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RISKMETRICS
Dr Philip Symes
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1. Introduction
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.
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2. Contents
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.
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3. Risk Factors
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.
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4. Risk Factors (cont)
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
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5. Risk Factors (cont)
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.
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6. Risk Factors (cont)
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.
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7. Empirical Models
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.
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8. Empirical Models (cont.)
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.
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9. Empirical Models (cont.)
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 P0 and scenario price PT. The portfolio P&L is given by
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10. Theoretical Models
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.
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11. Theoretical Models (cont.)
The return on the risk factor with these assumptions is:
Volatility estimated from exponentially weighted moving average:
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12. Theoretical Models (cont.)
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 99.9% of information is contained in the last days
λln10ln 3−
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13. Theoretical Models (cont.)
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.
One day returns are:– Conditioned on the current
level of volatility;– Independent across time;– Normally distributed.
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14. Theoretical Models (cont.)
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:
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15. Theoretical Models (cont.)
The covariance matrix is most easily written as:
Where the mxn matrix of weighted returns is:
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16. Theoretical Models (cont.)
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.
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17. Theoretical Models (cont.)
The cij used in the formula are not unique:− These coefficients satisfy certain requirements.− They build up a vector C of units [cij]. − The covariance matrix can then be written as:
− And the vector of returns as:
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18. Theoretical Models (cont.)
Independent standard normal variables (ISNV) are used to generate random scenarios:
− L'Ecuyer method with 2x1018 period;− Will take 1010 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.
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19. Theoretical Models (cont.)
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 cij 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).
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20. Theoretical Models (cont.)
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.
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21. Theoretical Models (cont.)
The present value V is given by a 1st order Taylor expansion:
There is a simple expression for P&L where δ are “delta equivalents”:
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22. Theoretical Models (cont.)
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.
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23. Stress Testing
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.
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24. Stress Testing (cont.)
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:
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25. Stress Testing (cont.)
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).
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26. Stress Testing (cont.)
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%.
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27. Pricing Framework: Basic Concepts
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).
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28. Pricing Framework Examples
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:
− sum of discountedcashflows: $98.03
Interpolation of interest rates from term structure
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29. Pricing Framework Examples (cont.)
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:
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30. Pricing Framework Examples (cont.)
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.
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31. Pricing Framework Examples (cont.)
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
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32. Risk Measures
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.
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33. Risk Measures (cont.)
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.
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34. Risk Measures (cont.)
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.
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35. Risk Measures (cont.)
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.
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36. Risk Reporting
At the simplest level, reporting is just a P&L histogram − Shows VAR and expected shortfall
MC shows lowest figures
© RiskMetrics
Historical simulation shows most conservative figures
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37. Risk Reporting (cont.)
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.
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38. Risk Reporting (cont.)
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.
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37. Summary
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.