Mafinrisk 2010
Market Risk course
Value at Risk Models: the
parametric approach
Andrea Sironi
Sessions 5 & 6
Agenda







Market Risks
VaR Models
Volatility estimation
The confidence level
Correlation & Portfolio Diversification
Mapping
Problems of the parametric approach
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Market Risks
 The risk of losses resulting from
unexpected changes in market factors’





Interest rate risk (trading & banking book)
Equity risk
FX risk
Volatility risk
Commodity risk
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Market Risks
 Increasingly important because of:




Securitization
Diffusion of mark-to-market approaches
Huge losses (LTCM, Barings, 2008 crisis, etc.)
Basel Capital requirements
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VaR models
 Question: which is the maximum loss that could be
suffered in a given time horizon, such that there is only
a very small probability, e.g. 1%, that the actual loss is
then larger than this amount?
 Definition of risk based on 3 elements:
 maximum potential loss that a position could suffer
 with a certain confidence level,
 in a given time horizon
PrL  VaR  1  c
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Value at Risk (VaR) Models
Risk
Maximum Potential Loss ...
1. ... with a predetermined confidence level
2. ... within a given time horizon
VaR = Market Value x Sensitivity x Volatility
Three main approaches:
1. Variance-covariance (parametric)
2. Historical Simulations
3. Monte Carlo Simulations
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VaR models: an example
10 yrs Treasury Bond
Market Value:
Holding period:
YTM volatility:
Worst case:
Modified Duration:
€ 10 mln
1 month
30 b.p. (0,30%)
60 b.p.
6
VaR = € 10m x 6 x 0.6% = € 360,000
The probability of losing more than € 360,000 in
the next month, investing € 10 mln in a 10 yrs
Treasury bond, is lower than 2.5%
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VaR models: an example
VaR = € 10 mln x 6 x (2*0.3%) = 360,000 Euro
Market Value
(Mark to Market)
A proxy of the sensitivity
of the bond price to
changes in its yield to
maturity (for a stock it
would be the beta)
An estimate of the future
variability of interest
rates (for a stock it would
be the volatility of the
equity market)
A scaling factor needed to obtain the
desired confidence level under the
assumption of a normal distribution
of market factors’ returns
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Estimating Volatility of Market
Factors’ Returns
Three main alternative criteria
• Historical Volatility
Backward looking
• Implied Volatility
Option prices: forward looking
• Garch models (econometric)
Volatility changes over time autoregressive
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Estimating Volatility of Market
Factors’ Returns
Historical Volatility: monthly changes of the Morgan Stanley Italian
equity index (10/96-10/98)
01/10/96
01/11/96
01/12/96
01/01/97
01/02/97
01/03/97
01/04/97
01/05/97
01/06/97
01/07/97
01/08/97
01/09/97
6,74%
-5,38%
6,92%
0,89%
14,42%
-3,76%
-1,93%
5,34%
-1,47%
10,66%
7,76%
-2,37%
01/10/97
01/11/97
01/12/97
01/01/98
01/02/98
01/03/98
01/04/98
01/05/98
01/06/98
01/07/98
01/08/98
01/09/98
6,87%
-3,20%
4,05%
7,68%
11,27%
4,84%
20,14%
-7,65%
1,86%
1,33%
3,07%
-16,69%
n
t 
2
(
R

