1 RISKMETRICS Dr Philip Symes
1. Introduction 2 RiskMetrics is JP Morgan's risk management methodology. It was released in 1994 This was to standardise risk analysis in the industry. Scenarios are generated using: Historical simulation; Theoretical modelling; Stress testing scenarios. Metholodolgies are discussed in the short term limit Collateral is not modelled.
2. Contents 3 This presenation will focus on these topics. Risk Factors in the RiskMetrics approach. Methodologies for risk management. Products and pricing frameworks. Risk analysis and reporting.
3. Risk Factors 4 The main factors affecting portfolio value are modelled in RiskMetrics. Equities: Individual prices (absolute or relative to an index (β)); Index levels, e.g. FTSE 100; Affects equities and equity futures/options. FX rates: Affects cash positions, FX forwards/options and currency swaps. Commodity prices: Construct constant maturity curves; Affects spot and future prices.
4. Risk Factors (cont) 5 Interest rates are the fourth major factor. Yield curves are constructed from zero coupon and coupon bond prices; interest rate swap prices. Continuously compounded interest rate is used for simplicity other IR payments must be converted
5. Risk Factors (cont) 6 Coupon bonds are priced in terms of zero coupon bonds. Example: Bond maturing in 1 year; Semi-annual coupon of 10%: Same process is applied to swaps. IR are used for pricing swaps, options and fixed income.
6. Risk Factors (cont) 7 RiskMetrics also deals with less major factors that affect price. Credit spread: Construct yield curves with similar quality instruments; Calibrate: add a spread to each security. Implied volatility: Used for pricing options; Assume constant implied volatility if no historic data.
7. Empirical Models 8 Distribution of returns is given by past performance No theoretical models are used. The historical simulation method: Uses observations of actual changes in risk factors; Events are scaled with their frequency of occurrence; Models these changes to generate scenarios. Past observations must be scaled according to their volatility (Hull & White Model). Method includes extreme returns that occurred during the historical period.
8. Empirical Models (cont.) 9 Changes in asset prices are converted to risk factors. Formalise ideas in a matrix R of historical returns using of n risk factors with m daily returns: So each row of R corresponds to a specific scenario r.
9. Empirical Models (cont.) 10 Obtain a T-day P&L scenario from R: Take row/scenario r from R; This gives a vector of prices P (for each risk factor). Obtain price P of risk factor T days from now using Price each instrument using P 0 and scenario price P T. The portfolio P&L is given by
10. Theoretical Models 11 The multivariate normal model is used to predict returns: This model assumes lognormal returns; Geometric random walk; This is standard - see Hull or Wilmott for more details. Drifts are assumed to be zero (volatility dominates): No accurate predictions available for time horizons below 3 months; Zero assumption as good as any prediction.
11. Theoretical Models (cont.) 12 The return on the risk factor with these assumptions is: Volatility estimated from exponentially weighted moving average:
12. Theoretical Models (cont.) 13 An exponentially weighting moving average scheme is used to determine the decay factors: The optimal value was found by finding the minimum mean square difference between the variance estimate and the actual squared return on each day. Decay factors were set at: 0.94 (1-day) from 112 days of data; 0.97 (1-month) from 227 days of data. The number of days included comes from the fact that 3 99.9% of information is contained in the last ln10 ln λ days
13. Theoretical Models (cont.) 14 One day returns are: Conditioned on the current level of volatility; Independent across time; Normally distributed. RiskMetrics This does not preclude a heavy tailed unconditional distribution E.g. if volatilities dependent on the day of the week, then days could be dealt with separately.
14. Theoretical Models (cont.) 15 Multivariate method can be generalised to include multiple risk factors: these are correlated with a covariance matrix. In this case, the return for each asset i is now given by: And the covariance between i and j by:
15. Theoretical Models (cont.) 16 The covariance matrix is most easily written as: Where the mxn matrix of weighted returns is:
16. Theoretical Models (cont.) 17 Monte Carlo (MC) simulation: Generates scenarios from of random numbers; See MC in Finance presentation for more details. Generating random scenarios: Use Principle Component Analysis to derive formula.
