1. EF506 Corporate Treasury Management 10/11/10 Value-at-Risk (VaR) References: Hull, ch. 20; Cuthbertson & Nitzsche, ch. 31 (handout); Dowd (handout)

2. Origins of Value-at-Risk (VaR) Measuring & managing risk is a key issue for: fund managers; bank regulators; hedge funds; corporate treasurers etc. Various categories of risk: Business; Market; Credit; Liquidity; Operational; Legal Financial institutions wanted an aggregate risk measure across the entire organisation JP Morgan’s Chairman, Weatherstone, got his staff to deliver a one-page summary – the ‘4.15 report’ Since 1994, JP Morgan has provided its RiskMetrics system public & data freely available  industry standard

3. Origins of VaR If at 4.15pm the reported daily VaR is $10m then: Maximum amount I expect to lose in 19 out of next 20 days, is $10m OR I expect to lose more than $10m only 1 day in every 20 days (ie. 5% of the time) VaR of $10m assumes my portfolio of assets fixed Exactly how much will I lose on any one day? Unknown !!!

4. Question Being Asked in VaR Progression from portfolio theory VaR  following statement: “We are X% certain that we will not lose more than V euro in the next N business days” V = portfolio VaR  function of 2 parameters: 1) N: time horizon + 2) X: confidence level i.e. VaR = loss corresponding to (100 – X)th percentile of the distribution of the change in the value of the portfolio over the next N days

5. Example: One-Day 90% USD VaR Value-at-risk equals the amount of money such that there is a 90% probability of the portfolio losing less than that amount over the next trading day.

6. Time Horizon Instead of calculating the 10-day, 99% VaR directly analysts usually calculate a 1-day 99% VaR and assume This is exactly true when portfolio changes on successive days come from independent identically distributed normal distributions Regulators base capital requirements for banks on VaR, i.e. market-risk capital is k times the 10-day 99% VaR where k is at least 3.0

7. VaR: A More Complete Definition VaR can be used in 4 different ways: a) Amount of money: max. loss over some period at some confidence level b) VaR estimation procedure – 3 standard methods Next week: c) VaR methodology – applicable to other risks, e.g. cash-flow-at-risk (CFaR) + credit-at-risk d) VaR approach to risk management: e.g. ERM (Enterprise Risk Management)

8. Rapid Expansion - VaR now extended to many other decisions, risks & even firm-wide risk management…why? Public availability of RiskMetrics system Increased use + complexity of derivatives Complementary factors: volatility; cross-market linkages; IT developments VaR now industry standard for banks + financial houses  1996 Market Risk Amendment to Basel I Accord: charge for market risk based on bank’s internal VaR models Also used by large non-financial corporates (may not be appropriate here?)

9. Characteristics of VaR VaR has a number of features that make it a useful tool: VaR is a single number that can be easily understood by senior management. VaR can be calculated for any product. VaR can be calculated at any level, from position up to firm. VaR can be aggregated. When aggregated, VaR is sub-additive – e.g. aggregate VaR for a trading desk should be less than the sum of the VaRs of each strategy on the desk.  Hedges, or offsetting exposures should be effective in a VaR model.

10. Attractions & Uses of VaR Intuitive: describes magnitude of likely portfolio losses in a single number & asks a simple question: “How bad can things get?” VaR figure has 2 main characteristics: a) Common consistent measure of risk across different positions + risk factors b) Accounts for correlations b/e different risk factors Uses: set risk targets + position limits; internal capital allocation + requirements ; investment appraisal + performance evaluation; reporting

11. 3 Approaches to VaR To calculate VaR: - common measurement unit; time horizon; & probability (confidence interval) Estimating statistical distributions of returns: 1) Historical simulation 2) Monte Carlo simulation 3) Variance-covariance (model-building) Stocks (handout) – compares 3 methodologies

12. 1) Historical Simulation Cuthbertson & Nitzsche: 706-711 Uses past data directly as a guide to what might happen in the future Makes no assumptions about linearity or normality of asset returns - fits with empirical findings of skewness + kurtosis No explicit assumptions about volatilities of returns + correlations between them

13. Implementing Historical Simulation Create a database of the daily movements in all market variables. First simulation trial assumes that the percentage changes in all market variables are as on the first day Second simulation trial assumes that the percentage changes in all market variables are as on the second day and so on