R
)
 i
t 1
n 1
Standard Deviation = 7,77%
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Estimating Volatility of Market
Factors’ Returns
 Most VaR models use historical volatility
 It is available for every market factor
 Implied vol. is itself derived from historical
 Which historical sample?
 Long (i.e. 1 year)  high information content, does
not reflect current market conditions
 Short (1 month)  poor information content
 Solution: long but more weight to recent data
(exponentially weighted moving average)
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Example of simple moving averages
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Example of simple moving averages
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Example of simple moving averages
Figure 3 – The ”Echo Effect” Problem
12/31/2001
12/17/2001
12/03/2001
-8,0%
11/19/2001
0,0%
11/05/2001
-4,0%
10/22/2001
0,4%
10/08/2001
0,0%
9/24/2001
0,8%
9/10/2001
4,0%
8/27/2001
1,2%
8/13/2001
8,0%
7/30/2001
1,6%
7/16/2001
12,0%
7/02/2001
2,0%
Daily returns (right hand scale)
23-days moving standard deviation (left hand scale)
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Estimating Volatility of Market
Factors’ Returns
Exponentially weighted moving average (EWMA)
xt = return of day t
 = decay factor (higher , higher persistence,
lower decay)
 x
 2 x
 3x ...n1xt n 0    1
t 1
t 2
t 3
t 4
1    2  3 ...n1
0 x

 i 1
1     xt i
i 1
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Figure 4 – An Example of Volatility Estimation Based Upon an Exponential
Moving Average
2,4%
12,0%
2,0%
8,0%
1,6%
4,0%
1,2%
0,0%
0,8%
-4,0%
12/31/2001
12/17/2001
12/03/2001
11/19/2001
11/05/2001
10/22/2001
-8,0%
10/08/2001
9/24/2001
9/10/2001
8/27/2001
8/13/2001
7/30/2001
7/16/2001
0,0%
7/02/2001
0,4%
Daily returns (right hand scale)
23-days simple moving standard deviation (left hand scale)
23-days exp. weighted moving standard deviation (left hand scale)
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Figure 5 – An Example of Historical Volatility Estimation Based Upon Different
Decay Factors
S&P 500 equally-weighted index daily returns
Moving standard deviations based on different decay factors
12/31/2004
12/24/2004
12/17/2004
-2,0%
12/10/2004
0,4%
12/03/2004
0,0%
11/26/2004
0,5%
11/19/2004
2,0%
11/12/2004
0,6%
11/05/2004
4,0%
10/29/2004
0,7%
10/22/2004
6,0%
10/15/2004
0,8%
10/08/2004
8,0%
10/01/2004
0,9%
Daily returns (right hand scale)
23-days exp. weighted moving standard deviation (l =0,94)
23-days exp. weighted moving standard deviation (l =0,90)
23-days exp. weighted moving standard deviation (l =0,99)
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Estimating Volatility of Market
Factors’ Returns
 Which time horizon (daily volatility,
weekly, monthly, yearly, etc.)?
 Two main factors:
 Holding period  subjective
 Liquidity of the position  objective
 However:
T   d  T
 Implied hp.: no serial correlation
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Estimating Volatility of Market
Factors’ Returns
Daily
Volatility
EFFECTIVE
ESTIMATED
ERROR
1,02%
-
EFFECTIVE
ESTIMATED
ERROR
0,63%
-
EFFECTIVE
ESTIMATED
ERROR
0,96%
-
EFFECTIVE
ESTIMATED
ERROR
1,23%
-
EFFECTIVE
ESTIMATED
ERROR
0,61%
-
Weekly
Volatility
MIB 30
2,64%
2,28%
0,37%
S&P 500
1,40%
1,40%
0,00%
CAC 40
2,07%
2,14%
-0,07%
Nikkei
2,68%
2,75%
-0,07%
FTSE 100
1,52%
1,35%
0,16%
Monthly
Volatility
6,01%
4,78%
1,24%
2,40%
2,94%
-0,54%
4,00%
4,49%
-0,50%
6,30%
5,76%
0,54%
 Test of the non-serial
correlation assumption
 Two years data of
daily returns for five
major equity markets
(1/1/95-31/12/96)
 It only holds for very
liquid markets and
from daily to weekly
5,16%
2,84%
2,31%
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The confidence level
 In estimating potential losses (VaR), i.e.
economic capital, one has to define the
confidence level, i.e. the probability of not
not recording higher than VaR losses
 In the variance-covariance approach, this is
done by assuming a zero-mean normal
distribution of market factors’ returns
 The zero-mean assumption is justified by the
short time horizon (1 day)  the best
forecast of tomorrow’s price is today’s one
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The confidence level
 Hp. Market factor returns std. dev. = 1%
 If the returns distribution is normal, then
 68% prob. return between -1% and + 1%
 16% probability of a loss higher than 1%
(only loose one side)  84% confidence level
 95% prob. return between -2% and + 2%
 2.5% probability of a loss higher than 2% 
97.5% confidence level
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The normal distribution assumption
Probabilità = 5%
α = 1,65σ
VaR(95%)
Profitto atteso
(VM x δ x µ)
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The confidence level
The higher the scaling factor, the higher is VaR,
the higher is the confidence level
Scaling
Confidence
Factor (# of
level
std.dev.s)
99,5%
3
99,0%
2,323
97,5%
2
95,0%
1,65
90,0%
1,28
84,0%
1
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Potential losses
(Treasury bond
example)
540.000
418.140
360.000
297.000
230.400
180.000
23
The confidence level
 More risk-averse banks would choose a
higher confidence level
 Most int.l banks derive it from their rating





