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bondbyzero

Price bond from set of zero curves

Syntax

[Price,DirtyPrice,CFlowAmounts,CFlowDates] = bondbyzero(RateSpec,CouponRate,Settle,Maturity)
[Price,DirtyPrice,CFlowAmounts,CFlowDates] = bondbyzero(___,Name,Value)

Description

example

[Price,DirtyPrice,CFlowAmounts,CFlowDates] = bondbyzero(RateSpec,CouponRate,Settle,Maturity) prices a bond from a set of zero curves. bondbyzero computes prices of vanilla bonds, stepped coupon bonds and amortizing bonds.

example

[Price,DirtyPrice,CFlowAmounts,CFlowDates] = bondbyzero(___,Name,Value) adds additional name-value pair arguments.

Examples

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Price a 4% bond using a zero curve.

Load deriv.mat, which provides ZeroRateSpec, the interest-rate term structure, needed to price the bond.

load deriv.mat; 
CouponRate = 0.04;
Settle = '01-Jan-2000';
Maturity = '01-Jan-2004';
Price = bondbyzero(ZeroRateSpec, CouponRate, Settle, Maturity)
Price = 97.5334

Price single stepped coupon bonds using market data.

Define data for the interest-rate term structure.

Rates = [0.035; 0.042147; 0.047345; 0.052707];
ValuationDate = 'Jan-1-2010';
StartDates = ValuationDate;
EndDates = {'Jan-1-2011'; 'Jan-1-2012'; 'Jan-1-2013'; 'Jan-1-2014'};
Compounding = 1;

Create the RateSpec.

RS = intenvset('ValuationDate', ValuationDate, 'StartDates', StartDates,...
'EndDates', EndDates,'Rates', Rates, 'Compounding', Compounding)
RS = struct with fields:
           FinObj: 'RateSpec'
      Compounding: 1
             Disc: [4x1 double]
            Rates: [4x1 double]
         EndTimes: [4x1 double]
       StartTimes: [4x1 double]
         EndDates: [4x1 double]
       StartDates: 734139
    ValuationDate: 734139
            Basis: 0
     EndMonthRule: 1

Create the stepped bond instrument.

Settle = '01-Jan-2010';
Maturity = {'01-Jan-2011';'01-Jan-2012';'01-Jan-2013';'01-Jan-2014'};
CouponRate = {{'01-Jan-2012' .0425;'01-Jan-2014' .0750}};
Period = 1;

Compute the price of the stepped coupon bonds.

PZero= bondbyzero(RS, CouponRate, Settle, Maturity ,Period)
PZero = 

  100.7246
  100.0945
  101.5900
  102.0820

Price a bond with an amortizing schedule using the Face input argument to define the schedule.

Define data for the interest-rate term structure.

Rates = 0.065;
ValuationDate = '1-Jan-2011';
StartDates = ValuationDate;
EndDates=  '1-Jan-2017';
Compounding = 1;

Create the RateSpec.

RateSpec = intenvset('ValuationDate', ValuationDate,'StartDates', StartDates,...
'EndDates', EndDates,'Rates', Rates, 'Compounding', Compounding)
RateSpec = struct with fields:
           FinObj: 'RateSpec'
      Compounding: 1
             Disc: 0.6853
            Rates: 0.0650
         EndTimes: 6
       StartTimes: 0
         EndDates: 736696
       StartDates: 734504
    ValuationDate: 734504
            Basis: 0
     EndMonthRule: 1

Create and price the amortizing bond instrument. The bond has a coupon rate of 7%, a period of one year, and matures on 1-Jan-2017.

CouponRate = 0.07;
Settle ='1-Jan-2011';
Maturity = '1-Jan-2017';
Period = 1;
Face = {{'1-Jan-2015' 100;'1-Jan-2016' 90;'1-Jan-2017' 80}};
Price = bondbyzero(RateSpec, CouponRate, Settle, Maturity, 'Period',...
Period, 'Face', Face)
Price = 102.3155

Compare the results with price of a vanilla bond.

PriceVanilla = bondbyzero(RateSpec, CouponRate, Settle, Maturity,Period)
PriceVanilla = 102.4205

Price both the amortizing and vanilla bonds.

Face = {{'1-Jan-2015' 100;'1-Jan-2016' 90;'1-Jan-2017' 80};
         100};
PriceBonds = bondbyzero(RateSpec, CouponRate, Settle, Maturity, 'Period',...
               Period, 'Face', Face)
PriceBonds = 

  102.3155
  102.4205

When a bond is first issued, it can be priced with bondbyzero on that day by setting the Settle date to the issue date. Later on, if the bond needs to be traded someday between the issue date and the maturity date, its new price can be computed by updating the Settle date, as well as the RateSpec input.

