INSIDE THE DNC: American Tax Dollars and Counterparty Risk

Can American taxpayers afford to subject our economy to risky over the counter derivative trades which are leveraging out future?

Charlotte DNC are you listening?

Does the current and future President of the United States understand the complex financial modeling that global banks who accepted taxpayer money are using for their own profit?

I don't want my tax money spent padding the pockets of Goldman Sachs, Morgan Stanley, JP Morgan and Bank of America while they print money using elastic trade execution protocols on sketchy bets.

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Counterparty Risk Management

Christopher Finger

msci.comAgenda

 Introduction and definitions  What is counterparty risk? Why is it different from other credit risks?  Simple use cases  Definitions of exposure  Discussion of credit mitigation

 Applications of exposure measures  Banking models – reserve and market  Banking regulation  Investment firm applications

 Modeling topics

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What is Counterparty Risk?

 Traditionally, the risk that the counterparty in a bilateral contract fails to meet its obligations at some point in the future

 Typically associated with  Over-the-counter swaps, forwards, derivatives  Securities lending  Repurchase agreements  Settlement  Central clearing and exchanges

 Can also be considered as part of more “natural” credit risk taking  Bond investments  Structured finance investments  Loans

 Guarantees

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What Distinguishes These Risks?

 Risk horizon  Period of risk  Contingencies in exposure  Time profiles of exposure  Pricing and accounting for credit  Collateral  Credit mitigation (netting, mark-to-market), enforceability  Legal entity hierarchies  Active or passive risk taking

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Basic Measures of Exposure

 How much can be lost if the counterparty defaulted now, or at some future date?

 Loss is relative to the value of the contract under the assumption that the counterparty is riskless, i.e. the value from a standard swap or derivative pricer

 Assumption is that any contracts lost via a counterparty default are replaced with an equivalent contract with a new (riskless) counterparty

 Also assume that there is no gain from a counterparty default, i.e. obligations to the defaulting counterparty are honored

 Therefore counterparty losses are only possible on contracts that have positive value

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Simple Example – Interest Rate Swap

 I contract to receive fixed rate against Libor over 30 years  At the outset, the swap is fairly priced, so Present Value (PV) is 0 to me ...

no loss in the case of an immediate counterparty default  After one week, swap rates fall (rise) ...

 PV becomes positive (negative) to me,  A counterparty default causes (does not cause) a loss,  I have positive (zero) exposure to the counterparty

 Over the life of the contract ...  Rates could fall by a lot ... PV and exposure become positive  Rates could rise by a lot ... PV becomes negative, exposure zero  Cashflows are paid, remaining duration shrinks, swings in exposure are reduced

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Exposure Calculations Involve Simulating the Evolution of a Netting Set Through the Life Of All Trades

Source: Modeling, Pricing and Hedging Counterparty Credit Exposure – Giovanni Cesari et Al., Spring 2010

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Simple Example – Repurchase Agreement, Securities Lending

 I lend 100 to a counterparty, and accept a security worth 105 as collateral. Counterparty commits to repurchase security in 10 days

 At outset (ignoring discounting)  PV(riskless)=100-105=-5 to me  Zero exposure ... if default occurs today, loan is fully collateralized  Some risk (e.g. liquidity concerns) that I cannot actually sell the security for 105

 At maturity, if security value falls to 95  PV(riskless)=100-95=+5 to me  Positive exposure ... I stand to lose if the counterparty defaults

 Fundamental concepts are the same, but emphasis is on short-horizon volatility and not on aging of the contract

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Credit Mitigation

 Techniques to reduce exposure, particularly across multiple transactions  Netting

 Governed by standardized ISDA Master Agreements between counterparties  In the event of a default, counterparty cannot refuse to pay on negative PV contracts

while making claims for positive PV contracts

 Eligible contracts are valued and netted, with a single claim for the “netting set”

 For exposure measurement, distinguish between  Non-netted exposure ...  Netted exposure ...