(i) bank’s economic capital = VaR
(ii) VaR confidence level = 99%
 bank’s PD = 1%
If PD of a single-A company= 0,3% (Moodys)
 A single-A bank should have a 99.7% c.l.
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The confidence level
Moody’s Rating Class
Aaa
Aa1
Aa2
Aa3
A1
A2
A3
Baa1
Baa2
Baa3
Ba1
Ba2
Ba3
B1
B2
B3
1-Year Probability of Insolvency
0.001%
0.01%
0.02%
0.03%
0.05%
0.06%
0.09%
0.13%
0.16%
0.70%
1.25%
1.79%
3.96%
6.14%
8.31%
15.08%
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Confidence Level
99.999%
99.99%
99.98%
99.97%
99.95%
99.94%
99.91%
99.87%
99.84%
99.30%
98.75%
98.21%
96.04%
93.86%
91.69%
84.92%
25
The confidence level
Rabobank
Rating (Standard & Poor's)
AAA
Better rated banks should
have a higher Tier 1
capital

AA+
BoS
AA
Bnp
ING
HSBC BBVASG
HBOS
Intesa SP
RBS Lloyds
AAA+
Unicredit
A
The empirical
relationship is not
precisely true for a group
of major European
banking groups
Santander
Natixis
Calyon
Deutsche