14. Historical Simulation cont. Suppose we use m days of historical data Let v i be the value of a variable on day i There are m-1 simulation trials The i th trial assumes that the value of the market variable tomorrow (i.e., on day m+1) is

15. NASDAQ ETF: Historical Simulation

16. Cuthbertson & Nitzsche’s approach  3 steps (p 708): 1. Calculate $ change in value of current portfolio over each of last 1000 days 2. Order daily profits/losses from lowest to highest 3. VaR at 1% probability is 10 th data point from left  C&N: figure 31.3 (p. 707) & example using Excel (p.710) – in class

17. Historical Simulation: Final Issues Limitations: - Lacks flexibility: no sensitivity analysis - Costly: requires access to, & maintenance of, historic market databases - Data may be limited or even non-existent - Historical path may not be representative of future events - Requires use of valuation models

18. 2) Monte Carlo Simulation Stages are as follows Value portfolio today Sample once from the multivariate distributions of the  x i Use the  x i to determine market variables at end of one day Revalue the portfolio at the end of day

19. Monte Carlo Simulation Calculate  P Repeat many times to build up a probability distribution for  P VaR is the appropriate fractile of the distribution x  N E.g. - with 1,000 trials, the 1 percentile is the 10th worst case.

20. NASDAQ ETF: MCS

21. Monte Carlo: Final Issues Advantages: - Large number of simulated paths - No assumptions made about returns Limitations: - can be slow (although partial simulation can be used to speed it up simulation) - lot of computations; requires mathematical modelling

22. 3) Variance-Covariance Approach C&N: 699 – 706 Aka ‘model-building’ & ‘parametric’ Based on estimate of var-cov matrix of asset returns, using historical time series of asset returns to calculate standard deviations + correlations Available data (e.g. RiskMetrics): quick + easy to update Based on modern portfolio theory Assumes normal distribution of position returns(may be unlikely in practice given empirical evidence of skewness + kurtosis)

23. Daily Volatilities In VaR calculations we express volatility as volatility per day Should define  day as standard deviation of the continuously compounded return in one day - in practice, assume it is the standard deviation of the proportional change in one day

24. VaR of a Single Asset Are Daily Returns Normally Distributed?-not quite but close fat tails (excess kurtosis) peak is higher and narrower negative skewness small (positive) autocorrelations squared returns have strong autocorrelation, ARCH But niid is a (good) approx for equities, long term bonds, spot FX, and futures (but not for short term interest rates or options)

25. Microsoft Example: Hull Position worth $10 million in Microsoft shares Volatility of Microsoft is 2% per day (about 32% per year) Use N=1 and X=99 Assume the expected change in the value of the portfolio is zero (OK for short time periods) Assume that the change in the value of the portfolio is normally distributed

26. Microsoft Example continued VAR = $V 0 (2.33  ) From the normal table, N(–2.33)=0.01  2.33  = 2.33 x 0.02 = 0.0466  1-day 99% VaR = 0.0466 x 10,000,000 = $466,000 Q: What does this mean?  C&N: Table 31.1 (p. 701) uses 95% VaR

27. AT&T Example Consider a position of $5 million in AT&T Daily volatility of AT&T is 1% (approx 16% per year) Assuming change is normally distributed, the 1-day 99% VaR = 5,000000 x 2.33 x.01= $116,500

28. VaR : Portfolio of Assets Text Book Approach 95% VaR p = V p (1.65  p ) V p = total €’s held in whole portfolio Can we express the above formula in terms of VaR of each individual asset? (VaR 1, VaR 2 etc) Yes!

29. VaR : Portfolio of Assets Market uses € ($) held in each asset, V i Note that w i = V i / V p (substitute in above equation) Then where VaR i = V i (1.65  i ) for single asset-i Var (Portfolio) =

30. VaR for Many Assets Can we put VaR in matrix form? - Yes Let Z = [ VaR 1, VaR 2 ] (2 x 1 vector) C = [ 1  ;  1 ] = correlation matrix (2x2) THEN: VaR p = [Z C Z’] 1/2

31. VaR: Long & Short An arithmetical nuance If asset-1 is held long and asset-2 has been ‘short sold’ then for example V 1 = +€100 and V 2 = -€50 So when constructing the “Z” vector then: VaR 1 = (€100)1.65  1 and VaR 2 = (-450)1.65  2 still have Z = [VaR 1, VaR 2 ] and VaR p = [Z C Z’] 1/2

