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Bond Yields

The investment return of a bond is the difference

between what an investor pays for a bond and what is ultimately
received over the term of the bond. The bond yield is the
annualized return of the bond. Thus, bond yield will depend on the
purchase price of the bond, its stated interest rate — which is equal
to the annual payments by the issuer to the bondholder divided by
the par value of the bond — plus the amount paid at maturity.
Because the stated interest rate and par value are stipulated in
the bond indenture, the price of the bond will vary inversely to
prevailing interest rates. If interest rates rise, then the price of the
bond must decrease to remain competitive with other investments,
and vice versa.

Bond prices — not including accrued interest — vary inversely to market interest rates: bond
prices will decline with rising interest rates, and vice versa. Bonds with longer effective
maturities, or durations, are more sensitive to changes in interest rates, as can be seen in the
diagram below, showing the price/yield curves per $100 of nominal value, as the market interest
rate varies from 1% to 16%, for a bond with 3 years left until maturity and one with 10 years left,
both with the same 6% coupon rate and paying interest semi-annually. Note that both curves
intersect at $100 when the market yield = coupon rate of 6%.

The price of the bond will also depend on the creditworthiness of

the issuer, which indicates the risk of the investment. The higher
the credit rating of the issuer, the less interest the issuer must offer
to sell its bonds. The prevailing interest rate—the cost of money—
is determined by the supply and demand of money. As for virtually
 everything else, supply and demand determine price, so for bonds,
the greater the supply and the lower the demand, the lower the
price of the bond and, correspondingly, the higher the interest rate,
and vice versa. An often used measure of the prevailing interest
rate is the prime rate charged by banks to their best customers.

Most bonds pay interest semi-annually until maturity, when the

bondholder receives the par value, or bond principal, of the
bond back. Zero coupon bonds pay no interest, but are sold
at a discount to par value, so the interest, which is the difference
between par value and the discounted issue price, is paid when the
bond matures. Nonetheless, the yield of the zero coupon bond is
the annualized return, which allows it to be compared to coupon
bonds.

Nominal Yield, Coupon Rate

Nominal yield, or the coupon rate, is the stated

interest rate of the bond. This yield percentage is the
percentage of par value—$5,000 for municipal bonds, and
$1,000 for most other bonds—that is usually paid semiannually.
Thus, a bond with a $1,000 par value that pays 5% interest pays
$50 dollars per year in 2 semi-annual payments of $25. The return
of a bond is the return/investment, or in the example just cited,
$50/$1,000 = 5%.

Nominal Yield Formula

Annual Interest Payment

Nominal Yield =
Par Value

Current Yield

Because bonds trade in the secondary market, they may sell for
less or more than par value, which will yield an interest rate that is
different from the nominal yield, called the current yield,
or current return. Since the price of bonds moves in the
opposite direction of interest rates, bond prices decrease when
interest rates increase, and vice versa. To see why, consider this
simple example. You buy a bond when it is issued for $1,000 that
 pays 8% interest. Suppose you want to sell the bond, but since you
bought it, the interest rate has risen to 10%. You will have to sell
your bond for less than what you paid, because why is somebody
going to pay you $1,000 for a bond that pays 8% when they can
buy a similar bond with an equal credit rating and get 10%. So to
sell your bond, you would have to sell it so that the $80 that is
received per year in interest will be 10% of the selling price—in
this case, $800, $200 less than what you paid for it. (Actually, the
price probably wouldn't go this low, because the yield-to-maturity
is greater in such a case, since if the bondholder keeps the bond
until maturity, he will receive a price appreciation which is the
difference between the bond's par value of $1,000 and what he
paid for it.) Bonds selling for less than par value are said to be
selling at a discount. If the market interest rate of a new bond
issue is lower than what you are getting, then you will be able to
sell your bond for more than par value—you will be selling your
bond at a premium. Note, however, that the bond price is based
on the clean price, meaning that any accrued interest is
excluded, since that will be paid to the bond owner on the next
interest payment.

