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### Mafinrisk 2010Market 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

Mafinrisk - Sironi

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

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

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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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10 yrs Treasury Bond

Market 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%

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

Market Value (Mark to Market)

An estimate of the future variability of interest rates (for a stock it would be the volatility of the equity 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)

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)

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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

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Estimating Volatility of Market Factors’ Returns

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

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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:
- Implied hp.: no serial correlation

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Estimating Volatility of Market Factors’ Returns

- 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

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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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VaR(95%)

Profitto atteso (VM x δ x µ)

α = 1,65σ

The normal distribution assumption

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The higher the scaling factor, the higher is VaR,

the higher is the confidence level

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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

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The confidence level

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)

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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

Sum of VaRs: € 1,340,000

If correl. €/$-€/Yen = 0.54

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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

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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

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Mapping FX forward

Example: Buy USD 1 mln 6 m forward

FX and interest rates

1. Debt in EUR

2. Buy USD spot

3. USD investment

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Mapping FX forward

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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

Mafinrisk - Sironi

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.

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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

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Mapping of a stock portfolio

Example with individual stocks volatilities and correlations

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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

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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)

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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

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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:
- 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
- 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

Mafinrisk - Sironi

Questions & Exercises

- 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:
- the modified duration;
- the price volatility;
- the daily VaR with a confidence level of 95%, computed based on the parametric (delta-normal) approach

Mafinrisk - Sironi

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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Questions & Exercises

3. Using the parametric approach, find the VaR of the following portfolio:

- assuming zero correlations;
- assuming perfect correlations;
- using the correlations shown in the Table

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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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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.

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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

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.

Mafinrisk - Sironi

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

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 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

Mafinrisk - Sironi

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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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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