Bill#

class rateslib.instruments.Bill(effective=NoInput.blank, termination=NoInput.blank, frequency=NoInput.blank, modifier=NoInput.blank, calendar=NoInput.blank, payment_lag=NoInput.blank, notional=NoInput.blank, currency=NoInput.blank, convention=NoInput.blank, settle=NoInput.blank, calc_mode=NoInput.blank, curves=NoInput.blank, spec=NoInput.blank, metric='price')#

Bases: FixedRateBond

Create a discount security.

Parameters:

effective (datetime) – The adjusted or unadjusted effective date.
termination (datetime or str) – The adjusted or unadjusted termination date. If a string, then a tenor must be given expressed in days (“D”), months (“M”) or years (“Y”), e.g. “7M”.
frequency (str in {"M", "B", "Q", "T", "S", "A"}, optional) – The frequency used only by the ytm() method. All Bills have an implicit frequency of “Z” for schedule construction.
modifier (str, optional) – The modification rule, in {“F”, “MF”, “P”, “MP”}
calendar (calendar or str, optional) – The holiday calendar object to use. If str, looks up named calendar from static data.
payment_lag (int, optional) – The number of business days to lag payments by.
notional (float, optional) – The leg notional, which is applied to each period.
currency (str, optional) – The currency of the leg (3-digit code).
convention (str, optional) – The day count convention applied to calculations of period accrual dates. See dcf().
settle (int) – The number of business days for regular settlement time, i.e, 1 is T+1.
calc_mode (str, optional (defaults.calc_mode["Bill"])) – A calculation mode for dealing with bonds that are in short stub or accrual periods. All modes give the same value for YTM at issue date for regular bonds but differ slightly for bonds with stubs or with accrued.
curves (CurveType, str or list of such, optional) –
A single Curve or string id or a list of such.

A list defines the following curves in the order:
- Forecasting Curve for leg1.
- Discounting Curve for leg1.
spec (str, optional) – An identifier to pre-populate many field with conventional values. See here for more info and available values.
metric (str, optional) – The pricing metric returned by the rate method of the Instrument.

Examples

This example is taken from the US Treasury Federal website. A copy of which is available here.

We demonstrate the use of analogue methods which do not need Curves or Solvers, price(), simple_rate(), discount_rate(), ytm(), ex_div(), accrued(), repo_from_fwd() fwd_from_repo() duration(), convexity().

In [1]: bill = Bill(
   ...:     effective=dt(2004, 1, 22),
   ...:     termination=dt(2004, 2, 19),
   ...:     calendar="nyc",
   ...:     modifier="NONE",
   ...:     currency="usd",
   ...:     convention="Act360",
   ...:     settle=1,
   ...:     notional=-1e6,  # negative notional receives fixed, i.e. buys a bill
   ...:     curves="bill_curve",
   ...:     calc_mode="us_gbb",
   ...: )
   ...: 

In [2]: bill.ex_div(dt(2004, 1, 22))
Out[2]: False

In [3]: bill.price(rate=0.80, settlement=dt(2004, 1, 22))
Out[3]: 99.93777777777778

In [4]: bill.simple_rate(price=99.937778, settlement=dt(2004, 1, 22))
Out[4]: 0.8004952269972267

In [5]: bill.discount_rate(price=99.937778, settlement=dt(2004, 1, 22))
Out[5]: 0.7999971428571513

In [6]: bill.ytm(price=99.937778, settlement=dt(2004, 1, 22))
Out[6]: 0.819704602724924

In [7]: bill.accrued(dt(2004, 1, 22))
Out[7]: 0.0

In [8]: bill.fwd_from_repo(
   ...:     price=99.937778,
   ...:     settlement=dt(2004, 1, 22),
   ...:     forward_settlement=dt(2004, 2, 19),
   ...:     repo_rate=0.8005,
   ...:     convention="Act360",
   ...: )
   ...: 
Out[8]: 100.00000037100256

In [9]: bill.repo_from_fwd(
   ...:     price=99.937778,
   ...:     settlement=dt(2004, 1, 22),
   ...:     forward_settlement=dt(2004, 2, 19),
   ...:     forward_price=100.00,
   ...:     convention="Act360",
   ...: )
   ...: 
Out[9]: 0.8004952269973565

