Chapter 3
Introduction to Fixed-Income Valuation
105
3. PRICES AND YIELDS: CONVENTIONS FOR QUOTES AND
CALCULATIONS
When investors purchase shares, they pay the quoted price. For bonds, however, there can
be a difference between the quoted price and the price paid. This section explains why this
difference occurs and how to calculate the quoted price and the price that will be paid. It also
describes how prices are estimated for bonds that are not actively traded, and demonstrates
how yield measures are calculated for fixed-rate bonds, floating-rate notes, and money market
instruments.
3.1. Flat Price, Accrued Interest, and the Full Price
When a bond is between coupon payment dates, its price has two parts: the flat price (PV
VFlatt)
Full
l
and the accrued interestt (AII). The sum of the parts is the full price (PV
V ), which also is
called the invoice or “dirty” price. The flat price, which is the full price minus the accrued
interest, is also called the quoted or “clean” price.
Copyright © 2015. John Wiley & Sons, Incorporated. All rights reserved.
PV
VFull = PV
VFlat + AII
(3)
The flat price usually is quoted by bond dealers. If a trade takes place, the accrued interest is
added to the flat price to obtain the full price paid by the buyer and received by the seller on
the settlement date. The settlement date is when the bond buyer makes cash payment and the
seller delivers the security.
The reason for using the flat price for quotation is to avoid misleading investors about the
market price trend for the bond. If the full price were to be quoted by dealers, investors would
see the price rise day after day even if the yield-to-maturity did not change. That is because the
amount of accrued interest increases each day. Then, after the coupon payment is made, the
quoted price would drop dramatically. Using the flat price for quotation avoids that misrepresentation. It is the flat price that is “pulled to par” along the constant-yield price trajectory
shown in Exhibit 3.
Accrued interest is the proportional share of the next coupon payment. Assume that the
coupon period has “T” days between payment dates and that “t” days have gone by since the
last payment. The accrued interest is calculated using Equation 4:
AI =
t
× PMT
T
(4)
where
t = number of days from the last coupon payment to the settlement date
T = number of days in the coupon period
t/T = fraction of the coupon period that has gone by since the last payment
PMT = coupon payment per period
Notice that the accrued interest part of the full price does not depend on the yield-to-maturity.
Therefore, it is the flat price that is affected by a market discount rate change.
There are different conventions used in bond markets to count days. The two most common day-count conventions are actual/actual and 30/360. For the actual/actual method, the
actual number of days is used, including weekends, holidays, and leap days. For example, a
Pinto, J. E., Petitt, B. S. P., Pirie, W. L., & CFA, I. (2015). Fixed income analysis. John Wiley & Sons, Incorporated.
Created from curtin on 2022-04-27 06:18:46.
106
Part I: Fixed-Income Essentials
semiannual payment bond pays interest on 15 May and 15 November of each year. The accrued interest for settlement on 27 June would be the actual number of days between 15 May
and 27 June (t = 43 days) divided by the actual number of days between 15 May and 15 November (T = 184 days), times the coupon payment. If the stated coupon rate is 4.375%, the
accrued interest is 0.511209 per 100 of par value.
AI =
43 4.375
×
= 0.511209
184
2
Day-count conventions vary from market to market. However, actual/actual is most common
for government bonds.
The 30/360 day-count convention often is used on corporate bonds. It assumess that each
month has 30 days and that a full year has 360 days. Therefore, for this method, there are assumed to be 42 days between 15 May and 27 June: 15 days between 15 May and 30 May and
27 days between 1 June and 27 June. There are assumed to be 180 days in the six-month period
between 15 May and 15 November. The accrued interest on a 4.375% semiannual payment
corporate bond is 0.510417 per 100 of par value.
AI =
42 4.375
×
= 0.510417
180
2
The full price of a fixed-rate bond between coupon payments given the market discount
rate per period (rr) can be calculated with Equation 5:
Copyright © 2015. John Wiley & Sons, Incorporated. All rights reserved.
