2.8Bond Prices, Yields to Maturity, and Bond Market Conventions

Now that you are familiar with the varieties of bonds and the nature of the bond mar-

ket, it is important to understand how bond prices are quoted. There are two languages

for talking about bonds: the language of pricesand the language of yields. It is impor-

tant to know how to speak both of these languages and how to translate one language

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58Part IFinancial Markets and Financial Instruments

easily into the other. While people refer to a number of yields when discussing bonds,

our focus is primarily on the yield to maturity. The yield to maturityis the discount

rate (as defined later in Chapter 9, a rate of return applied with the arrow of time in

reverse) that makes the discounted value of the promised future bond payments equal

to the market price of the bond. For example, consider the straight-coupon bond on the

left-hand side of Exhibit 2.13. The yield-to-maturity is the discount rate rthat solves

$10,000$10,000$10,000$110,000

P . . .

1 r(1 r)²²⁹³⁰

(1 r)(1 r)

where Pis the price of the bond. The annuity on the right side of Exhibit 2.13 has a

yield-to-maturity, r,that

$10,607.925$10,607.925$10,607.925

P . ..

r r)^{2 30}

1(1(1 r)

if Pis the price of the annuity. If both these bonds have prices equal to $100,000, their

yields-to-maturity are both 10 percent per annum.²⁶

Exhibit 2.18 graphs the relation between the yield to maturity and the price of the

bond. Since the promised cash flows of a bond are fixed, and the price of the bond is

the discounted value of the promised cash flows using the yield to maturity as a dis-

count rate, increasing the yield to maturity decreases the present value (or current mar-

ket price) of the bond’s cash flows. Hence, there is an inverse relationship between

bond price and yield.

The price-yield curve also has a particular type of curvature. The curvature in

Exhibit 2.18, known as convex curvature, occurs because the curve must always decline

but always at a slower rate as the yield increases. (See Chapter 23 for more detail.)

The results of this subsection can be summarized as follows:

Result 2.1

For straight-coupon, deferred-coupon, zero-coupon, perpetuity, and annuity bonds, the bondprice is a downward sloping convex function of the bond’s yield to maturity.

Settlement Dates

Knowing the date to which the bond’s future cash flows should be discounted is essen-

tial when computing a bond yield. The critical date is the date of legal exchange of

cash for bonds, known as the settlement date. An investor who purchases a bond does

not begin to accrue interest or receive coupons until the settlement date. As of the late

²⁶A

yield to maturity is an ambiguous concept without knowing the compounding convention. The

convention in debt markets is to use the yield to maturity that has a compounding frequency equal to the

number of times per year a coupon is paid. For instance, the yield quotes for residential mortgages,

which have monthly payments, represent monthly compounded interest. Most U.S. government and cor-

porate bond coupons are paid twice a year, so the yield to maturity is a rate compounded semiannually.

For debt instruments with one year or more to maturity, the bond-equivalent yieldis the yield to matu-

rity stated as a semiannually compounded rate.

There are a few exceptions to this convention. U.S. Treasury bills and short-term agency securities

have a type of quoted yield related to simple interest. Treasury strips, zero-coupon bonds that are obli-

gations of the U.S. Treasury, are closely tied to the coupon-paying Treasury market and thus have yield

quotes compounded semiannually. In addition, bids for Treasury bonds and notes at Treasury auctions are

based on Treasury yields, which combine semiannually compounded interest and simple interest if the

first coupon is not due exactly one-half year from the issue date. Finally, Eurobonds, with annual

coupons, have a yield that is generally compounded annually.See Chapter 9 for a discussion of

compounding frequencies for interest rates and yields.

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Chapter 2Debt Financing	59
EXHIBIT2.18Bond Price/Yield Relationship
Price/Yield Relationship

Bond Price

Yield to Maturity

1990s, the conventional settlement date for U.S. Treasury bonds is one trading day after

a trade is executed; for U.S. government agency securities, settlement is two trading

days after a trade is executed; for corporate bonds, settlement is three trading days after

an order is executed.

