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Bootstrap default probability curve from bond prices
[ProbabilityData,HazardData]
= bondDefaultBootstrap(ZeroData,MarketData,Settle)
[ProbabilityData,HazardData]
= bondDefaultBootstrap(___,Name,Value)
[
bootstraps
the default probability curve from bond prices.ProbabilityData
,HazardData
]
= bondDefaultBootstrap(ZeroData
,MarketData
,Settle
)
Using bondDefaultBootstrap
, you can:
Extract discrete default probabilities for a certain period from market bond data.
Interpolate these default probabilities to get the default probability curve for pricing and risk management purposes.
[
adds
optional name-value pair arguments.ProbabilityData
,HazardData
]
= bondDefaultBootstrap(___,Name,Value
)
Use the following bond data.
Settle = datenum('08-Jul-2016'); MarketDate = datenum({'06/15/2018', '01/08/2019', '02/01/2021', '03/18/2021', '08/04/2025'}','mm/dd/yyyy'); CouponRate = [2.240 2.943 5.750 3.336 4.134]'/100; MarketPrice = [101.300 103.020 115.423 104.683 108.642]'; MarketData = [MarketDate,MarketPrice,CouponRate];
Calculate the ProbabilityData
and HazardData
.
TreasuryParYield = [0.26 0.28 0.36 0.48 0.61 0.71 0.95 1.19 1.37 1.69 2.11]'/100;
TreasuryDates = datemnth(Settle, [[1 3 6], 12 * [1 2 3 5 7 10 20 30]]');
[ZeroRates, CurveDates] = pyld2zero(TreasuryParYield, TreasuryDates, Settle);
ZeroData = [CurveDates, ZeroRates];
format longg
[ProbabilityData,HazardData]=bondDefaultBootstrap(ZeroData,MarketData,Settle)
ProbabilityData = 5×2
737226 0.0299675399937613
737433 0.0418832295824674
738188 0.0905183328842618
738233 0.101248065083714
739833 0.233002708031915
HazardData = 5×2
737226 0.0157077745460246
737433 0.0217939816590399
738188 0.0251849128247209
738233 0.096260871864083
739833 0.0361632398787917
format
Reprice one of the bonds from bonds list based on the default probability curve. The expected result of this repricing is a perfect match with the market quote.
Use the following Treasury data from US Department of the Treasury.
Settle = datetime('08-Jul-2016','Locale','en_US'); TreasuryParYield = [0.26 0.28 0.36 0.48 0.61 0.71 0.95 1.19 1.37 1.69 2.11]'/100; TreasuryDates = datemnth(Settle, [[1 3 6], 12 * [1 2 3 5 7 10 20 30]]');
Preview the bond date using semiannual coupon bonds with market quotes, coupon rates, and a settle date of July-08-2016.
MarketDate = datenum({'06/01/2017','06/01/2019','06/01/2020','06/01/2022'}','mm/dd/yyyy'); CouponRate = [7 8 9 10]'/100; MarketPrice = [101.300 109.020 114.42 118.62]'; MarketData = [MarketDate, MarketPrice, CouponRate]; BondList = array2table(MarketData, 'VariableNames', {'Maturity', 'Price','Coupon'}); BondList.Maturity = datetime(BondList.Maturity,'Locale','en_US','ConvertFrom','datenum'); BondList.Maturity.Format = 'MMM-dd-yyyy'
BondList=4×3 table
Maturity Price Coupon
___________ ______ ______
Jun-01-2017 101.3 0.07
Jun-01-2019 109.02 0.08
Jun-01-2020 114.42 0.09
Jun-01-2022 118.62 0.1
Choose the second coupon bond as the one to be priced.
number = 2; TestCase = BondList(number, :);
Preview the risk-free rate data provided here that is based on a continuous compound rate.
[ZeroRates, CurveDates] = pyld2zero(TreasuryParYield, TreasuryDates, Settle); ZeroData = [datenum(CurveDates), ZeroRates]; RiskFreeRate = array2table(ZeroData, 'VariableNames', {'Date', 'Rate'}); RiskFreeRate.Date = datetime(RiskFreeRate.Date,'Locale','en_US','ConvertFrom','datenum'); RiskFreeRate.Date.Format = 'MMM-dd-yyyy'
RiskFreeRate=11×2 table
Date Rate
___________ _________
Aug-08-2016 0.0026057
Oct-08-2016 0.0027914
Jan-08-2017 0.0035706
Jul-08-2017 0.0048014
Jul-08-2018 0.0061053
Jul-08-2019 0.0071115
Jul-08-2021 0.0095416
Jul-08-2023 0.012014
Jul-08-2026 0.013883
Jul-08-2036 0.017359
Jul-08-2046 0.022704
Bootstrap the probability of default (PD) curve from the bonds.