 Complicated by complex parent-subsidiary structures, multiple agreements, leading to expressions like

 Non-netted exposure may still be of interest if enforceability is questionable (legal risk)

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Credit Mitigation

 Collateral  Counterparty posts cash or securities, to which creditor has access in the event of a

counterparty default

 So collateralized exposure is

 Legal risk – Do I really have access to collateral in a default event? Where is the collateral?

 Liquidity risk – In a large counterparty event, do many creditors try to sell the same types of collateral securities at the same time?

 Market risk – Could the collateral securities lose value, leading to undercollateralization?  To mitigate liquidity and market risk, lenders insist on overcollateralization by applying a

haircut to the collateral value. Recall repo example  The haircut should be related to the volatility and liquidity of the collateral, and to the

period of risk

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Credit Mitigation

 Mark-to-Market, Variation Margins  Collateral for derivatives is governed by the Credit Support Annex (CSA) to the Master

Agreement

 The threshold amount is the amount of credit exposure that will be left uncollateralized

 Variation margin is collateral that is posted against valuations above the threshold amount

 Margin period is the time between valuations and collateral posting

 This is similar to mark-to-market on futures contracts

 A counterparty default will be seen through a failed collateral posting. The position will then be closed out, and collateral seized against the overall claim

 For modeling purposes, assume that future collateral postings are cash, or track the haircut collateral value. Future market risk in the collateral is not modeled

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Variation Margin Can Change the Period Over Which We Are Exposed to Market Fluctuations

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Period of risk

Exposure

Variation Margin Can Change the Period Over Which We Are Exposed to Market Fluctuations

Last successful collateral call

Threshold

Margin period

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Variation Margin Can Change the Period Over Which We Are Exposed to Market Fluctuations

Default

Last successful collateral call

Threshold

Margin period

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Variation Margin Can Change the Period Over Which We Are Exposed to Market Fluctuations

Last successful collateral call

Threshold

Margin period

Default

Close out period

Period of risk

Exposure

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Measures of Exposure

 Exposure measures are understood to apply at the netting set (or higher) level, accounting for credit mitigation

 Current Exposure – positive present value (replacement cost) of claims  Potential Exposure or Potential Future Exposure (PFE) refer generally to

statistical measures for exposure at a future date, accounting for

 Market fluctuations  Collateral posting failures  Liquidity of collateral  Long-term evolution of positions

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Measures of Potential Future Exposure

 Specific definitions ... fixed horizon  Expected Exposure (EE) ...  Peak Exposure (also known as Potential Exposure or Maximum Exposure) ...

 Consider Expected Loss over full time horizon of the netting set, assuming independence of default and exposure processes

 Sampling points t0,...,tn. Default time ¿

EL =

' '

EI¿2(t0;tn] maxfX0;V(¿)g n

Pf¿ 2 (t0; tn]g Pf¿ 2 (ti¡1; ti]j¿ 2 (t0; tn]gEE(ti) i=1

Pf¿ 2 (t0; tn]g Xn ti ¡ ti¡1 EE(ti) i=1 tn¡t0

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Measures of Potential Future Exposure

 Compare previous expression to the expected loss on a term loan of size E

 So from the point of view of expected loss, the expression

may be considered as a loan equivalent amount

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Measures of Potential FutuXre Exposure n ti¡ti¡1

EPEeff = EEeff(ti)  Effective Expected Exposure i=1 tn ¡ t0

maxPE(ti) i

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 Expected Positive Exposure (aka Average Exposure)

 Effective Expected Positive Exposure  Peak of peaks (aka Maximum

Reserve Model – Loans

 The most basic way to price and manage credit risk  Positions are held (accounted for) at face value  How to distinguish between two loans with different credit quality?

 Valuation is equal

 Hold reserves based on expected loss due to default through the maturity of the loan  Hold capital to cover “unexpected losses”. How to define losses?