17. Theoretical Models (cont.) 18 The c ij used in the formula are not unique: These coefficients satisfy certain requirements. They build up a vector C of units [c ij ]. The covariance matrix can then be written as: And the vector of returns as:
18. Theoretical Models (cont.) 19 Independent standard normal variables (ISNV) are used to generate random scenarios: L'Ecuyer method with 2x10 18 period; Will take 10 10 years to repeat scenarios. Matrix decomposition by Cholesky or Single Value decomposition methods: See FIDES presentation for details on matrix decomposition; Note that Cholesky decomposition only works for positive definite matrices; But any negative terms are redundant anyway.
19. Theoretical Models (cont.) 20 The scheme to generate the MC variables is: 1) Generate a set z of ISNV; 2) Transform ISNV to set of returns r, correlated to each risk factor using matrix C from c ij so 3) Obtain the price of each risk factor (as for historical simulation); 4) Price each instrument at current price and 1-day price scenario; 5) Get portfolio P&L (as for historical simulation).
20. Theoretical Models (cont.) 21 Parametric methods (PM) are an alternative to MC. The method uses approximate pricing for every instrument to get analytic formulae: Assumes lognormality of returns. PM uses a δ-method : It models changes in asset values in a portfolio; This is based on a linear approximation. This makes PM faster than MC MC is still often preferred as it is more accurate.
21. Theoretical Models (cont.) 22 The present value V is given by a 1 st order Taylor expansion: There is a simple expression for P&L where δ are delta equivalents :
22. Theoretical Models (cont.) 23 Assume the lognormality of returns, because: Lognormal returns aggregate nicely across time (temporal additive); One period returns are independent; This implies that the volatility scales with root of time consistent with MC; Average P&L from this method is 0 since instrument prices and risk levels are linear. The alternative is percentage returns These aggregate across assets.
23. Stress Testing 24 Stress tests are needed to complement statistical models: Stress tests and models predict different types of scenarios; Stress tests need certain types of credible scenarios. Selection of stress events is important, and can be: Historical events E.g. Tequila crisis in 1995; User defined simple scenarios E.g. interest rate steepeners; User defined predictive models These take account of correlations, etc.
24. Stress Testing (cont.) 25 Using historical events is a useful way of creating meaningful scenarios What would happen to my portfolio if the events that caused x crash happened again? In general, between times t and T, the historical returns are given by: The P&L for the portfolio based on this is:
25. Stress Testing (cont.) 26 The portfolio must be revalued based on the events in the stress scenario. The RiskMetrics framework: Defines changes for a subset of core factors; Uses these to predict the effect on peripheral factors. Covariance matrices are used for multiple core factors Approach corresponds to multivariate regression (as before).
26. Stress Testing (cont.) 27 Example with 1 core factor: $1,000 in Indonesian JSE equity index; Scenario of 10% currency devaluation (IDR): With β=0.2, JSE index drops by an average 2%.
27. Pricing Framework: Basic Concepts 28 Cashflows are the building blocks for describing positions in RiskMetrics. Cashflows must always be mapped and discounted: The NPV of a cashflow is the product of cashflow amount and discount factor; Cashflow mapping means that principal and coupon payments are converted to their equivalent zero coupon rates at the payoff date. Yield curves are treated in RiskMetrics as piecewise linear. Points between vertices are joined with straight lines. RiskMetrics uses continuous compounding (see earlier).