Current Yield Formula

Annual Interest Payment

Current Yield =
Current Market Price of Bond

Current Yield Example

$60 Annual Interest Payment

= 8% Current Yield
$800 for Bond

Note that if the market price for the bond is equal to its par value,
then:

Current Yield = Nominal Yield

Taxable Equivalent Yield (TEY) for Munis and Treasuries

 The interest from municipal bonds is not taxed by the federal
government, and U.S. Treasury bonds, notes, and T-Bills do not
incur state or local taxes. Hence, these bonds can pay a lower
interest rate than a corporation with a comparable credit rating. To
compare municipal bonds or Treasuries with taxable bonds, the
yield is converted to a taxable equivalent yield (TEY),
sometimes called equivalent taxable yield. The taxable
equivalent yield is the yield that a taxable bond would have to
pay to be equivalent to the tax-free bond.

Taxable Equivalent Yield Formula for Municipal Bonds

Muni Yield
Taxable Equivalent Yield (TEY) =
100% - Your Federal Tax Bracket %

Taxable Equivalent Yield Example

4% Muni Yield
= 5.5% TEY
100% - 28% Federal Tax Bracket

We can call this the federal taxable equivalent yield, but note that
if you live in the municipality of the bond issuer, then the bond
may be free of state and local taxes as well. To take in
consideration all taxes saved, the above formula can be extended
for any tax situation by simply adding up the percentages to arrive
at a composite tax bracket and use that in the above equation to get
the tax-free yield.

Taxable Equivalent Yield Example for Municipal Bond Exempt of All Taxes

6.1% Muni Yield

= 10% TEY
100% - 28% federal tax - 10% state tax - 1% local tax

To look at it from a different angle, suppose a bond pays 10%, as

in the above example. That's $100 per year for a par value of
 $1,000. If you pay 28% of your income in federal taxes, 10% in
state taxes, and 1% in local taxes, and the bond is taxable, then the
federal tax will be $28, the state tax will be $10, and the local tax
will be $1—that would leave you with a net of $61. A tax-free
municipal bond yielding 6.1% would net you the same amount.
U.S. Treasuries do not incur state or local taxes, but federal taxes
have to be paid on the interest, so the taxable equivalent yield for
Treasuries is calculated using the same formula, but only the state
and local tax rate is deducted from 100%.

Taxable Equivalent Yield Formula for U.S. Treasury Bonds, Notes, and T-Bills

Treasury Yield
Taxable Equivalent Yield (TEY) =
100% - State Tax Rate % – Local Tax Rate %

Taxable Equivalent Yield Example for U.S. Treasuries

4% Treasury Yield
= 4.5% TEY
100% - 10% state tax - 1% local tax

Thus, a corporate bond that is taxable by the federal, state and local
government would have to pay 4.5% to net the same amount that a
U.S. Treasury paying 4% would net. Note, also, that U.S.
Treasuries are considered the safest investment, so the corporate
bond would have to pay a little more—even if it had the highest
credit rating—than the Treasury, to compensate the investor for the
additional risk. The lower the credit rating of the corporate bond,
the greater the interest the corporate bond would have to pay to
entice investors away from safe Treasuries, which are considered
risk-free investments.

Yield to Maturity or True Yield

If an investor buys a bond in the secondary market and pays a price

different from par value, then not only will the current yield differ
from the nominal yield, but there will be a gain or loss when the
bond matures and the bondholder receives the par value of the
bond. Like the calculation for current yield, yield to maturity and
other yields based on the purchase price of the bond in the
 secondary market is based on the clean bond price,
excluding accrued interest. If the investor holds the bond until
maturity, he will lose money if he paid a premium for the bond, or
he will earn money if it was bought at a discount. The yield-to-
maturity (YTM) (aka true yield, effective yield) of a
bond held to maturity accounts for the gain or loss that occurs
when the par value is repaid, so it is a better measure of the
investment return.