 Default losses until maturity for every loan ... capital is well defined, but not risk horizon  Default losses over a single (one-year) risk horizon ... no maturity effect  Default plus migration losses over a single horizon

 How to think about migration risk when there is no mark-to-market?  One option is to consider the risk of having to increase reserves on a long maturity loan

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Reserve Model – Loans

 Reserves (actuarial expected loss) plays the role of valuation  Hurdle rate of return for a specific loan is

 Cost of Funding + Reserves + Cost of Incremental Capital  To manage risks and create correct incentives, loan originators are

assessed a capital charge, in addition to a funding and reserve charge

 In addition, or instead of a capital charge, borrower limits are enforced in order to cap the maximum loss in the event of a specific credit event

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Reserve Model – Derivatives

 Motivation – a derivatives trader must choose between two trades 1. Swap contract with a high quality counterparty, where counterparty pays a fixed rate of

5% annually

2. Same contract (maturity, underlying, other terms), but to a poor counterparty. Because of credit concerns, trader can charge 6% annually

 Using traditional pricing models, the trader achieves an instant profit (enters a contract paying him 6%, while the market rate is 5%). Incentive is clearly to trade with poor quality counterparties

 How to make sure that derivatives traders are rewarded for trading derivatives?

 Would like to create a similar reserve and capital charge

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How to Calculate Reserves For Swaps/Derivatives?

 Complication is that future exposure is unknown  Under independence of credit and exposure processes,

 Where X is the default indicator and E is the future exposure at time of default  EPE denotes the Expected Potential Exposure

 Reserves are computed by summing over possible future times to default  Note that reserves formula is the same as for loans (with known

exposure), where EPE plays the role of a loan equivalent amount

 Setting reserves in this way implicitly assumes replacement of any derivative that is lost to a default, and continued benefit from mark-to- market gains

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Limits Under The Reserve Model

 The purpose of limits makes EPE less relevant  A quantile-based, or PFE metric is more typical and appropriate  A simple approach is to apply the “peak” PFE to the overall borrower limit  It is also common to enforce different limits on different maturity bands  Simplest practices include limits only on current exposure  Pre-trade compliance – “Can I do this deal with this counterparty?”

 Non-trivial, as interaction with existing positions in the netting set is important  Calculations must be fast enough for trading decisions

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Capital Under The Reserve Model

 Reserves cover expected losses, but unexpected losses come from two categories of effects

 Loan-like effects  Borrower/counterparty correlation  Counterparty concentration

 Derivative effects

 Uncertainty of default timing

 Uncertainty of exposure amount

 Interplay between counterparty exposures (e.g. long EUR with one counterparty, short with a second)

 Standard approach is to use EPE as loan equivalent in the capital model. This captures first category, but not second

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Market Model – Loans/Bonds

 Now consider loans or bonds in the trading book, subject to mark-to- market

 Market valuation can be thought of as the difference of  Value of the bond if it were risk-free,  Less the market-implied expected loss on the bond

Or  Value of the bond if it were risk-free,  Less the cost of default protection on the bond

 This decomposition isolates the effects of interest rates and spreads, and makes the reserves analogy clear

 Spread risk is a new source of market risk to be managed  A “jump-to-default” component of risk remains, which is different from

traditional market risk exposures  Credit traders must manage both of these

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Market Model – Derivatives

 The first step is to account for counterparty credit in the valuation  In simple terms, value a derivative similarly to the bond before:

 Value assuming counterparty is risk-free,

 Less market expected loss or cost of protection

 The second term is the Credit Valuation Adjustment (CVA)

 Technically, CVA should be an expectation over the risk-neutral measure covering both default and the derivative underlying

 Again assuming market-credit independence, CVA reduces to  In many practical cases, standard EPE is used

 First two terms are essentially the CDS spread, meaning there is a new source of P&L volatility

 So a loan equivalent view is that CVA is the cost of buying protection on the derivative’s EPE

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CVA Complications – Wrong-way Exposure

 Typically assume independence of the default and derivative underlying processes

 Plenty of examples challenge this assumption ... consider a contract with an emerging market counterparty who has liability in a developed market currency

 As described in Finger (2000), the same risk management policies may be used, but with expected exposure replaced by conditional expected exposure, given a default

 From a modeling point of view, it is natural to describe the opposite conditioning ... probability of default conditional on a market move

