lec_7_review_questions

7/25/2019 Lec_7_Review_questions

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CLEFIN A.A. 2012/2013

Financial Econometrics — Prof. Massimo Guidolin

A Few Review Questions and Problems on Lecture 7,

“Simulation-Based Methods in Risk Management”

SUGGESTION: try to approach the questions first, without looking at the answers. It may be VERY

useful to use the questions to make sure you have read and understood the material in the references listed

at the end of every question.

1. K-step Ahead Forecasts of Variance and VaR measures Under Three Alternative Mod-

els.

Call 6= 0 the mean of a portfolio return series, which is serially uncorrelated. Consider three

alternative models, as examined in the lectures, of the variance portfolio returns:

2+1 = 2 with +1 ∼ IID (0 2) (Homoskedastic Gaussian IID Model)

2+1 = 2

+ (1

−)2

, +1

∼ (0 2

+1) (Gaussian RiskMetrics)

2+1 = + ( − )2 + 2

, +1 ∼ (0 2+1) (Gaussian GARCH(1,1)).

1a. Compute the time , one-step ahead forecasts of variance and the corresponding -percent VaR under

each of the three models.

Answer. This is material of the first part of the course:

[2+1] = 2 (Homoskedastic Gaussian IID Model)

[2+1] = 2

+ (1 − )2 (Gaussian RiskMetrics)

[2+1] = + ( − )2 + 2

(Gaussian GARCH(1,1)),

so that

+1 = − − Φ−1( ) (Homoskedastic Model)

+1 = − − [2

+1]Φ−1( ) = − −q

2 + (1 − )2

Φ−1( ) (Gaussian RiskMetrics)

+1 = − − [2

+1]Φ−1( ) = − −q

+ ( − )2 + 2Φ−1( ) (Gaussian GARCH(1,1)).

Notice that when 6= 0 the GARCH(1,1) model must be written linking 2+1 to the squared shocks

to portfolio returns, in this case ( − )2. However, as long as is small and especially (as

assumed here) not time-varying, it will make a minor diff erence to the results that follow, although

it may be a good exercise to pay attention at the formulas involved.1

1This is not the case in Guidolin and Timmermann (2007), cited in chapter 7, where as we have seen Markov switching

means do matter considerably, also because they model monthly portfolio returns.

1



1b. Repeat the same task as in 1a for the case of ≥ 2 step forecasts of total portfolio return variance,

i.e., the variance of

:+ ≡ X =1

+

Answer. Here each case needs to examined separately. Let’s start with variance forecasts and with

= 2 just to gain some intuition. Notice that if :+2 ≡ +1 + +2, then

[:+2] = [+1] + [+2] + 2[+1 +2]

= 2+1 + 2

+2

because of the assumption of zero serial correlation. At this point:

[2+1 + 2

+2] = 22 (Homoskedastic Model)

[2+1 + 2

+2] = 2 + (1 − )2

+ [2+1] + (1 − ) [

2+1]

= 2 + [2

+1] + (2

−2 ) + ( [2

+1]

− [2

+1])

= 2 + 2

+ (1 − )2 + (2

− 2 ) + 2

= 2(1 − )2 + 22

+ 2 = 2 [2+1] + 2

(Gaussian RiskMetrics)

[2+1 + 2

+2] = 2 + (-)2 + 2 + [(+1-)2] + [2

+1]

= 2 +P

2

=1( + ) −1(2+1 − )

(Gaussian GARCH(1,1)),

where the last expression comes from

[2+1 + 2

+2] = 2 + ( − )2 + 2 + [(+1 − )2] + [

2+1]

= 22 +

©( − )2 + [(+1 − )2] − 22

ª+

©2 + [

2+1] − 22

ª= 2

2

+ (2

+1 − 2

) + ( + )( [2

+1] − 2

)

= 22 +2X

=1

( + ) −1(2+1 − 2)

where 2 ≡ (1 − − ). In the RiskMetrics expression, [2+1] 6= [2

+1] because 6= 0; in

fact, [2+1] = [2

+1] + 2.2 The expressions for the VaR measures are at this point obvious,

+1:+2 = −2 −

q [:+2]Φ−1( )

with one caveat: while in the homoskedastic case it is correct to use the Gaussian inverse CDF

Φ−1

( ), in the GARCH and RiskMetrics cases (RiskMetrics is an integrated/non-covariance station-ary GARCH(1,1) model with = 0 and + = 1) we have no presumption that even at = 2,

+1:+2 = −2 −

p [:+2]Φ−1( ) as for ≥ 2 the marginal normality assumption may

not be warranted (see question 1c below).