In [10]: bill.duration(settlement=dt(2004, 1, 22), ytm=0.8005, metric="risk")
Out[10]: np.float64(0.07777777777777778)

In [11]: bill.duration(settlement=dt(2004, 1, 22), ytm=0.8005, metric="modified")
Out[11]: np.float64(0.07782623325533848)

In [12]: bill.convexity(settlement=dt(2004, 1, 22), ytm=0.8005)
Out[12]: np.float64(-0.0)

The following digital methods consistent with the library’s ecosystem are also available, rate(), npv(), analytic_delta(), cashflows(), delta(), gamma(),

In [13]: bill_curve = Curve({dt(2004, 1, 21): 1.0, dt(2004, 3, 21): 1.0}, id="bill_curve")

In [14]: instruments = [
   ....:     (bill, (), {"metric": "ytm"}),
   ....: ]
   ....: 

In [15]: solver = Solver(
   ....:     curves=[bill_curve],
   ....:     instruments=instruments,
   ....:     s=[0.8005],
   ....:     instrument_labels=["Feb04 Tbill"],
   ....:     id="bill_solver",
   ....: )
   ....: 
SUCCESS: `func_tol` reached after 2 iterations (levenberg_marquardt), `f_val`: 7.317488829914903e-14, `time`: 0.0097s

In [16]: bill.npv(solver=solver)
Out[16]: <Dual: 999370.628215, (bill_curve0, bill_curve1), [516341.5, 483658.7]>

In [17]: bill.analytic_delta(disc_curve=bill_curve)
Out[17]: <Dual: -7.772883, (bill_curve0, bill_curve1), [-4.0, -3.8]>

In [18]: bill.rate(solver=solver, metric="price")
Out[18]: <Dual: 99.939232, (bill_curve0, bill_curve1), [-46.6, 46.7]>

The sensitivities are also available. In this case the Solver is calibrated with instruments priced in yield terms so sensitivities are measured in basis points (bps).

In [19]: bill.delta(solver=solver)
Out[19]: 
local_ccy                                 usd
display_ccy                               usd
type        solver      label                
instruments bill_solver Feb04 Tbill -7.844079

In [20]: bill.gamma(solver=solver)
Out[20]: 
type                                                      instruments
solver                                                    bill_solver
label                                                     Feb04 Tbill
local_ccy display_ccy type        solver      label                  
usd       usd         instruments bill_solver Feb04 Tbill    0.000452

The DataFrame of cashflows.

In [21]: bill.cashflows(solver=solver)
Out[21]: 
          Type    Period  Ccy  Acc Start    Acc End    Payment Convention       DCF   Notional        DF Collateral  Rate  Spread   Cashflow            NPV  FX Rate        NPV Ccy
0  FixedPeriod      Stub  USD 2004-01-22 2004-02-19 2004-02-19     Act360  0.077778 -1000000.0  0.999371       None   0.0     NaN        0.0       0.000000      1.0       0.000000
1     Cashflow  Exchange  USD        NaT        NaT 2004-02-19        NaN       NaN -1000000.0  0.999371       None   NaN     NaN  1000000.0  999370.628215      1.0  999370.628215

Attributes Summary

`dcf`
`fixed_rate`	If set will also set the `fixed_rate` of the contained leg1.
`float_spread`	If set will also set the `float_spread` of contained leg1.
`index_base`	If set will also set the `index_base` of the contained leg1.
`leg2_fixed_rate`	If set will also set the `fixed_rate` of the contained leg2.
`leg2_float_spread`	If set will also set the `float_spread` of contained leg2.
`leg2_index_base`	If set will also set the `index_base` of the contained leg1.

