USA: +1(669)289-8683 
Sign in
Forum Topic Responses: One comprehensive forum topic response is assigned weekly. Students are required to select and research one of the forum topics listed below using a minimum of 3 reference sources in addition to the textbook and then write a 1,000-word or more response to the forum topic. APA format is required. Also submit your forum topic response to Turnitin.
Comprehensive forum topic response contributions will be critically graded on the thought quality of the response, work effort, research, APA format, and analysis. Please see attachments for further clarification.
REMEMBER TO:
Use APA format – title page, running head, citations (MUST HAVE), references
Do not
plagiarize
. Quote, paraphrase or summarize the data that you take from a source. Include a citation. Plagiarism will result in a serious loss of points.
Make sure your paper (the text) is 1,000 words or more.
Do not use Wikipedia or Investopedia or any ~pedia sources. The text may be used but will not count toward the required 3 sources.
Select one of the following forum topics to research and write about.
-Bond Investments and Risk
-Interest Rates
-Interest Rates and the Federal Reserve
-Bond Rating
-Bond Duration (including Web Extension 5C)
-The Term Structure of Interest Rates
Forum Topic Responses:
One comprehensive forum topic response is assigned weekly. Students are required to select and research one of the forum topics listed below using a minimum of 3 reference sources in addition to the textbook and then write a 1,000-word or more response to the forum topic. APA format is required. Also submit your forum topic response to Turnitin.
Comprehensive forum topic response contributions will be critically graded on the thought quality of the response, work effort, research, APA format, and analysis.
Remember to:
Use APA format – title page, running head, citations (MUST HAVE), references
Do not plagiarize. Quote, paraphrase or summarize the data that you take from a source. Include a citation. Plagiarism will result in a serious loss of points.
Make sure your paper (the text) is 1,000 words or more.
Do not use Wikipedia or Investopedia or any ~pedia sources. The text may be used but will not count toward the required 3 sources.
Select one of the following forum topics to research and write about.
-Bond Investments and Risk
-Interest Rates
-Interest Rates and the Federal Reserve
-Bond Rating
-Bond Duration (including Web Extension 5C)
-The Term Structure of Interest Rates
WE B E X T E N S I O N 5A
ACloser Look at Zero
Coupon Bonds
Some bonds pay no interest but are offered at a substantial discount below theirpar values and hence provide capital appreciation rather than interest income.These securities are called zero coupon bonds (“zeros”), or original issue
discount bonds (OIDs). Some corporations use these bonds to manage their matu-
rity structure. In addition, these bonds provide some desirable tax features for cor-
porations, as we discuss later in this extension.
Corporations first used zeros in a major way in
1
981. IBM, Alcoa, JCPenney,
ITT, Cities Service, GMAC, Martin-Marietta, and many other companies have
used them to raise billions of dollars. Municipal governments also sell “zero munis.”
Shortly after corporations began to issue zeros, investment bankers figured out a way
to create zeros from U.S. Treasury bonds, which were issued only in coupon form.
In 1983 Salomon Brothers bought $1 billion of 7%, 30-year Treasuries. Each bond
had 60 coupons worth $35 each, which represented the interest payments due every 6
months. Salomon then, in effect, clipped the coupons and placed them in 60 piles;
the last pile also contained the now “stripped” bond itself, which represented a prom-
ise of $1,000 in the year 2013. These 60 piles of U.S. Treasury promises were then
placed with the trust department of a bank and used as collateral for “zero coupon
U.S. Treasury Trust Certificates,” which are, in essence, zero coupon Treasury
bonds. A pension fund that, in 1984, expected to need money in 2009 could have
bought 25-year certificates backed by the interest the Treasury will pay in 2009.
In 1985, the Treasury Department began allowing investors to strip long-term
U.S. Treasury bonds and directly register the newly created zero coupon bonds,
called STRIPs, with the Treasury Department. This bypasses the role formerly
played by investment banks. Now virtually all U.S. Treasury zeros are held in the
form of STRIPs. These STRIPs are, of course, safer than corporate zeros, so they
are very popular with pension fund managers.
