Treasury Indexed Bonds are quoted and traded on a real yield to maturity basis rather than on a price basis. This means the price is calculated by inputting the real yield to maturity into the appropriate pricing formula.
The price per $100 face value is calculated using the following pricing formulae:
(1) Basic Formula
\(\Large P = v^\frac{f}{d}\left(g \left( 1 + \require{enclose}a_{\enclose{actuarial}{n}}\right) + 100 v^{n} \right) \LARGE \frac{K_{t}(1+\frac{p}{100})^{\frac{-f}{d}}}{100}\)
(2) Ex-Interest Formula
\( \Large P = v^\frac{f}{d}\left(g\require{enclose}a_{\enclose{actuarial}{n}} + 100 v^{n} \right) \LARGE \frac{K_{t}(1+\frac{p}{100})^{\frac{-f}{d}}}{100} \)
In these formulae:
\(\large P=\) the price per $100 face value. \( P \) is rounded to three decimal places in all cases except during the final ex-interest period for the bond, where it is unrounded.
\(\large i=\) the annual real yield (per cent) to maturity divided by 400.
\(\large v=\LARGE\frac{1}{1 + i} \)
\(\large f=\) the number of days from the date of settlement to the next interest payment date.
\(\large d=\) the number of days in the quarter ending on the next interest payment date.
\(\large g=\) the fixed quarterly interest rate payable (equal to the annual fixed rate divided by 4).
\(\large n=\) the number of full quarters between the next interest payment date and the date of maturity.
\(\large \require{enclose}a_{\enclose{actuarial}{n}}=\large v + v^2 + ... + v^n = \Large \frac{1 - v^n}{i} \) \(.\ \mathrm{Except \, if\ \,} i = 0 \ \mathrm{\,then\,}\ \require{enclose}a_{\enclose{actuarial}{n}} = n \)
\( \large p=\) half the semi-annual change in the Consumer Price Index over the two quarters ending in the quarter which is two quarters prior to that in which the next interest payment falls (for example, if the next interest payment is in November, is based on the movement in the Consumer Price Index over the two quarters ending the June quarter preceding).
\(\Large =\frac{100}{2}\left[ \frac{CPI_t}{CPI_{t-2}}-1\right] \)
where \( CPI_{t} \) is the Consumer Price Index for the second quarter of the relevant two quarter period; and \( CPI_{t-2} \) is the Consumer Price Index for the quarter immediately prior to the relevant two quarter period. It is rounded to two decimal places.
\(\large K\)s are indexation factors (also known in the market as 'the nominal value of the principal' or 'capital value').
\( \large K_t= \) nominal value of the principal at the next interest payment date.
\( \large K_{t-1}= \) nominal value of the principal at the previous interest payment date.
The relationship between successive \( K \) values is as follows:
\( \large K_t= \) \( \large K_{t-1}\left[1+ \Large \frac{p}{100}\right] \)
Worked Examples
(1) Basic Formula
Consider the 1.25% 21 August 2040 Treasury Indexed Bond, with a real yield to maturity of 0.10 per cent and settlement date of 15 September 2019.