When a bond is bought at a discount, yield to maturity will always

be greater than the current yield because there will be a gain when
the bond matures, and the bondholder receives par value back, thus
raising the true yield; when a bond is bought at a premium, the
yield to maturity will always be less than the current yield because
there will be a loss when par value is received, which lowers the
true yield.

Summary of Bond Yield Relationships: Nominal Yield, Current Yield, YTM

When the bondholder pays... Bond Yield Relationships

less than par value (discount). Yield to Maturity > Current Yield > Nominal Yield

par value. Nominal Yield = Current Yield = Yield to Maturity

more than par value (premium). Nominal Yield > Current Yield > Yield to Maturity

Yield to Call

Because some bonds are callable, these bonds will also have
a yield to call (YTC), which is calculated exactly the same as
yield to maturity, but the call date is substituted for the maturity
date and the call price or call premium is substituted for
par value. When a bond is bought at a premium, the yield to call is
always the lowest yield of the bond.
Yield to Sinker

Some bonds are redeemed periodically by a sinking fund—

also called a mandatory redemption fund—that the issuer
establishes to retire debt periodically at sinking fund
dates specified in the redemption schedule of the bond contract
for specified sinking fund prices, which are often just par
value. Such bonds are usually selected at random for redemption
on such dates, so yield to sinker is calculated as if the bond
will be retired at the next sinking fund date. If the bond is retired,
then the bondholder simply receives the sinking fund price, and so
the yield to sinker is calculated like the yield to maturity,
substituting the sinking fund date for the maturity date, and, if
different, substituting the sinking fund price for the par value.

Note, however, that yield to call and yield to sinker may not be
pertinent if interest rates have risen since the bonds were first
issued, because these bonds will be selling for less than par value
in the secondary market, and it would save the issuer money to
simply buy back the bonds in the secondary market, which helps to
support bond prices for bondholders who want to sell.

Yield to Average Life

The yield to average life calculates the yield using the

average life of a sinking bond issue. So for a 20-year bond with
an indenture that specifies that 10% of an issue must be retired
each year from the 10th year to the 20th year of the bond's term,
the average life would be 15 years.

The yield to average life is also used for asset-backed securities,

especially mortgage-backed securities, because their lifetime
depends on prepayment speeds of the underlying asset pool.

Yield to Put Option

Some bonds have a put option, which allows the bondholder to

receive the principal of the bond from the issuer when the
bondholder exercises the put. This yield to put would be
calculated like the yield to maturity, except that the date that the
put is exercised is substituted for the maturity date, because the
bondholder receives the par value on the exercise date just as if the
bond matured.
 Yield to Worst

Finally, there is the yield to worst, which simply calculates

the bond's yield if the bond is retired at the earliest possible date
allowed by the bond's indenture.

The Formula Relating a Bond's Price to its Yield to Maturity, Yield to

Call, or Yield to Put

The formula below shows the relationship between the bond's price
in the secondary market (excluding accrued interest) and its yield
to maturity, or other yields, depending on the maturity date chosen.
In this equation, which assumes a single annual coupon
payment, YTM would be the bond's yield to maturity, but this is
difficult to solve, so bond traders usually read the yield to maturity
from a table that can be generated from this equation, or they use a
special calculator or software, such as Excel as shown further
below. Yield to call is determined in the same way, but n would
equal the number of years until the call date instead of the maturity
date, and P would be the call price. Similarly, the yield to put, or
any of the other yields, is calculated by substituting the appropriate
date when the principal will be received for the maturity date.

