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

[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.

Note

Alternatively, you can use the FixedBond object to price fixed-rate bond instruments. For more information, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.

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

Examples

Price a Vanilla Bond

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 = datetime(2000,1,1);
Maturity = datetime(2004,1,1);
Price = bondbyzero(ZeroRateSpec, CouponRate, Settle, Maturity)

Price = 
97.5334

Price a Stepped Coupon Bond

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 = datetime(2010,1,1);
StartDates = ValuationDate;
EndDates = [datetime(2011,1,1) ; datetime(20102,1,1) ; datetime(2013,1,1) ; datetime(2014,1,1)];
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 = datetime(2010,1,1);
Maturity = [datetime(2011,1,1) ; datetime(2012,1,1) ; datetime(2013,1,1) ; datetime(2014,1,1)];
CouponRate = {{datetime(2012,1,1) .0425;datetime(2014,1,1) .0750}};
Period = 1;

Compute the price of the stepped coupon bonds.

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

PZero = 4×1

  100.7246
  101.4247
  101.6442
  102.1362

Price a Bond with an Amortizing Schedule

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 = datetime(2011,1,1);
StartDates = ValuationDate;
EndDates=  datetime(2017,1,1);
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 = datetime(2011,1,1);
Maturity = datetime(2017,1,1);
Period = 1;
Face = {{datetime(2015,1,1) 100;datetime(2016,1,1) 90;datetime(2017,1,1) 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 = {{datetime(2015,1,1) 100;datetime(2016,1,1) 90;datetime(2017,1,1) 80};
         100};
PriceBonds = bondbyzero(RateSpec, CouponRate, Settle, Maturity, 'Period',...
               Period, 'Face', Face)

PriceBonds = 2×1

  102.3155
  102.4205

Price a Bond in a Holding Period

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 = datetime(2014,5,20);
CouponRate = 0.01;
Maturity = datetime(2019,5,20);

Determine the bond price on 20-May-2014.

Settle1 = datetime(2014,5,20);
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 = datetime(2015,8,10);
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

Price Three Bonds Using Two Different Curves

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

StartDates = datetime(2016,4,1);
EndDates = [datetime(2017,4,1) ; datetime(2018,4,1) ; datetime(2019,4,1) ; datetime(2020,4,1)];
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 = datetime(2016,4,1);
Maturity = datetime(2020,4,1);
Price = bondbyzero(RateSpec,[0.025;0.028;0.035],Settle,Maturity)

Price = 3×2

   92.0766   92.4888
   93.1680   93.5823
   95.7145   96.1338

Price a Vanilla Bond Using the Optional Input Argument AdjustCashFlowsBasis

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 = datetime(2011,1,1);
StartDates = ValuationDate;
EndDates=  datetime(2017,1,1);
Compounding = 1;
RateSpec = intenvset('ValuationDate',ValuationDate,'StartDates',StartDates,...
'EndDates', EndDates,'Rates',Rates,'Compounding',Compounding);
CouponRate = 0.07;
Settle = datetime(2011,1,1);
Maturity = datetime(2017,1,1);
Period = 1;
Face = {{datetime(2015,1,1) 100;datetime(2016,1,1) 90;datetime(2017,1,1) 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 = 14×7

         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 = 14×1

  102.4205
  102.4205
  102.9216
  102.4506
  102.4205
  102.4205
  102.4205
  102.4205
  102.4205
  102.9216
      ⋮

Input Arguments

RateSpec — Interest-rate structure

structure

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

Data Types: struct

CouponRate — Bond coupon rate

positive decimal value

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

Settle — Settlement date

datetime array | string array | date character vector

Settlement date, specified either as a scalar or NINST-by-1 vector using a datetime array, string array, or date character vectors.

To support existing code, bondbyzero also accepts serial date numbers as inputs, but they are not recommended.

Settle must be earlier than Maturity.

Maturity — Maturity date

datetime array | string array | date character vector

Maturity date, specified as a NINST-by-1 vector using a datetime array, string array, or date character vectors representing the maturity date for each bond.

