# Pricing Formulae for Commonwealth Government Securities

In common with most fixed income trading in Australia, Commonwealth Government Securities are both quoted and traded on a yield to maturity basis rather than on a price basis. This means the price is calculated after agreeing on the yield to maturity. The price is calculated by inputting the yield to maturity into the appropriate pricing formula:

## Pricing Formula for Treasury Notes

Treasury Notes are a short-term discount security redeemable at face value on maturity. As such this security provides the purchaser with a single payment on maturity.

Treasury Notes are traded on a yield to maturity basis with the price per \$100 face value calculated using the following pricing formula:

Where:

 P = the price per \$100 face value. f = the number of days from the date of settlement to the maturity date. i = the annual yield (per cent) to maturity divided by 100.

Settlement amounts are rounded to the nearest cent (0.50 cent being rounded up).

### Working Example

As an example of the working of the formula, the price of the 26 November 2010 Treasury Note for settlement on 7 May 2010 assuming a yield to maturity of 4.73% is calculated to be \$97.4367722107122 per \$100 face value.

In this example, f = 203 and i = .0473.

If the trade was for Treasury Notes with a face value of \$100 million the settlement amount would be \$97,436,772.21.

## Pricing Formulae for Treasury Bonds

Treasury Bonds are medium to long-term debt securities with a fixed coupon paid semi-annually in arrears, redeemable at face value on the maturity date.

The Australian Financial Markets Association (AFMA) has published conventions that apply to trading in the over-the-counter market of long-dated debt securities such as Treasury Bonds.

Treasury Bonds are traded on a yield to maturity basis with the price per \$100 face value calculated using the following pricing formulae:

 (a) Basic formula: (1) (b) Ex interest bonds: (2) (c) Near-maturing bonds (between the record date for the second last coupon and the record date for the final coupon): (3) (d) Near-maturing bonds (between the record date for the final coupon and maturity of the bond): (4)

In these formulae:

 P = the price per \$100 face value (the computed price is rounded to three decimal places in formulae (1) and (2)) v = i = the annual percentage yield to maturity divided by 200 in formulae (1) and (2), or the annual percentage yield to maturity divided by 100 in formula (3) f = the number of days from the date of settlement to the next interest payment date in formulae (1) and (2) or to the maturity date in formula (3). In formula (3), if the maturity date falls on a non-business day, the next good business day (defined as a day, not being a Saturday or Sunday, on which banks are open for business in Melbourne or Sydney) is used in the calculation of f. d = the number of days in the half year ending on the next interest payment date g = the half-yearly rate of coupon payment per \$100 face value n = the term in half years from the next interest-payment date to maturity

Settlement amounts are rounded to the nearest cent (0.50 cent being rounded up).

### Working Example

As an example of the working of the basic formula, the price of the 5.75% 15 July 2022 Treasury Bond, assuming a yield to maturity of 5.855% per annum and settlement date of 10 May 2010, is calculated to be \$100.912.

In this example, i = 0.029275 (i.e. 5.855 divided by 200), f = 66, d = 181, g = 2.875 (i.e. half of 5.75) and n = 24.

If the trade was for Treasury Bonds with a face value of \$50,000,000 the settlement amount would be \$50,456,000.00.

### Ex-Interest Treasury Bonds

The ex-interest period for Treasury Bonds is seven calendar days. With ex-interest Treasury Bonds the next coupon payment is not payable to a purchaser of the bonds. In this case, calculation of an ex-interest price is effected by the removal of the ‘1' from the term in formula (1), thereby adjusting for the fact that the purchaser will not receive a coupon payment at the next interest payment date.

### Near-Maturing Treasury Bonds

When a Treasury Bond goes ex-interest for the second last time (i.e. six months plus seven days before maturity) it is treated as a special case. In this case formula (3) applies up until the record date for the final interest payment and formula (4) applies from the time the bond goes ex-interest for the final time. There may be a slight discontinuity in the progress of the price of the bond around the time the bond goes ex-interest for the second last time but market participants can, if they wish, allow for this in their trading.

Where the maturity date coincides with a weekend or public holiday, the commonly accepted practice is to price near-maturing Treasury Bonds according to the actual date the principal and final interest are paid (and not the nominal maturity date).

## Pricing Formulae For Treasury Indexed Bonds

Treasury Indexed Bonds are medium to long-term debt securities. The nominal value of the security, on which a fixed rate of interest applies, varies over time according to movements in the Consumer Price Index (CPI). Interest coupons are paid quarterly in arrears. At maturity, the adjusted capital value of the bond is paid.

The basic pricing formula used for Treasury Indexed Bonds per \$100 face value, rounded to the third decimal place in all except the last interest period when there is no rounding, is as follows:

 (1)

where:

 v = i = the annual real yield (per cent) to maturity divided by 400. f = the number of days from the date of settlement to the next interest payment date. d = the number of days in the quarter ending on the next interest payment date. g = the fixed quarterly interest rate payable (equal to the annual fixed rate divided by 4). n = the number of full quarters between the next interest payment date and the date of maturity. 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, p is based on the movement in the Consumer Price Index over the two quarters ending the June quarter preceding). rounded to two decimal places, where CPIt is the Consumer Price Index for the second quarter of the relevant two quarter period; and CPIt-2 is the Consumer Price Index for the quarter immediately prior to the relevant two quarter period. The Ks are indexation factors (also known in the market as 'the nominal value of the principal' or 'capital value'): Kt = nominal value of the principal at the next interest payment date. Kt-1 = nominal value of the principal at the previous interest payment date. Kt-1 is equal to \$100 (the face value of the stock) at the date one quarter before the date on which the stock pays its first coupon. The relationship between successive K values is as follows:

Settlement amounts are rounded to the nearest cent (0.50 cent being rounded up).

### Working Example

As an example of the working of the formula consider the 4.0% 20 August 2020 Treasury Indexed Bond for a trade settling on 31 May 2010. Assuming a real yield to maturity of 2.65 per cent per annum the price per \$100 face value is calculated to be \$160.144.

In this example, i = 0.006625 (i.e. 2.65 divided by 400), f = 81, d = 92, g = 1.0 (i.e. 4 divided by 4) and n = 40. The K value of this bond () on 20 May 2010 (the previous interest payment date) was 142.65 and the K value () for 20 August 2010 (the next interest payment date) is 143.66. The 0.71 per cent increase in the K value reflects the average increase in the Consumer Price Index over the two quarters to the March quarter 2010.

If the trade was for Treasury Indexed Bonds with a face value of \$20,000,000 the settlement amount would be \$32,028,800.00.

### Ex-Interest Treasury Indexed Bonds

The ex-interest period for Treasury Indexed Bonds is seven calendar days. With ex-interest Treasury Indexed Bonds the next coupon payment is not payable to a purchaser of the bonds. In this case, calculation of an ex-interest price is effected by the removal of the ‘1' from the term in formula (1), thereby adjusting for the fact that the purchaser will not receive a coupon payment at the next interest payment date. The formula in this instance is therefore:

 (2)

Note that the in formula (2) is still the indexation factor on the next interest payment date, even though there is no interest payable to the subscriber or purchaser on that date. That is, this continues to apply in the ex-interest period.