zbtyield (Financial Toolbox)

Zero curve bootstrapping from coupon bond data given yield

Syntax

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

Arguments

Bonds
Coupon bond information used to generate the zero curve. An n-by-2 to n-by-6 matrix where each row describes a bond. The first two columns are required; the rest are optional but must be added in order. All rows in Bonds must have the same number of columns. Columns are [Maturity CouponRate Face Period Basis EndMonthRule] where

Maturity
Maturity date of the bond, as a serial date number. Use datenum to convert date strings to serial date numbers.

CouponRate
Coupon rate of the bond, as a decimal fraction.

Face
(Optional) Redemption or face value of the bond. Default = 100.

Period
(Optional) Coupons per year of the bond, as an integer. Allowed values are 0, 1, 2, 3, 4, 6, and 12. Default = 2.

Basis
(Optional) Day-count basis of the bond: 0 = actual/actual (default), 1 = 30/360, 2 = actual/360, 3 = actual/365.

EndMonthRule
(Optional) End-of-month flag. This flag applies only when Maturity is an end-of-month date for a month having 30 or fewer days. 0 = ignore flag, meaning that a bond's coupon payment date is always the same day of the month. 1 = set flag (default), meaning that a bond's coupon payment date is always the last day of the month.

Yields
A column vector containing the yield to maturity of each bond in Bonds, respectively. The number of rows (n) must match the number of rows in Bonds.

Settle
Settlement date, as a scalar serial date number. This represents time zero for deriving the zero curve, and it is normally the common settlement date for all the bonds.

OutputCompounding
(Optional) A scalar that sets the compounding frequency per year for the output zero rates in ZeroRates. Allowed values are:

1
annual compounding

2
semi-annual compounding (default)

3
compounding three times per year

4
quarterly compounding

6
bimonthly compounding

12
monthly compounding

OutputBasis
(Optional) Output day-count basis for mapping cash-flow dates to years, in generating the output zero rates in ZeroRates. A scalar. 0 = actual/actual (default), 1 = 30/360, 2 = actual/360, 3 = actual/365.

MaxIterations
(Optional) Maximum number of iterations for deriving the zero rates in ZeroRates. A scalar. Default = 50. A value greater than 50 may slow processing.

`Bonds`	Coupon bond information used to generate the zero curve. An n-by-2 to n-by-6 matrix where each row describes a bond. The first two columns are required; the rest are optional but must be added in order. All rows in `Bonds` must have the same number of columns. Columns are `[Maturity CouponRate Face Period Basis EndMonthRule]` where

	`Maturity`	Maturity date of the bond, as a serial date number. Use `datenum` to convert date strings to serial date numbers.
	`CouponRate`	Coupon rate of the bond, as a decimal fraction.
	`Face`	(Optional) Redemption or face value of the bond. Default = 100.
	`Period`	(Optional) Coupons per year of the bond, as an integer. Allowed values are 0, 1, 2, 3, 4, 6, and 12. Default = 2.
	`Basis`	(Optional) Day-count basis of the bond: 0 = actual/actual (default), 1 = 30/360, 2 = actual/360, 3 = actual/365.
	`EndMonthRule`	(Optional) End-of-month flag. This flag applies only when `Maturity` is an end-of-month date for a month having 30 or fewer days. 0 = ignore flag, meaning that a bond's coupon payment date is always the same day of the month. 1 = set flag (default), meaning that a bond's coupon payment date is always the last day of the month.
`Yields`	A column vector containing the yield to maturity of each bond in `Bonds`, respectively. The number of rows (n) must match the number of rows in `Bonds`.
`Settle`	Settlement date, as a scalar serial date number. This represents time zero for deriving the zero curve, and it is normally the common settlement date for all the bonds.
`OutputCompounding`	(Optional) A scalar that sets the compounding frequency per year for the output zero rates in `ZeroRates`. Allowed values are:
	`1`	annual compounding
	2	semi-annual compounding (default)
	3	compounding three times per year
	4	quarterly compounding
	6	bimonthly compounding
	12	monthly compounding
`OutputBasis`	(Optional) Output day-count basis for mapping cash-flow dates to years, in generating the output zero rates in `ZeroRates`. A scalar. 0 = actual/actual (default), 1 = 30/360, 2 = actual/360, 3 = actual/365.
`MaxIterations`	(Optional) Maximum number of iterations for deriving the zero rates in `ZeroRates`. A scalar. Default = 50. A value greater than 50 may slow processing.

Description

[ZeroRates, CurveDates] = zbtyield(Bonds, Yields, Settle, OutputCompounding, OutputBasis, MaxIterations) 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. For best results, use a portfolio of at least 30 bonds evenly spaced across the investment horizon.

ZeroRates
An m-by-1 vector of decimal fractions that are the implied zero rates for each point along the investment horizon represented by CurveDates; m is the number of bonds of different 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.

CurveDates
An m-by-1 vector of unique maturity dates (as serial date numbers) that correspond to the zero rates in ZeroRates; m is the number of bonds of different maturity dates. These dates begin with the earliest maturity date and end with the latest maturity date Maturity in the Bonds matrix. Use datestr to convert serial date numbers to date strings.

`ZeroRates`	An m-by-1 vector of decimal fractions that are the implied zero rates for each point along the investment horizon represented by `CurveDates`; m is the number of bonds of different 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.
`CurveDates`	An m-by-1 vector of unique maturity dates (as serial date numbers) that correspond to the zero rates in `ZeroRates`; m is the number of bonds of different maturity dates. These dates begin with the earliest maturity date and end with the latest maturity date `Maturity` in the `Bonds` matrix. Use `datestr` to convert serial date numbers to date strings.

Examples

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 = [0.048;
          0.06 ;
          0.089;
          0.053;
          0.069;
          0.064;
          0.078;
          0.059;
          0.071;
          0.069;
          0.057;
          0.068];

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

Set semi-annual compounding for the zero curve, on an actual/365 basis. Derive the zero curve within 50 iterations.

OutputCompounding = 2;
OutputBasis = 3;
MaxIterations = 50;

Execute the function

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

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

ZeroRates =

            0.0480
            0.0577
            0.0909
            0.0529
            0.0699
            0.0724*
            0.0584
            0.0716
            0.0696
            0.0526
            0.0687

CurveDates =

             729907  (serial date number for 01-Jun-1998)
             730364  (01-Sep-1999)
             730439  (15-Nov-1999)
             730500  (15-Jan-2000)
             730667  (30-Jun-2000)
             730668  (01-Jul-2000)*
             730971  (30-Apr-2001)
             731032  (30-Jun-2001)
             731033  (01-Jul-2001)
             731321  (15-Apr-2002)
             731336  (30-Apr-2002)

See Also

zbtprice and other functions for Term Structure of Interest Rates

References

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.

McEnally, Richard W. and James V. Jordan. "The Term Structure of Interest Rates." Ch. 37 in Fabozzi and Fabozzi, ibid.

Das, Satyajit. "Calculating Zero Coupon Rates." Swap and Derivative Financing. Appendix to Ch. 8, pp. 219-225. New York: Irwin Professional Publishing. 1994.

zbtprice zero2disc