Yield to Call, Yield to Put, or Yield to Maturity Formula

C1 Cn P
Bond Price = + ... + +
(1+YTM)1 (1+YTM)n (1+YTM)n

or, expressed in summation, or sigma, notation:

n
Ck

∑
P
B= +
(1+YTM) k
(1+YTM)n
k=1

Note that if the bond pays a semiannual coupon, as most US bonds

do, then the following formula applies:
 Yield to Call, Yield to Put, or Yield to Maturity Formula for Bonds that Pay Coupons
Semiannually

C1 Cn P
Bond Price = + ... + +
(1+YTM/2)1 (1+YTM/2)2 n (1+YTM/2)2 n

This equation shows that the bond price is equal to the present
value of all bond payments with the interest rate equal to the yield
to maturity. Although it is difficult to solve for the yield using the
above equation, it can be approximated by this formula:

Yield-to-Maturity Approximation Formula for Bonds

AIP + (PV - CBP)/Years

Approximate Yield-to-Maturity Yield Percentage =
(PV+ CBP)/2

 AIP = Annual Interest Payment

 PV = Par Value
 CBP = Current Bond Price
 Years = Number of Years until Maturity

Yield-to-Maturity Example

$60 + ($1,000 - $800)/3

≈ 14%
($1,000 + $800)/2
 Annual Interest=$60
 Par Value = $1,000
 Current Bond Price=$800
 Bond matures in 3 years.)

Note that if a premium was paid for the bond, then the term (Par
Value - Current Bond Price)/Number of Years until
Maturity would be subtracted from the annual interest payment
rather than added to it, since it would be a negative number. A
good way to remember this formula is that it is simply taking the
difference between the par value and the current bond price and
dividing it by the remaining term of the bond. This is the profit or
loss per year, which is then added to or subtracted from the annual
 interest payment. The resulting sum, in turn, is divided by the
average of the par value and the current bond price. Once the bond
is bought, then the yield to maturity is fixed, so the current bond
price is replaced with the purchase price in the above formula.

A simplification of the YTM formula can be made if the bond has

no coupon payments, since all the terms involving coupon
payments become zero, and the yield to maturity reduces to the
present value of the principal payment (Formula #1 below):

Yield to Call, Yield to Put, or Yield to Maturity Formula for a Zero Coupon Bond

Principal
Payment
1. Discounted Bond Price =

(1+YTM)n

Future
Value
2. Present Value =
(1+YTM)n

Note that equations #1 and #2 above are the same, since the
discounted bond price is the present value of the investment and
the principal payment is the future value, so we can find a simple
way to calculate YTM by using a basic formula for the present
value and future value of money. To find the yield to maturity, we
transpose the equation for the future value of money to equal the
yield to maturity. The equation for future value, which can be
obtained by multiplying both sides of Equation #2 by (1 + YTM)n ,
is:

Present Value * (1 + YTM)n = Future Value

Divide both sides by the Present Value:

(1 + YTM)n = Future Payment/Present Value

Take the nt h root of both sides:

1 + YTM = (Future Payment/Present Value)1 / n

 Then subtract 1 from both sides, to arrive at YTM, the yield to
maturity for the discount:

YTM Formula for a Zero Coupon Bond

FV 1/n

r=
( PV ) -1

r = YTM per time period

n = number of time periods
FV = Future Value
PV = Present Value

or

YTM Example for a Zero Coupon Bond

If

 Settlement date = 3/31/2008

 Maturity = 3/31/2018 (10 year bond)
 Price = 60.00 (as a percent of par value which equals 60% × $1,000 = $600.00 Bond Price)
 Redemption = Value received at maturity as a percentage of par value = 100 (100% × par
value = $1,000)

Then

 YTM = (100/60)1 / 1 0 - 1 = 0.05241 ≈ 5.241%

To check our result, we plug the YTM into the YTM formula:

Principal
Discounted Payment
Bond =
Price
(1+YTM)n
 $100
=
(1+0.05241)1 0

$100
= = $60 (rounded)
1.66667

Note that the above example is compounded annually. To find the

YTM compounded biannually, simply set n=20 and multiply the
resulting YTM for a 6-month period by 2:
YTM = (100/60)1/20
- 1 = 0.025870255 × 2 = 0.051741 ≈ 5.17%

Note that the YTM is slightly lower because it is compounded

twice a year instead of once a year, so it must be lower to yield the
same payment amount of $40 at maturity. To check the result:

Discounted Bond Price = 100/1.025872 0 = 100/1.666666667 = $60

YTM can also be readily solved by using Microsoft Excel, as

shown below.

