it Risk Modeling and Pricing
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V ( t )2 VEgk + V$i'
Chapter 8
, even if the contractual collateral amount
CVco,( t )= max V t - V e k ,0) can be positive, there is no actual collat-
( 0
eral transfer as V g , ( t ) < V,;'
.
Thus, the actual collateral amount
VCOl(t)= 0 *Then, at the first time when V ( t )2 V$zi + VEi' , the counterpartywill be requested or called to post collaterals in the amount ofV& ( t ) - Vco,( t ) = V ( t ) - V&& 2 VEi' .Typically, the collateralscalled will be received at a later time, since there is a time delay Atco,between the time when the collaterals are called and the time when thecollaterals are received.Due to this time delay, even if both the threshold and the MTA arezero, the exposures would not be zero. Such exposures come from thevariance of the portfolio market value in the time period of AtCo, (whichis typically assumed to be 2 weeks). Such exposures are also similar to aseries of forward starting options with time-to-expiry of AtCo,.
8.9.5
Net Credit Charge and Funding Spread Calculation
The net credit charge (or net credit premium) PVNcc is the total PV ofthe two default options, one due to the counterparty default and the otherdue to one's own default, and is given by:
(8.97)The counterparty portfolio is equivalent to a corresponding portfoliowithout the counterparty credit risks plus the two counterparty defaultoptions. Thus, the PV of the counterparty portfolio (from one's ownperspective) is given by(8.98)where PV, is the PV of the corresponding portfolio without the counterparty credit risks. It is important to point out that, even though this corresponding portfolio has no counterparty credit risks, it can have creditrisk to the underling reference obligations (such as in the case of a CDStrade) that, unlike the counterparty credit risks, are typically priced in bythe market and by the micro models.
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Sometimes, especially for swaps, the net credit charge is not chargedas an up-front fee (in the amount of -PVNc-), rather it is charged as anadditional funding spread A S .In this case, we need to solve for the spread AS that is equivalent toPVNcc by solving the following equationpvO
(O) = pvO
(AS) f p v N C , (hs)
7
(8.99)
and the trade will be transacted at the price of PV,(O) that can be thesame as PVo in the previous case.It is important to point out that the spread payments are actually subject to default risk and, thus, the net credit charge PV,,,depends on
As.A simple approximated solution can be easily obtained for the aboveequation as typically AS is mathematically very small (or a very smallpercentage of the corresponding credit spread and the coupon rate).More specifically, the following approximation can be obtained througha Taylor expansion of the above equation, i.e.,
thus,AS =
-“NCC
(O)
I
ap‘NCC (AsdASIAs=o
(8.101)
As an example, if the counterparty credit spread (continuously compounded) AS = lOObp p a . , the duration of the derivatives trade
D = 1OYeurs , the counterparty bond or CDS recovery R (0, T ) = 50%,the counterparty recovery on the derivatives trade R, = 50%, thenPD(0,T)=19% and Vcc =-9S%V(O). We have used Eq. (8.10) inderiving these results.
Credit Risk Modeling and Pricing
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8.9.6
Chapter 8
Martingale Relationships in Credit Charge Calculations
In addition to the general martingale relationships that we have presentedin the previous chapters and sections, here we present additional martingale relationships. The key is to have the portfolio-based or macromodel in the credit charge calculations produce trade-level results consistent with thosc from thc trade-level or micro models. These are the necessary conditions for accurate credit charge calculations and can be usedin martingale accuracy testing and martingale control variate or resampling.From Eqs. (8.88) and (8.95), we can see that the default exposuresare options on the value of the counterparty portfolio. Thus, we firstneed to make sure that the portfolio-based or macro model can accuratelyprice the underlying derivatives instruments, i.e., each trade in the portfolio or match the price from the trade-level or micro model.More specifically, the trade-level overall exposure E$
[%]
is the
PV of the cash or dirty price of the tradc i for all cashflows with paymentdate after t (from the portfolio-based or macro model) and, thus, we needto test and ensure the following (first order) martingale relationship,(8.102)where 5(0,t) is the corresponding price from the trade-level or micromodel. This price is also termed as forward time t PV.For instance, for a simple swap, the trade-level overall exposure isthe PV of a corresponding forward starting swap. For path-dependentderivatives instruments (such as those with callable, barrier, and averaging features), the situation is more complicated and we need to explicitlyhandle the path-dependency in the above equation. For instance, for acancelable swap, when we look at its value at a future time, we need tohandle the probability that the swap has been cancelled before that time(in which case there will be no default risks).Furthermore, we need to test and ensure the following higher ordermartingale relationships,