Methods Summary

`accrued`(settlement)	Calculate the accrued amount per nominal par value of 100.
`analytic_delta`([curve, disc_curve, fx, base])	Return the analytic delta of the security via summing all periods.
`cashflows`([curves, solver, fx, base, settlement])	Return the properties of the security used in calculating cashflows.
`cashflows_table`([curves, solver, fx, base])	Aggregate the values derived from a `cashflows()` method on an Instrument.
`convexity`(ytm, settlement)	Return the second derivative of `price` w.r.t.
`delta`(args, *kwargs)	Calculate the delta of the Instrument.
`discount_rate`(price, settlement)	Return the discount rate of the security from its `price`.
`duration`(ytm, settlement[, metric])	Return the duration of the Bill.
`ex_div`(settlement)	Return a boolean whether the security is ex-div at the given settlement.
`exo_delta`(vars[, curves, solver, fx, base, ...])	Calculate delta risk of an Instrument against some exogenous user created Variables.
`fwd_from_repo`(price, settlement, ...[, ...])	Return a forward price implied by a given repo rate.
`gamma`(args, *kwargs)	Calculate the gamma of the Instrument.
`npv`([curves, solver, fx, base, local])	Return the NPV of the security by summing cashflow valuations.
`oaspread`([curves, solver, fx, base, price, ...])	The option adjusted spread added to the discounting Curve to value the security at `price`.
`price`(rate, settlement[, dirty, calc_mode])	Return the price of the bill given the `discount_rate`.
`rate`([curves, solver, fx, base, metric])	Return various pricing metrics of the security calculated from `Curve` s.
`repo_from_fwd`(price, settlement, ...[, ...])	Return an implied repo rate from a forward price.
`simple_rate`(price, settlement)	Return the simple rate of the security from its `price`.
`ytm`(price, settlement[, calc_mode])	Calculate the yield-to-maturity on an equivalent bond with a coupon of 0%.

Attributes Documentation

dcf#

fixed_rate#

If set will also set the fixed_rate of the contained leg1.

Note

fixed_rate, float_spread, leg2_fixed_rate and leg2_float_spread are attributes only applicable to certain Instruments. AttributeErrors are raised if calling or setting these is invalid.

Type:: float or None

float_spread#

If set will also set the float_spread of contained leg1.

Type:: float or None

index_base#

If set will also set the index_base of the contained leg1.

Note

index_base and leg2_index_base are attributes only applicable to certain Instruments. AttributeErrors are raised if calling or setting these is invalid.

Type:: float or None

leg2_fixed_rate#

If set will also set the fixed_rate of the contained leg2.

Type:: float or None

leg2_float_spread#

If set will also set the float_spread of contained leg2.

Type:: float or None

leg2_index_base#

If set will also set the index_base of the contained leg1.

Note

index_base and leg2_index_base are attributes only applicable to certain Instruments. AttributeErrors are raised if calling or setting these is invalid.

Type:: float or None

Methods Documentation

accrued(settlement)#

Calculate the accrued amount per nominal par value of 100.

Parameters:: settlement (datetime) – The settlement date which to measure accrued interest against.

Notes

Fractionally apportions the coupon payment based on calendar days.

\[\text{Accrued} = \text{Coupon} \times \frac{\text{Settle - Last Coupon}}{\text{Next Coupon - Last Coupon}}\]

analytic_delta(curve=NoInput.blank, disc_curve=NoInput.blank, fx=NoInput.blank, base=NoInput.blank)#

Return the analytic delta of the security via summing all periods.

For arguments see analytic_delta().

cashflows(curves=NoInput.blank, solver=NoInput.blank, fx=NoInput.blank, base=NoInput.blank, settlement=NoInput.blank)#

Return the properties of the security used in calculating cashflows.

Parameters:

curves (Curve, str or list of such) –
A single Curve or id or a list of such. A list defines the following curves in the order:
- Forecasting Curve for leg1.
- Discounting Curve for leg1.
solver (Solver, optional) – The numerical Solver that constructs Curves from calibrating instruments.
fx (float, FXRates, FXForwards, optional) – The immediate settlement FX rate that will be used to convert values into another currency. A given float is used directly. If giving a FXRates or FXForwards object, converts from local currency into base.
base (str, optional) – The base currency to convert cashflows into (3-digit code), set by default. Only used if fx_rate is an FXRates or FXForwards object.
settlement (datetime, optional) – The settlement date of the security. If None adds the regular settle time to the initial node date of the given discount curves.

Return type:

DataFrame

cashflows_table(curves=NoInput.blank, solver=NoInput.blank, fx=NoInput.blank, base=NoInput.blank, **kwargs)#

Aggregate the values derived from a cashflows() method on an Instrument.