format longg
[defaultProb1, hazard1] = bondDefaultBootstrap(ZeroData, MarketData, Settle)
defaultProb1 = 4×2
736847 0.0704863142317494
737577 0.162569420050034
737943 0.217308133826187
738673 0.38956773145021
hazard1 = 4×2
736847 0.0813390794774647
737577 0.0521615800986284
737943 0.0674145844133175
738673 0.12428587278862
format
Reformat the default probability and hazard rate for a better representation.
DefProbHazard = [defaultProb1, hazard1(:,2)]; DefProbHazardTable = array2table(DefProbHazard, 'VariableNames', {'Date', 'DefaultProbability', 'HazardRate'}); DefProbHazardTable.Date = datetime(DefProbHazardTable.Date,'Locale','en_US','ConvertFrom','datenum'); DefProbHazardTable.Date.Format = 'MMM-dd-yyyy'
DefProbHazardTable=4×3 table
Date DefaultProbability HazardRate
___________ __________________ __________
Jun-01-2017 0.070486 0.081339
Jun-01-2019 0.16257 0.052162
Jun-01-2020 0.21731 0.067415
Jun-01-2022 0.38957 0.12429
Preview the selected bond to reprice based on the PD curve.
TestCase
TestCase=1×3 table
Maturity Price Coupon
___________ ______ ______
Jun-01-2019 109.02 0.08
To reprice the bond, first generate cash flows and payment dates.
[Payments, PaymentDates] = cfamounts(TestCase.Coupon, Settle, TestCase.Maturity); AccInt=-Payments(1); % Truncate the payments as well as payment dates for calculation % PaymentDates(1) is the settle date, no need for following calculations PaymentDates = PaymentDates(2:end)
PaymentDates = 1x6 datetime array
Columns 1 through 5
01-Dec-2016 01-Jun-2017 01-Dec-2017 01-Jun-2018 01-Dec-2018
Column 6
01-Jun-2019
Payments = Payments(2:end)
Payments = 1×6
4 4 4 4 4 104
Calculate the discount factors on the payment dates.
DF = zero2disc(interp1(RiskFreeRate.Date, RiskFreeRate.Rate, PaymentDates, 'linear', 'extrap'), PaymentDates, Settle, -1)
DF = 1×6
0.9987 0.9959 0.9926 0.9887 0.9845 0.9799
Assume that the recovery amount is a fixed proportion of bond's face value. The bond’s face value is 100
, and the recovery ratio is set to 40% as assumed in bondDefaultBootstrap
.
Num = length(Payments); RecoveryAmount = repmat(100*0.4, 1, Num)
RecoveryAmount = 1×6
40 40 40 40 40 40
Calculate the probability of default based on the default curve.
DefaultProb1 = bondDefaultBootstrap(ZeroData, MarketData, Settle, 'ZeroCompounding', -1, 'ProbabilityDates', PaymentDates'); SurvivalProb = 1 - DefaultProb1(:,2)
SurvivalProb = 6×1
0.9680
0.9295
0.9055
0.8823
0.8595
0.8375
Calculate the model-based clean bond price.
DirtyPrice = DF * (SurvivalProb.*Payments') + (RecoveryAmount.*DF) * (-diff([1;SurvivalProb])); ModelPrice = DirtyPrice - AccInt
ModelPrice = 109.0200
Compare the repriced bond to the market quote.
ResultTable = TestCase; ResultTable.ModelPrice = ModelPrice; ResultTable.Difference = ModelPrice - TestCase.Price
ResultTable=1×5 table
Maturity Price Coupon ModelPrice Difference
___________ ______ ______ __________ __________
Jun-01-2019 109.02 0.08 109.02 0
ZeroData
— Zero rate dataIRDataCurve
objectZero rate data, specified as an M
-by-2
matrix
of dates and zero rates or an IRDataCurve
object
of zero rates. For array input, the dates must be entered as serial
date numbers, and discount rate must be in decimal form.
When ZeroData
is an IRDataCurve
object, ZeroCompounding
and ZeroBasis
are
implicit in ZeroData
and are redundant inside this
function. In this case, specify these optional parameters when constructing
the IRDataCurve
object before using this bondDefaultBootstrap
function.
For more information on an IRDataCurve
object,
see Creating an IRDataCurve Object (Financial Instruments Toolbox).