Note that the bond's price is determined by its remaining cash flows and the zero-rate term structure, which can both change as the bond matures. While bondbyzero automatically updates the bond's remaining cash flows with respect to the new Settle date, you must supply a new RateSpec input in order to reflect the new zero-rate term structure for that new Settle date.

Use the following Bond information.

IssueDate = datenum('20-May-2014');
CouponRate = 0.01;
Maturity = datenum('20-May-2019');

Determine the bond price on 20-May-2014.

Settle1 = datenum('20-May-2014');
ZeroDates1 = datemnth(Settle1,12*[1 2 3 5 7 10 20]');
ZeroRates1 = [0.23 0.63 1.01 1.60 2.01 2.27 2.79]'/100;
RateSpec1 = intenvset('StartDate',Settle1,'EndDates',ZeroDates1,'Rates',ZeroRates1);
[Price1, ~, CFlowAmounts1, CFlowDates1] = bondbyzero(RateSpec1, ...
    CouponRate, Settle1, Maturity, 'IssueDate', IssueDate);
Price1
Price1 = 97.1899

Determine the bond price on 10-Aug-2015.

Settle2 = datenum('10-Aug-2015');
ZeroDates2 = datemnth(Settle2,12*[1 2 3 5 7 10 20]');
ZeroRates2 = [0.40 0.73 1.09 1.62 1.98 2.24 2.58]'/100;
RateSpec2 = intenvset('StartDate',Settle2,'EndDates',ZeroDates2,'Rates',ZeroRates2);
[Price2, ~, CFlowAmounts2, CFlowDates2] = bondbyzero(RateSpec2, ...
    CouponRate, Settle2, Maturity, 'IssueDate', IssueDate);
Price2
Price2 = 98.9384

To price three bonds using two different curves, define the RateSpec:

StartDates = '01-April-2016';
EndDates = ['01-April-2017'; '01-April-2018';'01-April-2019';'01-April-2020'];
Rates = [[0.0356;0.041185;0.04489;0.047741],[0.0325;0.0423;0.0437;0.0465]];
RateSpec = intenvset('Rates', Rates, 'StartDates',StartDates,...
'EndDates', EndDates, 'Compounding', 1)
RateSpec = struct with fields:
           FinObj: 'RateSpec'
      Compounding: 1
             Disc: [4x2 double]
            Rates: [4x2 double]
         EndTimes: [4x1 double]
       StartTimes: [4x1 double]
         EndDates: [4x1 double]
       StartDates: 736421
    ValuationDate: 736421
            Basis: 0
     EndMonthRule: 1

Price three bonds with the same Maturity and different coupons.

Settle = '01-April-2016';
Maturity = '01-April-2020';
Price = bondbyzero(RateSpec,[0.025;0.028;0.035],Settle,Maturity)
Price = 

   92.0766   92.4888
   93.1680   93.5823
   95.7145   96.1338

To adjust the cash flows according to the accrual amount, use the optional input argument AdjustCashFlowsBasis when calling bondbyzero.

Use the following data to define the interest-rate term structure and to create a RateSpec.

Rates = 0.065;
ValuationDate = '1-Jan-2011';
StartDates = ValuationDate;
EndDates=  '1-Jan-2017';
Compounding = 1;
RateSpec = intenvset('ValuationDate',ValuationDate,'StartDates',StartDates,...
'EndDates', EndDates,'Rates',Rates,'Compounding',Compounding);
CouponRate = 0.07;
Settle ='1-Jan-2011';
Maturity = '1-Jan-2017';
Period = 1;
Face = {{'1-Jan-2015' 100;'1-Jan-2016' 90;'1-Jan-2017' 80}};

Use cfamounts and cycle through the Basis of 0 to 13, using the optional argument AdjustCashFlowsBasis to determine the cash flow amounts for accrued interest due at settlement.

AdjustCashFlowsBasis = true;
CFlowAmounts =  cfamounts(CouponRate,Settle,Maturity,'Period',Period,'Basis',0:13,'AdjustCashFlowsBasis',AdjustCashFlowsBasis)
CFlowAmounts = 

         0    7.0000    7.0000    7.0000    7.0000    7.0000  107.0000
         0    7.0000    7.0000    7.0000    7.0000    7.0000  107.0000
         0    7.0972    7.1167    7.0972    7.0972    7.0972  107.1167
         0    7.0000    7.0192    7.0000    7.0000    7.0000  107.0192
         0    7.0000    7.0000    7.0000    7.0000    7.0000  107.0000
         0    7.0000    7.0000    7.0000    7.0000    7.0000  107.0000
         0    7.0000    7.0000    7.0000    7.0000    7.0000  107.0000
         0    7.0000    7.0000    7.0000    7.0000    7.0000  107.0000
         0    7.0000    7.0000    7.0000    7.0000    7.0000  107.0000
         0    7.0972    7.1167    7.0972    7.0972    7.0972  107.1167

Notice that the cash flow amounts have been adjusted according to Basis.