Alternatives to these settlement conventions are possible. However, requesting an

unconventional settlement typically generates extra transaction costs for the buyer or

seller making the request. These additional transaction costs usually are manifested in

a disadvantageous price—higher for the buyer, lower for the seller—relative to the

transaction price for conventional settlement.

Accrued Interest

Yield computations also require knowledge of which price to use. This would seem to

be a simple matter except that, for bonds not in default, the price paid for a bond is

not the same as its quoted price. The price actually paid for an interest-paying bond is

understood by all bond market participants to be its quoted price plus accrued interest.

Accrued interestis the amount of interest owed on the bond when the bond is

purchased between coupon payment dates. For example, halfway between payment

dates accrued interest is half the bond coupon. The sum of the bond’s quoted price and

the accrued interest is the amount of cash required to obtain the bond. This sum is the

appropriate price to use when computing the bond’s yield to maturity.

The quoted price of a bond is called its flat price. The price actually paid for a

bond is its full price. Thus, for a bond not in default, the full price is the flat price plus

accrued interest. Aprice quote for a bond represents a quote per $100 of face value.

The accrued interest quotation convention prevents a quoted bond price from falling

by the amount of the coupon on the ex-coupon date. The ex-coupon date (ex-date) is

the date on which the bondholder becomes entitled to the coupon. If the bondholder

sells the bond the day before the ex-date, the coupon goes to the new bondholder. If

the bond is sold on or after the ex-date, the coupon goes to the old bondholder. In con-

trast, stocks do not follow this convention when a dividend is paid. Therefore, stock

prices drop abruptly at the ex-date of a dividend, also called the ex-dividend date.

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60Part IFinancial Markets and Financial Instruments

EXHIBIT2.19Methods forCalculating Accrued Interest

	Accrual		Days of
Securities	Method		Accrued Interest	Divided by	Times

U.S. Treasury bonds and notes	Actual/actual	Number of days	Number of days	Semiannual
		since last	in current	coupon
		coupon date	coupon period

Eurobonds, and Euro-floating	Actual/365	Number of days	365	Annual
rate notes (FRNs), many		since last		coupon
foreign (non-U.S.) govt. bonds		coupon date

Eurodollar deposits, commercial	Actual/360	Number of days	360	Annual
paper, banker’s acceptances,		since last		coupon
repo transactions, many FRNs		coupon date
and LIBOR-based transactions

Corporate bonds, U.S. agency	30/360	Number of days	360	Annual
securities, municipal bonds,		since last		coupon
mortgages		coupon date,
		assuming
		30-day months

Accrued interest calculations are based on simple interest. Accrued interest is zero

immediately on the ex-date. The coupon payment to the bondholder as of the ex-date

reflects the payment in full of the bond interest owed. Just before the ex-date, accrued

interest is the amount of the coupon to be paid. On days between ex-dates, accrued

interest is the full coupon for the current coupon period times the number of “days”

elapsed between the last coupon date and the settlement date of the transaction divided

by the number of “days” in the current coupon period, that is:

“Days” elapsedCoupon per

Accured interest“Days” in current coupon period

coupon period

“Days” appears in quotes because several day-count conventions for computing

accrued interest exist in the bond market. These vary from bond to bond. Among these

are actual/actual, actual/365, actual/360, and 30/360. Exhibit 2.19 outlines the various

interest rate computations. The two most difficult accrued calculations are actual/actual

and 30/360.

Accrued Interest and Coupons forMost U.S. Treasury Notes and Bonds: Actual/

Actual.As noted earlier, many corporate securities, particularly those about to be

issued, have prices quoted as a spread to Treasury securities of comparable maturity.

