Challenges In Modelling Inflation For Counterparty Risk

Challenges In Modelling Inflation For Counterparty Risk Vinay Kotecha, Head of Rates/Commodities, Market and Counterparty Risk Analytics Vladimir Chorniy, Head of Market & Counterparty Risk Analytics Quant Congress Europe 2010 Group Risk Management Risk Capital Markets

Outline Inflation background Relationship between nominal rates, real rates and CPI index Inflation products Typical vanilla products and examples of exotic structures Counterparty risk on inflation products Computing potential future exposure on inflation derivatives Inflation simulation model Requirements for modelling for pricing versus counterparty risk purposes 3-factor HJM model for ratio of zero-coupon nominal and real rate bonds Simulation and back-testing results Simulation of percentile envelopes for inflation indices and year-on-year rates Performance of long-term simulations against back-testing 2

Inflation background (I) Inflation is monitored by means of an index e.g., Consumer Price Index (CPI), Retail Price Index (RPI): The index measures the cost of a representative basket of consumer goods in an economy in nominal terms Example: Assume CPI index at time zero is 100 and in 1Y it changes to 102 100 invested in risk free account becomes 105 Today: nominal 100 CPI 100 basket value 1 1Y: nominal 105 CPI 102 basket value 105/102 Over 1Y: nominal change: 105 / 100 1 = 5% inflation rate: 102 / 100-1 = 2% real change: [ nominal(1y) / CPI(1Y) ] / [ nominal(0) / CPI(0) ] (105 / 102) / (100 / 100) 1 = 2.94% 3

Inflation background (II) The link between the inflation index, nominal, real and inflation rates is analogous to FX market and can be explained via an inflation-linked zero-coupon bond An inflation linked zero-coupon bond has a single payment at maturity T; the notional grows with the index (e.g. CPI) For an inflation linked zero-coupon bond starting at time t and maturing at T: Its nominal value at maturity T is the CPI level at T: I(T) Its real value (valued relative to CPI) at maturity T Its real value at time t is the real zero-coupon bond price: Its nominal value at t is then The CPI is effectively the exchange rate between the nominal and real economy 4

Inflation background (III) This gives a simple relationship between inflation, nominal and real rates: Define the nominal and real continuously compounded zero rate, and : We have the relationship between the inflation/nominal/real zero rates: If necessary, real rates can be extracted from nominal and inflation rates using the above relationship For pricing purposes we need forward inflation index The forward index can be derived from the no-arbitrage relation: where is the nominal short rate. From the FX analogy 5

Standard inflation products Zero coupon swap A one-off exchange of an inflation linked cash-flow against a fixed-rate cash-flow Year-on-Year swap Yearly payment of realised inflation rate against a fixed or floating rate payment Cap/Floor on year-on-year rate Yearly payment of realised inflation rate which is capped or floored Cap/Floor on zero-coupon rate Inflation linked cash-flow which is capped or floored 6

Some popular exotic structures Pay-as-you-go swaps Payment of periodic or zero-coupon fixed (or floating) rate against zero-coupon inflation at maturity plus intermediate periodic cash-flows of the form: where j denotes the coupon index Limited price indexation (LPI) swaps Exotic structure composed of a strip of caps and floors where payoff for coupon i depends on previous coupon value: 7

Counterparty credit risk on inflation products Counterparty credit risk is linked to the replacement cost of a derivative in the event the counterpart defaults Measured by computing the potential future values of the derivative The potential future value of the derivative can result in exposure A positive value generates exposure to the counterpart (i.e. the counterpart owes us) A negative value generate no exposure (we owe to the counterpart) Typically, PFE on a derivative is computed using a Monte-Carlo simulation: Step 1: simulate possible future evolution of the market parameters or risk drivers, e.g., nominal rate, CPI, spot FX, etc, at various time-points Step 2: calculate the future value of the transaction at each time-point under each future market scenario Step 3: calculate the required measure from the resulting distribution of deal PVs Typically, potential future exposure on a derivative is computed using a Monte-Carlo simulation: Various measures can be computed from the distribution of future deal PVs A high percentile (e.g., 95%, 99%) for exposure limit monitoring purposes Expected positive exposure for CVA purposes Effective expected positive exposure for regulatory capital purposes 8