\(\Large P = v^\frac{f}{d}\left(g \left( 1 + \require{enclose}a_{\enclose{actuarial}{n}}\right) + 100 v^{n} \right) \LARGE \frac{K_{t}(1+\frac{p}{100})^{\frac{-f}{d}}}{100}\)
where:
\( \large\ i=\Large\frac{0.10}{400}= \) \(\large 0.00025\)
\( \large\ v=\Large\frac{1}{1+i}=\frac{1}{1+0.00025}= \) \(\large 0.99975\)
\(\large \require{enclose}a_{\enclose{actuarial}{n}}=\Large \frac{1 - v^n}{i}=\frac{1-0.99975^{83}}{0.00025}=\)\(\large 82.1346\)
\( \large\ f = 67 \), the number of days from 15 September 2019 to 21 November 2019
\( \large\ d = 92 \), the number of days from 21 August 2019 to 21 November 2019
\( \large\ g =\Large\frac{1.25}{4}= \) \(\large 0.3125\)
\( \large\ n = 83\), the number of full quarters from 21 November 2019 to 21 August 2040
\( CPI_{t-2} = 114.1\), the Consumer Price Index for the December 2018 quarter
\( CPI_{t} = 114.8\), the Consumer Price Index for the June 2019 quarter
\( \large p =\)\(\Large \frac{100}{2}\left[ \frac{CPI_t}{CPI_{t-2}}-1\right] = \frac{100}{2}\left[ \frac{114.8}{114.1}-1\right] = 0.31\)
\( \large K_{t-1}= 107.12\), the \( K \) value of this bond on 21 August 2019 (the previous interest payment date)
\( \large K_t= \) \( \large K_{t-1}\left[1+\frac{p}{100}\right] = 107.12\left[1+\frac{0.31}{100}\right] = 107.45 \), the \( K \) value for 21 November 2019 (the next interest payment date)
\(\Large P = v^\frac{f}{d}\left(g \left( 1 + \require{enclose}a_{\enclose{actuarial}{n}}\right) + 100 v^{n} \right) \LARGE \frac{K_{t}(1+\frac{p}{100})^{\frac{-f}{d}}}{100} \)
\(\Large P = 0.99975^\frac{67}{92}\left(0.3125 \left( 1 + 82.1346\right) + 100 \times 0.99975^{83} \right) \LARGE \frac{107.45(1+\frac{0.31}{100})^{\frac{-67}{92}}}{100} \)
\(\Large P = 132.835\)
(2) Ex-Interest Formula
Consider the 1.25% 21 August 2040 Treasury Indexed Bond, with a real yield to maturity of 0.10 per cent and settlement date of 15 November 2019.
\( \Large P = v^\frac{f}{d}\left(g\require{enclose}a_{\enclose{actuarial}{n}} + 100 v^{n} \right) \LARGE \frac{K_{t}(1+\frac{p}{100})^{\frac{-f}{d}}}{100} \)
where:
\( \large\ i=\Large\frac{0.10}{400}= \) \(\large 0.00025\)
\( \large\ v=\Large\frac{1}{1+i}=\frac{1}{1+0.00025}= \) \(\large 0.99975\)
\(\large \require{enclose}a_{\enclose{actuarial}{n}}=\Large \frac{1 - v^n}{i}=\frac{1-0.99975^{83}}{0.00025}=\)\(\large 82.1346\)
\( \large\ f = 6 \), the number of days from 15 November 2019 to 21 November 2019
\( \large\ d = 92 \), the number of days from 21 August 2019 to 21 November 2019
\( \large\ g =\Large\frac{1.25}{4}= \) \(\large 0.3125\)
\( \large\ n = 83\), the number of full quarters from 21 November 2019 to 21 August 2040
\( CPI_{t-2} = 114.1\), the Consumer Price Index for the December 2018 quarter
\( CPI_{t} = 114.8\), the Consumer Price Index for the June 2019 quarter
\( \large p =\)\(\Large \frac{100}{2}\left[ \frac{CPI_t}{CPI_{t-2}}-1\right] = \frac{100}{2}\left[ \frac{114.8}{114.1}-1\right] = 0.31\)
\( \large K_{t-1}= 107.12\), the \( K \) value of this bond on 21 August 2019 (the previous interest payment date)
\( \large K_t= \) \( \large K_{t-1}\left[1+\frac{p}{100}\right] = 107.12\left[1+\frac{0.31}{100}\right] = 107.45 \), the \( K \) value for 21 November 2019 (the next interest payment date)
\(\Large P = v^\frac{f}{d}\left(g \left( \require{enclose}a_{\enclose{actuarial}{n}}\right) + 100 v^{n} \right) \LARGE \frac{K_{t}(1+\frac{p}{100})^{\frac{-f}{d}}}{100} \)
\(\Large P = 0.99975^\frac{6}{92}\left(0.3125 \times 82.1346 + 100 \times 0.99975^{83} \right) \LARGE \frac{107.45(1+\frac{0.31}{100})^{\frac{-6}{92}}}{100} \)
\(\Large P = 132.794\)
Record Date Examples
Example 1
The 2.50% 20 September 2030 Treasury Indexed Bond makes a Coupon Interest Payment on Thursday, 20 June 2024. The Record Date for this Coupon Interest Payment is Tuesday, 12 June 2024.
Example 2
The 1.25% 21 August 2040 Treasury Indexed Bond makes a Coupon Interest Payment on Monday, 21 August 2023. The Record Date for this Coupon Interest Payment is Friday, 11 August 2023 (ten days prior to the Coupon Interest Payment Date, since the date eight days prior to the Coupon Interest Payment Date falls on a weekend).