 Merton model ... linking credit to equity

 Intensity models ... linking default rate to spread process  In practice, market participants focus on obvious wrong-way transactions,

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CVA Complications – Credit Benefit

 To enter a new contract, I should require the counterparty to compensate me for the CVA

 But if I am risky, then the counterparty will require a similar compensation from me. So at contract inception, the value to me is the risk-free value less the net CVA of the counterparty and myself:

 Can embed the net CVA into the contract premium in order to clear at the desired price (i.e. V=0)

 If my credit deteriorates, then the term CVA(Me) increases. This term is known as a “credit benefit”, or a Debt Valuation Adjustment (DVA)

 Should I recognize increases in the credit benefit as “profit”?

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Market Model – Derivatives

 One approach to risk management is to impose CVA valuation on traders, and insist that credit risk management is their responsibility

 This places credit decisions outside groups with credit expertise, and is not aware of other positions to the same counterparties

 In a “CVA desk” concept, derivative traders must purchase default protection from an internal credit portfolio group

 Credit portfolio owns the credit-related P&L, and is free to hedge, make bets, utilize offsets, etc.

 Derivative traders retain the market-related P&L  Risk management is segregated by expertise  The concept is a good start, but issues remain ...

 Who sets market prices? Little standardization in exposure measures

 Who owns the risk in exposure size? Does the CVA desk sell contingent protection?

 What if the institution does not apply CVA valuation?

 Does it make sense to use risk-neutral valuation when there is no market or hedge for the contingent credit protection?

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“Basel Model” – Banking Book

 The Basel Minimum Capital Standards provide a floor for the amount of loss-bearing capital a bank must hold to support risky activities

 Treatments are different for assets in the banking book (hold to maturity, no mark-to-market accounting) versus the trading book (possibility to trade, mark-to-market)

 The Basel model for the banking book is mostly unchanged since Basel 2  Loan effects – the Internal Ratings-Based (IRB) formula

 Capital driven by a model of one-year portfolio default loss  Maturity adjustment is empirical ... effectively linear in maturity, with greater sensitivity

for best credits  Model captures broad market correlations, but not obligor concentrations  Broad correlations are a function of probability of default (PD)

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Internal Ratings-Based (IRB) formula

Loss Given Default Contribution to Expected Loss Maturity adjustment portfolio loss at

99.9% worst case

Exposure At Default

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Maturity factor (empirical)

Correlation function (political)

“Basel Model” – Banking Book

 Counterparty risk for derivatives is handled with a loan equivalent approach

 EAD is Effective EPE from before with some twists

 Special treatment in EPE formula for netting sets with short final maturity

 An alpha multiplier is added to reflect the additional risks due to exposure uncertainty. Supervisory value is 1.2, but banks may petition for lower levels based on their own analysis.

 Under Basel 3, correlations for financial obligors (meaning most derivatives counterparties) are raised

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Estimating Alpha

 Full model

 Simulate exposure amounts based on market risk model for underlying factors. Exposures across counterparties will be correlated through their joint dependence on the factors

 Independent of the exposures, simulate counterparty defaults (binary random variables), with correlations across counterparties as in the economic capital (e.g. CreditMetrics) model

 Simulated losses are the product of the default and exposure scenarios

 Compute worst case loss at 99.9% confidence  Loan equivalent model

 Treat exposures as fixed, and equal to their expected value for each counterparty  Simulate defaults and losses as above  Compute worst case loss at 99.9% confidence

 Alpha is the ratio of capital under the full model to capital under the simplified model

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Simple Alpha Factor Model Results

3 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1

-1

PD=0.01, rho=0.3

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Exposure correlation

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Alpha factor

Simple Alpha Factor Model Results

3 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1

-1

PD=0.001, rho=0.3

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Exposure correlation

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Alpha factor

Simple Alpha Factor Model Results

3 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1

-1

PD=0.01, rho=0.1

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Exposure correlation

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Alpha factor

“Basel Model” – Trading Book

 Under Basel 3, market risks are covered by multiple models  For traded loans/bonds, there are three additive capital charges:

 Value at risk (VaR), based on interest rate and spread (i.e. specific) risks, 10-day VaR at 99% confidence, multiplied by three

 Stress VaR – same charge, but based on volatility/correlation from a stressed period  Incremental Risk Charge (IRC), covering default and migration risks over a one-year

horizon at 99.9% confidence  If a bond is in the trading book, no banking (IRB) charge

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“Basel Model” – Trading Book

 For derivatives, there are a total of five charges  The first two are the VaR and Stress VaR charges related to risks from the

derivative underlying, with no counterparty risk

 The banking book charge, based on Effective EPE, is retained to cover default and migration risks

 Basel 3 adds two new charges to cover CVA-related spread volatility

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The CVA Capital Charge (updated December 2010)

 Apply the following “valuation formula”

Loss-given-default (possibly different from level used for banking book)

Credit spread for counterparty at maturity t_i

Market-implied probability of default

Expected exposure of netting set at maturity t_i

Average discounted exposure over ith sampling period

 Based on estimates of EE, apply standard market risk model, using counterparty spreads as the only risk factors

 Compute VaR and Stress Var charges  This is an update of the original “bond equivalent” proposal

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Investment Firms Are Different, So Which Model Applies?

 What differences exist?  Product mix – shorter horizons, more standard products  Number of counterparties  Concentration of counterparties  Decentralized structure ... funds as distinct legal entities  Distinction between investor risks and firm risks  Price takers or price makers  Valuation process  Active versus passive extension of credit

 This leads to most risk management operating at the fund level

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A Typical Breakdown of Responsibilities

 Fund level  Current exposure measurement based on valuations, application of fund-level netting  Limits setting and enforcement  Valuations often provided by counterparty ... CVA not included

 Firm level  Counterparty approval  Fundamental credit analysis on counterparties  Some guidelines on limits

 Neither the reserves or market model applies, though daily mark-to- market should lead to the market model

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Why Measure at an Enterprise Level?

 Consistency and discipline on  Regulation  Non-core investment activities  Valuation

 Limits  Manage enterprise exposure to a specific event (headline risk)

 Common performance impact across many funds  Do competitors suffer from the same event?  Liquidity needs?  Investor flight?

 Reliance on backstop facilities?

 Does the firm’s reputation suffer?  Exposure to a systemic event

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Regulations in Investment Space Focus On Current Exposure

 UCITS  Counterparty concentration limits based on current exposure only  No capital or CVA types of restrictions (no impact of counterparty credit quality)  Overall limits are enforced by limits on how much overall derivatives activity is permitted

 Form PF (proposed US hedge fund disclosure)  Current exposure to and from top five counterparties  Details on collateral arrangements  Focus is on “connectedness” rather than potential loss of a specific fund  No potential future exposure questions

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Modeling Needs

 Current exposure is largely a valuation exercise, but allowing for  Legal entity hierarchy  Netting  Initial collateral

 For short-horizon exposures (e.g. repos), a standard risk model suffices, provided it can accommodate the notions above

 For long-horizon exposures (e.g. OTC derivatives), the model also should cover

 Realistic long-term market scenarios  Path-dependent position effects (e.g. resets, calls, etc.)  Evolution (aging) of positions over time  Variation margin and close-out

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Generating Exposure Scenarios

 Requirements for long-horizon simulations  Troubles with short-term and pricing models  Building blocks

 Principal Component Analysis (PCA)

 Mean reversion  A non-parametric approach – Rebonato et al (2005)

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Requirements for Long-Horizon Simulations

 Bridge from current risk climate to a reflection of the unconditional distribution

 For interest rates, unlike for equities, it is reasonable to assume that an unconditional (or long-term) distribution exists

 Capture the unconditional distributions of levels, slopes and curvatures  Corollary: keep interest rates positive  Model an appropriate choice of returns  Capture the correct scaling from short- to long-term return variance  Produce plausible curve scenarios, even at long horizons

 Capture the correct eigenvector structure of short-term returns, with appropriate richness of curve moves

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Unconditional Distributions

 Examine distribution of interest rate levels and slopes over long histories  Two data sets:

 USD Swap daily data (1997-2011), RiskMetrics  USD Treasury monthly data (1953-2011), St. Louis Federal Reserve database

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Unconditional Distribution of USD Swap Rates, 1997-2011

350 300 250 200 150 100

50 0

350 300 250 200 150 100

50 0

350 300 250 200 150 100

50 0

0 0.01 0.02

12mo

0.03 0.04 0.05

60mo

0.03 0.04 0.05

240mo

0.06 0.07 0.08

0.03 0.04 0.05

0.06 0.07 0.08

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Unconditional Distribution of USD Swap Slopes, 1997-2011

250 200 150 100

0 -0.005

250 200 150 100

0 -0.005

250 200 150 100

0 -0.005

0 0.005 0.01

60mo-12mo

120mo-24mo

360mo-60mo

0.015

0.02 0.025 0.03

0.015

0.02 0.025 0.03

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Unconditional distribution of USD Treasury rates, 1953-2011

50 40 30 20 10

0 0.02

0.04 0.06 0.08

12mo

60mo

240mo

0.1

0.12 0.14 0.16 0.18

0.1

0.12 0.14 0.16 0.18

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Unconditional distribution of USD Treasury slopes, 1953-2011

100

0 -0.025

100

0 -0.025

100 80 60 40 20

0 -0.025

-0.02

-0.015

-0.01

-0.005 0

60mo-12mo

120mo-60mo

240mo-120mo

0.005

0.01 0.015

0.02 0.025

-0.01

-0.005 0

0.005

0.01 0.015

0.02 0.025

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Unconditional Distributions

 Examine distribution of interest rate levels and slopes over long histories  Two data sets:

 USD Swap daily data (1997-2011), RiskMetrics

 USD Treasury monthly data (1953-2011), St. Louis Federal Reserve database  Observations

 Swap rates since 1997 have ranged from close to zero to about eight percent. Longer maturity rates have not been lower than two percent. The distribution is quite flat

 Treasury rates (over a much longer history) display a more skewed distribution, with similar minimum levels as the swap rates, but maximum levels around 16%

 Swap curves are almost always upward sloping, with maximum slopes of about three percent

 Treasury curves have displayed some stronger inversion (of about two percent) historically, with similar maximal levels as the swap slopes

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Rebonato et al (2005) Figure4: StandardDeviationOfCurvaturesByMaturity

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Which Returns To Model?

 Consdier equities (ignoring dividends)  Price process is well modeled by a Geometric Brownian Motion (or variations thereof)  For relative returns (changes in log of price), magnitude is independent of price level  For simple returns (changes in price), magnitude is positively correlated with level  No controversy in modeling relative returns

 Consider a simple random walk price process  For simple returns, magnitude is independent of price level  For relative returns, magnitude is negatively correlated with level  Process is not guaranteed to stay positive

 So how to model interest rates?  Simple returns? Relative returns?  Changes in a different price transformation – Zumbach (2006)  Constant elasticity of variance (CEV) process – Rebonato (2005)

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US Swap Curve – Return Magnitude Versus Level

Simple returns

Relative returns

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Rebonato et al (2005) Figure20: CorrelationofReturnMagnitudetoRateLevel

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Variance Scaling

 Long-horizon returns are sums of small-horizon returns  For a simple random walk (constant volatility, independent increments),

this implies that variance of returns scales linearly

 Examine returns over different horizons using non-overlapping data. Does variance scale linearly?