Price a vanilla bond using the input argument AdjustCashFlowsBasis.

PriceVanilla = bondbyzero(RateSpec,CouponRate,Settle,Maturity,'Period',Period,'Basis',0:13,'AdjustCashFlowsBasis',AdjustCashFlowsBasis)
PriceVanilla = 

  102.4205
  102.4205
  102.9216
  102.4506
  102.4205
  102.4205
  102.4205
  102.4205
  102.4205
  102.9216

Input Arguments

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Interest-rate structure, specified using intenvset to create a RateSpec for an annualized zero rate term structure.

Data Types: struct

Bond coupon rate, specified as an NINST-by-1 decimal annual rate or NINST-by-1 cell array, where each element is a NumDates-by-2 cell array. The first column of the NumDates-by-2 cell array is dates and the second column is associated rates. The date indicates the last day that the coupon rate is valid.

Data Types: double | cell

Settlement date, specified either as a scalar or NINST-by-1 vector of serial date numbers or date character vectors.

Settle must be earlier than Maturity.

Data Types: char | double

Maturity date, specified as a NINST-by-1 vector of serial date numbers or date character vectors representing the maturity date for each bond.

Data Types: char | double

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: Price = bondbyzero(RateSpec,CouponRate,Settle,Maturity,'Period',4,'Face',10000)

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Coupons per year, specified as an NINST-by-1 vector. Values for Period are 1, 2, 3, 4, 6, and 12.

Data Types: double

Day-count basis of the instrument, specified as an NINST-by-1 vector.

  • 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

End-of-month rule flag for generating dates when Maturity is an end-of-month date for a month having 30 or fewer days, specified as nonnegative integer [0, 1] using a NINST-by-1 vector.

  • 0 = Ignore rule, meaning that a payment date is always the same numerical day of the month.

  • 1 = Set rule on, meaning that a payment date is always the last actual day of the month.

Data Types: logical

Bond issue date, specified as an NINST-by-1 vector using a serial nonnegative date number or date character vector.

Data Types: double | char

Irregular first coupon date, specified as an NINST-by-1 vector using a serial nonnegative date number or date character vector.

Data Types: double | char

Irregular last coupon date, specified as an NINST-by-1 vector using a serial nonnegative date number or date character vector.

Data Types: double | char

Forward starting date of payments (the date from which a bond cash flow is considered), specified as a NINST-by-1 vector using serial date numbers or date character vectors.

If you do not specify StartDate, the effective start date is the Settle date.

Data Types: char | double

Face value, specified as an NINST-by-1 scalar of nonnegative face values or an NINST-by-1 cell array, where each element is a NumDates-by-2 cell array. The first column of the NumDates-by-2 cell array is dates and the second column is the associated face value. The date indicates the last day that the face value is valid.

Data Types: cell | double

Derivatives pricing options, specified as structure that is created with derivset.

Data Types: struct

Flag to adjust cash flows based on actual period day count, specified as a NINST-by-1 vector of logicals with values of 0 (false) or 1 (true).

Data Types: logical

Business day conventions, specified by a character vector or a N-by-1 (or NINST-by-2 if BusDayConvention is different for each leg) cell array of character vectors of business day conventions. The selection for business day convention determines how non-business days are treated. Non-business days are defined as weekends plus any other date that businesses are not open (e.g. statutory holidays). Values are:

  • actual — Non-business 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 non-business 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 non-business day are assumed to be distributed on the previous business day.

  • modifiedprevious — Cash flows that fall on a non-business 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

Holidays used in computing business days, specified as MATLAB date numbers using a NHolidays-by-1 vector.

Data Types: double

Output Arguments

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Floating-rate note prices, returned as a (NINST) by number of curves (NUMCURVES) matrix. Each column arises from one of the zero curves.

Dirty bond price (clean + accrued interest), returned as a NINST- by-NUMCURVES matrix. Each column arises from one of the zero curves.

Cash flow amounts, returned as a NINST- by-NUMCFS matrix of cash flows for each bond.

Cash flow dates, returned as a NINST- by-NUMCFS matrix of payment dates for each bond.

More About

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Vanilla Bond

A vanilla coupon bond is a security representing an obligation to repay a borrowed amount at a designated time and to make periodic interest payments until that time.

The issuer of a bond makes the periodic interest payments until the bond matures. At maturity, the issuer pays to the holder of the bond the principal amount owed (face value) and the last interest payment.

Stepped Coupon Bond

A step-up and step-down bond is a debt security with a predetermined coupon structure over time.

With these instruments, coupons increase (step up) or decrease (step down) at specific times during the life of the bond.

Bond with an Amortization Schedule

An amortized bond is treated as an asset, with the discount amount being amortized to interest expense over the life of the bond.

Introduced before R2006a

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