Sec. 8.9 Counterparty Credit Risk Pricing in OTC Derivatives
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(8.103)and
(8.104)where Y'(0,t) is the price of the call option with expiry of t on theoriginal trades from the trade-level or micro model and V,- (0,t) is thenegative price of the corresponding put option. Both of these optionprices should be obtained from the trade-level or micro model.In other words, the trade-level positive (negative) exposure curves orprofiles correspond to a series of call (put) options on the original trade.For instance, the exposure profiles of a simple swap correspond to a series of swaptions. The exposures of a simple credit default swap (CDS)correspond to a series of extinguishing forward starting default swaptions.Thus, as we pointed out before, even if the underlying trades may notbe sensitive to volatilities, their exposures and credit charges are typically sensitive to volatilities, which is a typical challenge in the creditcharge pricing. For instance, pricing a simple CDS and a simpleCDO/CSO tranche does not require the modeling of the credit spreadprocess, but pricing their credit charges requires the modeling of jointdefault and credit spread processes.If the underlying trades are exotic derivatives, then pricing creditcharges becomes even more challenging.Figure 8.2 shows a simple example of such martingale relationships.This simple example does not capture the effect of the mismatch of thecashflow payment dates of the floating and the fixed legs. This effect, ofsecond order, can be important.
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Exposures and Martingale Targets of a Payer Swap10%-m
c
.-ij0
50
Pay. Swapti
8%
6%4%
$
2%
2
0%
(Short) RecSwaptn-POS. Exp.A
z0 -2%:: -4%W
. ...... Neg. Exp.
-6%
0
10
20
30
Time (Years)
Figure 8.2. The exposure profiles of a simple ATM payer IR swap as comparedwith the corresponding martingale targets or the PV of a series of swaptions.More specifically, the positive exposures (Pos. Exp.) should match the PV of aseries payer swaptions (Pay. Swaptn) and the negative exposures (Neg. Exp.)should match the PV of a series short receiver swaptions ((Short) Rec. Swaptn).They typically do not match exactly, unless we perform martingale resampling.
If the underlying trade is a long position of a simple option with onlyone cashflow at the final payment date T, then we have even simplermartingale relationship, i.e.,
6- (0,t ) = E$
min(Y(t),o)
.
(8.105)
(0 I t I T )In this case, the exposure profiles are constant with the positive orasset-side exposure equal to the PV of the underlying trade and the negative or the liability-side of the exposure equal to zero.
8.9.7
Closed-form Solutions and Approximations
Under various special cases, we can obtain closed-form solutions or approximations for the credit charge calculations. Here, we provide a few
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such examples.Let's start with the case of zero correlation among the market process, the default probability process, and the recovery process. In thiscase, the PV of the credit charge and the credit benefit or Eq. (8.87) and(8.94) can be written as
and
where
[m]7
yLiab
(8.108)
(O, I ) EO Q ' L i a b ( I )
and we have hrther assumed that expected recovery rates are constant(which is a usual assumption).In this case, finding the closed-form solutions or approximations tothe PV of the credit charge and the credit benefit is equivalent to findingthe closed-form solutions or approximations to the exposures VAssef(0, t )
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Credit Risk Modeling and Pricing
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and vLiab (O, t ) .If the portfolio has only one simple trade and has no collaterals, weoften can find closed-form solutions or approximations to the exposuresVAsset(0, t ) and vLiab (0, t ) . For instance, as we pointed out in the previous section, the exposures of a simple swap are a series of swaptions andthe exposures of a simple option (with only one cashflow at the finalpayment date) are either zero or the PV of the option itself.More specifically, if one has only one simple option with PV ofV (0) and the final payment time of T in the counterparty portfolio withno collaterals, then
(8.109)
and
v,, = 0 .
(8.110)
As an example, if the counterparty credit spread (continuously compounded) S = lOObp p.a. (per annum), the duration of the derivativestrade D=lOYears, the counterparty bond or CDS recoveryR (0, T ) = 50%, the counterparty recovery on the derivatives tradeRcP = 50%, then PD( 0 , T ) = 19% and V,, z -9S%V(O). We haveused Eq. (8.10) in deriving these results.If the bond (or CDS) recovery and the derivatives recovery are thesame, then we can use much simpler (but more crude) approximation asfollows
v,, z - V ( O ) S D .
