mafinrisk 2010 market risk course
DESCRIPTION
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. Market Risks. - PowerPoint PPT PresentationTRANSCRIPT
Mafinrisk 2010Market Risk course
Value at Risk Models: the parametric approach
Andrea SironiSessions 5 & 6
Mafinrisk - Sironi 2
Agenda
Market Risks VaR Models Volatility estimation The confidence level Correlation & Portfolio Diversification Mapping Problems of the parametric approach
Mafinrisk - Sironi 3
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
Mafinrisk - Sironi 4
Market Risks
Increasingly important because of: Securitization Diffusion of mark-to-market approaches Huge losses (LTCM, Barings, 2008 crisis,
etc.) Basel Capital requirements
Mafinrisk - Sironi 5
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
cVaRL 1Pr
Mafinrisk - Sironi 6
Value at Risk (VaR) Models
Risk
Maximum Potential Loss ... 1. ... with a predetermined confidence level2. ... within a given time horizon
VaR = Market Value x Sensitivity x Volatility
Three main approaches:1. Variance-covariance (parametric)2. Historical Simulations3. Monte Carlo Simulations
Mafinrisk - Sironi 7
10 yrs Treasury BondMarket Value: € 10 mlnHolding period: 1 month YTM volatility: 30 b.p. (0,30%)Worst case: 60 b.p.Modified Duration: 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%
VaR models: an example
Mafinrisk - Sironi 8
VaR models: an exampleVaR = € 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
Mafinrisk - Sironi 9
Estimating Volatility of Market Factors’ Returns
• Historical VolatilityBackward looking
• Implied VolatilityOption prices: forward looking
Three main alternative criteria
• Garch models (econometric)
Volatility changes over time autoregressive
Mafinrisk - Sironi 10
Estimating Volatility of Market Factors’ Returns
1
)(1
2
n
RRn
ti
t
01/10/96 6,74% 01/10/97 6,87%01/11/96 -5,38% 01/11/97 -3,20%01/12/96 6,92% 01/12/97 4,05%01/01/97 0,89% 01/01/98 7,68%01/02/97 14,42% 01/02/98 11,27%01/03/97 -3,76% 01/03/98 4,84%01/04/97 -1,93% 01/04/98 20,14%01/05/97 5,34% 01/05/98 -7,65%01/06/97 -1,47% 01/06/98 1,86%01/07/97 10,66% 01/07/98 1,33%01/08/97 7,76% 01/08/98 3,07%01/09/97 -2,37% 01/09/98 -16,69%
Standard Deviation = 7,77%
Historical Volatility: monthly changes of the Morgan Stanley Italian equity index (10/96-10/98)
Mafinrisk - Sironi 11
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)
Mafinrisk - Sironi 12
Example of simple moving averages
Mafinrisk - Sironi 13
Example of simple moving averages
Mafinrisk - Sironi 14
Example of simple moving averagesFigure 3 – The ”Echo Effect” Problem
0,0%
0,4%
0,8%
1,2%
1,6%
2,0%
7/02
/200
1
7/16
/200
1
7/30
/200
1
8/13
/200
1
8/27
/200
1
9/10
/200
1
9/24
/200
1
10/0
8/20
01
10/2
2/20
01
11/0
5/20
01
11/1
9/20
01
12/0
3/20
01
12/1
7/20
01
12/3
1/20
01
-8,0%
-4,0%
0,0%
4,0%
8,0%
12,0%
Daily returns (right hand scale)23-days moving standard deviation (left hand scale)
Mafinrisk - Sironi 15
01 2
23
34
1
1 2 3 1xt xt xt xt
n xt nn
...