32. Worst-Case VaR Assume all correlations are +1 and all assets are held “long” then with all  =+1 gives VaRp = { VaR 1 + VaR 2 } -i.e. no “diversification effect”

33. Question 1 A US investor holds $10m of AT&T shares and has short sold $5m of IBM shares. Daily volatility of AT&T, 1σ = 1.5% and daily volatility of IBM, 2σ = 1.0%. The correlation between AT&T returns and IBM returns is - 0.1. Q: What is the $-VaR for this portfolio? What is the worst-case VaR?

34. Using Excel to Calculate VaR C&N: Excel example Table 31.3 Question 2 You have a portfolio consisting of £10,000 in each of 3 assets, 1, 2 and 3. You have calculated the daily standard deviations to be 5.418%, 3.0424%, 3.6363%. The correlation between returns on assets 1 and 2 is 0.962, between assets 1 and 3 is 0.403, and between assets 2 and 3 is 0.610. (a.) What is the VaR for this portfolio? (b.) What would be the worse-case VaR if returns on all assets were perfectly positively correlated?

35. RiskGrades TM (JP Morgan) www.riskgrades.com www.riskgrades.com RiskGrades TM consistent and reliable way to measure all market risks; standardised measure of volatility - RiskGrade = 0  asset has no price volatility (cash); RiskGrade = 1000  asset is 10 times as volatile as an asset or index with a RiskGrade of 100. - Very useful for comparing risks of assets & portfoliosSimplified way of measuring risk of investment portfolios based on var-cov approach Target - ‘small investors’; limited set of assets

36. RiskGrades: Lots of Functions RiskGrades are dynamic and vary over time  prices for financial instruments are constantly changing RiskGrades are updated daily in order to capture the changing level of risk over time. www.riskgrades.com (link on moodle) www.riskgrades.com 1. RiskMaps of Various Markets - Useful for benchmarking shares 2. RiskCharts for various assets (click Riskchart) http://www.riskgrades.com/retail/quote-riskchart.cgi - Plot changes over time (examples follow) 3. RiskGrades for portfolios - You can construct your own portfolios + track them - Examples in class: Microsoft & ISEQ

37. Calculating RiskGrades TM C&N (611-5): summarises methodology Formulas for RG of asset (22.20) + portfolio (22.21) where RiskGrade(i) is the RiskGrade for asset i, si is the current estimate for the daily return volatility of asset i, and sbase is the average daily return volatility of the international basket (which is set to 20% on an annualised basis).

38. RiskGrades TM cont. Diversification benefit = URG P – RG P % risk impact = marginal contribution to risk %RI = change in RG as assets are excluded / initial RG (formula 22.24) C&N illustrate methodology for stocks, bonds + options Example – in class

39. Calculating VaR from RiskGrades Consider a portfolio consisting of these 3 assets: A currency swap: Because of changes in exchange rates since the swap was first entered into, the swap now has a value of €2 million, or 8.7% of the portfolio’s total value; A bond: The market value of the bond is €17 million, which is 73.9% of the portfolio’s total value; A stock: The 10,000 shares are worth €4 million, or 17.4% of the portfolio’s total value.

40. VaR: Using RiskGrades From RiskGrades, the variance-covariance matrix of the assets’ daily returns is: Swap Bond Stock Swap 0.0090 -0.00008 0.00007 Bond 0.00040 -0.00010 Stock 0.00300 Q: Calculate the variance of the portfolio’s daily returns distribution, and use this to calculate the portfolio’s 1-day 95% Value-at-Risk (VaR).

41. VaR & Foreign Assets Question Suppose you have an inventory of 1,000 barrels of crude oil which is priced in US dollars but you are UK company. The price volatility of your inventory is therefore expressed in sterling (since your operating costs etc are also in sterling). The current spot price of oil is Soil = 70 $/barrel and the current sterling-USD exchange rate is $1.7 per £ (SFX= 0.5882 £’s per $) and the daily volatility of oil prices is 2.0% per day and the daily volatility of the exchange rate is 0.5% per day.

