21 Valuation of Bonds Between Coupon Dates

21.1 Full Price of a Bond

Suppose that at time $t$, a noncallable bond has $n$ coupon payments remaining, and the next coupon payment occurs in one unit of time. Then the bond prices at time $t$ and time $(t+1)$ are:

\[\begin{align} P_t &= Fr\times a_{\overline{n|}} + Cv^n \\ P_{t+1} &= Fr\times a_{\overline{n-1|}} + Cv^{n-1}\end{align}\]

Now suppose that the bond is purchased between coupon payments, at time $(t+h)$, where $0<h<1$ and:

\[h = \frac{\text{Number of days from last coupon paymentot setlement date}}{\text{Number of days between coupon payments}}\]

The full price of a bond, sometimes called the price-plus-accrued or dirty price, is the price that is paid for the bond when it is purchased. The time of purchase is called the settlement date. The trade date, which comes a few days before the settlement date, is the date that the terms of the purchase are agreed upon. We use the settlement date, not the trade date, when calculating $h$.

The full price of the bond is found by taking the present value of all future payments (coupons plus redemption). We have: \[\begin{align} FP_{t+h} &= (P_{t+1} + Fr) \times v_j^{1-t} \\ FP_{t+h} &= P_t \times (1+j)^t\end{align}\]

21.2 Quoted Price and Accrued Interest

The full price can also be written as the sum of the quoted price (QP) and the accrued interest (AI). The quoted price is also known as the clean price, the flat price, or the market price: \[FP_{t+h} = QP_{t+h} + AI\] where the accrued interest is the pro rata portion of the coupon that is earned since the last coupon payment date, based on a simple interest calculation \[AI = Fr\times h\]

The quoted price is useful for monitoring price movements in the bond that are due to factors other than the passage of time since the last coupon payment. If the yield of a bond does not change over time, then its quoted price doesn’t change much either.

21.3 Day Counts

The calculation of $h$ above is based on the number of days in both the numerator and the denominator. The two most common ways to count days are referred to as actual/actual and 30/360.

Actual/actual is usually used for government bonds, and as the name implies, the actual number of days from the last coupon payment date (or issue date) to the settlement date is used in the numerator, and the actual number of days between coupon payments is used in the denominator.

The 30/360 method is usually applied for corporate bonds. Using this method, months are assumed to have 30 days each, and each year is assumed to have 360 days.

Example: A bond with a par value of $\$100$ makes semiannual coupon payments at an annual rate $8\%$ per year. The bond makes coupon payments on June 1 and December 1 of each year, and the bond matures on December 1, 2025.

The bond was purchased on September 10, 2015 to yield $6\%$ per year compounded semiannually.

Calculate the number of days from the last coupon date to the settlement date. Use the 30/360 day count method
Calculate $h$
Calculate the full price, the accrued interest and the quoted price

Solution:
We need to find the number of days between June 1, 2015 and Sept 10, 2015. There are 29 days remaining in June after June 1. We treat July and August as if they have 30 days each. The first 10 days of September are fully counted, since 10 does not exceed 30: \[29 + 30 + 30 + 10 = 99\] We can also count the days using the BA II calculator: (US Date)
[2nd] [DATE] 6.0115 ↓ 9.1015 ↓↓
(Use ↓ [2nd] [SET] to switch to 360 if necessary)
↑ [CPT] yields day between dates equals to 99
Under the 30/360 method, the number of days between semiannual coupon payments is 180, hence $h = 99/180 = 0.55$.

The remaining number of coupons are $21$. On June 1, 2015 the price-plus-accrued of the bond after the coupon payment is: \[P_t = 4 \times a_{\overline{21|}0.03} + 100 \times (1.03)^{-21} = 115.4150\] The full price of the bond on September 10, 2015 is: \[FP_{t+h} = (1.03)^{0.55} \times 115.4150 = 117.3067\] The accrued interest is: \[AI = 0.55 \times 4 = 2.20\] The quoted price is: \[QP_{t+h} = FP_{t+h} - AI = 115.1067\]

Using the bond worksheet on the BA II Calculator, we have: [2nd] [BOND] 9.1015 [ENTER] ↓
8 [ENTER] ↓
12.0125 [ENTER] ↓
100 [ENTER] ↓
(Use [2nd] [SET] as necessary so that 360 appears) ↓
(Use [2nd] [SET] as necessary so that 2Y/ appears) ↓
6 [ENTER] ↓
[CPT] ↓ ↓
$115.1067$ is the quoted price and $2.20$ is the accrued interest.