Computation of counterparty credit risk High-level prescription: For PFE purposes, simulation of risk drivers is done typically in real-world measure $/ FX rate Possible paths the $/ FX rate might take 6M $ LIBOR Today s 6M LIBOR 2% Possible paths the 6M $ LIBOR rate might take Today s Today Spot FX 5Y Time co-dependence Today 5Y Time PFE / EUR m PFE profile Today s portfolio PV Today 5Y Time Possible paths the counterparty portfolio PV might take 9

Inflation models in literature Various models are available in the literature Jarrow-Yildrim (2003) 3-factor model based on the FX analogy: nominal rate as domestic currency, real rate as foreign and inflation index as the spot FX rate Calibration of real rate process is problematic Mercurio market model (2004) Two market models alternative to Jarrow-Yildrim The first model results in lognormal Libor model for the nominal and real rates and GBM for forward inflation index Computationally expensive as formulas are non-analytic and also real rate volatility is difficult to calibrate The second model based on process for inflation index and circumvents problem of real rate calibration Based on an approximation that affects long maturities when correlation between forward nominal and inflation rates is non-zero Belgrade et al (2004) Market model considering forward inflation index as a GBM with deterministic drift and volatility Computationally expensive 10

Modelling for front-office vs. risk - lost in translation? Front office and Risk both aim at correct representation of behaviour is it just different language or different definition of right behaviour? Front office concerns Calibration instruments & model choices right behaviour is to match market prices Exotics vs model: pricing exotics (for example, P. Hagan*) natural hedging instruments versus calibration instruments right behaviour is to allow stable hedging Emergence of stochastic volatility models versus local volatility models Again hedging needs needs right behaviour So right behaviours most common denominator: match the market prices and allow good hedging What is considered right behaviour for counterparty risk? Adjusters:Turning Good Pricers into great Pricers, P. Hagan, Wilmott Magazine, 2002 11

Inflation modelling for counterparty risk To compute potential future exposure on inflation deals, we need an inflation simulation model able to capture long-term behaviour (~ 50Y) of inflation rates Over long term (50y) we need also to capture short term (10 days) dynamics of the rates - driver for collateralised risk Need to capture joint dynamics of nominal, real and inflation rates for both - 50y and 10d. [Simulation of either two gives the third (Fisher relation)] Need to ensure model reflects economic reality: Mean reversion of rates over long simulation horizons Nominal rates are typically positive; inflation may be negative; real-rates can become negative but not for long periods of time the pattern should hold Over range of time Range of percentiles with correct trends (e.g. longer or deeper negative real rates only for higher percentiles) Both for cumulative and forward measures Evolution of inflation reference index (e.g. CPI) is tied to inflation/nominal/real rates, but has its own dynamics Realised vs forwards relationship (special topics addressed further in the presentation) 12

Inflation model for counterparty risk (I) Recall that the forward inflation index is given by We define the ratio of nominal and real zero coupon bonds by such that Propose to simulate inflation forward rate process via the dynamics of the ratio and spot index process Nominal and real zero-coupon bond prices are modelled with a 3-factor HJM model under the nominal and real spot measures respectively 13

Inflation model for counterparty risk (II) The quantities are martingales under the domestic risk-neutral measure. We have then and The process for the ratio is then given by 14

Inflation model for counterparty risk (III) The expression can be simplified to For a Monte-Carlo simulation we use the closed-form solution to evolve over the time interval where We make the approximation The impact of the approximation on simulation results is minimal The calibration is significantly simplified 15

Model calibration The inflation process is simulated jointly with the nominal (3-factor HJM) and index (log-normal) processes At each time-point we construct the simulated forward index Calibration of correlations and volatilities driving the models is done using historical data For counterparty risk, calibration is done in real-world measure For inflation process Calibration is done typically to historical zero-coupon inflation rates Given the requirement to risk long dated (50Y) inflation deals, we may choose to calibrate using a long historical data window For index process Long time-series better capture historical cycles of inflation movement. Long time-series better capture possibilities of regime switches over long time-horizon Calibration is done to historical index time-series with long historical window 16