It is possible (although a bit tedious, even though you can proceed recursively, by induction) to

generalize these expressions to the general ≥ 2 case:

[P

=1 + ] = 2 (Homoskedastic Model)

[

P =1 + ] = [2

+1] +

P =1 −12 (Gaussian RiskMetrics)

[P

=1 + ] =

2

+P

=1( + )

−1

(

2

+1 −

2

) (Gaussian GARCH(1,1)),2If we were to write the RiskMetrics model as 2+1 = ( − )2 +(1− )2 then the need for this correction would

disappear and [2+1] = 2 would obtain.

2



which yields

+1:+ = − −

√ Φ−1( ) (Homoskedastic Model)

+1:+ = − −

q [2

+1] +P

=1 −12 Φ−1( ) (Gaussian RiskMetrics)

+1:+ =

−

−q 2 +P =1( + ) −1(2

+1

−2)Φ−1( ) (Gaussian GARCH(1,1)),

where the caveats we have expressed before on the validity of the expression +1:+ = − −p

[:+ ]Φ−1( ) gets stronger as grows large.3

1c. Using again the simple case of = 2 explain why in the Gaussian IID case, :+2 ≡ +1 +

+2 ∼ (2 22) in the RiskMetrics case

:+2 ¿ (2 2 [2+1] + 2)

Also explain why instead in the GARCH(1,1) case it is true that

:+ ¿

Ã2 +

X =1

( + ) −1(2+1 − 2)

!

but such an approximation becomes increasingly appropriate (i.e., :+ → (2 +P

=1( + ) −1 (2+1 − 2)) as → ∞). What are the implications of these conclusions for

the use of Monte Carlo simulations in financial econometrics?

Answer. Under the assumption of Gaussian IID portfolio returns, it is trivial to see that if

+ ∼ ( 2) ( = 1 2), the sum of two Gaussian random variables will be Gaussian, while their

independence and identical distribution over time will imply that +1 + +2 ∼ (2 22).

However, in the RiskMetrics case, even though it is true that the current forecast of [:+2] =

[+1]+ [+2] is

[2+1 + 2

+2] = 2 [2+1] + 2

the independence is trivially lost because

2+2 = 2

+1 + (1 − )2+1

so that the variance of +2 depends on +1. Moreover, while +1 ∼ (2 + (1 −

)2 ) +2 comes from a mixture of normal distributions. It is a mixture in the following sense:

if +1 = 1 then +2 ∼ (21 + (1 − )2

+ (1 − )22 ); if +1 = 2

then +2 ∼ (22 + (1 − )2

+ (1 − )22 ); if +1 = 3 then +2 ∼

(23 + (1 − )2

+ (1 − )22 ), etc. Now, can we restrict +1 to either come from

1, 2, or 3? Obviously not, as we know that because +1 ∼ (2+(1−)2

)

+1 may actually take a continuum, an infinity of possible values. Moreover, notice that to state

that +2 ∼ ( ( [+1])2 + (1 − ) + (1 − )22 ) would be incorrect. Bottom line

3

However, in a mean-reverting (in variance, i.e., in a covariance stationary) GARCH model, as gets large the returndistribution does approach a normal distribution, see question 1c to follow.

3



is that :+2 ¿ (2 2(2 + 2)). Finally, without getting lost in tedious details, a similar

argument applies to showing that in the case of GARCH(1,1),

:+ ¿

Ã2 +

X =1

( + ) −1(2+1 − 2)

!

also because you have established in the first part of the course that the RiskMetrics volatility

model corresponds to a GARCH(1,1) in which = 0 and + = 1. However, in this case it is

possible to add that under the covariance stationarity assumption that + 1, because 2+ =

2 + ( + ) −1(2+1 − 2) so that

lim →∞

2+ = 2

a Gaussian approximation will become increasingly appropriate as → ∞, because the very long

horizon forecast of variance is actually the unconditional, long-run variance 2 ≡ (1 − − ).

Equivalently, notice that as grows large

[ X =1

+ ] = 2 + X =1

( + ) −1(2+1 − 2)

' 2 + 1

1 − − (2

+1 − 2)

which is a term that grows linearly in similarly to the Gaussian IID case.

The implication of this facts for the use of the simple but incorrect formula

+1:+ = − −v uut [

X =1

+ ]Φ−1

( )

in risk management applications is that we are faced with the challenge of computing risk measures

such as VaR at multiple horizons, without knowing the analytical form for the distribution of returns

at those horizons. Fortunately, this challenge can be met through the use of Monte Carlo simulation

techniques.

References

Christoff ersen P. F. (2012) Elements of Financial Risk Management , Academic Press 2nd edition,

chapter 6.4.Christoff ersen, P., F. Diebold, and T. Schuermann. (1998, October). “Horizon Problems and Extreme

Events in Financial Risk Management,” Economic Policy Review , Federal Reserve Bank of New York,

109-118.

