PRICING FORMULAE FOR COMMONWEALTH GOVERNMENT SECURITIES

Price and Yield Formulae used by the Reserve Bank of Australia

This operational note is based on an attachment to an October 1992 Media Release which outlined the Bank's formulae for the pricing of Commonwealth Government securities.

Introduction

A fixed interest security consists of a series of future coupon payments, usually of equal size, and the repayment of the principal at maturity. Any formula used to put a value on these payments must rest on certain assumptions. Once these are established, the calculations required to derive a price are relatively simple.

The yield to maturity, which represents the annual rate of return of the security, is derived by solving an equation with the purchase price on one side and the present (discounted) value of the series of payments on the other. However, obtaining the calculated yield to maturity assumes that a buyer is able to reinvest all coupon interest payments at this yield. In practice, this will rarely be the case but people's views about re-investment rates during the life of the security being priced can influence their desired yield as calculated by the relevant formula. The main purpose of the formula is to ensure that buyers and sellers are ‘talking the same language’.

A yield could be expressed for any defined period but it is convenient to quote yearly rates. With bonds, it is conventional in Australia that these annual rates are obtained by doubling an effective or true half-yearly rate. That is, the usual calculations involve rates of return earned over a half year. The yearly rate is simply double this half-yearly rate and should properly be regarded as a nominal rate as it ignores the compounding (or reinvestment effect) of the half-yearly rate over the year. The use of effective half-yearly rates accords with the fact that, in Australia, interest is normally paid half-yearly. Some correction may be needed for comparison of yields e.g. with those overseas, where interest periods are different from this.

There is a minor practical difficulty with the use of the half year as a unit of measurement, as the number of days in a half year varies from 181 to 184. As interest payments are made half-yearly, they will generally accumulate over alternating periods of 181 and 184 days (or 182 and 183 days). Acknowledgment of these different rates of accrual is necessary to avoid discontinuities in the progress of the price and the Bank's price and yield formulae take account of this fact. Obviously, this impairs the idea of the redemption yield as a mathematically precise concept and emphasises that it is more a matter of what is conventionally accepted.

The foregoing should make plain that certain questions concerning price and yield calculations may have no single right answer. The aim should be to accord with the facts and to produce convenient, consistent and generally acceptable formulae. The formulae described below are presented with this aim in mind.

Treasury Bonds

The formulae which the Bank uses are:

(a) Basic formula:		(1)
(b) Ex interest securities:		(2)
(c) Near-maturing bonds (specifically, those entitling a purchaser to only the final coupon payment and repayment of principal).
		(3)

In these formulae:

P =	the price per $100 face value (the computed price is rounded to three decimal places)
v =
	where 100i = the half-yearly yield (per cent) to maturity in formulae (1) and (2), or the annual yield (per cent) to maturity 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 formulae (3).1
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.

The following notes provide further explanation of the above formulae and give some examples of their application.

(a) Basic formula

As an example of the working of the formula, the price per $100 face value on 24 October 2003 to yield 5.60% p.a. to maturity for a 6.25% 15 Apr 2015 bond is calculated (with i = .028, f = 174, d = 183, g = 3.125 (i.e. half of 6.25) and n = 22) as $105.600.

The problem of finding the yield consistent with a given price must be approached indirectly, as equations such as (1) above cannot be solved directly. To find the required yield, an iterative process may be used successively to approximate the given price. Alternatively, straight-line interpolation using prices near the given price enables the yield to be derived accurately.

As mentioned, while the yield to maturity is expressed as an annual rate, the calculations are in terms of returns for a half year i.e. effective half-yearly rates. Also, to accord with the fact that the half year is treated as the basic accounting period, the price of a bond between interest-payment (coupon) dates is calculated by discounting back using , being the fraction of the half year to the next payment.

There are modifications required in using this formula in some situations, as discussed in the following paragraphs.

(b) Ex interest securities

With these securities, either there is no coupon payable by the issuer at the next half-yearly interest date or the next coupon payment is not available to a purchaser of the securities on the market because, for example, they have gone ‘ex interest’ in the period leading up to distribution of coupon payments. In either 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.

In January 1993, the ‘ex interest’ period for Treasury bonds was reduced from 14 to seven calendar days. Trading of these securities in the Registry is permitted until they are within seven days of maturity. However, trading in Austraclear is possible until the day before maturity.

