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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.

© RiskMetrics

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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

© RiskMetrics

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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.