Mafinrisk 2010 Market Risk course Value at Risk Models: the parametric approach Andrea Sironi Sessions 5 & 6

Mafinrisk - Sironi2 Agenda  Market Risks  VaR Models  Volatility estimation  The confidence level  Correlation & Portfolio Diversification  Mapping  Problems of the parametric approach

Mafinrisk - Sironi3 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 - Sironi4 Market Risks  Increasingly important because of:  Securitization  Diffusion of mark-to-market approaches  Huge losses (LTCM, Barings, 2008 crisis, etc.)  Basel Capital requirements

Mafinrisk - Sironi5 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

Mafinrisk - Sironi6 Value at Risk (VaR) Models Risk Maximum Potential Loss with a predetermined confidence level 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

Mafinrisk - Sironi7 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% VaR models: an example

Mafinrisk - Sironi8 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

Mafinrisk - Sironi9 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

Mafinrisk - Sironi10 Estimating Volatility of Market Factors’ Returns Historical Volatility: monthly changes of the Morgan Stanley Italian equity index (10/96-10/98)

Mafinrisk - Sironi11 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 - Sironi12 Example of simple moving averages

Mafinrisk - Sironi13 Example of simple moving averages

Mafinrisk - Sironi14 Example of simple moving averages

Mafinrisk - Sironi15 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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Mafinrisk - Sironi18 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

Mafinrisk - Sironi19 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

Mafinrisk - Sironi20 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 - Sironi21 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 - Sironi22 The normal distribution assumption Probabilità = 5% Profitto atteso (VM x δ x µ) α = 1,65σ VaR (95%)

Mafinrisk - Sironi23 The confidence level The higher the scaling factor, the higher is VaR, the higher is the confidence level

Mafinrisk - Sironi24 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 - Sironi25 The confidence level

Mafinrisk - Sironi26 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)

Mafinrisk - Sironi27 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 - Sironi28 An example Sum of VaRs: € 1,340,000 Diversification & correlations If correl. €/$-€/Yen = 0.54

Mafinrisk - Sironi29 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

Mafinrisk - Sironi30 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 - Sironi31 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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Mafinrisk - Sironi33 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

Mafinrisk - Sironi34 Mapping FX forward

Mafinrisk - Sironi35 Mapping FX forward Total VaR of the USD 6 m forward position

Mafinrisk - Sironi36 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 - Sironi37 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 *( * 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 - Sironi38 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

Mafinrisk - Sironi39 Mapping of a stock portfolio Example

Mafinrisk - Sironi40 Mapping of a stock portfolio Example with individual stocks volatilities and correlations

Mafinrisk - Sironi41 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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Mafinrisk - Sironi43 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 - Sironi44 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

Mafinrisk - Sironi45 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

Mafinrisk - Sironi46 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”

Mafinrisk - Sironi47 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 - Sironi48 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 - Sironi49 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.

Mafinrisk - Sironi50 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

Mafinrisk - Sironi51 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

Mafinrisk - Sironi52 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.

Mafinrisk - Sironi53 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 - Sironi54 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

Mafinrisk - Sironi55 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 - Sironi56 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 - Sironi57 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 - Sironi58 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

Mafinrisk - Sironi59 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.