2. Step-by-Step, Hand-Made Calculations of HS and WHS VaR Measures.

Consider the following record of monthly returns concerning a rarely traded collateralized debt obliga-

tion (CDO) written on residential mortgage backed securities issued between 2006 and 2007:

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Date Return

February 2008 +1.38%

March 2008 -8.49%

April 2008 +0.37%

May 2008 +4.58%June 2008 -3.21%

July 2008 -1.03%

August 2008 -4.59%

September 2008 -28.10%

October 2008 -7.89%

November 2008 -3.07%

December 2008 -6.88%

January 2009 -8.63%

February 2009 -0.63%

After the end of February 2009 the CDO market completely dried up and the security has not been

traded since then.

2a. Using a moving window of 5 months ( = 5), use HS to compute the 20% VaR of a position long one

dollar into this CDO between June 2008 and February 2009. What are the problems with the values

of the VaR obtained? (Note : please do not waste time commenting on the fact that computing a

20% VaR on a security that is no longer traded and using just a dozen monthly return data points

may be futile or misleading; this is obvious, but here we are after the simplest possible example inwhich you can handle the data “with paper and pencil”)

Answer. It is almost trivial:

Date 20% VaR

February 2008

March 2008

April 2008

May 2008

June 2008 8.49%July 2008 8.49%

August 2008 4.59%

September 2008 28.10%

October 2008 28.10%

November 2008 28.10%

December 2008 28.10%

January 2009 28.10%

February 2009 8.63%

1. For instance, the 20% VaR as of February 2009 comes from the fact that in the = 5 (five-month)

data record {−789%, − 307%, − 688%, − 863%, − 063%} 20% of the probability mass in the

5



left tail of the realized returns (the right right tail of realized losses) is left by -8.63% which also

happens to be the lowest of the 5 observations taken into consideration. The problem with the VaR

estimates obtained is the one discussed during the lectures: if you plot the 20% VaR measure, its path

follows a curious box-shaped pattern in which the risk measures get stuck in the infamous 28.10%

realized loss from September 2008, when Lehman Bros. went bust for 5 months, i.e., until that loss

leaves the record, which only occurs in February 2009.

2b. Using a moving window of 5 months ( = 5) and a decay parameter = 09 use WHS to compute

the 20% VaR of a position long one dollar into this CDO between June 2008 and February 2009.

Answer. In this case a useful set of calculations are the weights to be assigned in the WHS scheme:

1 = 091−1(1 − 09)(1 − 095) = 0244

2 = 092−1(1 − 09)(1 − 095) = 0220 3 = 092−1(1 − 09)(1 − 095) = 0198

4 = 092−1

(1 − 09)(1 − 095

) = 0178 5 = 092−1

(1 − 09)(1 − 095

) = 0160

Because the WHS rule states that the 20% VaR is calculated by accumulating the weights of the

ascending returns until 20% is reached, the observations, along with their assigned weights, are

sorted in ascending order. At this point, the 20%VaR is calculated by accumulating the weights of

the ascending returns until 20% is reached. In practice, we obtain:

Date 20% VaR

February 2008

March 2008April 2008

May 2008

June 2008 3.21%

July 2008 3.21%

August 2008 4.59%

September 2008 28.10%

October 2008 28.10%

November 2008 7.89%

December 2008 7.89%

January 2009 8.63%

February 2009 8.63%

which is quite diff erent than the HS 20% VaR, although it seems to be subject to similar disadvantages.

2c. Using a moving window of 5 months ( = 5), use HS to compute the 20% VaR of a position short

one dollar into this CDO between June 2008 and February 2009. Compare the 20% VaR obtained

with those in 2a for a long position of one dollar in the CDO: are the di ff erences sensible?

Answer. As trivial as the answer to 2a, but for the fact that now the calculations must be performed

in terms of the largest gain (i.e., the highest possible return) on the CDO:

6



Date 20% VaR/short position 20% VaR/long position

February 2008

March 2008

April 2008

May 2008June 2008 4.58% 8.49%

July 2008 4.58% 8.49%

August 2008 4.58% 4.59%

September 2008 4.58% 28.10%

October 2008 -1.03% 28.10%

November 2008 -1.03% 28.10%

December 2008 -3.07% 28.10%

January 2009 -3.07% 28.10%

February 2009 -0.63% 8.63%

1. As commented in the lectures, these results are not particularly sensible because in spite of the

dramatic volatility surge of September 2008, this is not reflected in the VaR for the short position

which actually becomes very benign entering and staying in the negative range after October 2008.

More generally, it is rather unexpected and shocking to find such huge asymmetries in the thickness of

the two tails of the empirical distribution and–even though asymmetric GARCH models do exist–it

is unlikely that an investor with a short position in the RMBS CDOs really felt that safe, with a

negative VaR, in the Fall of 2008.

References

Christoff ersen P. F. (2012) Elements of Financial Risk Management , Academic Press 2nd edition,

chapter 2.

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