Commerz
6,00
7,00
8,00
9,00
10,00
Tier 1 capital
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Rating agencies
evaluations are also
affected by other factors
(e.g. contingent
guarantee in case of a
11,00
crisis)
26
Diversification & correlations
• VaR must be estimated for every single position
and for the portfolio as a whole
• This requires to “aggregate” positions together to
get a risk measure for the portfolio
• This can be done by:
– mapping each position to its market factors;
– estimating correlations between market
factors’ returns;
– measuring portfolio risk through standard
portfolio theory.
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Diversification & correlations
An example
Currency
USD
Yen
Position (€
mln)
-50
50
Worst case
(1.65*std.dev.)
0.92%
1.76%
VaR (Euro)
460.000
880.000
Sum of VaRs: € 1,340,000
If correl. €/$-€/Yen = 0.54
2
VarTot  Var$2  VarYen
 2 Var$ VarYen  $,Yen 
 460m2  880m2  2  (460m)  880m  0.54  € 740,821
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Diversification & correlations
 Three main issues
 1) A 2 positions portfolio VaR may be lower than the
more risky position VaR  natural hedge
 1) Correlations tend to shoot up when market
shocks/crises occur  day-to-day RM is different
from stress-testing/crises mgmt
 2) A relatively simple portfolio has approx.ly 250
market factors  large matrices  computationally
complex  an assumption of independence between
different types of market factors is often made
VarTot  Var  Var  Var
2
FX
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2
IR
2
Equity
29
Mapping
 Estimating VaR requires that each individual
position gets associated to its relevant market
factors
 Example: a long position in a US Treasury bond
is equivalent to:
 a long position on the USD exchange rate
 a short position on the US dollar
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Mapping FX forward
 A long position in a USD forward 6 month
contract is equivalent to:
 A long position in USD spot
 A short deposit (liability) in EUR with maturity 6 m
 A long deposit (asset) in USD with maturity 6 m
1  id  t
Ft  S 
1 if t
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Figure 4 – Mapping of a 6-Month Forward Dollar Purchase
inflows
6-month forward dollar purchase
0
€
outflows
€
$
time
6-month EUR-denominated debt
1
€
6-month USD-denominated investment
$
2
$
$
Spot dollar purchase
3
€
$
1+2+3
€
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Mapping FX forward
Example: Buy USD 1 mln 6 m forward
FX and interest rates
EUR/USD Spot
6 m EUR interest rate
6 m USD interest rate
EUR/USD 6 m forward
1. Debt in EUR
2. Buy USD spot
3. USD investment
1,20
3,50%
2,00%
1,209
DEUR  990.0991,2  1.118.119
USDspot  990.0991,2
1.000.000
I USD 
 $990.099
1  0,02  0,5
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Mapping FX forward
Volatilities and correlations - Forward position market factors
Correlation with…
Market factor
Volatility
EUR/USD EUR 6 m IR USD 6 m IR
EUR/USD Spot
3%
1
-0,2
0,4
EUR 6 m IR
1,50%
-0,2
1
0,6
USD 6m IR
1,20%
0,4
0,6
1
VaRiEUR6m  1.118.1191,5%  2,326 0,483 EUR18.849
VaRiUSD 6m  990.0991,2%  2,326 0,490  13.549  EUR16.259
VaRUSDspot  990.099 3%  2,326  69.099  EUR82.919
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Mapping FX forward
VaRUSD 6 m 
2
2
2
VaRiEUR
6 m  VaRiUSD 6 m  VaRUSDspot  2  VaRiEUR 6 mVaRiUSD 6 m  iEUR ,iUSD 
 2  VaRiEUR 6 mVaRUSDspot  iEUR 6 m,USDspot  2  VaRiUSD 6 mVaRUSDspot  iUSD 6 m,USDspot
18.8492  16.2592  82.9192  2 18.849 (16.259)  0,6 

 83.646
 2 18.849 82.919 (0,2)  2  (16.259)  82.919 0,4
Total VaR of the USD 6 m forward
position
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
Mapping of a FRA
 An FRA is an agreement locking in the interest rate on an
investment (or on a debt) running for a pre-determined
 A FRA is a notional contract  no exchange of principal
at the expiry date; the value of the contract (based on
the difference between the pre-determined rate and the
current spot rates) is settled in cash at the start of the
FRA period.
 A FRA can be seen as an investment/debt taking place in
the future: e.g. a 3m 1 m Euro FRA effective in 3 month’s
time can be seen as an agreement binding a party to pay
– in three month’s time – a sum of 1 million Euros to the
other party, which undertakes to return it, three months
later, increased by interest at the forward rate agreed
upon
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Mapping of a FRA
 Example: 1st August 2000, FRA rate 5.136%
 Investment from 1st November to 1st February 2001 with
delivery: 1,000,000 *(1 + 0.05136 * 92/360) = 1,013,125 Euros.
 Equivalent to:
 a three-month debt with final principal and interest of one
million Euros;
 A six-month investment of the principal obtained from the
transaction as per 1.
investment
1m
1.013m
1,013m
1m
1-8-2000
1-11-2000
1-2-2001
f
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m
37
Mapping stock portfolio
 Equity positions can be mapped to their stock
index through their beta coefficient
 In this case beta represents a sensitivity
coefficient to the return of the market index
 Individual stock VaR VaRi  VM i   i   j  
N