...0 1
1 11
i xt ii
Estimating Volatility of Market Factors’ ReturnsExponentially weighted moving average (EWMA) = return of day t = decay factor (higher , higher persistence, lower decay)
tx
Mafinrisk - Sironi 16
Figure 4 – An Example of Volatility Estimation Based Upon an Exponential Moving Average
0,0%
0,4%
0,8%
1,2%
1,6%
2,0%
2,4%
7/02
/200
1
7/16
/200
1
7/30
/200
1
8/13
/200
1
8/27
/200
1
9/10
/200
1
9/24
/200
1
10/0
8/20
01
10/2
2/20
01
11/0
5/20
01
11/1
9/20
01
12/0
3/20
01
12/1
7/20
01
12/3
1/20
01
-8,0%
-4,0%
0,0%
4,0%
8,0%
12,0%
Daily returns (right hand scale)23-days simple moving standard deviation (left hand scale)23-days exp. weighted moving standard deviation (left hand scale)
Mafinrisk - Sironi 17
Figure 5 – An Example of Historical Volatility Estimation Based Upon Different Decay Factors
S&P 500 equally-weighted index daily returnsMoving standard deviations based on different decay factors
0,4%
0,5%
0,6%
0,7%
0,8%
0,9%10
/01/
2004
10/0
8/20
04
10/1
5/20
04
10/2
2/20
04
10/2
9/20
04
11/0
5/20
04
11/1
2/20
04
11/1
9/20
04
11/2
6/20
04
12/0
3/20
04
12/1
0/20
04
12/1
7/20
04
12/2
4/20
04
12/3
1/20
04
-2,0%
0,0%
2,0%
4,0%
6,0%
8,0%
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)
Mafinrisk - Sironi 18
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:
Implied hp.: no serial correlationTdT
Mafinrisk - Sironi 19
Estimating Volatility of Market Factors’ Returns
DailyVolatility
WeeklyVolatility
MonthlyVolatility
MIB 30EFFECTIVE 1,02% 2,64% 6,01%ESTIMATED - 2,28% 4,78%ERROR - 0,37% 1,24%
S&P 500EFFECTIVE 0,63% 1,40% 2,40%ESTIMATED - 1,40% 2,94%ERROR - 0,00% -0,54%
CAC 40EFFECTIVE 0,96% 2,07% 4,00%ESTIMATED - 2,14% 4,49%ERROR - -0,07% -0,50%
NikkeiEFFECTIVE 1,23% 2,68% 6,30%ESTIMATED - 2,75% 5,76%ERROR - -0,07% 0,54%
FTSE 100EFFECTIVE 0,61% 1,52% 5,16%ESTIMATED - 1,35% 2,84%ERROR - 0,16% 2,31%
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
Mafinrisk - Sironi 20
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
Mafinrisk - Sironi 21
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
Mafinrisk - Sironi 22
The normal distribution assumption
Probabilità = 5%
Profitto atteso (VM
x δ x µ)α =
1,65σ
VaR(95%)
Mafinrisk - Sironi 23
The confidence level
Confidence level
Scaling Factor (# of std.dev.s)
Potential losses (Treasury bond
example)99,5% 3 540.00099,0% 2,323 418.14097,5% 2 360.00095,0% 1,65 297.00090,0% 1,28 230.40084,0% 1 180.000
The higher the scaling factor, the higher is VaR, the higher is the confidence level
Mafinrisk - Sironi 24
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.
Mafinrisk - Sironi 25
The confidence levelMoody’s Rating Class 1-Year Probability of Insolvency Confidence Level Aaa 0.001% 99.999% Aa1 0.01% 99.99% Aa2 0.02% 99.98% Aa3 0.03% 99.97% A1 0.05% 99.95% A2 0.06% 99.94% A3 0.09% 99.91% Baa1 0.13% 99.87% Baa2 0.16% 99.84% Baa3 0.70% 99.30% Ba1 1.25% 98.75% Ba2 1.79% 98.21% Ba3 3.96% 96.04% B1 6.14% 93.86% B2 8.31% 91.69% B3 15.08% 84.92%
Mafinrisk - Sironi 26
The confidence level
Bnp
RBSDeutsche
SGING
HBOS
Santander
Unicredit
BoSHSBC
Commerz
Rabobank
Lloyds
Calyon
NatixisBBVA
Intesa SP
6,00 7,00 8,00 9,00 10,00 11,00
Ratin
g (S
tand
ard
& P
oor's
)
Tier 1 capital
AA+
AAA
AA
AA-
A+
A
Better rated banks should have a higher
Tier 1 capital
The empirical relationship is not precisely true for a
group of major European banking
groups
Rating agencies evaluations are also
affected by other factors (e.g.
contingent guarantee in case of a crisis)
Mafinrisk - Sironi 27
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.
Mafinrisk - Sironi 28
An example
Currency Position (€ mln)
Worst case (1.65*std.dev.)
