zbtyield

Zero curve bootstrapping from coupon bond data given yield

Syntax

[ZeroRates,CurveDates] = zbtyield(Bonds,YieldsSettle)

ZeroRates,CurveDates = zbtyield(___,OutputCompounding)

Description

[ZeroRates,CurveDates] = zbtyield(Bonds,YieldsSettle) uses the bootstrap method to return a zero curve given a portfolio of coupon bonds and their yields.

A zero curve consists of the yields to maturity for a portfolio of theoretical zero-coupon bonds that are derived from the input Bonds portfolio. The bootstrap method that this function uses does not require alignment among the cash-flow dates of the bonds in the input portfolio. It uses theoretical par bond arbitrage and yield interpolation to derive all zero rates; specifically, the interest rates for cash flows are determined using linear interpolation. For best results, use a portfolio of at least 30 bonds evenly spaced across the investment horizon.

ZeroRates,CurveDates = zbtyield(___,OutputCompounding) adds an optional argument for OutputCompounding.

Examples

Compute a Zero Curve Given a Portfolio of Coupon Bonds and Their Yields

Given data and yields to maturity for 12 coupon bonds, two with the same maturity date; and given the common settlement date.

Bonds = [datenum('6/1/1998')   0.0475   100  2  0  0;
         datenum('7/1/2000')   0.06     100  2  0  0;
         datenum('7/1/2000')   0.09375  100  6  1  0;
         datenum('6/30/2001')  0.05125  100  1  3  1;
         datenum('4/15/2002')  0.07125  100  4  1  0;
         datenum('1/15/2000')  0.065    100  2  0  0;
         datenum('9/1/1999')   0.08     100  3  3  0;
         datenum('4/30/2001')  0.05875  100  2  0  0;
         datenum('11/15/1999') 0.07125  100  2  0  0;
         datenum('6/30/2000')  0.07     100  2  3  1;
         datenum('7/1/2001')   0.0525   100  2  3  0;
         datenum('4/30/2002')  0.07     100  2  0  0];

Yields = [0.0616
          0.0605
          0.0687
          0.0612
          0.0615
          0.0591
          0.0603
          0.0608
          0.0655
          0.0646
          0.0641
          0.0627];

Settle = datenum('12/18/1997');

Set semiannual compounding for the zero curve.

OutputCompounding = 2;

Execute the function zbtyield which returns the zero curve at the maturity dates. Note the mean zero rate for the two bonds with the same maturity date.

[ZeroRates, CurveDates] = zbtyield(Bonds, Yields, Settle,... 
OutputCompounding)

ZeroRates = 11×1

    0.0616
    0.0603
    0.0657
    0.0590
    0.0649
    0.0650
    0.0606
    0.0611
    0.0643
    0.0614
      ⋮

CurveDates = 11×1

      729907
      730364
      730439
      730500
      730667
      730668
      730971
      731032
      731033
      731321
      ⋮

Compute a Zero Curve Given a Portfolio of Coupon Bonds and Their Yields Using datetime Inputs

Given data and yields to maturity for 12 coupon bonds (two with the same maturity date), and given the common settlement date, compute the zero curve using datetime inputs.

Bonds = [datenum('6/1/1998')   0.0475   100  2  0  0;
         datenum('7/1/2000')   0.06     100  2  0  0;
         datenum('7/1/2000')   0.09375  100  6  1  0;
         datenum('6/30/2001')  0.05125  100  1  3  1;
         datenum('4/15/2002')  0.07125  100  4  1  0;
         datenum('1/15/2000')  0.065    100  2  0  0;
         datenum('9/1/1999')   0.08     100  3  3  0;
         datenum('4/30/2001')  0.05875  100  2  0  0;
         datenum('11/15/1999') 0.07125  100  2  0  0;
         datenum('6/30/2000')  0.07     100  2  3  1;
         datenum('7/1/2001')   0.0525   100  2  3  0;
         datenum('4/30/2002')  0.07     100  2  0  0];

Yields = [0.0616
          0.0605
          0.0687
          0.0612
          0.0615
          0.0591
          0.0603
          0.0608
          0.0655
          0.0646
          0.0641
          0.0627];

Settle = datenum('12/18/1997');
OutputCompounding = 2;
t = array2table(Bonds,'VariableNames',{'Maturity','CouponRate', 'Face' ,'Period', 'Basis', 'EndMonthRule'});
disp(t)

     Maturity     CouponRate    Face    Period    Basis    EndMonthRule
    __________    __________    ____    ______    _____    ____________

    7.2991e+05      0.0475      100       2         0           0      
    7.3067e+05        0.06      100       2         0           0      
    7.3067e+05     0.09375      100       6         1           0      
    7.3103e+05     0.05125      100       1         3           1      
    7.3132e+05     0.07125      100       4         1           0      
     7.305e+05       0.065      100       2         0           0      
    7.3036e+05        0.08      100       3         3           0      
    7.3097e+05     0.05875      100       2         0           0      
    7.3044e+05     0.07125      100       2         0           0      
    7.3067e+05        0.07      100       2         3           1      
    7.3103e+05      0.0525      100       2         3           0      
    7.3134e+05        0.07      100       2         0           0      

t.Maturity = datetime(t.Maturity,'ConvertFrom','datenum','Locale','en_US');
Settle = datetime(Settle,'ConvertFrom','datenum','Locale','en_US');
[ZeroRates, CurveDates] = zbtyield(t, Yields, Settle,...
OutputCompounding)

ZeroRates = 11×1

    0.0616
    0.0603
    0.0657
    0.0590
    0.0649
    0.0650
    0.0606
    0.0611
    0.0643
    0.0614
      ⋮

CurveDates = 11x1 datetime array
   01-Jun-1998 00:00:00
   01-Sep-1999 00:00:00
   15-Nov-1999 00:00:00
   15-Jan-2000 00:00:00
   30-Jun-2000 00:00:00
   01-Jul-2000 00:00:00
   30-Apr-2001 00:00:00
   30-Jun-2001 00:00:00
   01-Jul-2001 00:00:00
   15-Apr-2002 00:00:00
   30-Apr-2002 00:00:00

Compute the Real Zero Rates From the Real Yields of Inflation-Linked Bonds

Use zbtyield to compute the real zero rates from the real yields of inflation-linked bonds.