(8.111)
The above equation can be generalized to
v,, = -FSD ,
v
(8.112)
whereis an average asset side exposure.According to Eq. (8.101), the additional funding spread AS is given
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by
As=
FSD(8.113)
aPVo(AS)
aA,s
dAS
M=O
Oftentimes, the denominator in the above equation can be approximatedby
where N is the notional. Thus,(8.115)
which is typically a few percent of S.The approximations that we discussed so far are very helpful for thereader to build intuitions on the credit charge calculation and its magnitude.If the portfolio has multiple trades, but no collaterals, then
v (0,t)=
VAsse, (O, t ) + VLjab (0,t)
(8.1 16)
In general, the forward time t PV V ( 0 , t ) is much easier to obtain (and isoften available from the underlying trade-level pricer) than the exposuresVAsser(0, t ) and VLiab(0, t ) . Thus, we can rewriteVNCC = Vcc + VCBT
=-
Iv (0, t ) (1 - R C P ) d,P$ (0, t )0
(8.1 17)
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Credit Risk Modeling and Pricing
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(8.118)
TvNCC z -
( 0 9
t )(1 - 'self
)dtp& (0,t )
(8.119)
0
is a conservative approximation.In general, with or without collaterals, we can use quadratic approximations to obtain exposures V,,,,, (0,t ) and V,, (0,t ) by matching the first two moments.The above procedures can also be applied to pricing CCDS and extinguishers.If needed, we can also perform adjustments due to the diffusion andjump correlations between the credit and market processes.
8.10
Framework for Counterparty Credit Risk Modeling andPricing
The prevailing derivatives pricing models or micro (or trade-level) models normally do not incorporate specific counterparty credit risk. As amatter of fact, it is quite difficult for micro (or trade-level) models to directly price the specific counterparty credit risk, as there is a significantportfolio effect. More specifically, for the same trade with the samecounterparty, the counterparty credit risk price can vary significantly depending on other trades in the same portfolio, the netting agreement, thecollateral terms, and the credit contingency terms. Even more challenging is the fact that the same counterparty can have trades modeled bydifferent pricers, booked in different systems, or even belonging to different desks and product areas. This is because, as we discussed earlier,that the counterparty credit risk price is the price of credit contingent options on each of the netting node of the counterparty portfolio.Micro or trade-level models do incorporate some generic counter-
Sec. 8.10 Frameworkfor Counterparty Credit Risk Modeling and Pricing
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party credit risk. For instance, they normally use LIBOR to discount thederivatives payoffs. This takes into account LIBOR credit quality.Micro models (even non-credit models) can also handle, to a certainextent, the credit quality of the underlying assets when one can buy (orsell) them through repo, i.e., with the rep0 rate as the funding cost (orgain). Thus, for derivatives modeling in this case, the future price of theunderlying asset grows at the rep0 rates or has market observable forward prices based on the rep0 rates when a liquid rep0 market is available for the asset. For instance, in pricing an option on a Treasury issue,the Treasury issue future payoff is PVed by discounting at the Treasuryrep0 rate (and the Treasury forward yield at longer term) and the optionfuture payoff is PVed by discounting at LIBOR. The rep0 rate on Treasury is lower than the LIBOR rate (of the same currency) except for someemerging market countries. The rep0 rate is less sensitive to the creditquality of the counterparties of the derivatives transaction, but is moresensitive to the credit quality and the market risk of the underlying asset.In general, various underlying assets can have market observablepricing information, such as forward prices, forward yields, par coupons,and par swap rates. Many of them are priced at a spread (positive ornegative) or a ratio relative to LIBOR due to their credit quality, taxbenefit, and other factors, such as liquidity. Examples of these assets,aside from Treasuries, include agency debt and swaps, municipal debtand swaps, mortgage instruments and swaps.There are assets, such as corporate bonds, emerging market sovereign bonds (denominated in foreign currencies), and the correspondingcredit default swaps, which can be significantly more credit risky thanthe ones just mentioned. These credit risky assets can have significantspread over LIBOR and credit models are needed to model such spreadsusing additional factors.Counterparty credit risk pricing is aimed at pricing the credit risk thatis not priced in (or not priced accurately) by the micro models, whichrequires a macro model. The micro and macro models differ in manyways as follows.Micro models are trade-level derivatives pricing models that oftenhave trade specific features, such as trade specific model parameters.For instance, depending on a particular trade, a micro model can calibrate its volatilities to different calibration instruments, e.g., caps andswaptions, and with different skew volatilities. It can also have normal,lognormal, CEV (Constant Elasticity of Variance), or shifted lognormal