Data Types: double
MarketData
— Bond market dataBond market data, specified as an N
-by-3
matrix
of maturity dates, market prices, and coupon rates for bonds. The
dates must be entered as serial date numbers, market prices must be
numeric values, and coupon rate must be in decimal form.
A warning is displayed when MarketData
is
not sorted in ascending order by time.
Data Types: double
Settle
— Settlement dateSettlement date, specified as a serial date number, a date character
vector, a datetime object, or a date string object. Settle
must
be earlier than or equal to the maturity dates in MarketData
.
Data Types: double
| char
| datetime
| string
Specify optional
comma-separated pairs of Name,Value
arguments. Name
is
the argument name and Value
is the corresponding value.
Name
must appear inside quotes. You can specify several name and value
pair arguments in any order as Name1,Value1,...,NameN,ValueN
.
[ProbabilityData,HazardData] = bondDefaultBootstrap(ZeroData,MarketData,Settle,'RecoveryRate',Recovery,'ZeroCompounding',-1)
Any optional input of size N
-by-1
is
also acceptable as an array of size 1
-by-N
,
or as a single value applicable to all contracts.
'RecoveryRate'
— Recovery rate0.4
(default) | decimalRecovery rate, specified as the comma-separated pair consisting
of 'RecoveryRate'
and a N
-by-1
vector
of recovery rates, expressed as a decimal from 0
through 1
.
Data Types: double
'ProbabilityDates'
— Dates for output of default probability dataMarketData
(default) | serial date number | date character vector | datetime object | date string objectDates for the output of default probability data, specified
as the comma-separated pair consisting of 'ProbabilityDates'
and
a P
-by-1
vector, given as serial
date numbers, datetime objects, date character vectors, or date string
objects.
Data Types: double
| char
| datetime
| string
'ZeroCompounding'
— Compounding frequency of the zero curve2
(semiannual) (default) | integer with value of 1
,2
,3
,4
,6
,12
,
or –1
Compounding frequency of the zero curve, specified as the comma-separated
pair consisting of 'ZeroCompounding'
and a N
-by-1
vector.
Values are:
1
— Annual compounding
2
— Semiannual compounding
3
— Compounding three times
per year
4
— Quarterly compounding
6
— Bimonthly compounding
12
— Monthly compounding
−1
— Continuous compounding
Data Types: double
'ZeroBasis'
— Basis of the zero curve0
(actual/actual) (default) | integer with value of 0
to 13
Basis of the zero curve, specified as the comma-separated pair
consisting of 'ZeroBasis'
and the same values listed
for Basis
.
Data Types: double
'RecoveryMethod'
— Recovery method'facevalue'
(default) | character vector with value of 'presentvalue'
or 'facevalue'
| string object with value of 'presentvalue'
or 'facevalue'
Recovery method, specified as the comma-separated pair consisting
of 'RecoveryMethod'
and a character vector or a
string with a value of 'presentvalue'
or 'facevalue'
.
'presentvalue'
assumes that upon
default, a bond is valued at a given fraction to the hypothetical
present value of its remaining cash flows, discounted at risk-free
rate.
'facevalue'
assumes that a bond
recovers a given fraction of its face value upon recovery.
Data Types: char
| string
'Face'
— Face or par value100
(default) | numericFace or par value, specified as the comma-separated pair consisting
of 'Face'
and a NINST
-by-1
vector
of bonds.
Data Types: double
'Period'
— Payment frequency2
(default) | numeric with values 0
, 1
, 2
, 3
, 4
, 6
or 12
Payment frequency, specified as the comma-separated pair consisting
of 'Period'
and a N
-by-1
vector
with values of 0
, 1
, 2
, 3
, 4
, 6
,
or 12
.
Data Types: double
'Basis'
— Day-count basis of the instrument0
(actual/actual) (default) | integers of the set [0...13]
| vector of integers of the set [0...13]
Day-count basis of the instrument, specified as the comma-separated
pair consisting of 'Basis'
and a positive integer
using a NINST
-by-1
vector. Values
are:
0 = actual/actual
1 = 30/360 (SIA)
2 = actual/360
3 = actual/365
4 = 30/360 (PSA)
5 = 30/360 (ISDA)
6 = 30/360 (European)
7 = actual/365 (Japanese)
8 = actual/actual (ICMA)
9 = actual/360 (ICMA)
10 = actual/365 (ICMA)
11 = 30/360E (ICMA)
12 = actual/365 (ISDA)
13 = BUS/252
For more information, see basis.