F
PV Full
=
PMT
1−tt T
( +r)
+
PMT
( +r)
2 −t T
+
+
PMT
(1 + r )
F
FV
N tT
(5)
This is very similar to Equation 1. The difference is that the next coupon payment (PMT
T) is
discounted for the remainder of the coupon period, which is 1 – t/T
T. The second coupon payment is discounted for that fraction plus another full period, 2 – t/T.
T
Equation 5 is simplified by multiplying the numerator and denominator by the expression
(1 + r)t/TT. The result is Equation 6:
⎡ PMT
PMT
F
PV Full
=⎢
+
+
1
2
⎢⎣ ( + r ) ( + r )
= PV × ( +
+
F
FV ⎤
⎥×( +
+r)
⎥⎦
PMT
(
N
)t T
(6)
tT
)
An advantage to Equation 6 is that PV
V, the expression in the brackets, is easily obtained using
the time-value-of-money keys on a financial calculator because there are N evenly spaced periods. PV
V here is identical to Equation 1 and is not the same as PV
VFlat.
For example, consider a 5% semiannual coupon payment government bond that matures
on 15 February 2024. Accrued interest on this bond uses the actual/actual day-count convention. The coupon payments are made on 15 February and 15 August of each year. The bond
is to be priced for settlement on 14 May 2015. That date is 88 days into the 181-day period.
Pinto, J. E., Petitt, B. S. P., Pirie, W. L., & CFA, I. (2015). Fixed income analysis. John Wiley & Sons, Incorporated.
Created from curtin on 2022-04-27 06:18:46.
Chapter 3
107
Introduction to Fixed-Income Valuation
There are actually 88 days from the last coupon on 15 February to 14 May and 181 days between 15 February and the next coupon on 15 August. The annual yield-to-maturity is stated
to be 4.80%. That corresponds to a market discount rate of 2.40% per semiannual period. As
of the beginning of the coupon period on 15 February 2015, there would be 18 evenly spaced
semiannual periods until maturity. The first step is to solve for PV
V using Equation 1, whereby
PMT = 2.5, N = 18, FV = 100, and r = 0.0240.
PV =
25
(
)
1
+
25
(
)
2
+
+
102.5
(1.0240 )18
= 101.447790
The price of the bond would be 101.447790 per 100 of par value if its yield-to-maturity is
2.40% per period on the last coupon payment date. This is not the actual price for the bond
on that date. It is a “what-if ” price using the required yield that corresponds to the settlement
date of 14 May 2015.
Equation 6 can be used to get the full price for the bond.
PV
VFull = 101.447790 × (1.0240)88/181 = 102.624323
The full price is 102.624323 per 100 of par value. The accrued interest is 1.215470 per 100
of par value.
AI =
88
× 2 5 = 1.215470
181
The flat price is 101.408853 per 100 of par value.1
Copyright © 2015. John Wiley & Sons, Incorporated. All rights reserved.
PV
VFlat = PV
VFull – AI = 102.624323 – 1.215470 = 101.408853
EXAMPLE 5 Calculating the Full Price, Accrued Interest, and Flat
Price for a Bond
A 6% German corporate bond is priced for settlement on 18 June 2015. The bond
makes semiannual coupon payments on 19 March and 19 September of each year and
matures on 19 September 2026. The corporate bond uses the 30/360 day-count convention for accrued interest. Calculate the full price, the accrued interest, and the flat
price per EUR100 of par value for three stated annual yields-to-maturity: (A) 5.80%,
(B) 6.00%, and (C) 6.20%.
1 Microsoft Excel users can obtain the flat price using the PRICE financial function: PRICE (“5/14/2015,”
“2/15/2024,” 0.05, 0.0480, 100, 2, 1). The inputs are the settlement date, maturity date, annual coupon
rate as a decimal, annual yield-to-maturity as a decimal, par value, number of periods in the year, and the
code for the day-count (0 for 30/360, 1 for actual/actual).
Pinto, J. E., Petitt, B. S. P., Pirie, W. L., & CFA, I. (2015). Fixed income analysis. John Wiley & Sons, Incorporated.
Created from curtin on 2022-04-27 06:18:46.