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Credit Risk Modeling and Pricing
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processes. In other words, in this paradigm, each trade is priced with itsown model. Consequently, while being able to price a particular tradeaccurately, the same micro model can badly price other trades, and thuscannot be used to price a portfolio accurately.One of the most important requirements for the macro model is to beable to price portfolios consistently with desired accuracy. Furthermore,the macro model needs to price options on the portfolios with desiredaccuracy. This is the most challenging task in counterparty credit riskpricing, but it is necessary as the payoffs of the default options in thecounterparty credit risk are options on the counterparty portfolio. Thisrequires the global calibration of volatilities among other requirements,including various resamplings.The macro model is essentially a framework consisting of the following modules or engines.8.10.1
Centralized Market Process Modeling and Scenario GenerationEngine
It can be based on the arbitrage-free view or other views (such as historical views)."' If the arbitrage-free view is used, then, for IR derivatives,global calibration of volatility to the entire caphwaption matrix is neededin order to price the counterparty portfolio with desired accuracy. If theproduct area is credit derivatives, then it should also include the modeling of the credit processes of the underlying reference obligors.It needs to correctly model the correlation among all the state variables or risk factors. Such correlation can be either historical correlationor market implied correlation.It is normally implemented using Monte Carlo simulations. Numerical techniques, such as martingale resampling, are needed for achievingthe desired accuracy.The output of this engine is the market scenarios (or states or simulation paths) of all the risk factors that the counterparty portfolio depend onat various future dates.8.10.2 Exposure or MTM Modeling Engine
It involves pricing each of the trades in the portfolios at future market' I 9 For arbitrage pricing of the counterparty credit risk, we need to use the arbitrage-freeview. For risk analysis in general, such as scenario/horizon analysis and stress testing,we can choose other views.
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381
states and future dates, i.e., under the centralized market scenarios generated by the previous engine.In order for this engine to be scalable and to have fast turn-aroundtime for handling new trades, it should delegate the trade MTM pricingto the micro model of each trade. This can be achieved through full repricing (with calibration to both the volatility and the yield curve), partialre-pricing (with calibration to the yield curve only), interpolation in gridsor trees, or state stratification in Monte Carlo simulations. Path dependency of the trades, such as early exercise, physical exercise, knock-in,and knock-out, needs to be handled properly. Cashflow timing and itsmismatch, such as the reset and payment time lags also need to be handled properly. Accrued interest at risk can be handled by using the cashprice rather than the clean price.The specific requirements on micro models in order to achieve theabove delegation include the following steps (in a phased approach).1. Full re-pricing (with calibration to both the volatility and theyield curve) or partial re-pricing (with calibration to the yieldcurve only) as of future dates for given future states. The micromodel needs to properly handle the resets based on the given future dates and history. In order to reduce the computation time,the micro model should have a quick pricing mode (with less accuracy).2. Interpolation in grids or trees based on the given future statesand history. The independent variable(s) in the interpolation of atrade can be chosen to be the value of its underlying asset(s) orstate variable(s), or principal components of the future yieldcurves. The micro models need to properly handle the cashflowtiming, such as the reset and payment time lags, which requiresforward induction after the backward induction.3 . State stratification in Monte Carlo simulations based on thegiven future states and history. The independent variable(s) inthe state stratification of a trade can be chosen to be the value ofits underlying asset(s) or state variable(s), or principal components of the future yield curves. The micro model needs to properly handle the cashflow timing, such as the reset and paymenttime lags, which requires forward induction after the backwardinduction.4. Path dependencies, such as early exercise, trigger, and physicalsettlement, also need to be handled properly. The Brownian
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Credit Risk Modeling and Pricing