Data Types: double
'EndMonthRule'
— End-of-month rule flag1
(in effect) (default) | nonnegative integer 0
or 1
End-of-month rule flag, specified as the comma-separated pair
consisting of 'EndMonthRule'
and a nonnegative
integer, 0
or 1
, using a NINST
-by-1
vector.
This rule applies only when Maturity
is an end-of-month
date for a month having 30 or fewer days.
0
= Ignore rule, meaning that a
bond coupon payment date is always the same numerical day of the month.
1
= Set rule on, meaning that a
bond coupon payment date is always the last actual day of the month.
Data Types: double
'IssueDate'
— Bond issue dateIssueDate
not specified,
cash flow payment dates determined from other inputs (default) | serial date number | date character vector | datetime object | date string objectBond issue date, specified as the comma-separated pair consisting
of 'IssueDate'
and a N
-by-1
vector,
given as serial date numbers, datetime objects, date character vectors,
or date string objects.
Data Types: double
| char
| datetime
| string
'FirstCouponDate'
— First actual coupon dateFirstCouponDate
,
cash flow payment dates are determined from other inputs (default) | serial date numberFirst actual coupon date, specified as the comma-separated pair
consisting of 'FirstCouponDate'
and a serial date
number. FirstCouponDate
is used when a bond has
an irregular first coupon period. When FirstCouponDate
and LastCouponDate
are
both specified, FirstCouponDate
takes precedence
in determining the coupon payment structure.
Data Types: double
'LastCouponDate'
— Last actual coupon dateLastCouponDate
,
cash flow payment dates are determined from other inputs (default) | scalar for serial date numberLast actual coupon date, specified as the comma-separated pair
consisting of 'LastCouponDate'
and a serial date
number. LastCouponDate
is used when a bond has
an irregular last coupon period. In the absence of a specified FirstCouponDate
,
a specified LastCouponDate
determines the coupon
structure of the bond. The coupon structure of a bond is truncated
at the LastCouponDate
, regardless of where it falls,
and is followed only by the bond's maturity cash flow date.
Data Types: double
'StartDate'
— Forward starting date of paymentsStartDate
,
effective start date is Settle
date (default) | serial date numberForward starting date of payments, specified as the comma-separated
pair consisting of 'StartDate'
and a serial date
number. StartDate
is when a bond actually starts
(the date from which a bond cash flow is considered). To make an instrument
forward-starting, specify this date as a future date.
Data Types: double
'BusinessDayConvention'
— Business day conventions'actual'
(default) | character vector or string object with
values'actual'
, 'follow'
,
'modifiedfollow'
, 'previous'
or
'modifiedprevious'
Business day conventions, specified as the comma-separated pair consisting of
'BusinessDayConvention'
and a character
vector or a string object. The selection for business day convention
determines how nonbusiness days are treated. Nonbusiness days are
defined as weekends plus any other date that businesses are not open
(for example, statutory holidays). Values are:
'actual'
— Nonbusiness days
are effectively ignored. Cash flows that fall on
non-business days are assumed to be distributed on the
actual date.
'follow'
— Cash flows that
fall on a nonbusiness day are assumed to be distributed
on the following business day.
'modifiedfollow'
— Cash
flows that fall on a non-business day are assumed to be
distributed on the following business day. However if
the following business day is in a different month, the
previous business day is adopted instead.
'previous'
— Cash flows that
fall on a nonbusiness day are assumed to be distributed
on the previous business day.
'modifiedprevious'
— Cash
flows that fall on a nonbusiness day are assumed to be
distributed on the previous business day. However if the
previous business day is in a different month, the
following business day is adopted instead.
Data Types: char
| cell
| string
ProbabilityData
— Default probability valuesDefault probability values, returned as a P
-by-2
matrix
with dates and corresponding cumulative default probability values.
The dates match those in MarketData
, unless the
optional input parameter ProbabilityDates
is
provided.
HazardData
— Hazard rate valuesHazard rate values, returned as an N
-by-2
matrix
with dates and corresponding hazard rate values for the survival probability
model. The dates match those in MarketData
.
A warning is displayed when nonmonotone default probabilities (that is, negative hazard rates) are found.
[1] Jarrow, Robert A., and Stuart Turnbull. "Pricing Derivatives on Financial Securities Subject to Credit Risk." Journal of Finance. 50.1, 1995, pp. 53–85.
[2] Berd, A., Mashal, R. and Peili Wang. “Defining, Estimating and Using Credit Term Structures.” Research report, Lehman Brothers, 2004.
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