor?

MC simulations MC do not capture, as historical simulations, the correlation structure

We need to introduce a method to simulate the different market factors taking into account their correlations


Monte Carlo Simulations 5 steps

Estimate variance-covariance matrix Decompose the original matrix into two symmetric

matrices, A and AT “Cholesky decomposition” Generate scenarios for the different market factors

multiplying matrix AT, which reflects the historical correlations of market factors returns, for a vector z of random numers

Calculate the market value change corresponding to each of the simulated scenarios

Calculate VaR cutting the empirical probability distribution at the desired confidence level


Monte Carlo Simulations Example: 2 positions

Buy a call on MIB 30 (same data as before) Sell an at the money call on DAX with ne year

maturity Hp. 3) The DAX returns distribution is normal

with mean 0.18% and standard deviation 1.24%

Hp. 4) The returns correlation between the two market indices is 0.75



Un esempio di simulazione Monte Carlo di due posizioni indipendenti

Num.Cas.

InversaNormaleStand.

Distrib.NormaleEffettiva

ValoreMIB30

VMCall

MIB30

ΔVMCall

MIB30Num.Cas.



ValoreDAX

VMCallDAX

ΔVMCallDAX

ΔVMPort.

1 0,273 -0,601 -0,75% 99,25 8,971 -0,443 0,094 -1,316 -1,45% 98,55 8,571 -0,843 0,400

2 0,515 0,038 0,21% 100,20 9,538 0,124 0,649 0,382 0,65% 100,65 9,810 0,397 -0,272

3 0,425 -0,188 -0,13% 99,86 9,335 -0,079 0,238 -0,712 -0,70% 99,29 8,998 -0,415 0,336

4 0,404 -0,242 -0,21% 99,78 9,286 -0,127 0,924 1,434 1,96% 101,97 10,634 1,221 -1,348

5 0,544 0,111 0,32% 100,31 9,604 0,191 0,162 -0,986 -1,04% 98,96 8,803 -0,610 0,801

6 0,577 0,197 0,44% 100,44 9,682 0,269 0,175 -0,936 -0,98% 99,02 8,838 -0,575 0,844

7 0,680 0,470 0,85% 100,85 9,934 0,521 0,792 0,812 1,19% 101,19 10,142 0,729 -0,208

8 0,829 0,951 1,58% 101,59 10,389 0,976 0,948 1,628 2,20% 102,22 10,791 1,378 -0,402

9 0,606 0,269 0,55% 100,55 9,749 0,336 0,586 0,217 0,45% 100,45 9,685 0,271 0,064

10 0,146 -1,051 -1,43% 98,58 8,585 -0,828 0,200 -0,843 -0,86% 99,13 8,905 -0,508 -0,320

… … … … … … … … … … … … … …


Variazioni valori di mercato delle due call in ipotesi di indipendenza

-3,000

-2,000

-1,000

0,000

1,000

2,000

3,000

-3,000 -2,000 -1,000 0,000 1,000 2,000 3,000 4,000

Variazione VM Call MIB 30

Va

ria

zio

ne

VM

Ca

ll D

AX


Monte Carlo SimulationsUn esempio di simulazione Monte Carlo di due posizioni correlate

Num.Cas.(1)


(2)


(3)

ValoreMIB30

(4)

VMCall

MIB30(5)

ΔVMCall

MIB30(6)

Num.Cas.(7)


(8)


(9)

ValoreDAX(10)

VMCallDAX(11)

ΔVMCallDAX(12)

ΔVMPort.(13)

0,19 -0,88 -1,17% 98,84 8,73 -0,68 0,76 0,72 -0,04% 99,96 9,39 -0,03 -0,65

0,57 0,18 0,42% 100,42 9,67 0,25 0,27 -0,62 -0,16% 99,84 9,32 -0,10 0,35

0,72 0,58 1,02% 101,02 10,04 0,62 0,33 -0,45 0,35% 100,35 9,62 0,21 0,41

0,94 1,52 2,43% 102,46 10,94 1,53 0,42 -0,21 1,42% 101,43 10,29 0,87 0,65

0,85 1,04 1,71% 101,73 10,48 1,06 0,92 1,41 2,31% 102,33 10,86 1,45 -0,39

0,60 0,26 0,54% 100,54 9,74 0,33 0,52 0,05 0,46% 100,46 9,69 0,28 0,05

0,74 0,64 1,11% 101,11 10,09 0,68 0,69 0,50 1,18% 101,19 10,14 0,73 -0,05

0,11 -1,25 -1,72% 98,29 8,42 -0,99 0,15 -1,04 -1,83% 98,18 8,36 -1,06 0,06

0,26 -0,65 -0,83% 99,17 8,93 -0,49 0,50 -0,01 -0,44% 99,56 9,15 -0,26 -0,23

0,16 -0,99 -1,34% 98,67 8,63 -0,78 0,408 -0,234 -0,94% 99,069 8,864 -0,549 -0,23

… … … … … … … … … … … … …


Monte Carlo SimulationsLa scomposizione della matrice varianze-covarianze

Dati di inputMIB 30 DAX

Media 0,15% 0,18%Deviazione Standard 1,50% 1,24%

Correlazione 0,75Matrice Varianze-Covarianze

0,023% 0,014%0,014% 0,015%

Scomposizione di CholeskyMatrice A

1,500% 0,000%0,930% 0,820%

Matrice AT

1,500% 0,930%0,000% 0,820%


Esempio di simulazione congiunta di due posizioni correlate ( = 0,75)Variazioni valori mercato delle due call in ipotesi di correlazione