% Load the data
load usbond_02Sep2008
Settle = datenum('02-Sep-2008');

Compute the real yields and then compute the real zero rates.

RealYields = bndyield(TIPSPrice,TIPSCoupon,Settle,TIPSMaturity);
TIPSBonds = [TIPSMaturity TIPSCoupon];
[RealZeroRates, CurveDates] = zbtyield(TIPSBonds, RealYields, Settle)

RealZeroRates = 26×1

    0.0069
    0.0094
    0.0092
    0.0111
    0.0110
    0.0119
    0.0116
    0.0128
    0.0126
    0.0136
      ⋮

CurveDates = 26×1

      734153
      734243
      734518
      734608
      734883
      734974
      735065
      735339
      735430
      735614
      ⋮

Input Arguments

Bonds — Coupon bond information to generate zero curve

table | matrix

Coupon bond information to generate zero curve, specified as a 6-column table or a n-by-2 to n-by-6 matrix of bond information, where the table columns or matrix columns contains:

• Maturity (Column 1, Required) Maturity date of the bond, as a serial date number. Use datenum to convert date character vectors to serial date numbers. If the input Bonds is a table, the Maturity dates can be serial date numbers, date character vectors, or datetime arrays.

• CouponRate (Column 2, Required) Decimal fraction indicating the coupon rate of the bond.

• Face (Column 3, Optional) Redemption or face value of the bond. Default = 100.

• Period (Column 4, Optional) Coupons per year of the bond. Allowed values are 0, 1, 2 (default), 3, 4, 6, and 12.

• Basis (Column 5, Optional) Day-count basis of the bond. A vector of integers.

  • 0 = actual/actual (default)

  • 1 = 30/360 (SIA)

  • 2 = actual/360

  • 3 = actual/365

  • 4 = 30/360 (BMA)

  • 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.

• EndMonthRule (Column 6, Optional) End-of-month rule. 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's coupon payment date is always the same numerical day of the month. 1 = set rule on (default), meaning that a bond's coupon payment date is always the last actual day of the month

:

Note

• If Bonds is a table, the Maturity dates can be serial date numbers, date character vectors, or datetime arrays.

• If Bonds is a matrix, is an n-by-2 to n-by-6 matrix where each row describes a bond, the first two columns (Maturity and CouponRate) are required. The remainder of the columns are optional but must be added in order. All rows in Bonds must have the same number of columns.

.

Data Types: double | table

Yields — Yield to maturity of each bond in Bonds

numeric

Yield to maturity of each bond in Bonds, specified as a N-by-1 column vector. The number of rows (n) must match the number of rows in Bonds.

Note

Yield to maturity must be compounded semiannually.

Data Types: double

Settle — Settlement date representing time zero in derivation of zero curve

serial date number | date character vector | datetime

Settlement date representing time zero in derivation of zero curve, specified as serial date number, date character vector, or datetime array. Settle represents time zero for deriving the zero curve, and it is normally the common settlement date for all the bonds.

Data Types: double | char | datetime

OutputCompounding — Compounding frequency of output ZeroRates

2 (default) | numeric values: 0,1, 2, 3, 4, 6, 12, 365, -1

(Optional) Compounding frequency of output ZeroRates, specified using the allowed values:

• 0 — Simple interest (no compounding)

• 1 — Annual compounding

• 2 — Semiannual compounding (default)

• 3 — Compounding three times per year

• 4 — Quarterly compounding

• 6 — Bimonthly compounding

• 12 — Monthly compounding

• -1 — Continuous compounding

Data Types: double

Output Arguments

ZeroRates — Implied zero rates for each point along the investment horizon defined by maturity date

decimal fractions

Implied zero rates for each point along the investment horizon defined by a maturity date, returned as a m-by-1 vector of decimal fractions where m is the number of bonds with unique maturity dates. In aggregate, the rates in ZeroRates constitute a zero curve.

If more than one bond has the same Maturity date, zbtyield returns the mean zero rate for that Maturity. Any rates before the first Maturity are assumed to be equal to the rate at the first Maturity, that is, the curve is assumed to be flat before the first Maturity.

CurveDates — Maturity dates that correspond to ZeroRates

serial date number | date character vector | datetime

Maturity dates that correspond to the ZeroRates, returned as a m-by-1 vector of unique maturity dates, where m is the number of bonds of different maturity dates. These dates begin with the earliest Maturity date and end with the latest Maturitydate in the Bonds table or matrix.

If either inputs for Bonds or Settle have datetime values, then CurveDatesCurveDates is datetimes. Otherwise CurveDates is serial date numbers.

References

[1] Fabozzi, Frank J. “The Structure of Interest Rates.” Ch. 6 in Fabozzi, Frank J. and T. Dessa Fabozzi, eds. The Handbook of Fixed Income Securities. 4th ed. New York, Irwin Professional Publishing, 1995.

[2] McEnally, Richard W. and James V. Jordan. “The Term Structure of Interest Rates.” in Ch. 37 in Fabozzi and Fabozzi, ibid

[3] Das, Satyajit. “Calculating Zero Coupon Rates.” in Swap and Derivative Financing. Appendix to Ch. 8, pp. 219–225. New York, Irwin Professional Publishing, 1994.