To understand how zeros are used and analyzed, consider the zeros to be issued
by Vandenberg Corporation, a shopping center developer. Vandenberg is developing
a new shopping center in San Diego, California, and it needs $50 million. The com-
pany does not anticipate major cash flows from the project for about 5 years; how-
ever, Pieter Vandenberg, the president, plans to sell the center once it is fully
developed and rented, which should take about 5 years. Therefore, Vandenberg
wants to use a financing vehicle that will not require cash outflows for 5 years. He
has decided on a 5-year zero coupon bond issue, with each bond having a maturity
value of $1,000.
Vandenberg Corporation is an A-rated company, and A-rated zeros with 5-year
maturities yield 6% at this time. (5-year coupon bonds also yield 6%.) The company
is in the 40% federal-plus-state tax bracket. Pieter Vandenberg wants to know the
1
firm’s after-tax cost of debt if it uses 6%, 5-year maturity zeros, and he also wants to
know what the bond’s cash flows will be. Figure 5A-1 provides an analysis of the sit-
uation, and the subsequent numbered list explains the table itself. See the worksheet
“Web 5A” in Ch05 Tool Kit.xls for all calculations. Because the table was calculated
in Excel, there is no rounding in intermediate steps.
1. The information in the “Input Data” section, except the issue price, was given in
the preceding paragraph, and the information in the “Analysis” section was calcu-
lated using the known data. The maturity value of the bond is always set at
$1,000 or some multiple thereof.
2. The issue price is the PV of $1,000, discounted back 5 years at the rate rd =
6%
with annual compounding. Using a financial calculator, we input N = 5, I/YR = 6,
PMT = 0, and FV = 1000, then press the PV key to find PV = $747.26.1 Note
that $747.26, when compounded annually for 5 years at 6%, will grow to
$1,000, as shown in Figure 5A-1.
3. The year-end accrued values, as shown on Line 2 in the analysis section, rep-
resent the compounded value of the bond at the end of each year.2 The
F IGURE 5A-1 Analysis of a Zero Coupon Bond from Issuer’s Perspective
Input Data
Analysis:
Amount needed = $50,000,000
$747.26
$1,000
6%
40%
0%
$0
5
Maturity value =
Pre-tax market interest rate, rd =
Maturity (in years) =
Corporate tax rate =
Coupon rate =
Coupon payment (assuming annual payments) =
Years
(1) Remaining years
After-tax-cost of debt = 3.60%
Number of $1,000 zeros the
company must issue to raise $50 million Amount needed/Price per bond
66,911.279 bonds.
$66,911,279Face amount of bonds = # bonds × $1,000 =
=
=
0
5 4 3 2 1 0
$890.00 $943.40 $1,000.00$839.62
$0.00 $0.00 $0.00
$0.00
$792.09$747.26
$747.26
$44.84
$17.93
$17.93
$47.53
$19.01
$19.01
$50.38
$20.15
$20.15
$53.40
$21.36
$21.26
$56.60
$22.64
($977.36)
$0.00
1 2 3 4 5
(2) Year-end accrued value
(3) Interest payment
(4) Implied interest
deduction on discount
(5) Tax savings
(6) Cash flow
Issue Price = PV of payments at rd =
1This value is reported to two decimal places, but all significant digits will be used in the subsequent
calculations.
2Line 1 is included in the Excel table because it facilitates some calculations.
2 Web Extension 5A: A Closer Look at Zero Coupon Bonds
accrued value for Year 0 is the issue price; the accrued value for Year 1 is
found as $747.26(1.06) = $792.09; and the accrued value at the end of Year 2
is $747.26(1.06)2 = $839.62. In general, the value at the end of any Year N is
calculated as follows:
Accrued value at the end of Year N ¼ ðIssue priceÞð1 þ rdÞN (5A-1)
4. The interest deduction as shown on Line 4 represents the increase in accrued
value during the year.3 Thus, interest in Year 1 = $792.09 − $747.26 = $44.84.4
The general equation is
Interest in Year N ¼ ðAccrued valueÞN − ðAccrued valueÞN – 1 (5A-2)
This method of calculating taxable interest is specified in the Tax Code.