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bridge technique can be used. Additional forward and backwardinductions may be needed.5. Provides the targets for martingale resampling techniques discussed before. Such targets are comprised of a vector of the PVof the trade by eliminating various cashflows at forward dates,and a vector of PV of the options on these cashflow eliminatedtrades.The micro models are often mutually inconsistent as internally theyoften use mutually inconsistent market scenarios or evolutions. Therefore, in order to minimize such inconsistency, it is critical to use the micro models to price based on the centralized market scenarios as described in the previous steps.In order to achieve the desired accuracy, one also needs to ensureconsistencies between the macro model and the micro models in thesense of expected values (and possibly certain tail distributions). In otherwords, one needs to ensure that for any given trade, the macro modelgives the same price on the trade itself and on options on the trade as themicro model. Numerical techniques, such as the martingale resamplingtechniques, are needed to ensure that for any given trade, the macromodel gives the same price on the trade or even on an option on the tradeas the micro model in order to achieve the desired accuracy.The output of this engine is the MTM grids (or simulation prices) ofeach trade at all the market scenarios and all the hture dates.8.70.3 New Trade and Real-time Exposure or MTM Modeling Engine
The previous engine is for daily over-night batch runs. This engine is fornew trade modeling and for real-time MTM modeling as the marketchanges intraday. A typical use case for this engine is the real-time pricing for a new trade in a portfolio, which requires the updated MTM values for all the trades in the portfolio.One way to implement this engine is that one can start by saving themarket scenarios and trade prices from the previous day’s batch runs.Then one can generate the MTM grids for the new trade under the savedmarket scenarios. This is essential for consistently aggregating the MTMgrids of the new trade with the saved ones. These MTM grids can beused in the counterparty credit risk pricing in real-time or close to realtime in the sense of computation speed, but not in the sense of using updated market.In order to further achieve real-time or close to real-time modeling
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using the updated market, one can use various resampling techniques tobring the MTM grids inline with the updated market.Another (brute force) way to implement this engine is to call the previous engine in real-time with the updated market. Parallel computing isneeded in order to handle the extensive computations involved so as toachieve real-time or close to real-time modeling.8.70.4 Counterparty Credit Process Modeling and Scenario GenerafionEngineIn the simplest approach, one can use deterministic default probabilitiesand recoveries for estimations of the counterparty credit risk.However, in order to price the counterparty credit risk more accurately, and to measure and manage unexpected risks (such as the economic capital) and diversification benefits (for, e.g., the reduction ofregulatory capital), one needs to model the stochastic processes of default event, default intensity, credit rating, and recovery of the counterparty.In order to measure and manage the wrong-way and right-way exposures, one needs to handle the correlation among the counterparty creditprocesses and the market processes.One can also handle the correlation among the credit processes of allthe counterparties and all the reference obligors. This is not needed forpricing the expected counterparty credit risk for a product area that is notcredit sensitive, such as the interest derivatives area. This is needed formeasuring unexpected counterparty credit risks, which can be achievedby using a factor model with factors from each industry, each credit rating, each geographical region, plus a global factor and idiosyncratic factor.The output of this engine is the counterparty credit scenarios at various future dates.8.70.5 Portfolio Effect Handling and Aggregation Engine
This engine handles proper netting among the trades of each entity. Itcan happen that multiple netting nodes can exist in a single counterpartyportfolio. This can happen when an entity has sub-entities and nettingcan be legally enforced within sub-entities, but across sub-entities.This engine also models collaterals received and posted at futuremarket states and future dates. The amount of the collaterals is state dependent and depends on both the market state or MTM and the counter-
384
Credit Risk Modeling and Pricing
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party credit state, such as the counterparty credit rating.It also needs to handle the time lag (normally assumed to be 2weeks) between the collateral call and the time when the collaterals areactually received. This creates a path dependency and results in additional risks including the collateral at risk.The output of this engine is the netted and collateralized exposures ofeach portfolio at each of the market scenarios and each of the futuredates. None-netted and/or un-collateralized exposures are often usefulalso.
