Australian Treasury Bond Pricing, Accrued Interest, and Dirty Price
Summary
The document examines a present value formula for an Australian Commonwealth Treasury bond with semiannual coupons. It asks how to interpret the fractional discount factor applied before the first coupon and the coupon term that appears separately from the later coupon sum, in relation to accrued interest. The answers explain that the fractional exponent discounts cash flows over the settlement-to-next-coupon stub period, using the day-count fraction of a semiannual period.
The first coupon is included in the bracket as the zero-index coupon; later coupons are represented by the sum beginning with the next period. The formula therefore produces the bond's dirty price, the value including accrued interest. To obtain the clean price, accrued interest must be subtracted. The explanation follows the stated discounting convention and does not address alternative market conventions, day-count choices, or yield quotation methods.
Key ideas
- The fractional discount exponent accounts for the time from settlement to the next coupon date.
- The first coupon can be viewed as the coupon-sum term at period zero.
- The stated cash-flow formula returns the dirty bond price, including accrued interest.
- Subtract accrued interest from the dirty price to obtain the clean price.
Tags
Full text
# Australian Treasury Bonds - Price Calculation with Accrual
# Australian Treasury Bonds - Price Calculation with Accrual
In this document ASX Interest Rate Derivatives (on page 7) the Australian Commonwealth Treasury Bond (paying semi-anually) is valued as
$$ P = v^{f/d} \cdot \left(\frac{c}{2} + \frac{c}{2}\cdot\sum_{k=1}^n v^k + 1\cdot v^n\right)\cdot 100$$ where $v=\frac{1}{1+y/2}$ (the "one-period" DF), $c=$ annual coupon, $f=$ number of days from settlement date of to next interest payment date (ranging from $0$ to ~$184$), $d=$ number of days in the half year ending on next interest payment date (usually ~$184$). So $f$ will decrease from $d$ to $0$ as we approach the next payment date.
The middle summand $\frac{c}{2}\cdot\sum_{k=1}^n v^k$ (PV of coupons) and last summand $1\cdot v^n$ (PV of nominal) are clear.
What confuses me is $v^{f/d}$ and the standalone $\frac{c}{2}$ at the front, which I assume account for accrued interest? But shouldn't accrued interest be calculated as $$\text{acrr} = \frac{c}{2}\cdot \frac{d-f}{d}$$ ?
Would anybody know how to interpret the first parts of this formula?
## Answer by dm63 (score 1, accepted)
https://quant.stackexchange.com/a/41102
Adding to @Delsim answer, the formula gives the total value of all cash flows, or the dirty price of the bond. If you want the clean price you need to subtract accrued interest from the formula.
## Answer by delsim (score 1)
https://quant.stackexchange.com/a/41100
The factor $v^{f/d}$ multiplying the whole equation represents discounting the future bond flows over the period between the settlement date and the first coupon payment - it is not a whole period, so some fraction of the semi-annual DF factor is needed. With this convention, the fraction is obtained by raising the DF to the power of the ratio of number of days.
The first coupon is then the $\frac{c}2$ term inside the bracket. You can see that it can be incorporated into the sum over the other coupons simply by starting the summation at $k=0$ rather than $k=1$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.