5. The company can deduct interest each year, even though the payment is not made
in cash. This deduction lowers the taxes that would otherwise be paid, producing
the following tax savings:
Tax savings ¼ ðInterest deductionÞðTÞ (5A-3)
For example, in Year 1, the tax savings are
Tax savings ¼ $44:84ð0:4Þ
¼ $17:93 in Year 1
6. Line 5 reports the tax savings as just calculated, and Line 6 reports the cash flows;
it shows the cash flow at the end of Years 0 through 5. At Year 0, the company
receives the $747.26 issue price. The company also has positive cash inflows
equal to the tax savings during Years 1 through 4. Finally, in Year 5, it must pay
the $1,000 maturity value, but it receives one more year of interest tax savings.
Therefore, the net cash flow in Year 5 is −$1,000 + $22.64 = −$977.36.
7. Next, we can determine the after-tax cost (or after-tax yield to maturity) of issuing
the bonds. Since the cash flow stream is uneven, the after-tax yield to maturity is
found by entering the after-tax cash flows, shown on Line 6 of Figure 5A-1, into
the cash flow register and then pressing the IRR key on the financial calculator.
The IRR is the after-tax cost of zero coupon debt to the company. Conceptually,
here is the situation:
∑
N
t¼1
CFN
ð1þ rdðATÞÞN
¼ 0 (5A-4)
3Line 3 is included for the case of an original issue discount bond with a coupon rate greater than zero
but not as high as the going rate in the market. Line 3 is not required for the analysis of a zero coupon
bond, such as the one in this example, but the spreadsheet can be used for any OID bond.
4The reported numbers have been rounded, but all significant digits are used in the actual calculations.
Web Extension 5A: A Closer Look at Zero Coupon Bonds 3
For the bond in this example, we have
$747:26
ð1þrdðATÞÞ0
þ $17:93ð1þrdðATÞÞ1
þ $19:01ð1þrdðATÞÞ2
þ $20:15ð1þrdðATÞÞ3
þ $21:36ð1þrdðATÞÞ4
þ −$977:36ð1þrdðATÞÞ5
¼0
The value of rd(AT) = 0.036 = 3.6%, found with a financial calculator, produces
the equality, and it is the after-tax cost of this debt. (Input in the cash flow regis-
ter CF0 = 747.26, CF1 = 17.94, and so forth, out to CF5 = −977.36; then press the
IRR key to find rd(AT) = 3.6%.) The IRR function was used in the Excel model.
8. Note that rd(1 – T) = 6%(0.6) = 3.6%. As we will see in Chapter 9, the cost
of capital for regular coupon debt is found using the formula rd(1 − T). Thus,
for tax purposes there is symmetrical treatment for zero coupon and regular
coupon debt; that is, both types of debt use the same after-tax cost formula.
This was the intent of Congress, and it is why the Tax Code specifies the
treatment set forth in Figure 5A-1.5
Not all original issue discount bonds (OIDs) have zero coupons. For example,
Vandenberg might have sold an issue of 5-year bonds with a 5% coupon at a time
when other bonds with similar ratings and maturities were yielding 6%. Such bonds
would have had a value of $957.88.
Bond value ¼∑
5
t¼1
$50
ð1:06Þt þ
$1; 000
ð1:06Þ5 ¼ $957:88
If an investor had purchased these bonds at a price of $957.88, the yield to maturity
would have been 6%. The discount of $1,000 – $957.88 = $42.12 would have been
amortized over the bond’s 5-year life, and it would have been handled by both
Vandenberg and the bondholders exactly as the discount on the zeros was handled.
Thus, zero coupon bonds are just one type of original issue discount bond. Any
nonconvertible bond whose coupon rate is set below the going market rate at the
time of its issue will sell at a discount, and it will be classified (for tax and other pur-
poses) as an OID bond.
The purchaser of a zero coupon bond must calculate interest income on the bond
in the same manner as the issuer calculates the interest deduction. Given this tax
treatment, investors pay taxes in each year even though they don’t receive any cash
flows until the bond is sold or matures, a situation that many investors find unattrac-
tive. Consequently, pension funds and other tax-exempt entities buy most zero cou-
pon bonds. Individuals do, however, buy taxable zeros for their Individual Retirement
Accounts (IRAs). Also, state and local governments issue “tax-exempt muni zeros”
that are purchased by individuals in high tax brackets.
Questions
(5A–1) Do all original issue discount (OID) bonds have zero coupon payments? Explain.