Parameters:

curves (CurveType, str or list of such, optional) – Argument input to the underlying cashflows method of the Instrument.
solver (Solver, optional) – Argument input to the underlying cashflows method of the Instrument.
fx (float, FXRates, FXForwards, optional) – Argument input to the underlying cashflows method of the Instrument.
base (str, optional) – Argument input to the underlying cashflows method of the Instrument.
kwargs (dict) – Additional arguments input the underlying cashflows method of the Instrument.

Return type:

DataFrame

convexity(ytm, settlement)#

Return the second derivative of price w.r.t. ytm.

Parameters:

ytm (float) – The yield-to-maturity for the bond.
settlement (datetime) – The settlement date of the bond.

Return type:

float

Examples

In [1]: gilt = FixedRateBond(
   ...:     effective=dt(1998, 12, 7),
   ...:     termination=dt(2015, 12, 7),
   ...:     frequency="S",
   ...:     calendar="ldn",
   ...:     currency="gbp",
   ...:     convention="ActActICMA",
   ...:     ex_div=7,
   ...:     fixed_rate=8.0
   ...: )
   ...: 

In [2]: gilt.convexity(4.445, dt(1999, 5, 27))
Out[2]: np.float64(2.0367301586109305)

This number is interpreted as hundredths of a cent. For a 1bp increase in yield the duration will decrease by 2 hundredths of a cent.

In [3]: gilt.duration(4.445, dt(1999, 5, 27))
Out[3]: np.float64(14.659753980778154)

In [4]: gilt.duration(4.455, dt(1999, 5, 27))
Out[4]: np.float64(14.63940251353963)

delta(*args, **kwargs)#

Calculate the delta of the Instrument.

For arguments see Sensitivities.delta().

discount_rate(price, settlement)#

Return the discount rate of the security from its price.

Parameters:

price (float, Dual, or Dual2) – The price of the security.
settlement (datetime) – The settlement date of the security.

Return type:

float, Dual, or Dual2

duration(ytm, settlement, metric='risk')#

Return the duration of the Bill. See duration for arguments.

Notes

Warning

This function returns a duration that is consistent with a FixedRateBond yield-to-maturity definition. It currently does not use the specified convention of the Bill, and can be sensitive to the frequency of the representative FixedRateBond equivalent.

In [1]: bill = Bill(effective=dt(2024, 2, 29), termination=dt(2024, 8, 29), spec="us_gbb")

In [2]: bill.duration(settlement=dt(2024, 5, 30), ytm=5.2525, metric="duration")
Out[2]: np.float64(0.2629070107142689)

In [3]: bill = Bill(effective=dt(2024, 2, 29), termination=dt(2024, 8, 29), spec="us_gbb", frequency="A")

In [4]: bill.duration(settlement=dt(2024, 5, 30), ytm=5.2525, metric="duration")
Out[4]: np.float64(0.2696349145097242)

ex_div(settlement)#

Return a boolean whether the security is ex-div at the given settlement.

Parameters:: settlement (datetime) – The settlement date to test.
Return type:: bool

Notes

By default uses the UK DMO convention of returning False if settlement is on or before the ex-div date.

Ex-div dates are determined as measured by the number of ex_div business days prior to the unadjusted coupon end date.

With an ex_div of 1, a settlement that occurs on the coupon payment date will be classified as ex-dividend and not receive that coupon.

With an ex_div of 0, a settlement that occurs on the coupon payment date will not be classified as ex-dividend and will receive that coupon.

Most bonds under this convention have a ex_div parameter of 1. They will still receive the coupon if they settle 1 business day before the coupon end date, and will not receive the coupon if they settle on or after the coupon end date.

exo_delta(vars, curves=NoInput.blank, solver=NoInput.blank, fx=NoInput.blank, base=NoInput.blank, local=False, vars_scalar=NoInput.blank, vars_labels=NoInput.blank, **kwargs)#

Calculate delta risk of an Instrument against some exogenous user created Variables.

See What are exogenous variables? in the cookbook.