(c) Near-maturing bonds

When a 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, using the principles embodied in the pricing of Treasury notes (see below). There is a slight discontinuity in the progress of the price of the bond at the point the formula is introduced but market participants can, if they wish, allow for this in their trading around this time.

Where the maturity date coincides with a weekend or public holiday, the Bank prices such bonds in this final period according to the date funds become available (and not the nominal maturity date).

Treasury Capital Indexed Bonds

Indexed bonds are essentially the same as non-indexed bonds except that the yield is expressed in real terms and the future interest and maturity payments are expressed in constant (base-date) dollars. This requires the use of an indexation factor to convert the bond price from base-date dollars to current (settlement) date dollars. Otherwise, the calculation of yields/prices for indexed bonds is the same as for their non-indexed counterparts.

Thus, the basic formula used by the Bank for Treasury Capital Indexed Bonds throughout its life, including the last interest period, is as follows:

(4)

where P, v, f, d, g, , and n all have meanings analogous to those given previously:

v =	where 100i = the quarterly real yield (per cent) to maturity.
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 rate of quarterly coupon payment per $100 face value.
n =	the term in quarters from the next interest-payment date to maturity.

The Ks are indexation factors (also known in the market as 'the nominal value of the principal' or 'capital value') supplied by the issuer:

is the indexation factor at the previous interest payment date.

is the indexation factor at the next interest-payment date.

The relationship between successive K values is as follows:

	where
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 preceding June quarter).
=	rounded to two decimal places.
	where is the Consumer Price Index for the second quarter of the relevant two quarter period, and
	is the Consumer Price Index two quarters previously.

The second part of the formula above simply interpolates values for K between the relevant interest-payment dates, so as to provide a basis for converting the base-date price given by the first part of the formula into dollars of the day.

As indicated, the K values are provided by the issuer. This is normally done according to a formula set out in the prospectus. Indexation factors are usually set at 100 on the day which is one interest-payment period (one quarter in the case of Treasury Capital Indexed Bonds) prior to the date of the first interest payment. While the first interest payment need not be a full quarterly payment, it is standard practice for it to be so; if it isn't, the formula can be adapted accordingly.

Another feature of Treasury Capital Indexed Bonds is that the coupon payments are made quarterly. Analogous to the case with the non-indexed bonds, the yield thus calculated is an effective quarterly yield which is expressed as a nominal annual yield after multiplication by four.

Once again, an example should help. Consider the Treasury Capital Indexed Bond which matures on 20 August 2005. The coupon is 4 per cent and interest payment dates are the 20th of August, November, February and May. On 20 August 2003, the K value of this bond () was 208.86 and the K value for 20 November () is 210.22. The increase of 0.65 per cent, reflects an average increase of this amount in the Consumer Price Index over the two quarters to the June quarter 2003. The price to produce a real yield of 3.0 per cent per annum when the settlement date is 24 October 2003 is calculated from formula (4) with i = 0.0075, f = 27, d = 92, g = 1.0 (i.e. one quarter of 4) and n = 7 as $215.011.

The treatment of indexed bonds when there is no interest payable to the subscriber or purchaser at the next interest-payment date is again analogous to that for non-indexed bonds. The formula is :

(5)

Note that the in formula (5) 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 so called 'ex-interest' period.2

Treasury Notes

These instruments offer the purchaser a single payment on maturity. The Bank prices them according to the following rules:

(a) Basic formula:

(6)

Where

As an example, the price of the 6 November 2003 Treasury note for settlement 24 October 2003 to yield 4.75% is calculated (with f = 13 and i = .0475) as $99.831 107 647 per $100 face value.

If formula (6) is used to calculate the yield for a given price, the resultant yield will be the nominal annual yield corresponding to the effective yield for the period f. This can be seen by recasting formula (6) to

i.e.	is the effective return during the term to maturity of the Treasury note, f, and multiplying
by	converts it to a nominal annual rate.

Footnotes

1. 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. (back to text)

2. This approach may seem at variance with that specified for calculating settlement prices in the prospectus under which Treasury Capital Indexed Bonds were issued between July 1985 and February 1988. However, that formula has to allow for the situation where there isn't a K value at the start of the interest-payment quarter in which the bonds are first issued. That was the case, for instance, with the inaugural issue of these bonds. (back to text)