### Flat Price, Accrued Interest, and Full Price of a Bond

When investors purchase shares, they pay the quoted price. For bonds, there can be a difference between the quoted price and the price paid.

## Full Price

When a bond is between coupon payment dates, the price has 2 components: the flat price (PVFlat) and the Accrued Interest (AI). The sum of these two is the full price (PVFull), also called invoice or dirty price:

$$PV^{ Full } = PV^{ Flat } + \text{Accrued interest}$$

## Flat Price

The flat price, on the other hand, is the full price minus the accrued interest. The flat price is generally the quoted price between bond dealers. It does not include any interest accrued between the scheduled coupon payments for the bond.

The reason for using the flat price is to avoid misleading investors as accrued interest does not change the yield-to-maturity. It is the flat price that is “pulled to par” along the life constant yield price line.

## Accrued Interest

The accrued interest is the proportional share of the next coupon payment. Supposing that coupon payment has “T” days between payment dates and “t” days have passed from the last payment date, then the accrued interest:

$$AI=\frac { t }{ T } \times PMT$$

Where:

t/T = fraction of the coupon period that has passed from the last payment; and

PMT = coupon payment for the period

## Day-count Conventions

### Actual/actual convention

Regarding bond day-counting, there are 2 major methods: Actual-actual and 30/360.

The first “actual” refers to the actual number of days since the last coupon date. The second refers to the actual number of days in a coupon period. This includes weekends, holidays and leap days.

For example, let’s assume that a semiannual payment bond pays interest on 15 June and 15 December of each year. To compute the accrued interest for settlement as of 27 July, we would take into account the actual number of days between 15 June and 27 July (t = 42 days). This number would be divided by the actual number of days between 15 June and 15 December (T = 183 days), and then multiplied by the coupon payment. With a coupon rate of, say, 5.25% the accrued interest would be 0.60246 per 100 of par value.

$$\text{Accrued interest}=\frac { 42 }{ 183 } \times \frac { 5.25% }{ 2 } =0.60246$$

### 30/360 day-count convention

The 30/360 convention is often used for corporate bonds. It assumes that each month has 30 days (even though some moths actually have 31 days and February has 28 or 29 days), and that a full year has 360 days.

Like we did before, let’s assume that a semiannual payment bond pays interest on 15 June and 15 December of each year. To compute the accrued interest for settlement as of 27 July, we would take into account a total of 15 days in June and 27 days in July (t = 42 days). This number would be divided by 180 – the number of days between 15 June and 15 December, assuming every month has 30 days (T = 6 × 30), and then multiplied by the coupon payment. With a coupon rate of 5.25%, the accrued interest would be 0.6125 per 100 of par value.

$$\text{Accrued interest}=\frac { 42 }{ 180 } \times \frac { 5.25% }{ 2 } =0.6125$$

The full price of a fixed-rate bond between coupon payments given the market discount rate per period (r) can be calculated as:

$$PV^{ Full }=PV×(1+r)^{ t/T }$$

## Question

Assuming a market discount rate of 4.5%, the full price of a semi-annual bond with a present value (PV) of 102 and 90 days of accrued interest is closest to:

A. 101.955

B. 103.135

C. 104.235

Solution

Since the bond coupons are paid semi-annually, the time (T) between coupon payments is 181 days.

PVFull = 102 × (1.0225)90/181 = 103.135

Describe and calculate the flat price, accrued interest, and the full price of a bond

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