Parameters:

vars (list[str]) – The variable tags which to determine sensitivities for.
curves (Curve, str or list of such, optional) –
A single Curve or id or a list of such. A list defines the following curves in the order:
- Forecasting Curve for leg1.
- Discounting Curve for leg1.
- Forecasting Curve for leg2.
- Discounting Curve for leg2.
solver (Solver, optional) – The Solver that calibrates Curves from given Instruments.
fx (float, FXRates, FXForwards, optional) – The immediate settlement FX rate that will be used to convert values into another currency. A given float is used directly. If giving a FXRates or FXForwards object, converts from local currency into base.
base (str, optional) – The base currency to convert cashflows into (3-digit code), set by default. Only used if fx_rate is an FXRates or FXForwards object.
local (bool, optional) – If True will ignore base - this is equivalent to setting base to None. Included only for argument signature consistent with npv.
vars_scalar (list[float], optional) – Scaling factors for each variable, for example converting rates to basis point etc. Defaults to ones.
vars_labels (list[str], optional) – Alternative names to relabel variables in DataFrames.

Return type:

DataFrame

fwd_from_repo(price, settlement, forward_settlement, repo_rate, convention=NoInput.blank, dirty=False, method='proceeds')#

Return a forward price implied by a given repo rate.

Parameters:

price (float, Dual, or Dual2) – The initial price of the security at settlement.
settlement (datetime) – The settlement date of the bond
forward_settlement (datetime) – The forward date for which to calculate the forward price.
repo_rate (float, Dual or Dual2) – The rate which is used to calculate values.
convention (str, optional) – The day count convention applied to the rate. If not given uses default values.
dirty (bool, optional) – Whether the input and output price are specified including accrued interest.
method (str in {"proceeds", "compounded"}, optional) – The method for determining the forward price.

Return type:

float, Dual or Dual2

Notes

Any intermediate (non ex-dividend) cashflows between settlement and forward_settlement will also be assumed to accrue at repo_rate.

gamma(*args, **kwargs)#

Calculate the gamma of the Instrument.

For arguments see Sensitivities.gamma().

npv(curves=NoInput.blank, solver=NoInput.blank, fx=NoInput.blank, base=NoInput.blank, local=False)#

Return the NPV of the security by summing cashflow valuations.

Parameters:

curves (Curve, str or list of such) –
A single Curve or id or a list of such. A list defines the following curves in the order:
- Forecasting Curve for leg1.
- Discounting Curve for leg1.
solver (Solver, optional) – The numerical Solver that constructs Curves from calibrating instruments.
fx (float, FXRates, FXForwards, optional) – The immediate settlement FX rate that will be used to convert values into another currency. A given float is used directly. If giving a FXRates or FXForwards object, converts from local currency into base.
base (str, optional) – The base currency to convert cashflows into (3-digit code), set by default. Only used if fx is an FXRates or FXForwards object.
local (bool, optional) – If True will ignore the base request and return a dict identifying local currency NPV.

Return type:

float, Dual, Dual2 or dict of such

Notes

The settlement date of the bond is inferred from the objects settle days parameter and the initial date of the supplied curves. The NPV returned is for immediate settlement.

If only one curve is given this is used as all four curves.

If two curves are given the forecasting curve is used as the forecasting curve on both legs and the discounting curve is used as the discounting curve for both legs.

oaspread(curves=NoInput.blank, solver=NoInput.blank, fx=NoInput.blank, base=NoInput.blank, price=NoInput.blank, metric=NoInput.blank, func_tol=NoInput.blank, conv_tol=NoInput.blank)#

The option adjusted spread added to the discounting Curve to value the security at price.

Parameters:

curves (Curve, str or list of such) –
A single Curve or id or a list of such. A list defines the following curves in the order:
- Forecasting Curve for leg1.
- Discounting Curve for leg1.
solver (Solver, optional) – The numerical Solver that constructs Curves from calibrating instruments.
fx (float, FXRates, FXForwards, optional) – The immediate settlement FX rate that will be used to convert values into another currency. A given float is used directly. If giving a FXRates or FXForwards object, converts from local currency into base.
base (str, optional) – The base currency to convert cashflows into (3-digit code), set by default. Only used if fx is an FXRates or FXForwards object.
price (float, Dual, Dual2) – The price of the bond to match.
metric (str, optional) – The metric to use when evaluating the price/rate of the instrument. If not given uses the instrument’s rate() method default.
func_tol (float, optional) – The tolerance for the objective function value when iteratively solving. If not given uses defaults.oaspread_func_tol.
conv_tol (float, optional) – The tolerance used for stopping criteria of successive iteration values. If not given uses defaults.oaspread_conv_tol.

