A closed-form formula for pricing bonds between coupon payments

We derive a closed-form formula for computing bond prices between coupon payments. Our results cover both the `Treasury' and the `Street' pricing methods used by sovereign and corporate issuers. We apply our formulas to two UK gilts, the 8% Treasury Gilt 2015, and the 0.5% Treasury Gilt 2022, and show that we can obtain the dirty price of these bonds at any date with a minimum of calculations, and without intensive computational resources.


Introduction
Bond pricing is a basic feature of fixed-income analytics, and is a direct application of the concept of time value of money. In the existing literature, most fixed-income securities are priced at the issuance date, t = 0 by convention. However, the current framework cannot be directly applied to pricing bonds traded after they were issued unless the date bonds exchange ownership coincide with a coupon payment date. When bonds are traded between coupon payments, the conventional formulas cannot be applied, and fixed-income analysts usually rely on 'backof-the-envelope' calculations of dirty and clean prices.
Advanced fixed income analytics do address the issue, albeit inconsistently. [6], [7], [8], [3], and even the very advanced [9] briefly cover the topic and present pricing formulas that are either complex or cumbersome. Further, the treatment of the calculations of dirty price and clean price is frequently counter-intuitive, since they are based on simple interest. This paper presents a simple closed-form formula for bond pricing between coupon payments that derives from first principles and is theoretically correct. Our results are more general than the current framework, and we prove that we can retrieve the conventional formula for pricing bonds at coupon dates as a special case. We also demonstrate that bond traders' 'dirty price' effectively assumes that interest between coupon payment is simple interest, when mathematical consistency requires that all interest should be coumpounded.
We illustrate our results with an application to two UK government bonds, the 8% Treasury Gilt 2015, and the 0 1 ⁄2% Treasury Gilt 2022. We show that the implmentation of our results involves very few steps irrespective of the maturity of the bonds. Existing approaches such as [1] and [8] become difficult to use for pricing bonds with a maturity longer than 2 years (semi-annual coupon payments) or four years (annual payments). Although absolutely correct, at longer maturities, they require the laborious calculation of several discount ratios and become rapidly computationally inefficient.
The main results of the paper will be derived in Section 2. A detailed application of our results to the UK 8% Treasury Gilt 2015 and the 0 1 ⁄2% Treasury Gilt 2022 in Section 3 shows that our formulas can replicate actual bond market practice. Section 4 concludes.

The price of bonds between interest payments
The fair price of a bond is the sum of the present value of the cash flow of coupon payments and the present value of the principal. The price is usually calculated at the issuance date (time t = 0 by convention). If a bond is bought at t = 0 and hold onto until maturity, t = N, the buyer receives all the coupon payments between t = 0 and t = N. However, bonds may be traded at any time before maturity, and should the transaction date fall between two coupon payments, the new buyer will receive the full interest payment at the next coupon date. Figure 1 illustrates the issue. The upper graph shows a coupon period split into two fractions by the settlement date, w, and 1 − w, with 0 < w ≤ 1. w is equal to the ratio of the number of days between the settlement date and the next coupon date to the total number of days in the coupon period (see Section 2.2 for details). The new owner receives the totality of the interest payment at time t=Next coupon, but is only entitled to the interest compounded over w-th of the coupon period. The lower graph shows that interest accrues daily until the coupon payment date, e.g., at times t = 1 and t=Next coupon, when it is then paid in full.
Two questions arise. The first concerns the fair price of a bond bought within the coupon period. The second relates to the compensation due to the seller of the bond for the loss of interest payment in the period between the previous coupon and the settlement date. The next two subsections address these questions.

Bond pricing
This section shows how to calculate the fair price of a coupon bond at time t = w. For the sake of simplicity, we will derive the main results assuming coupons are paid annually. The formulas for other frequencies are presented when needed.
Intuitively, one would want to take the extended formula for calculating the fair price at time t = 0 with first coupon payment at t = 1, C= coupon payment value N= number of years y= yield to maturity or discount rate M= face value of the bond at maturity and change the starting date to t = w, i.e., (2) captures the spirit of the formula we wish to obtain, it suffers from three shortcomings. First, the actual value of w depends on the problem at hand, i.e., w can be 1/5-th, or 0.7-th of the coupon period, or any value between 0 and 1. As a result, in (2), unlike in (1), the starting point of the sum of discounted cash flows is not known. Secondly, and relatedly, the counter of the sum is an integer by convention to make clear the sequence of powers in the discount factor. Specifically, if a coupon payment is made at time t = 10, and t can only be a whole number, the next coupon payment is necessarily at t = 11. If t = w = 0.8667, the date of the next payment is not immediately obvious. Finally, (1) and (2) are not equivalent. In (1), t = 1 is the date of the first coupon payment, whilst in (2) t = w is the date interest starts to accrue, as can be seen in the lower graph of Figure 1. This last point can be tackled by two distinct methods, which will be detailed below.
Theorem 2.1. Let y, M and i ∈ (0; ∞), w ∈ (0, 1], and let N ∈ N, then the fair price of a coupon bond between coupon payments is The sum can be rewritten as and can be expanded to and subtracting (7) from (6), we find (4) gives the closed form version of the bond price formula, Finally, by replacing q=1/(1 + y) into (4) yields (3).
and multiplying through q, we obtain factorising 1 − q as q (1/q − 1) and replacing this into (12) gives, cancelling out q, and replacing q ≡ 1 (1+y) , (10) is the price of a coupon bond bought at a coupon payment date. It is the closed-form of (1).