8.70.6
Counterparty Credit Risk Pricing Engine
This engine prices the expected credit loss (or credit charge) and expected credit gain (or credit benefit).It needs to properly handle various credit contingencies.It needs to handle wrong-way and right-way exposures.It needs to properly handle counterparties with credit quality betterthan LIBOR.It also needs to convert the net credit charge into proper fundingspread taking into account the expected loss of the spread payment.The output of this engine is the credit charge and credit benefit ofeach counterparty, as well as the marginal credit charge and credit benefit for new trades in terms of funding spread.8.70.7 Sensitivity and Scenario Analysis EngineThe counterparty credit risk needs to be managed in a similar way asother trades. This engine provides the sensitivities or hedge ratiosneeded for hedging the counterparty credit risk. Aside from shockingand reevaluation for the sensitivity analysis, various techniques, such asJacobian, PCA (principal component analysis), change of probabilitymeasure, Maliavin calculus, and quadratic approximations can also beused.As with other trades, the volatility skew effect on the sensitivities orhedge ratios, particularly Delta due to Vega, needs to be properly handled.It also provides scenario analysis for various market scenarios, creditscenarios (including the correlation between the credit and market processes for handling wrong-way and right-way exposures), and portfolioeffect scenarios (such as netting, collateral, credit contingency scenarios). It also provides stress testing for extreme market scenarios includ-
Sec. 8.10 Framework for Counterparty Credit Risk Modeling and Pricing
385
ing the scenarios produced by jumps.8.70.8
Unexpected Risk Modeling Engine
With all the above infrastructure, one can also establish the unexpectedrisk modeling engine for, e.g., PE, credit VAR, and economic capital.They are not needed for computing the credit charge and benefit (whichare from expected risks).These are helpful in deriving diversification strategies for reducingthe unexpected risks while keeping the expected risks more or less unchanged. This is important to a firm’s long-term economic soundnessand can also reduce the amount of the regulatory capital.
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PART II
INTEREST RATEMARKETFUNDAMENTALSANDPROPRIETARYT RADINGSTRATEGIES
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Chapter 9
9.1
Simple InterestRate Products
Treasury Issues
The U S . Treasury periodically issues Treasury bills, notes, and bonds,and they are all called Treasuries or simply bonds. A bond is a contractbetween the bond issuer and the bond buyer. The bond buyer pays anup-front price for the bond which obligates the bond issuer to pay backinterest and principal at a schedule specified by the bond contract. TheTreasuries that are issued with original maturities less than or equal toone year are called T-bills, those with original maturities of 2, 3, 5 , and10 years are called notes, and those with maturities longer than 10 yearsare called bonds. Primary dealers place bids in Treasury auctions to buyTreasuries for their own accounts or on behalf of their clients. The newissues of T-bills, notes, and bonds are called the currents, or the on-theruns, and because of their high liquidity and demand, they trade with apremium in price relative to the off-the-run issues. The purchase of aU.S. Treasury always settles on the next business day following the tradeday.9.1.1
Treasury Bills
A Treasury bill is a discount debt instrument with a maturity of 90 days,180 days or 1 year. The available face amount ranges from $10,000 to 1million dollars. An investor buys a T-bill at a discount price (below par),and on the maturity date, gets paid back the par amount. The discountprice is calculated by applying the ActuaV360 day count convention asfollows,
390
Simple Interest Rate Products
Chapter 9
P=lOO-- nY(9.1)360 'where P is the discount price, n the number of days from settlement tomaturity, and y the discount yield over the period [0, n].120For example, on trade date February 31d, 2003, the 6-month T-bill,which matures on August 6'h, 2003, is quoted at a discount yield of5.06%. The investor who buys the T-bill pays 97.447% of par amounton the next business day, and gets 100% back on August 6" 2003.
9.7.2
TreasuryNotes and Bonds
Currently the U.S. Treasury only issues new bonds with maturity up to30 years12'. T-notes and bonds pay interests semi-annually and redeemthe principal and the last interest payment on the maturity date. The interest payments are called coupons.The following chart, Figure 9.1 shows the cashflows of the 2-yearnote, T 5.375 1/31/2003 that is traded on February 3rdand settled on February 4th, 2003. The price is 100-4+, with a yield of 5.299%.
c+100
C
I
i
1
214102 713 1/02 1131/03 713 1/03 1131/04
Figure 9.1. Cash flows of a Treasury note.
In order to avoid price discontinuity, the market always quotes theclean price that is the full price (invoice price, or dirty or cash price) minus the accrued interest. The clean price is the one that reflects the economic value of a bond. The yield (y) and (clean) price ( P ) relationshipThe price-yield formula for T-bills is more complicated for maturities longer than 182days.On the run 30 year bonds stopped issuing in the late 90's. There are talks to bringback the 30 year