Simulation results (I) The graphs below show realised YoY inflation from model and empirical observations Simulated realised YoY inflation is computed from simulated index as Empirical realised UK RPI YoY inflation Envelope of realised YoY inflation for UK RPI 30.00% 25.00% 20.00% 20.00% 1st percentile 10.00% 15.00% 10.00% 10th percentile 0.00% YoY 5.00% mean -10.00% 0.00% 90th percentile -20.00% -30.00% Jan-16 Jan-24 Jan-32 Jan-40 Jan-48 Jan-56 Jan-64 Jan-72 Jan-80 Jan-88 Jan-96 Jan-04-5.00% -10.00% -15.00% 0 5 10 15 20 25 30 99th percentile measurement point (Y) Envelopes of realised YoY inflation shows possible target ranges What is true range? What is considered as conservative? 17

Simulation results (II) Comparison of realised inflation from model and empirical observations The envelope of simulated realised inflation should sufficiently cover empirical observations. 9.00% 8.00% 7.00% 6.00% 5.00% 4.00% 3.00% 2.00% 1.00% 0.00% -1.00% Jan-45 Empirical realised 30Y ZC inflation of UK RPI Jan-52 Jan-59 Jan-66 Jan-73 Jan-80 Jan-87 Jan-94 Jan-01 Jan-08 realised 30Y inflation 15.00% 10.00% 5.00% 0.00% -5.00% Envelope of realised (average YoY) ZC inflation of UK RPI -10.00% 0 10 20 30 measurement point (Y) ZC1 ZC10 ZC mean ZC90 ZC99 18

Simulation results (III) Illustration of 10-day moves for HICP and UK RPI and French CPI 30Y ZC inflation rates 10-day moves are used to measure add-on risk for collateralised counterparts 19

Simulation results (IV) Backtesting on forward inflation Empirical data is for 5Y fwd inflation since 1980. Historically the fwd UK inflation has been consistently above the realised inflation until recent years. Realised vs predicted which relationship should be targeted for risk management? 20

Simulation results (V) Correlation analysis Correlation is important for example to provide potential netting effect between inflation and nominal rate referenced deals in the same portfolio Correlation matrices of CPI/Fwd inflation rate/fwd nominal rates for HICPxT, French CPI and Euribor are compared - from empirical data and simulation. 21

Simulation results (VI) Practical problems Forward inflation/nominal/real rate Forward ZC inflation rates demonstrate mean-reversion. The envelope of fwd inflation reflects economically meaningful scenarios Envelope of forw ard 1Y ZC inflation rate of UK RPI Envelope of forward 30Y ZC inflation rate of UK RPI 20.00% 20.00% 15.00% 10.00% 5.00% 0.00% -5.00% 1st-percentile 10th-percentile mean 90th-percentile 99th-percentile 15.00% 10.00% 5.00% 0.00% 1st-percentile 10th-percentile mean 90th-percentile 99th-percentile -10.00% 0 10 20 30-5.00% 0 10 20 30 measurement point (Y) measurement point (Y) Real rate is not directly simulated but implied from simulated forward inflation and nominal rates. The envelope of real rate depends on the mean-reversion speeds for both inflation and nominal rates, and the correlation between the two. Note however that there are scenarios where forward real rates are consistently negative. These scenarios are not realistic especially for long-tenored real rates, e.g. 10Y fwd real rate. Unrealistic real rate scenarios can potentially lead to over-estimation of exposure. 22

Analysis on PAYG Swap (I) To illustrate the practical problems we consider the PFE profile for a 30Y Pay As You Go inflation swap Recall that for a PAYG swap, payment of fixed (or floating) rate against adjusted zerocoupon inflation annually, plus compounded inflation paid periodically (e.g. 5 years) PAYG swaps are exposed to both realised and forward inflation and also to forward nominal rates. Unrealistic real rate scenarios can lead to over-estimation of exposure. Profile below is for a 30Y UK RPI PAYG swap Profile of a 30Y PAYG swap on UK RPI 1,400,000,000 1,200,000,000 EUR 1,000,000,000 800,000,000 600,000,000 400,000,000 200,000,000 0 0 10 20 30 measurement point (Y) 90-percentile 99-percentile MNPV+ The scenario leading to the peak exposure contains unrealistic forward real rates. A more realistic path on the other hand gives much lower exposure 23