Also, the most popular interest rate derivative security in the U.S., the interest rate

swap(a contract to exchange fixed for floating interest rate payments),²⁷

has its price

quoted as a spread to the yield of the on-the-run U.S. Treasuries with the same matu-

rity as the swap. Thus, it is important to understand the pricing and cash flow con-

ventions of U.S. Treasuries because of their role as benchmark securities.

With the possible exception of the first coupon, U.S. Treasury notes and bonds pay

coupons every six months. The maturity date of the bond determines the semiannual

1

cycle for coupon payments. For example, the bond with the 8/coupon that matures

8

²⁷See

Chapter 7 for a detailed description of interest rate swaps.

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Chapter 2

Debt Financing

61

on May 15, 2021 (highlighted on Exhibit 2.17) pays interest every May 15 and Novem-

ber 15. The amount of interest that accrues over a full six-month period is half of the

1

8/coupon, or $4.0625 per $100 of face value. The number of days between May 15

8

and November 15 or between November 15 and May 15 is never half of 365, or 182.5.

For example, in 1999 there were 184 days between May 15 and November 15, and 182

days between November 1, 1999 and May 15, 2000. This means that the accrued inter-

est accumulating per day depends on the year and the relevant six-month period.

Example 2.2 shows an accrued interest calculation for this bond.

Example 2.2:Computing Accrued Interest fora Government Bond

1

Compute the accrued interest for the 8

8's maturing May 15, 2021, for a trade settling on

Monday, August 16, 1999.This generally means the trade was agreed to the previous busi-

ness day, or Friday the 13th of August 1999.

Answer:Use the formula from the U.S.Treasury bonds and notes row in Exhibit 2.19.

The number of days between May 15, 1999, and August 16, 1999 (days-of-accrued interest

column) is 93.Dividing 93 by 184 (days in current coupon period) yields the fraction of one

semiannual coupon due.Multiplying this fraction, 93/184, by the full semiannual coupon,

1

4.0625 (half of 88) yields $2.05333.Thus, $2.05333 is the amount of accrued interest per

$100 face value that must be added to the flat price agreed upon to buy the bond.

In Example 2.2, May 15, 1999, is a Saturday. Although all settlement dates are

business days, it is possible that one or both coupon dates may fall on a nonbusiness

day—a weekend or holiday. This does not alter the accrued interest calculation, but

does affect the day a coupon is received because the U.S. Treasury is closed on non-

business days. In this case, the May 15, 1999 coupon will be received on Monday, May

17, 1999. Based on convention in the bond trading industry, both the yield to maturity

and accrued interest calculations should assume that the coupon is received on Satur-

day. Example 2.3 gives a sample illustration.

Example 2.3:How Settlement Dates Affect Accrued Interest Calculations

Compute the accrued interest on a (hypothetical) 10 percent Treasury note maturing March

15, 2001, with a $100,000 face value if it is purchased on Wednesday, May 19, 1999.What

is the actual purchase price if the quoted price is $100.1875? (Note:U.S.Treasury notes

and bonds are quoted in 32nds.Hence, 100.1875 would appear as 100:6, meaning 100 plus

6/32nds.)

Answer:Since there is no legal holiday on May 20 (a weekday), the settlement date is

May 21, one trading day later.The prior coupon date is March 15.The subsequent coupon

date is September 15.Thus, the number of days since the last coupon is:

16(March) 30(April) 20(May) 66 days

The number of days between coupons is 184 days.The accrued interest is $5 (66/184)

per $100 of face value, or approximately $1793.48.Thus, the true purchase price is the sum

of $100,187.50 and $1793.48, or $101,980.98.

Accrued Interest forCorporate Securities: 30/360.As Exhibit 2.19 illustrates, U.S.

agency notes and bonds, municipal notes and bonds, and most corporate notes and

bonds pay interest on a 30/360 basis. Thus, to compute the number of days of accrued

interest on the basis of a 30-day month requires dividing by 360 and multiplying by

the annualized coupon.