'Treasury' method
Sovereign issuers, such as the UK Debt Management Office (DMO), tend to favour the procedure established by the International Securities Market Association (ISMA), (see [2] and [4]).
[7] and [3] refer to this variant as the 'Treasury' method. It is based on the assumption that the present value of cash flows should start at the time interest accrues rather than at the time of the first coupon payment.
The UK Debt Management Office has developed a variant of (15), A cursory comparison of (15) and (20) shows that these two formulas are related. In fact, (20) is also obtained from (16), by 'extracting' C 1 + C 2 q from the sum, which then starts at time t = 2. Replacing this sum by its closed-form results in (20). Nonetheless, the difference is not merely cosmetic. It has a financial justification that will be detailed in the next section, where we illustrate the application of the formulas above to two UK government bonds, the 8% Treasury Gilt 2015, and the 0 1 ⁄2% Treasury Gilt 2022.

Accrued interest
Having found the fair price of a bond, we now turn to evaluating the amount of accrued interest bond sellers should receive from the buyers to compensate for loss of interest. Figure 1 shows that it should be the amount of interest accrued during 1 − w days, and internal consistency requires that it should be compounded. We present below the universally accepted market practice and prove that this practice is based on simple interest.
Assuming coupons are paid annually, the number of days in the coupon period is 365 under the 'Actual/Actual' day count convention. If, for instance, the settlement date in Figure 1 Table 1 presents four scenarios that differ by their settlement dates, 24-May-99, 26-May-99, 27-May-99 and 07-June-99. The variables C 1 , C 2 , and C are the semi-annual coupon payments, calculated as 100 × (8%/2) = 4. N is the number of coupon payments, r is the number of days between the settlement date and the date of the next coupon payment, whilst s is the number of days in the coupon period. Our variable w is equal to the ratio r/s. The day count convention used by the DMO is 'Actual/Actual', signifying that the number of days in a month depends on the calendar month and that the number of days in the year is 365 (see [4] to the registered holder of the bond during these seven days. If the bond is sold during the ex-dividend period, the seller will receive the full amount of interest, but will have to refund

The UK 8% Treasury Gilt 2015
The press notices for these bonds are the files prosp160796a.pdf and pr110417.pdf, respectively. Both files can be downloaded from the DMO website. In this section we will use 'interest' and 'coupon' interchangeably. was changed to conform to the notation of our paper. some back to the buyer of the bond (see [5] pp.15-16) Finally, the 'quasi-coupon' date is the day compounding occurs, irrespective of whether a payment is made (see [4]). Given that coupon payments are made over seven days, it seems sensible to distinguish between the payment date and the compounding date. However, this distinction is particularly relevant for a gilt issued between the dates that will constitute its coupon period. For instance, the 0 1 ⁄2% Treasury Gilt 2022 was issued on 21 April 2017, but pays interest on 22 January and 22 July. As will be seen below, in order to calculate the accrued interest on any day before 22 July 2017, the DMO assumes that interest compounding started on 22 January 2017, i.e., before the gilt was actually issued. The 22nd January 2017 is thus a 'quasi-coupon' date. This is the dirty price for Scenario 1 seen in Table 1. Scenario 2 is merely a repetition of the above, with the minor difference that the settlement date coincides with the ex-dividend date.
The values to be inserted in (20) will be different but, unlike Scenario 3, there are no modifications of the formula.

Scenario 3
Under this scenario, the settlement date is within the ex-dividend period. As explained above, the seller has already been paid the coupon interest and has to refund it to the buyer. Consequently, the interest payment C 1 is excluded from (20), and the price of the gilt is

0 1 ⁄2% Treasury Gilt 2022
The results of the second case study are presented in Table 3, which summarizes the application of formulas (9)  demand for bonds on the day, and on other market conditions. Consequently, any price obtained from (23) will reflect those market conditions. Nonetheless, the two case studies presented in this section confirm that the 'Treasury' closed-form we derived is equivalent to that of the Debt Management Office.

Conclusion
This paper has presented an innovative and theoretically coherent way to price fixed-income securities between coupon payments. Our results cover the two main exisitng pricing methods, 'Street' and 'Treasury'. We proved that the industry's dirty price calculated by market fixedincome analysts, and commonly found on platforms such Bloomberg, is theoretically inconsistent. Our results were derived from first principles and replicate the more rigorous framework of the UK Debt Management Office.

Conflict of Interests
The authors declare that there is no conflict of interests.