Return type:

float, Dual, Dual2

Notes

The discount curve must be of type _BaseCurve with a provided shift() method available.

Warning

The sensitivity of variables is preserved for the input argument price, but this function does not preserve AD towards variables associated with the curves or solver.

Examples

In [1]: bond = FixedRateBond(dt(2000, 1, 1), "3Y", fixed_rate=2.5, spec="us_gb")

In [2]: curve = Curve({dt(2000, 7, 1): 1.0, dt(2005, 7, 1): 0.80})

# Add AD variables to the curve without a Solver
In [3]: curve._set_ad_order(1)

In [4]: bond.oaspread(curves=curve, price=Variable(95.0, ["price"], []))
Out[4]: <Dual: 11.809041, (price), [-42.6]>

This result excludes curve sensitivities but includes sensitivity to the constructed ‘price’ variable. Accuracy can be observed through numerical simulation.

In [5]: bond.oaspread(curves=curve, price=96.0)
Out[5]: -30.541959250658007

In [6]: bond.oaspread(curves=curve, price=94.0)
Out[6]: 54.615566044867904

price(rate, settlement, dirty=False, calc_mode=NoInput.blank)#

Return the price of the bill given the discount_rate.

Parameters:

rate (float) – The rate used by the pricing formula.
settlement (datetime) – The settlement date.
dirty (bool, not required) – Discount securities have no coupon, the concept of clean or dirty is not relevant. Argument is included for signature consistency with FixedRateBond.price.
calc_mode (str, optional) – A calculation mode to force, which is used instead of that attributed the Bill instance.

Return type:

float, Dual, Dual2

rate(curves=NoInput.blank, solver=NoInput.blank, fx=NoInput.blank, base=NoInput.blank, metric=NoInput.blank)#

Return various pricing metrics of the security calculated from Curve s.

Parameters:

curves (Curve, str or list of such) –
A single Curve or id or a list of such. A list defines the following curves in the order:
- Forecasting Curve for leg1.
- Discounting Curve for leg1.
solver (Solver, optional) – The numerical Solver that constructs Curves from calibrating instruments.
fx (float, FXRates, FXForwards, optional) – The immediate settlement FX rate that will be used to convert values into another currency. A given float is used directly. If giving a FXRates or FXForwards object, converts from local currency into base.
base (str, optional) – The base currency to convert cashflows into (3-digit code), set by default. Only used if fx is an FXRates or FXForwards object.
metric (str in {"price", "discount_rate", "ytm", "simple_rate"}) – Metric returned by the method. Uses the Instrument default if not given.

Return type:

float, Dual, Dual2

repo_from_fwd(price, settlement, forward_settlement, forward_price, convention=NoInput.blank, dirty=False)#

Return an implied repo rate from a forward price.

Parameters:

price (float, Dual, or Dual2) – The initial price of the security at settlement.
settlement (datetime) – The settlement date of the bond
forward_settlement (datetime) – The forward date for which to calculate the forward price.
forward_price (float, Dual or Dual2) – The forward price which iplies the repo rate
convention (str, optional) – The day count convention applied to the rate. If not given uses default values.
dirty (bool, optional) – Whether the input and output price are specified including accrued interest.

Return type:

float, Dual or Dual2

Notes

Any intermediate (non ex-dividend) cashflows between settlement and forward_settlement will also be assumed to accrue at repo_rate.

simple_rate(price, settlement)#

Return the simple rate of the security from its price.

Parameters:

price (float, Dual, or Dual2) – The price of the security.
settlement (datetime) – The settlement date of the security.

Return type:

float, Dual, or Dual2

ytm(price, settlement, calc_mode=NoInput.blank)#

Calculate the yield-to-maturity on an equivalent bond with a coupon of 0%.

Parameters:

price (float, Dual, Dual2) – The price of the Bill.
settlement (datetime) – The settlement date of the Bill.
calc_mode (str, optional) – A calculation mode to force, which is used instead of that attributed the Bill instance.

Notes

Maps the following Bill calc_mode to the following Bond specifications:

NoInput -> “ust”
“ustb” -> “ust”
“uktb” -> “ukt”
“sgbb” -> “sgb”

This method calculates by constructing a FixedRateBond with a regular 0% coupon measured from the termination date of the bill.

Bill#

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