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62Part IFinancial Markets and Financial Instruments

Here is the tricky part:To calculate the numbers of days that have elapsed since

the last coupon calculated on the basis of a 30-day month, assume that each month has

30 days. Begin by counting the last coupon date as day 1 and continue until reaching

the 30th day of the month. (Do not forget to assume that February has 30 days.) The

count continues for the following month until reaching the 30th day of the month. Then

it continues for the following month, and so on, until reaching the settlement date. The

settlement date is not counted for accrued interest because interest is assumed to be

paid at the beginning of the coupon date, that is, midnight, and only full days are

counted.²⁸

For example, there are 33 days of accrued interest if January 31 is the last coupon

date and March 3 is the settlement date in the same year, whether or not it is a leap

year. January 31 is day 1 even though the month exceeds 30 days, there are 30 days

in February, and 2 days in March before settlement. February 29 to March 2 in a leap

year contains three days of accrued interest—two days in February and one in March.

February 28 to March 1 also has three days of accrued interest, all in February, regard-

less of whether it is a leap year.

Example 2.4 provides a typical calculation.

Example 2.4:Computing Accrued Interest with 30/360 Day Counts

Compute the accrued interest on an 8 percent corporate bond that settles on June 30.The

last coupon date was February 25.

Answer:There are 25 days of accrued interest:6 days in February;30 in March, April,

and May;and 29 in June.125/360 .3472222.The product of .3472222 and $8 is $2.77778.

Thus, the bond has $2.77778 of accrued interest per $100 of face value.

Yields to Maturity and Coupon Yields

The coupon yieldof a bond is its annual interest payment divided by its current flat

price.²⁹

For example, a bond with a flat price (that is, net of accrued interest) of $92,

which pays coupons amounting to $10 each year, has a coupon yield of 10/92 10.87

percent.

It is easy of show that a straight-coupon bond with a yield to maturity of 10 per-

cent and a price of 100 has a coupon yield of 10 percent on a coupon payment date

and vice versa. Try it on a financial calculator. More generally, we have:

Result 2.2

Straight coupon bonds that (1) have flat prices of par (that is, $100 per $100 of face value)and (2) settle on a coupon date, have yields to maturity that equal their coupon yields.

Result 2.2 at first seems rather striking. The yield to maturity is a compound inter-

est rate while the coupon yield is a simple quotient. It appears to be a remarkable coin-

cidence that these two should be the same when the bond trades for $100 per $100 of

face value, or par. The obviousness of this “coincidence” only will become apparent

when you have mastered Chapter 9’s material on perpetuities and compounding

conventions.

When a bond is trading at par on a coupon date, discounting at the coupon yield

is the same as discounting at the yield to maturity. However, coupon yields can give

only approximate yields to maturity when a bond is trading at a premium or a discount.

²⁸It

is comforting to know that most financial calculators have internal date programs to compute the

number of days between two dates using the 30/360 day/count method.

²⁹Whenever

the term yieldis used alone, it refers to the yield to maturity, not the coupon yield.

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Chapter 2Debt Financing	63
EXHIBIT2.20Price of a Bond: Coupon Rate Equals Yield to Maturity overTime

Bond

price

100

					Time
Coupon	Coupon	Coupon	Coupon	Coupon
date	date	date	date	date
	1	2	3	4

When a bond is between coupon dates or if its first coupon differs in size from

other coupons because of its irregular timing—what is known as an odd first coupon—

the coupon yield of a bond trading at par is not the same as its yield to maturity.

Result 2.3

Abond between coupon dates has a flat price that is less than par when its yield to matu-rity is the same as its coupon rate.

The phenomenon described in Result 2.3, known as the scallop effect, is shown in

Exhibit 2.20. The scallop effect occurs because yields to maturity are geometric; that

is, they reflect compound interest. The prices at which bonds are quoted—the flat

prices—partly reflect compound interest and partly simple interest. The simple inter-

est part is due to accrued interest, which is generally subtracted from the bond’s traded

price to obtain the bond’s quoted price.