VaR (Euro)
USD -50 0.92% 460.000Yen 50 1.76% 880.000
Sum of VaRs: € 1,340,000
821,74054.0880)460(2880460
222
$,$22
$
€mmmm
VarVarVarVarVar YenYenYenTot
Diversification & correlations
If correl. €/$-€/Yen = 0.54
Mafinrisk - Sironi 29
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
222EquityIRFXTot VarVarVarVar
Mafinrisk - Sironi 30
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
Mafinrisk - Sironi 31
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
titi
SFf
dt
11
Mafinrisk - Sironi 32
Figure 4 – Mapping of a 6-Month Forward Dollar Purchase
6-month EUR-denominated debt
€
€ 6-month EUR-denominated debt
€
€
Spot dollar purchase
€
$ Spot dollar purchase
€
$
6-month USD-denominated investment
$
$6-month USD-denominated investment
$
$
€
$
€
$
6-month forward dollar purchase $
€
0time
6-month forward dollar purchase $
€
0time
outflows
inflows
1
2
3
1+2+3
Mafinrisk - Sironi 33
Mapping FX forwardExample: Buy USD 1 mln 6 m forward
FX and interest rates
099.990$5,002,01
000.000.1
USDI
119.118.12,1099.990 EURD1. Debt in EUR
2. Buy USD spot
3. USD investment
2,1099.990 spotUSD
EUR/USD Spot 1,206 m EUR interest rate 3,50%6 m USD interest rate 2,00%EUR/USD 6 m forward 1,209
Mafinrisk - Sironi 34
Mapping FX forward
849.18483,0326,2%5,1119.118.16 EURVaR miEUR
259.16549.13490,0326,2%2,1099.9906 EURVaR miUSD
919.82099.69326,2%3099.990 EURVaRUSDspot
Market factor Volatility EUR/USD EUR 6 m IR USD 6 m IREUR/USD Spot 3% 1 -0,2 0,4EUR 6 m IR 1,50% -0,2 1 0,6USD 6m IR 1,20% 0,4 0,6 1
Correlation with…Volatilities and correlations - Forward position market factors
Mafinrisk - Sironi 35
Mapping FX forward
USDspotmiUSDUSDspotmiUSDUSDspotmiEURUSDspotmiEUR
iUSDiEURmiUSDmiEURUSDspotmiUSDmiEURmUSD VaRVaRVaRVaR
VaRVaRVaRVaRVaRVaR
,66,66
,6622
62
66 22
2
646.834,0919.82)259.16(2)2,0(919.82849.182
6,0)259.16(849.182919.82259.16849.18 222
Total VaR of the USD 6 m forward position
Mafinrisk - Sironi 36
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
Mafinrisk - Sironi 37
Mapping of a FRA
1-11-2000 1-2-20011-8-2000
investment
1m
1,013m1m
mf
1.013m
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.
Mafinrisk - Sironi 38
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 Portfolio VaR
jiii VMVaR
j
N
iiij VMVaR
1
Mafinrisk - Sironi 39
Mapping of a stock portfolioExample
817,7326,207,0481
%99,
j
N
iiiP VMVaR
Stock A Stock B Stock C PortfolioMarket Value (EUR m) 10 15 20 45Beta 1,4 1,2 0,8Position in the Market Portfolio (EUR m) 14 18 16Volatility 15% 12% 10%Correlation with A 1 0,5 0,8Correlation with B 0,5 1 0Correlation with C 0,8 0 1
Mapping of equity positions
Mafinrisk - Sironi 40
Mapping of a stock portfolioExample with individual stocks volatilities and correlations
589,9222 ,,,222
%99, CBCBCACABABACBAP VaRVaRVaRVaRVaRVaRVaRVaRVaRVaR
Stock A Stock B Stock C PortfolioMarket Value (EUR m) 10 15 20 45Beta 1,4 1,2 0,8Position in the Market Portfolio (EUR m) 14 18 16Volatility 15% 12% 10%Correlation with A 1 0,5 0,8Correlation with B 0,5 1 0Correlation with C 0,8 0 1
Mapping of equity positions
Stock A Stock B Stock C MappingVolatilities & Correlations
VaR(99%) 3.490 4.187 4.653 7.817 9.589
VaR of an equity portfolio
Mafinrisk - Sironi 41
Mapping of a stock portfolio
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
Stock A Stock B Stock C MappingVolatilities & Correlations
VaR(99%) 3.490 4.187 4.653 7.817 9.589
VaR of an equity portfolio
Mafinrisk - Sironi 42
Figure 6 – Main Characteristics of the Parametric Approach
2. Portfolio: 3. Risk measures:
stocks
rates
commodities
fx
1. Risk factors:Are defined either as price changes (assetnormal) or as changesin market variables(delta normal) theirdistribution is thensupposed to benormal.