42. VaR & Foreign Assets Oil prices and the USD exchange rate are slightly negatively correlated, so when oil prices rise, then 1-USD buys less £-sterling, so that their correlation is -0.1. (a.) Intuitively, what determines the VaR? (b.) How would you calculate the VaR in sterling (£) over 25-days (using the variance-covariance method)? (c.) What is the ‘worse case VaR’? Note: Use method in C&N p. 704-6

43. When VaR Linear Model Can be Used If change in value of portfolio is linearly dependent on change in value of underlying market variables Various market variables can be used: - Portfolio of stocks - Portfolio of bonds: use modified duration formula: - Forward currency contract - Interest-rate swap

44. Q: VaR using Forward Contract Some time ago, a company entered into a six-month forward contract to buy £1 million for $1.5 million. The daily volatility of a six-month zero coupon sterling bond (when its price is translated into dollars) is 0.06% and the daily volatility of a six-month zero coupon dollar bond is 0.05%. The correlation between the returns from the two bonds is 0.8. The current exchange rate is 1.53 $/£. Calculate the standard deviation of the change in the dollar value of the forward contract in one day. What is the 10-day 95% VaR? Assume that the six-month interest rate in both sterling and dollars is 5% p.a. with continuous compounding.

45. Stress Testing + Back-Testing Stress Testing: involves testing how well a portfolio performs under some of the most extreme market moves seen in the last 10 to 20 years – e.g. 1987 Stock Market Crash; 1997/8 Asian Crisis - 2007-9 credit crunch: examine next week… - can link it to extreme value theory (EVT) Back-testing & tail probabilities (linked to Basel II): -e.g. How often was the loss > 10-day 99% VaR?  Related to current controversy: August credit crunch led to 1 in 10,000 year statistical losses for some banks  Was model wrong? Were assumptions incorrect?

46. Validation of Risk Measures 1.Individual Returns Series How many? Are about 5% of actual (individual) returns R t+1 ‘greater than’ the forecast of 1.65 s t+1|t ? Yes ! How big? Are the actual returns in lower 5 th percentile the same size as those for the normal distribution? “Yes” for equity, bond and spot FX - returns

47. Validation of Risk Measures 2. Portfolio of Assets  Portfolio Returns Take equal wtd portfolio of 200 assets. Forecast all the individual VaR i ’s = V i 1.65  t+1|t, calculate portfolio VaR for each day: VaRp = [Z C Z’] 1/2 then see if actual portfolio losses > only 5% of the time (over some historic period, eg. 252 days). C&N: fig. 31.6 (p. 714)

48. Other Statistical Issues with VaR Alternative forecasting techniques for parameter estimation (e.g forecasting volatility using EWMA, implied volatility, GARCH) Options: Linear model fails to capture skewness in the probability distribution of the portfolio value: if gamma is +ve (-ve), probability distribution of  P tends to have +ve (-ve) skewness VaR for a portfolio is critically dependent on the left tail of prob. distribution of  P: -ve gamma portfolio tends to have a fatter left tail than normal distribution More accurate VaR: use both delta + gamma measures Hull outlines quadratic model (not on EF506 syllabus)

49. VaR: Limitations 1. VaR forecasts likely future losses using past data – assumes stationarity 2. It does not tell you what the loss will be on the day that you lose more than the VaR value or does not tell you when the loss will happen. 3. Since VaR is a statistical measure, once you have had your one day’s big loss, still the same probability of losing more than your VaR figure on subsequent days.

50. VaR: Limitations 4. VaR deals with “expected” market conditions, not when things go badly wrong and markets move in a more extreme fashion (correlations break down; liquidity may disappear; price data may be unavailable) 5. VaR does not identify weaknesses in a portfolio. 6. Accuracy of VaR depends on data you feed in to it: particularly how quickly volatility can be incorporated.

51. VaR: Some Issues to Note Risk management as ‘much a craft as a science’ – VaR is only a tool VaR is not the only risk management tool a firm should use. VaR has to be used for regulatory and accounting disclosure purposes (e.g. Basel II) VaR should be used as a measure of aggregating risk across disparate product lines, businesses and locations. VaR should be used by people who know what it does and doesn’t provide. Dedicated website: www.gloriamundi.orgwww.gloriamundi.org

52. More on VaR Next Week a) Applications of VaR b) Pros and cons of VaR c) Cash-flow at Risk (CFaR) d) Enterprise Risk Management (ERM)