Analysis on PAYG Swap (II) Scenarios for realised YoY and ZC inflation leading to peak exposure Realised ZC inflation (averaged YoY inflation) good projection realised ZC inflation in the path w hich gives 470mio EUR exposure at 20Y 12% 10% 8% 6% ZC 4% 2% Realised YoY inflation good projection 0% 0 5 10 15 20 25 30 measurement point (Y) realised YoY inflation in the path w hich gives 470mio EUR exposure at 20Y 20% 15% 10% 5% YoY 0% -5% 0 5 10 15 20 25 30 measurement point (Y) 24

Analysis on PAYG Swap (III) Fwd 1Y ZC inflation/nominal/real rate - real rate projection is conservative in the first graph fwd 1Y ZC inflation/nominal/real rate in the path which gives 470mio EUR exposure at 20Y 15% fwd 1Y inflation/nominal/real rate in the corrected path 20% 15% 10% fw d1y_inflation 5% fw d1y_nominal 0% fwd1y_real -5% -10% 0 10 20 30 measurement point (Y) Should have turned positive. Similarly for Fwd 5Y ZC inflation/nominal/real rates: 10% 5% 0% -5% -10% -15% 0 5 10 15 20 25 30 measurement point (Y) fwd1y_inflation fwd1y_nominal fwd1y_real fwd 5Y ZC inflation/nominal/real rate in the path which gives 470mio EUR exposure at 20Y 12% fwd 5Y inflation/nominal/real rate in the corrected path 15% 10% 10% 8% 6% 5% 0% fwd5y_inflation fw d5y_nominal fw d5y_real 4% 2% 0% fwd5y_inflation fwd5y_nominal fwd5y_real -5% -2% -4% -10% 0 10 20 30 measurement point (Y) Should have turned positive. -6% -8% 0 5 10 15 20 25 measurement point (Y) 25

Analysis on PAYG Swap (IV) Fwd 10Y ZC inflation/nominal/real rate (real rate projection - conservative) fw d 10Y ZC inflation/nominal/real rate in the path w hich gives 470mio EUR exposure at 20Y 0.12 fwd 10Y inflation/nominal/real rate in the corrected path 15% 0.1 10% 5% fw d10y_inflation fw d10y_nominal fw d10y_real 0.08 0.06 0.04 0.02 fwd10y_inflation fwd10y_nominal fwd10y_real 0% -5% 0 10 20 30 0 0 5 10 15 20-0.02-0.04 measurement point (Y) -0.06 measurement point (Y) Should have turned positive. Problem is solvable in the model by constraining the mean reversion speeds for nominal and the inflation process Corrected path graphs illustrate the behaviour 26

Forward versus realised inflation (I) Forward versus realised inflation Do we believe that different dynamics reflect reality? Charts illustrating forward versus realised inflation for 5 and 10Y inflation: Difference is material for risk assessment and measurement 27

Forward versus realised inflation (II) Forward versus realised inflation for 5Y tenor at various measurement points Forward rate computed as Realised rate is computed as We want distribution of realised inflation to be narrower than forward inflation 28

Forward versus realised inflation (III) Suppressing stochastic drift in the inflation index process gives the right behaviour? Using deterministic drift in inflation process: The forward premium appears over time Correct dynamics is only partially reflected Relationship between realised versus forward Model parameter? Stable or decreasing? How to calibrate? Realised versus forward. alike but distinct worlds 29

Conclusions Model could be built realising most of requirements of risk management within HJM framework Historic versus implied (risk-neutral) calibration is not the only difference comparing risk to pricing view Different dynamics requirements Forward versus realised modelling needs interpretation of market premium Model choices affects risk and potentially CVA pricing and hedging (will CVA overprice?) 30

References Jarrow & Yildrim (2003): Pricing Treasury Inflation Protected Securities and Related Derivatives using HJM Model N. Belgrade, E. Benhamou and E. Koehler (2004): A Market Model for Inflation F. Mercurio (2005): Pricing Inflation-Indexed Derivatives Acknowledgement : We are grateful to Zaifei Liu for many fruitful discussions and contributions to this work. 31

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