 Portfolio VaR
VaR j   VM i   i    j  
 i 1
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
38
Mapping of a stock portfolio
Example
Mapping of equity positions
Stock A
Stock B
Market Value (EUR m)
10
15
Beta
1,4
1,2
Position in the Market Portfolio (EUR m)
14
18
Volatility
15%
12%
Correlation with A
1
0,5
Correlation with B
0,5
1
Correlation with C
0,8
0
VaRP,99%
Stock C
20
0,8
16
10%
0,8
0
1
Portfolio
45
 N

  VM i   i    j    48  0,07  2,326  7,817
 i 1

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39
Mapping of a stock portfolio
Example with individual stocks volatilities and correlations
Mapping of equity positions
Stock A
Stock B
Market Value (EUR m)
10
15
Beta
1,4
1,2
Position in the Market Portfolio (EUR m)
14
18
Volatility
15%
12%
Correlation with A
1
0,5
Correlation with B
0,5
1
Correlation with C
0,8
0
Stock C
20
0,8
16
10%
0,8
0
1
Portfolio
45
VaRP ,99%  VaRA2  VaRB2  VaRC2  2VaRAVaRB  A, B  2VaRAVaRC  A,C  2VaRBVaRC  B ,C  9,589
VaR of an equity portfolio
VaR(99%)
Stock A
3.490
Stock B
4.187
Stock C
4.653
Mafinrisk - Sironi
Mapping
7.817
Volatilities &
Correlations
9.589
40
Mapping of a stock portfolio
VaR of an equity portfolio
VaR(99%)
Stock A
3.490
Stock B
4.187
Stock C
4.653
Mapping
7.817
Volatilities &
Correlations
9.589
Mapping to betas:
 assumption of no specific risk
 the systematic risk is adequately captured by a CAPM type
model
 it only works for well diversified portfolios
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41
Figure
–
6
Main
Characteristics
of
the
Parametric
Approach
16%
CC
oo
nn
ff
ii
dd
eent
ii
nt
aal
l
R
14%
12%
stocks
10%
% di casi
commodities
8%
6%
4%
2%
Variazioni di valore del portafoglio (euro, valore centrale)
607
543
479
415
351
288
224
96
160
32
-32
-96
-160
-224
-288
-351
-415
-479
-543
-607
0%
Rep
fforepoorrtt
CCom or tthe
ompa he
CC.Epannyy’’s
.E..O. s
O.
rates
fx
1. Risk factors:
2. Portfolio:
Are defined either as
price changes (asset
normal) or as changes
in market variables
(delta normal) their
distribution is then
supposed to be
normal.
Risk factors are
mapped to individual
positions based on
virtual components
and linear coefficients
(deltas). Portfolio risk
is estimated based on
the correlation matrix
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3. Risk measures:
VaR is quickly
generated as a
multiple () of the
standard deviation.
42
Variance-covariance approach
 Assumptions and limits of the variancecovariance approach
 Normal distribution assumption of market
factor returns
 Stability of variance-covariance approach
 Assumption of serial indepence of market
factor returns
 linear sensitivity of positions (linear payoff)
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43
Normal distribution assumption
Possible solutions
1. Student t
 Entirely defined by mean, std. deviation and degrees of freedom
 Lower v (degrees of freedom)  fatter tails
Comparison between Normal and Student t distributions
Multiple of standard deviation
Student t with v degrees of freedom
Standardized
Confidence LevelNormal
v=10
v=9
v=8
v=7
v=6
99.99%
3.72
6.21
6.59
7.12
7.89
9.08
99.50%
2.58
3.58
3.69
3.83
4.03
4.32
99.00%
2.33
3.17
3.25
3.36
3.50
3.71
98.00%
2.05
2.76
2.82
2.90
3.00
3.14
97.50%
1.96
2.63
2.69
2.75
2.84
2.97
95.00%
1.64
2.23
2.26
2.31
2.36
2.45
90.00%
1.28
1.81
1.83
1.86
1.89
1.94
Mafinrisk - Sironi
v=5
11.18
4.77
4.03
3.36
3.16
2.57
2.02
v=4
15.53
5.60
4.60
3.75
3.50
2.78
2.13
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Normal distribution assumption
Possible solutions
2. Mixture of normals (RiskMetrics™)