To support existing code, bondbyzero also accepts serial date numbers as inputs, but they are not recommended.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

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

Period — Coupons per year

2 per year (default) | vector

Coupons per year, specified as the comma-separated pair consisting of 'Period' and a NINST-by-1 vector. Values for Period are 1, 2, 3, 4, 6, and 12.

Data Types: double

Basis — Day-count basis

0 (actual/actual) (default) | integer from 0 to 13

Day-count basis of the instrument, specified as the comma-separated pair consisting of 'Basis' and a 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

EndMonthRule — End-of-month rule flag for generating dates when Maturity is end-of-month date for month having 30 or fewer days

1 (in effect) (default) | nonnegative integer [0,1]

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 the comma-separated pair consisting of 'EndMonthRule' and a 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

IssueDate — Bond issue date

datetime array | string array | date character vector

Bond issue date, specified as the comma-separated pair consisting of 'IssueDate' and a NINST-by-1 vector using a datetime array, string array, or date character vectors.

To support existing code, bondbyzero also accepts serial date numbers as inputs, but they are not recommended.

FirstCouponDate — Irregular first coupon date

datetime array | string array | date character vector

Irregular first coupon date, specified as the comma-separated pair consisting of 'FirstCouponDate' and a NINST-by-1 vector using a datetime array, string array, or date character vectors.

To support existing code, bondbyzero also accepts serial date numbers as inputs, but they are not recommended.

LastCouponDate — Irregular last coupon date

datetime array | string array | date character vector

Irregular last coupon date, specified as the comma-separated pair consisting of 'LastCouponDate' and a NINST-by-1 vector using a datetime array, string array, or date character vectors.

To support existing code, bondbyzero also accepts serial date numbers as inputs, but they are not recommended.

StartDate — Forward starting date of payments

Settle date (default) | datetime array | string array | date character vector

Forward starting date of payments (the date from which a bond cash flow is considered), specified as the comma-separated pair consisting of 'StartDate' and a NINST-by-1 vector using a datetime array, string array, or date character vectors.

To support existing code, bondbyzero also accepts serial date numbers as inputs, but they are not recommended.

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

Face — Face value

100 (default) | scalar of nonnegative value | cell array of nonnegative values

Face value, specified as the comma-separated pair consisting of 'Face' and a 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

Options — Derivatives pricing options

structure

Derivatives pricing options, specified as the comma-separated pair consisting of 'Options' and a structure that is created with derivset.

Data Types: struct

AdjustCashFlowsBasis — Flag to adjust cash flows based on actual period day count

false (default) | value of 0 (false) or 1 (true)

Flag to adjust cash flows based on actual period day count, specified as the comma-separated pair consisting of 'AdjustCashFlowsBasis' and a NINST-by-1 vector of logicals with values of 0 (false) or 1 (true).

Data Types: logical

BusinessDayConvention — Business day conventions

actual (default) | character vector | cell array of character vectors

Business day conventions, specified as the comma-separated pair consisting of 'BusinessDayConvention' and a character vector or a N-by-1 (or NINST-by-2 if BusinessDayConvention 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 — Holidays used in computing business days

if not specified, the default is to use holidays.m (default) | MATLAB^® dates

Holidays used in computing business days, specified as the comma-separated pair consisting of 'Holidays' and MATLAB dates using a NHolidays-by-1 vector.

Data Types: datetime

Output Arguments

Price — Fixed-rate note prices

matrix

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

DirtyPrice — Dirty bond price

matrix

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

CFlowAmounts — Cash flow amounts

matrix

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

CFlowDates — Cash flow dates

matrix

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

More About

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.

Version History

Introduced before R2006a

R2022b: Serial date numbers not recommended

Although bondbyzero supports serial date numbers, datetime values are recommended instead. The datetime data type provides flexible date and time formats, storage out to nanosecond precision, and properties to account for time zones and daylight saving time.

To convert serial date numbers or text to datetime values, use the datetime function. For example:

t = datetime(738427.656845093,"ConvertFrom","datenum");
y = year(t)

y =

        2021

There are no plans to remove support for serial date number inputs.