mafinrisk 2010 market risk course value at risk models: the parametric approach andrea sironi...

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


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


Market Risks

Increasingly important because of: Securitization Diffusion of mark-to-market approaches Huge losses (LTCM, Barings, 2008 crisis,

etc.) Basel Capital requirements


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


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


10 yrs Treasury BondMarket Value: € 10 mln

Holding 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


VaR models: an example

VaR = € 10 mln x 6 x (2*0.3%) = 360,000 EuroVaR = € 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


Estimating Volatility of Market Factors’ Returns

• Historical Volatility

Backward looking

• Implied Volatility

Option prices: forward looking

Three main alternative criteria

• Garch models (econometric)

Volatility changes over time autoregressive



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)



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)


Example of simple moving averages



Figure 3 – The ”Echo Effect” Problem

0,0%

0,4%

0,8%

1,2%

1,6%

2,0%

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Daily returns (right hand scale)

23-days moving standard deviation (left hand scale)


01 2

23

34

1

1 2 3 1

xt xt xt xtn xt n

n

...

...0 1

1 11

i xt ii


Exponentially weighted moving average (EWMA) = return of day t = decay factor (higher , higher persistence, lower decay)

tx


Figure 4 – An Example of Volatility Estimation Based Upon an Exponential Moving Average

0,0%

0,4%

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1,2%

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-4,0%

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12,0%


23-days simple moving standard deviation (left hand scale)

23-days exp. weighted moving standard deviation (left hand scale)


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

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0,0%

2,0%

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23-days exp. weighted moving standard deviation (l =0,94)





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 correlation

TdT



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


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



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


The normal distribution assumption

Probabilità = 5%

Profitto atteso (VM

x δ x µ)

α = 1,65σ

VaR(95%)



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



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.


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%



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

Ra

tin

g (S

tan

da

rd &

Po

or'

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)


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.


An example

CurrencyPosition (€

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

2

22

$,$22

$

€mmmm

VarVarVarVarVar YenYenYenTot

Diversification & correlations

If correl. €/$-€/Yen = 0.54


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


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


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

ti

tiSF

f

dt

1

1


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 $

€

0

time

6-month forward dollar purchase $

€

0

time

outflows

inflows

1

2

3

1+2+3


Mapping FX forward

Example: Buy USD 1 mln 6 m forwardFX 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


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


Mapping FX forward

USDspotmiUSDUSDspotmiUSDUSDspotmiEURUSDspotmiEUR

iUSDiEURmiUSDmiEURUSDspotmiUSDmiEUR

mUSDVaRVaRVaRVaR

VaRVaRVaRVaRVaRVaR

,66,66

,6622

62

6

622

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


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


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.


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


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


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


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


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%

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16%

-607

-543

-47

9

-41

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-35

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-160 -9

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-32 32

96

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607

Variazioni di valore del portafoglio (euro, valore centrale)

% d

i cas

i


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)


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


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 , ,


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


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


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



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.



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



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 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



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.

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 - -



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.



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 IB) Only IIC) Only IIID) Only II and III



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%



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



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


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


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.

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