Returns are extracted by two normal distributions
with the same mean but different variance
Density function:
PDF  p1  N1  1 , 1   p2  N 2  2 ,  2 



The first distribution has a higher probability but
lower variance
Empirical argument: volatility is a fucntion of two
factors: (i) structural and (ii) cyclical
The first have a permanent effect on volatility
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Linear sensitivity
Assumption of linear payoffs

In reality many instruments have a non linear
sensitivity: bonds, options, swaps
Possible solution: delta-gamma approach



2
VARi  VM i   i     i      i 


2
This way you take into account “convexity”

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Linear sensitivity assumption
Assumption of linear payoffs



Problem: the distribution of portfolio changes
derives from a combination of a linear
approximation (delta) and a quadratic one
(gamma)  the functional form of the
distribution is not determined
Some option portfolios have a non monotonic
payoff  even the expansion to the second
term leads to significant errors
Possible alternative solution to delta-gamma:
full valuation  simulation approaches
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47
Questions & Exercises
1. An investment bank holds a zero-coupon bond
with a life-to-maturity of 5 years, a yield-tomaturity of 7% and a market value of 1 million
€. The historical average of daily changes in
the yield is 0%, and its volatility is 15 basis
points. Find:
(i) the modified duration;
(ii) the price volatility;
(iii) the daily VaR with a confidence level of 95%,
computed based on the parametric (deltanormal) approach
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48
Questions & Exercises
2. A trader in a French bank has just bought Japanese
yen, against euro, in a 6-month forward deal. Which of
the following alternatives correctly maps his/her
position?
A. Buy euro against yen spot, go short (make a debt) on
yen for 6 months, go long (make an investment) on
euro for 6 months.
B. Buy yen against euro spot, go short (make a debt) on
yen for 6 months, go long (make an investment) on
euro for 6 months.
C. Buy yen against euro spot, go short on euro for 6
months, go long on yen for 6 months.
D. Buy euro against yen spot, go short on euro for 6
months, go long on euro for 6 months.
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49
Questions & Exercises
3. Using the parametric approach, find the VaR of
the following portfolio:
(i) assuming zero correlations;
(ii) assuming perfect correlations;
(iii) using the correlations shown in the Table
Asset
Stocks (S)
Currencies (C)
Bonds (B)
VaR
50.000
20.000
80.000
 (S,C)
0,5
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 (S,B)
0
(C,B)
-0,2
50
Questions & Exercises
4. Which of the following facts may cause the VaR of a
stock, estimated using the volatility of the stock market
index, to underestimate actual risk?
A) Systematic risk is overlooked
B) Specific risk is overlooked
C) Unexpected market-wide shocks are overlooked
D) Changes in portfolio composition are overlooked
5. The daily VaR of the trading book of a bank is 10 million
euros. Find the 10-day VaR and show why, and based
on what hypotheses, the 10-day VaR is less than 10
times the daily VaR
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51
Questions & Exercises
6.
Using the data shown in the following table, find the parametric VaR, with a
confidence level of 99%, of a portfolio made of three stocks (A, B and C),
using the following three approaches: (1) using volatilities and correlations of
the returns on the individual stocks; (2) using the volatility of the rate of return
of the portfolio as a whole (portfolio-normal approach) (3) using the volatility
of the stock market index and the betas of the individual stocks (CAPM). Then,
comment the results and say why some VaRs are higher or lower than the
others.
Market value (€ million)
Beta
Volatility
Correlation with A
Correlation with B
Correlation with C
Stock A
Stock B
Stock C
Portfolio
15
1.4
15%
1
0,5
0,8
15
1.2
12%
0,5
1
0
20
0.8
10%
0,8
0
1
50
1.1
9%
-
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Market
index
1
7%
52
Questions & Exercises
7. In a parametric VaR model, the sensitivity coefficient of
a long position on Treasury bonds (expressing the
sensitivity of the position’s value to changes in the
