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Mafinrisk 2010 Market Risk course

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.

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Mafinrisk 2010 Market Risk course

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

  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 Mafinrisk - Sironi

  6. 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 Mafinrisk - Sironi

  7. VaR models: an example 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% Mafinrisk - Sironi

  8. VaR models: an example 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 Mafinrisk - Sironi

  9. 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 Mafinrisk - Sironi

  10. Estimating Volatility of Market Factors’ Returns 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 averages Mafinrisk - Sironi

  15. Estimating Volatility of Market Factors’ Returns Exponentially weighted moving average (EWMA) = return of day t = decay factor (higher , higher persistence, lower decay) Mafinrisk - Sironi

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  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 correlation Mafinrisk - Sironi

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

  23. The confidence level 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 level Mafinrisk - Sironi

  26. 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 - 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. Diversification & correlations An example Sum of VaRs: € 1,340,000 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 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 Mafinrisk - Sironi

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  33. 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 - Sironi

  34. Mapping FX forward Mafinrisk - Sironi

  35. Mapping FX forward 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 • 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 Mafinrisk - Sironi

  39. Mapping of a stock portfolio Example Mafinrisk - Sironi

  40. Mapping of a stock portfolio Example with individual stocks volatilities and correlations 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 Mafinrisk - Sironi

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  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 solutions 1. Student t • Entirely defined by mean, std. deviation and degrees of freedom • Lower v (degrees of freedom)  fatter tails Mafinrisk - Sironi

  45. 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 - 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” Mafinrisk - Sironi

  47. 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

  48. 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

  49. 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 - Sironi

  50. 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 Mafinrisk - Sironi

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