 Less than linear scaling implies negative autocorrelation in returns, or mean reversion

 Greater than linear scaling implies positive autocorrelation in returns, or trending

 Generally, over horizons where empirical analysis is possible, there is evidence of trending with shortest maturity rates and of mean reversion with longest maturity rates

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Rebonato et al (2005) Figure 2: Variance of n-Day Relative Changes

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Rebonato et al (2005) Figure 2: Variance of n-Day Relative Changes

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US Swap Curves, 1997-2011 Variance of n-Day Relative Changes

1m 3m 6m 0.25 0.2 0.14

0.2 0.15 0.1 0.05

0.1 0.08 0.06 0.04 0.02

0 20 40 60 80 100 12m

0.15 0.1 0.05

0.08 0.06 0.04 0.02

0 20 40 60 80 100

0.04 0.03 0.02 0.01

0 20 40 60 80 100 120m

0 20 40 60 80 100 240m

0 20 40 60 80 100 360m

0.03 0.025 0.02 0.015 0.01 0.005

0.025 0.02 0.015 0.01 0.005

0.025

0.02

0.015

0.01

0.005

0.12 0.1 0.08 0.06 0.04 0.02 000

000

000 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100

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24m 60m 0.05

Plausible Curve Scenarios?

 Do simple models agree with the unconditional distributions at long horizons?

 Do generated curves look plausible?  Rebonato (2005) proposes a simple random walk model based on

resampling historical daily returns

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Simulated Levels – 1-year Horizon, Simple Returns

500 400 300 200 100

0.01 0.02

12mo

0.03 0.04 0.05 0.06

60mo

0.03 0.04 0.05 0.06

240mo

0.07 0.08

0.09

0.1

0.03 0.04 0.05 0.06

0.07 0.08

0.09

0.1

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Simulated slopes – 1-year Horizon, Simple Returns

400 300 200 100

0 -0.02

400 300 200 100

0 -0.02

400 300 200 100

0 -0.02

-0.01

0 0.01

60mo-12mo

120mo-24mo

360mo-60mo

0.02

0.03 0.04

0.05

0 0.01

0.02

0.03 0.04

0.05

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Simulated Levels – 5-year Horizon, Simple Returns

500 400 300 200 100

-0.06

-0.04

-0.02

0 0.02

12mo

60mo

240mo

0.04

0.06 0.08

0.1

0.12

-0.04

-0.02

0 0.02

0.04

0.06 0.08

0.1

0.12

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Simulated Slopes – 5-year Horizon, Simple Returns

400 300 200 100

-0.04

-0.02 0

60mo-12mo

120mo-24mo

360mo-60mo

0.02

0.04 0.06

0.08

-0.02 0

0.02

0.04 0.06

0.08

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Simulated Levels – 5-year Horizon, Relative Returns

1000 800 600 400 200 0

0 0.05

12mo

0.1 0.15

60mo

0.1 0.15

240mo

0.2 0.25

0.3

0.1 0.15

0.2 0.25

0.3

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Simulated Slopes – 5-year Horizon, Relative Returns

1500

1000

500

0 -0.25

1500

1000

500

0 -0.25

1500

1000

500

0 -0.25

-0.2 -0.15 -0.1

-0.05

60mo-12mo

0 0.05 0.1

120mo-24mo

0 0.05 0.1

360mo-60mo

0.15 0.2

0.25

-0.05

0 0.05 0.1

0.15 0.2

0.25

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Rebonato et al (2005) Figure 6: Yield Curves Obtained From The Naïve Method

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What About Using Pricing Models?

 Most of the applications for credit exposure require a real-world (objective) estimate

 To use pricing models, either must believe the risk-neutral measure or extract out the risk premium

 Reduced-form models (e.g. Hull-White)

 Work directly in the pricing measure; no estimate of risk premium

 Under typical conditions, produce a long-term positive drift on the short rate, as well as a tendency for curves to flatten

 Very different from observations of actual curve evolution  More advanced models (e.g. Market Model with smiles) are more complex than

desirable for credit exposure purposes

 Fundamental models (e.g. Vasicek)

 Produce a restricted set of possible curves (which is an issue for pricing as well)

 Specification of risk aversion designed for tractability, not for a true estimate of the objective measure

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Building Blocks – Principal Component Analysis

 Characterize short-term movements in the yield curve through an eigenvalue/eigenvector decomposition of the covariance matrix

 Let § be the covariance matrix of one-day returns. Decompose 0 ̧1 0:::01

0 ::: 0 ̧n

where 0 ̧2 ::: 0 ¤ = B @ . . . . . . C A

ri(t)=Xn p ̧U Zj(t);whereZ(t)»N(0;I) j ij n

 Then returns can be characterized by j=1

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Building Blocks – Principal Component Analysis