-2,500

-2,000

-1,500

-1,000

-0,500

0,000

0,500

1,000

1,500

2,000

2,500

3,000

-3,000 -2,000 -1,000 0,000 1,000 2,000 3,000 4,000

Variazione VM Call MIB 30

Var

iazi

one

VM

Cal

l DA

X



Final resultEsempio di VaR di un portafoglio composto da due posizioni

VaR(95%) VaR(99%)Ipotesi Indipendenza 1,8983 2,7255Ipotesi correlazione = 0,75 0,9325 1,4000



Advantages of Monte Carlo simulations Full valuation: no problems with non

linear or non monotonic portfolios Flexibility: possibility to use any

probability distribution functional form Simulating not only final values but also

path: possibility to evaluate the risk profile of path dependent options



Limits of Monte Carlo simulationsNeed to estimate market factors’

returns correlations stability problemComputationally intensiveLarge number of scenarios one tends

to estimate VaR based on values which are not really extremes 10,000 simulations, VaR 99% = 100th worst change


Stress testing Estimate the effects, in terms of potential losses,

of extreme events The portfolio market value is revalued at the

market conditions of very pessimistic scenarios Similar to a simulation model based on

revaluing the portfolio at simulated conditions The extreme scenarios can be based on:

statistical techniques (e.g. 10 times standard deviation) subjective assumptions (e.g. 10% fall of the stock

market, 1% parallel shift of the yield curve, etc.) major historical events (e.g. 1987 stock market crash,

1992 currency crisis, 1994 bond markets collapse, 2000 equity markets, etc.)


Stress testingDerivative Policy Group (1995) Recommendations 100 basis points parallel shift, upwards or downwards, of

the yield curve 25 b.p. change in the yield curve slope 10% change in the stock market indices 6% changes in the FX rates 20% change in volatility


Stress testing Not a real VaR model discretionality They allow to overcome the restrictive

assumptions of VaR models They allow to simulate the impact of liquidity crisis They allow to capture the effects of crisis

episodes during which significant increases in correlations between different market factors tend to occur

They can be built on specific tailor made assumptions, based on the size, composition and sensitivity of the individual portfolio


Questions & Exercises1. Which of the following statements concerning Monte

Carlo simulations is correct? Monte Carlo simulations, unlike the parametric

approach, have the advantage of preserving the structure of correlations among market factor returns

Monte Carlo simulations have the advantage of not requiring any assumption on the shape of the of the probability distributions of market factor returns

Monte Carlo simulations allow to estimate the VaR of a portfolio, with the desired confidence level, using the percentile technique

Monte Carlo simulations allow to estimate the VaR of a portfolio, with the desired confidence level, using a multiple of standard deviation of market factor returns


Questions & Exercises2. A European bank computes the VaR associated to its overall

position in US dollars, based on parametric VaR and historical simulation. The two results are different (€100,000 and €102,000, respectively) regardless of the fact that they are based on the same data series and the same confidence level. Consider the following statements:

I. The distribution of the percent changes in the euro/dollar exchange rate is not normal

II. The distribution of the percent changes in the euro/dollar exchange rate is asymmetrical

III. The distribution of the percent changes in the euro/dollar exchange rates has a greater kurtosis than the normal distribution

Which ones would you agree with? A) Only IB) I, II and IIIC) Only I and IID) Only I and III


Questions & Exercises3. Read the following statements on Monte Carlo simulations:I. Monte Carlo simulations are more accurate than the

parametric approach when the value of the bank’s portfolio is a linear function of the risk factors, and the risk factor returns are normally distributed.

II. Monte Carlo simulations are quicker than the parametric approach.

III. Monte Carlo simulations can be made more precise through the delta/gamma approach.

IV. Monte Carlo simulations require the assumption that risk factor returns are uncorrelated with each other, since otherwise the Cholesky decomposition could not be computed.

Which one(s) would you agree with?A) Only II.B) Only III.C) I and IV.D) None of them


Questions & Exercises4. Consider the following statements: “historical simulations…i) …are totally distribution-free, meaning that users do not

have to make hypotheses on the shape of the probability distribution of market factor returns”;

ii) …are stationary, meaning that the variance of market factor returns is supposed to be constant”;

iii) …are equivalent to parametric models (including models where volatilities are exponentially-weighted) if the probability of past factor returns is close to normal”;

iv) …are extremely demanding in terms of past data, especially if VaR is based on a long holding period”.

Which ones would you agree with?A) all;B) ii and iv;C) i and iii;D) only iv.


Questions & Exercises

5. Following a brief period of sharp changes in market prices, a bank using historical simulations to estimate VaR decides to switch to a model based on hybrid simulations, adopting a decay factor of 0.95. Which of the following is true?

A) The new model is likely to lead to an increase in VaR, which can be mitigated by setting at 0.98;

B) The new model is likely to lead to an decrease in VaR, which can be mitigated by setting at 0.98;

C) The new model is likely to lead to an increase in VaR, which can be mitigated by setting at 0.90;

D) The new model is likely to lead to an decrease in VaR, which can be mitigated by setting at 0.90.

mafinrisk market risk course value at risk models: simulation approaches session 8 andrea sironi

Documents

simulation approaches

distribution percentile

historical values

historical conditions

returns distribution

andrea sironi slide

desired confidence level

position portfolio