28. Pricing Framework Examples 29 The first example is a fixed coupon bond: Duration 2 yr; Par value $100; Interest rate 5% p.a.; semi-annual coupons; first coupon 4.75% at 6 m: Interpolation of interest rates from term structure RiskMetrics sum of discounted cashflows: $98.03
29. Pricing Framework Examples (cont.) 30 E.g. a vanilla interest rate swap: Fixed for floating, with exchange of notionals; 1.25 y to maturity. Floating leg: Firm receives 6-mo LIBOR (next value 6.0%); Use cashflow mapping for 3, 9 & 15 months: Fixed leg: Firm pays 5% semi-annually on $100M notional: Value of swap:
30. Pricing Framework Examples (cont.) 31 Options can also be priced in this framework, e.g. a bond option. Black's Model is an extension of Black-Scholes: Assumes lognormal distribution of the value of the underlying at maturity; Can be used for Eu options, IR derivatives, caps & floors and swaptions.
31. Pricing Framework Examples (cont.) 32 The bond forward price, F, is given by: Consider a 10-month Eu bond option on: 9.75-year bond, $1,000 par value, r=10% semi-annual coupon; Dirty price $960 and clean price of X=$1,000; 3, 9 and 10 month risk free IR's are 9%, 9.5% and 10% p.a.; σ=9% annualised volatility of T=10 month bond price; $50 coupons in 3 months and 9 months; Bond forward price is: Option price is $9.49
32. Risk Measures 33 Value At Risk is the industry standard methodology: It states that, at a certain confidence limit (e.g. 99%) no more that x will be lost in a T day period; The current value of portfolio is used for predicting losses; VAR is the method specified in Basel 2. Marginal VAR (MVAR) is an extension to the VAR principle: It shows the amount of risk a particular position is adding to portfolio; It uses the parametric approach to separate out the risks and find correlations.
33. Risk Measures (cont.) 34 Incremental VAR (IVAR)is similar to MVAR: IVAR uses MVAR to adjust portfolio risk; It shows the sensitivity of VAR to portfolio changes. However, there are several drawbacks with VAR: There is no estimate of the size of losses once the VAR limit is exceeded; VAR is not a coherent measure of risk.
34. Risk Measures (cont.) 35 Coherent measures of risk have these properties: Translational invariance Adding cash to a portfolio decreases risk by the same amount; Subadditivity Risk of the sum of portfolios is smaller than the sum of their individual risks; Positive homogeneity of degree 1 If the size of the positions doubles, the risk will double; Monotonicity If portfolio A has higher losses than B for all risk factors, then A is riskier than B.
35. Risk Measures (cont.) 36 Expected shortfall (ES) provides more information than VAR on tail of the P&L distribution: It gives an average measure of how heavy the tail is; It is a convex function of portfolio weights useful for risk optimisation; The ES is always higher than the VAR. ES is a coherent risk measure. Combined with VAR, ES gives a measure of the cost of insuring portfolio losses These two methods are complementary.
36. Risk Reporting 37 At the simplest level, reporting is just a P&L histogram Shows VAR and expected shortfall MC shows lowest figures Historical simulation shows most conservative figures RiskMetrics
37. Risk Reporting (cont.) 38 Often need more detailed analysis to dissect risk and identify risk sources in a portfolio. Drilldowns slice-up portfolio risk to give more details. Drilldown dimensions are these sub-categories: Position; Portfolio; Asset type; Counterparty; Currency; Risk type (FX, IR, etc.); Yield curve maturity.
38. Risk Reporting (cont.) 39 Drilldown dimensions come in two main groups. Proper dimensions are groups of positions: Position assigned to one bucket so easy to calculate; E.g. region could assign VAR to different regions. Improper dimensions are groups of risk factors: Position might correspond to more than one bucket; E.g. an FX swap has IR risk, FX risk and two yield curves. Simulation or parametric methods must be used.
37. Summary 40 RiskMetrics is the industry standard risk analysis methodology: But does not include collateral. We have dealt only with non-collateralised trades in the short-term limit. RiskMetrics can handle trades in different asset classes Some examples have been shown. RiskMetrics handles risk by defining core risk factors, analyses the risk using 5 different methods and reports the risk using 2 metrics. RiskMetrics can be expanded to include non-normal distributions, copulas, etc.