(5A–2) What are Treasury STRIPs? Are they callable? Explain.
(5A–3) Do Treasury zeros face any interest rate (price) or reinvestment rate risk? Explain.
5Note, too, that we have analyzed the bond as if the cash flows accrued annually. Generally, to facilitate
comparisons with semiannual payment coupon bonds, the analysis is conducted on a semiannual basis.
4 Web Extension 5A: A Closer Look at Zero Coupon Bonds
Problems
(5A–1)
Zero Coupon Bonds
A company has just issued 4-year zero coupon bonds with a maturity value of $1,000
and a yield to maturity of 9%. Its tax rate is 40%. What is its after-tax cost of debt?
(5A–2)
Zero Coupon Bonds
An investor in the 35% tax bracket purchases the bond discussed in Problem 5A-1.
What is the investor’s after-tax return?
(5A–3)
Stripped U.S. Treasury
Bond
McGwire Company’s pension fund projected that a significant number of its employ-
ees would take advantage of an early retirement program the company plans to offer
in 5 years. Anticipating the need to fund these pensions, the firm bought zero coupon
U.S. Treasury Trust Certificates maturing in 5 years. When these instruments were
originally issued, they were 12% coupon, 30-year U.S. Treasury bonds. The stripped
Treasuries are currently priced to yield 10%. Their total maturity value is $6 million.
What is their total cost (price) to McGwire today?
(5A–4)
Zero Coupon Bond
At the beginning of the year, you purchased a 7-year, zero coupon bond with a yield
to maturity of 6.8%. The bond has a face value of $1,000. Your tax rate is 25%.
What is the total tax that you will have to pay on the bond during the first year?
(5A–5)
Zeros and Expectations
Theory
A 2-year, zero coupon Treasury bond with a maturity value of $1,000 has a price of
$873.4387. A 1-year, zero coupon Treasury bond with a maturity value of $1,000 has
a price of $938.9671. If the pure expectations theory is correct, for what price should
1-year, zero coupon Treasury bonds sell 1 year from now?
(5A–6)
Zero Coupon Bonds
and EAR
Assume that the city of Tampa sold tax-exempt (muni) zero coupon bonds 5 years
ago. The bonds had a 25-year maturity and a maturity value of $1,000 when they
were issued, and the interest rate built into the issue was a nominal 10% with semi-
annual compounding. The bonds are now callable at a premium of 10% over the ac-
crued value. If they were called today, what effective annual rate of return would be
earned by an investor who bought the bonds when they were issued and who still
owns them today?
Web Extension 5A: A Closer Look at Zero Coupon Bonds 5
W E B E X T E N S I O N 5B
A Closer Look at TIPS:
Treasury Inflation-
Protected Securities
I nvestors who purchase bonds must constantly worry about inflation. If inflation turnsout to be greater than expected, bonds will provide a lower than expected real return.To protect themselves against expected increases in inflation, investors build an infla-
tion risk premium into their required rate of return. This raises borrowers’ costs.
In order to provide investors with an inflation-protected bond, and possibly to
reduce the cost of debt to the government, on January 29,
1
997, the U.S. Treasury
issued $7 billion of 10-year inflation-indexed bonds called Treasury Inflation-
Protected Securities (TIPS). Since then, the Treasury has continued to offer TIPS
with original maturities up to 30 years. To see how TIPS work, let’s take a closer
look at the TIPS that were auctioned on April 7, 1999. These TIPS mature on April
15, 2029, and pay interest on April 15 and October 15 of each year. The bonds have
a fixed coupon rate of 3.875%, but they pay interest on a principal amount that
increases with inflation. At the end of each 6-month period, the principal (originally
set at par, or $1,000) is adjusted by the inflation rate. For example, on April 15, 1999,
the Reference CPI (as defined by the U.S. Treasury) was 164.39333. At the time of the
first coupon payment on October 15, 1999, the Reference CPI was 166.88065. The
Index Ratio is defined as the ratio of the current CPI and the original CPI:
Index Ratio ¼ 166:88065=164:39333 ¼ 1:01513
In essence, this Index Ratio measures the amount of inflation since the bond was first
issued. The inflation-adjusted principal is then calculated as $1,000 (Index Ratio) =
$1,000(1. 01513) = $1,015.13. So, on October 15, 1999, each bond paid interest of
(0.03875/2)($1,015.13) = $19.67. Note that the interest rate is divided by 2 because in-
terest on these (and most other) bonds is paid twice a year.
