Value at Risk Models: the parametric approach

1. Value at Risk Models: the parametric approach Giampaolo Gabbi

2. Agenda Market Risks VaR Models Volatility estimation The confidence level Correlation & Portfolio Diversification Mapping Problems of the parametric approach

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

4. Market Risks Increasingly important because of: Securitization Diffusion of mark-to-market approaches Huge losses (LTCM, Barings, etc.) Basel Capital requirements

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

7. Parametric VaR The VaR of a position is computed as the product of three factors: Market value (MV) Sensitivity of the position market value to changes in the relevant risk/market factor (?) The potential change in the market factor return, given by the product between: A volatility estimate of the market factor return (?) A scaling factor (?) corresponding to the desired confidence level

12. 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)

13. Example of simple moving averages

14. Example of simple moving averages

15. Example of simple moving averages

16. Estimating Volatility of Market Factors� Returns

19. Volatility: Garch models GARCH ? generalized autoregressive conditional heteroscedasticity Recognize that variance changes over time model volatility in an autoregressive way Variance in t is estimated based on 2 components: variance in t-1 and a market shock (surprise) in t-1

20. Garch model (p,q) Garch model (1,1): Variance is a function of three factors Constant ? should not be significantly different from zero Forecast of variance made in t-1 Forecasting error (surprise) Volatility: Garch models

21. Volatility: implied volatility Implied volatility Extract expected volatility from option prices Iterative process typically based on at the money options (1) Choose a pricing model (2) Compute theoretical option value (3) Modify volatility input until the option theoretical value equates the market value Not really used for RM purposes Need liquid markets for options ? not available for most market factors Counterparty risk may affect premium value Sometimes uncertainty about pricing model

22. 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

23. 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

24. 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

25. 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

28. 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.

29. The confidence level

30. The confidence level

31. Alternative approaches Asset normal vs. delta normal Asset normal Normal distribution assumption for positions market values (prices) Delta normal Normal distribution assumption for market factors returns The two approaches coincide if the sensitivity of positions is linear (e.g. foreign currency positions)

34. 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

35. 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 interest rates

36. 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

38. Mapping FX forward

39. Mapping FX forward

40. Mapping FX forward

41. 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 1 million FRA(3,6) can be seen as an agreement binding a party to pay � in 3 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

42. Mapping of a FRA

43. 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

44. Mapping of a stock portfolio

45. Mapping of a stock portfolio

46. Mapping of a stock portfolio

48. 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)

49. 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

50. 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

51. 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�

52. 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

53. 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

54. 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.

55. 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

56. 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

57. 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.

58. 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.

59. 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

60. 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.

61. 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

62. 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

63. 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

64. 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.