 In general, with most yield curves, most variance is “contained” in the first two principal components, that is

 So one candidate for smoothing yield curve moves is to remove some of the highest order principal components

 One problem with this approach is that it potentially ignores the risk of a partially hedged set of positions

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Building Blocks – Mean Reversion

 In discrete time, build mean reversion through a simple AR(1) process

 With μ<1, as t!1, the mean of Xt converges to 1, and the variance to

 Calibration

 3⁄4 comes from short-term risk estimates

 1 can come from unconditional distribution

 μ can be estimated through first order autocorrelation, but results are usually unstable ... true process operates over more actual lags

 Can set μ according to intuition on characteristic reversion time (e.g. five years), or by fitting the unconditional variance

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A Simple Parametric Approach

 Model yield curve returns as Gaussian

 Restrict to three principal components

 Set mean reversion on principal components to recover the long-term empirical distribution

 Criticisms:  Are resulting yield curve moves rich enough?  No coverage for trending behavior in short term rates

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yi(m)=yi(m¡1)+ri(u ); fori=1;:::n m

Rebonato (2005) Non-Parametric Model

 Begin with simple historical resampling  Initial yield curve denoted by yi(0) i=1,...,n  Denote historical yield curve returns by ri(t), t=1,...,T  To generate yield curve one day in the future:

 Pick a historical date u1 at random  Apply that date’s returns to intial yield curve

 Given yield curve after m-1 days, generate next day:  Pick a historical date um at random, independent of u1,...,um-1  Apply that date’s returns to the day m-1 yield curve

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¡¡

»i = Mi+1¡Mi ¡ Mi¡Mi¡1 Mi+1+Mi ¡ Mi+Mi¡1

Rebonato (2005) Non-Parametric Model

 Simple model suffers from problems of the random walk, as discussed, though it does exactly recover the short term risk properties, including all principal components

 To control the curves, introduce the notion of a “spring”

 At a particular maturity, examine the curvature iiiii

y(m)=y(m¡1)+hi(y1 ¡y(m¡1))+r(um); fori=1;n  If the curvature is too large, arbitrageurs will “push down” the middle yield relative to

the adjacent points

 Replace the “equation of motion” with  For the endpoints, introduce a mean reversion term

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Rebonato (2005) Non-Parametric Model

 Calibrate the “spring constants” to recover the unconditional distribution of curvatures

 The spring model recovers desirable unconditional distribution properties and produces sensible curve scenarios, while still approximating the short- term risk profile

 This model displays mean reversion (sub-linear variance scaling) at all maturity points

 To capture trending behavior, introduce a “box”  Pick an initial historical date v1 at random, as well as a length w (typical size of 50 days)  For m=1,...,w, set um= v1+m-1  Simulate equations of motion as before  At end of historical date range, pick another box, repeat

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Rebonato et al (2005) Figure 7: Eigenvalues of 1-Day Returns, Empirical and Simulated Data

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Rebonato et al (2005) Figure 8: First Eigenvector of 1-Day Returns

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Rebonato et al (2005) Figure 9: Second Eigenvector of 1-Day Returns

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Rebonato et al (2005) Figure 12: Variance of Curvatures, Empirical and Simulated Data

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Rebonato et al (2005) Figure 16: Variance of n-Day Relative Changes, Simulated Curves

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Rebonato et al (2005) Figure 16: Variance of n-Day Relative Changes, Simulated Curves

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Rebonato et al (2005) Figure 18-19: Sample Simulated Yield Curves, Seven-Year Horizon

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References

 Finger (2000). Toward a Better Estimation of Wrong-Way Credit Exposure. Journal of Risk Finance, Vol. 1, No. 3, pp. 43-51.

 Rebonato et al (2005). Evolving Yield Curves in the Real-World Measure: A Semi-Parametric Approach. Journal of Risk, Vol. 7, No. 3, pp. 29-62.

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Monday, February 27, 2012

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