Yield-to-Maturity (YTM) Formula for Bonds using Microsoft Excel

YTM = Yield(settlement, maturity, rate, price, redemption, frequency, basis)

All dates are expressed either as quotes or as cell references (e.g., "1/5/2013", A1).

 Settlement = Settlement date

 Maturity = Maturity date
 Rate = Nominal coupon interest rate.
 Price = Redemption value as a percent of par value (e.g., 96 = 96% of par value).
 Redemption = Price as a percent of par value.
 Frequency = Number of coupon payments per year.
o 1 = Annual
o 2 = Semiannual (the most common value)
o 4 = Quarterly
 Basis = Day count basis.
o 0 = 30/360 (U.S. basis, the default if basis is omitted in formula)
o 1 = actual/actual (actual number of days in month/actual number of days in
year)
o 2 = actual/360
o 3 = actual/365
 o 4 = European 30/360

Note that yield to call (YTC) and yield to put (YTP) can also be calculated using this formula.
To calculate the yield to call:

 Maturity = Date of earliest possible call.

 Redemption = Call price.

To calculate yield to put:

 Redemption = Date that put can be exercised.

Yield to Worst, Yield to Sinker, and Yield to Average Life can be calculated by substituting
the appropriate date for the maturity date.

Yield to Maturity (YTM) Example

If

 Settlement date = 3/31/2008

 Maturity = 3/31/2018 (10 year bond)
 Nominal coupon rate = 5%
 Price = 92.56 (as a percent of par value which equals 92.56% × $1,000 = $925.60 Bond
Price)
 Redemption = Value received at maturity as a percentage of par value = 100 (100% × par
value = $1,000)
 Frequency = 2 semi-annual coupon payments

Then

 YTM = Yield("3/31/2008","3/31/2018",0.05,92.56,100,2) = 6.00%

Using the YTM approximation formula for the above example

yields nearly the same result:

Yield-to-Maturity Example Using Approximation Formula

$50 + ($1,000 - $925.60)/10 $57.44

=
($1000 + $925.60)/2 $962.80

= 0.059659327 ≈ 6%
 Realized Compound Yield

The realized compound yield is the yield obtained by

reinvesting all coupon payments for additional interest income. It
will also depend on the bond price if it is sold before maturity.
What this yield ultimately is depends on how interest rates change
over the holding period of the bond. Although future interest rates
and bond prices cannot be predicted with certainty, horizon
analysis is often used to forecast interest rates and bond prices
over a specific time period to yield an expectation of the realized
compound yield.

Holding-Period Return

Yield to maturity is the average yield over the term of the bond. If
a bond is sold before maturity, then its actual yield will probably
be different from the yield to maturity. If interest rates rise during
the holding period, then the bond's sale price will be less than the
purchase price, decreasing the yield, and if interest rates, decrease,
then the bond's sale price will be greater. The holding-period
return is the actual yield earned during the holding period. It can
be calculated using the same formula for yield to maturity, but the
sale price would be substituted for the par value, and the term
would equal the actual holding period. Note that, unlike yield to
maturity, the holding-period return cannot be known ahead of time
because the sale price of the bond cannot be known before the sale,
although it could be estimated.

Bond Equivalent Yield (BEY)

Money market instruments are short-term discount instruments

with maturities of less than a year, so the interest is paid at
maturity. Because short-term instruments are issued at a discount,
their yield is often referred to as a discount yield, which is
often annualized as the bond equivalent yield (BEY)
(aka investment rate yield, equivalent coupon
yield), which simplifies the comparison of yields with other
financial returns:

BEY = Interest Rate per Term × Number of Terms per Year

 Bond Equivalent Yield (BEY) Formula

Interest Rate Number of Terms

Per Term per Year

Actual Number
Face Value - Price Paid
of Days in Year
BEY = ×
Price Paid Days Till Maturity

Although the BEY is not compounded, but is simply the discount

yield annualized, it can be converted directly to any compounded
rate of interest by using the formula for the present and future
value of a dollar. (See Calculating the Interest Rate of a
Discounted Financial Instrument for more info.)