underlying risk factor) is:
A) positive if we use an asset normal approach;
B) negative if we use an asset normal approach;
C) equal to convexity, if we use a delta normal approach;
D) it is not possible to measure VaR with a parametric
approach for Treasury bonds: this approach only works
with well diversifies equity portfolios.
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Questions & Exercises
8.
A bank finds that VaR estimated with the asset normal method is
lower than VaR estimated with the delta normal method. Consider
the following possible explanations.
I)
Because the position analysed has a sensitivity equal to one, as
for a currency position
II) Because the position analysed has a linear sensitivity, as for a
stock
III) Because the position analysed has a non-linear sensitivity, as for a
bond, which is being overestimated by its delta (the duration).
Which explanation(s) is/are correct?
A) Only I
B) Only II
C) Only III
D) Only II and III
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Questions & Exercises
9. An Italian bank has entered a 3-months forward
purchase of Swiss francs against euros. Using the
market data on exchange rates and interest rates
(simple compounding) reported in the following
Table, find the positions and the amounts into
which this forward purchase can be mapped.
Spot FX rate EURO/SWF
3-month EURO rate
3-month SWF rate
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0.75
4.25%
3.75%
55
Questions & Exercises
10. A stock, after being stable for some time, records a sudden,
sharp decrease in price. Which of the following techniques
for volatility estimation leads, all other things being equal,
to the largest increase in daily VaR?
A. Historical volatility based on a 100-day sample, based on an
exponentially-weighted moving average, with a  of 0.94
B. Historical volatility based on a 250-day sample, based on a
simple moving average
C. Historical volatility based on a 100-day sample, based on an
exponentially-weighted moving average, with a  of 0.97
D. Historical volatility based on a 250-day sample, based on an
exponentially-weighted moving average, with a  of 0.94
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Questions & Exercises
11. Consider the different techniques that can be used to
estimate the volatility of the market factor returns.
Which of the following problems represents the socalled “ghost features” or “echo effect” phenomenon?
A. A volatility estimate having low informational content
B. The fact that volatility cannot be estimated if markets
are illiquid
C. Sharp changes in the estimated volatility when the
returns of the market factor have just experienced a
strong change
D. Sharp changes in the estimated volatility when the
returns of the market factor have not experienced any
remarkable change
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Questions & Exercises
12. Here are some statements against the use of implied
volatility to estimate the volatility of market factor
returns within a VaR model. Which one is not correct?
A) Option prices may include a liquidity premium, when
traded on an illiquid market
B) Prices for options traded over the counter may include
a premium for counterparty risk, which cannot be
easily isolated
C) The volatility implied by option prices is the volatility in
price of the option, not the volatility in the price of the
underlying asset
D) The pricing model used to compute sigma can differ
from the one adopted by market participants to price
the option
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58
Questions & Exercises
13. Assuming market volatility has lately been decreasing,
which of the following represents a correct ranking - from
the largest to the lowest – of volatility estimates?
A) Equally weighted moving average, exponentially weighted
moving average with  = 0.94, exponentially weighted
moving average with  = 0.97;
B) Equally weighted moving average, exponentially weighted
moving average with  = 0.97, exponentially weighted
moving average with  = 0.94;
C) Exponentially weighted moving average with  = 0.94,
exponentially weighted moving average with  = 0.97,
equally weighted moving average;
D) Exponentially weighted moving average with  = 0.94,
equally weighted moving average, exponentially weighted
moving average with  = 0.97.
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