ConfidentialReportfor theCompany’sC.E.O.
ConfidentialReportfor theCompany’sC.E.O.
Risk factors are mapped to individualpositions based on virtual componentsand linear coefficients(deltas). Portfolio riskis estimated based on the correlation matrix
VaR is quicklygenerated as a multiple () of the standard deviation.
0%
2%
4%
6%
8%
10%
12%
14%
16%
-607
-543
-479
-415
-351
-288
-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
Mafinrisk - Sironi 43
Variance-covariance approach
Assumptions and limits of the variance-covariance 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)
Mafinrisk - Sironi 44
Normal distribution assumption
Possible solutions1. Student t
Entirely defined by mean, std. deviation and degrees of freedom Lower v (degrees of freedom) fatter tails
Confidence LevelStandardized
Normal v=10 v=9 v=8 v=7 v=6 v=5 v=499.99% 3.72 6.21 6.59 7.12 7.89 9.08 11.18 15.5399.50% 2.58 3.58 3.69 3.83 4.03 4.32 4.77 5.6099.00% 2.33 3.17 3.25 3.36 3.50 3.71 4.03 4.6098.00% 2.05 2.76 2.82 2.90 3.00 3.14 3.36 3.7597.50% 1.96 2.63 2.69 2.75 2.84 2.97 3.16 3.5095.00% 1.64 2.23 2.26 2.31 2.36 2.45 2.57 2.7890.00% 1.28 1.81 1.83 1.86 1.89 1.94 2.02 2.13
Student t with v degrees of freedomMultiple of standard deviation
Comparison between Normal and Student t distributions
Mafinrisk - Sironi 45
Normal distribution assumptionPossible solutions2. Mixture of normals (RiskMetrics™)
Returns are extracted by two normal distributions with the same mean but different variance
Density function:
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
PDF p N p N 1 1 1 1 2 2 2 2 , ,
Mafinrisk - Sironi 46
Linear sensitivity
Assumption of linear payoffs In reality many instruments have a non
linear sensitivity: bonds, options, swaps Possible solution: delta-gamma approach
This way you take into account “convexity”
VAR VMi i i i i
22
Mafinrisk - Sironi 47
Linear sensitivity assumptionAssumption 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
Mafinrisk - Sironi 48
Questions & Exercises
1. An investment bank holds a zero-coupon bond with a life-to-maturity of 5 years, a yield-to-maturity 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 (delta-normal) approach
Mafinrisk - Sironi 49
Questions & Exercises2. 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.
Mafinrisk - Sironi 50
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 VaR (S,C) (S,B) (C,B) Stocks (S) 50.000 0,5 0 -0,2 Currencies (C) 20.000 Bonds (B) 80.000
Mafinrisk - Sironi 51
Questions & Exercises4. 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 overlookedB) Specific risk is overlookedC) Unexpected market-wide shocks are overlookedD) 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
Mafinrisk - Sironi 52
Questions & Exercises6. 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.
Stock A Stock B Stock C Portfolio Market index
Market value (€ million) 15 15 20 50 - Beta 1.4 1.2 0.8 1.1 1 Volatility 15% 12% 10% 9% 7% Correlation with A 1 0,5 0,8 - - Correlation with B 0,5 1 0 - - Correlation with C 0,8 0 1 - -
Mafinrisk - Sironi 53
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.
Mafinrisk - Sironi 54
Questions & Exercises8. 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 IB) Only IIC) Only IIID) Only II and III
Mafinrisk - Sironi 55
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 0.75 3-month EURO rate 4.25% 3-month SWF rate 3.75%
Mafinrisk - Sironi 56
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
Mafinrisk - Sironi 57
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 so-called “ghost features” or “echo effect” phenomenon?
A. A volatility estimate having low informational contentB. The fact that volatility cannot be estimated if
markets are illiquidC. 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
Mafinrisk - Sironi 58
Questions & Exercises12. 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
Mafinrisk - Sironi 59
Questions & Exercises13. 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.