To find a compounded rate, add 1 to the discount yield and raise

the result to a power equal to the number of terms in the year, then
subtract the result from 1:

Compounded BEY Rate = (1 + Discount Yield) n – 1

 n = Number of Terms in the Year

So if the discount yield is 1% for 90 days, then, using a banker's

year of 360 days, the compounded BEY rate is calculated thus:

 Number of Terms in Year = 360/90 = 4

 Compounded BEY Rate = (1 + .01)4 – 1 = 1.014 – 1 = 1.0406
– 1 =.0406 = 4.06%

Example — Calculating the Bond Equivalent Yield of a T-Bill

If you buy a 4-week T-bill with a face value of $1,000 for $997, what is the bond equivalent yield, assuming it
is not a leap year?

($1,000-$997)/$997 × 365/28 = 3.92% (rounded)

Example—Formula for Finding the Annualized Effective Compounded Rate of

Interest for a Discounted Note

To find the compounded rate of interest for a discounted money market instrument:
 1. Divide the par value by the discounted price.
2. Raise the result by the number of terms in 1 year, then subtract 1.

If you bought a 4-week T-bill for $997 and receive $1,000 4 weeks later, what is the effective annual
compounded interest rate earned?

Solution:

1. $1,000/$997 = 1.003009 (rounded)

Since there are 13 4-week periods in a year, $1 compounded 13 times would equal:

1. (1.0035)1 3 - 1 = 1.040 - 1 = 4.0% (rounded)

(See how the future value of a dollar is calculated to understand the reasoning better.)

This formula can calculate the yields of any financial instrument sold at a discount.

Perpetuities (aka Perpetual Bonds, Annuity Bonds, Consols)

Perpetuities are bonds that are not redeemable and pay only
interest, but pay it indefinitely—hence the name. They do not
mature and, thus, the principal is never repaid. They were first
issued by the British government in the 1850's, and were
called consols, and some perpetuities were issued by the U.S.
Treasury, but perpetuities are very rare today.

The price of a perpetuity is equal to the present value of all future

payments. While this forms an infinite series, it does have a finite
limit, because successive terms become smaller and smaller, and
that limit is the following:

Price of Perpetuity = Annual Coupon Payment / Nominal Interest Rate

Consequently, the yield of a perpetuity is calculated as the current

yield:

Yield of Perpetuity = Current Yield = Annual Coupon Payment / Perpetuity Price

Note that because a perpetuity is not redeemable and pays no

principal, a perpetuity has no yield to maturity, since it never
matures.

Example — Calculating the Yield of a Perpetuity

If
  Perpetuity annual coupon payment = $1
 Perpetuity price = $20

Then

 Perpetuity Yield = 1 / 20 = 5/100 = 5%.

Risk Structure of Interest Rates

U.S. Treasuries are generally considered free of default risk, and

therefore generally have the lowest yield. All other bonds have
some risk of default—some more than others. To compensate
investors for the greater risk, these bonds pay a higher yield. This
difference in yield is known as the risk
premium (aka default premium), and how the risk
premium varies across different bonds and different maturities is
known as the risk structure of interest rates. The greater the risk of
default, the greater the risk premium.

During a recession, investors become more concerned that the risk

of default is greater than in good times, since recessions can cause
financial difficulties for companies. So many investors move their
investments to safer bonds—a flight to quality. This causes
the difference in yield between corporate bonds and riskless
government bonds to increase. As a result of the lower demand, the
default premium increases to compensate investors for the greater
risk.

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