By April 15, 2000, a bit more inflation had occurred, and the inflation-adjusted
principal was up to $1,029.04 (based on the Index Ratio at that time).1 On April 15,
2000, each bond paid interest of 0.03875/2 × $1,029.04 = $19.94. Thus, the total re-
turn during the first year consisted of $19.67 + $19.94 = $39.61 of interest and
$1,029.04 – $1,000.00 = $29.04 of “capital gains,” or $39.61 + $29.04 = $68.65 in to-
tal. Thus, the total rate of return, ignoring compounding, was $68.65/$1,000 =
6.865%. Therefore, if you had been able to buy this bond for $1,000, you would
have received a real rate of return of 3.875%.2
This same adjustment process will continue each year until the bonds mature on
April 15, 2029, at which time they will pay the adjusted maturity value. Thus, the
1The U.S. Treasury publishes the Reference CPIs and Index Ratios each month. For the April 2000 va-
lues, see http://www.treasurydirect.gov/instit/annceresult/tipscpi/2000/of042000cpi .
2The auction price usually differs slightly from the $1,000 par value.
1
cash income provided by the bonds rises by exactly enough to cover inflation, pro-
ducing a real, inflation-adjusted rate of 3.375%.3 Further, since the principal also
rises by the inflation rate, it too is protected from inflation.
The Treasury regularly conducts auctions to issue indexed bonds. The 3.375%
rate was based on the relative supply and demand for the issue, and it will remain
fixed over the life of the bond. However, new bonds are issued periodically, and
their “coupon” real rates depend on the market at the time the bond is auctioned.
In January 2009, 10-year indexed securities had a real rate of 1.86%.
Both the annual interest received and the increase in principal are taxed each year
as interest income, even though cash from the appreciation will not be received
until the bond matures. Therefore, these bonds are not good for accounts subject to
current income taxes but are excellent for individual retirement accounts (IRAs and
401(k) plans), which are not taxed until funds are withdrawn.
Keep in mind, though, that despite their protection against inflation, indexed
bonds are not completely riskless. The real interest rate can and often does change.
If real rates rise, then the prices of indexed bonds will decline. In fact, if you buy a
TIPS in the secondary market, its quoted yield is likely to be somewhat different
from the coupon rate because current real interest rates probably are different than
they were at the time of issue. Also, as the bond’s remaining time until maturity
declines, its maturity risk premium will also decline. Thus, the 3.875% coupon on
the bond we have been discussing was a good indicator of the real interest rate for a
30-year bond at the time it was issued in 1999, but that included the maturity risk
premium for a 30-year bond.
TIPS are also useful for providing estimates of the inflation premium for a given
maturity, which can be estimated by subtracting the TIPS’s yield from the yield of a
nonindexed Treasury bond of the same maturity.
3This assumes you had bought the bond at its auction for $1,000 and that you held the bond. It also does
not take into account the rate of return you might get on the coupon payments.
2 Web Extension 5B: A Closer Look at TIPS: Treasury Inflation-Protected Securities
W E B E X T E N S I O N 5C
A Closer Look at Bond Risk:
Duration
T his extension explains how to manage the risk of a bond portfolio using the con-cept of duration.
5.1 BOND RISK
In our discussion of bond valuation in Chapter 5, we discussed interest rate and rein-
vestment rate risk. Interest rate (price) risk is the risk that the price of a debt secu-
rity will fall as a result of increases in interest rates, and reinvestment rate risk is the
risk of earning a less than expected return when debt principal or interest payments
are reinvested at rates that are lower than the original yield to maturity.
To illustrate how to reduce interest rate and reinvestment rate risks, we will consider
a firm that is obligated to pay a worker a lump-sum retirement benefit of $10,000 at the
end of 10 years. Assume that the yield curve is horizontal, the current interest rate on all
Treasury securities is 9%, and the type of security used to fund the retirement benefit is
Treasury bonds. The present value of $10,000, discounted back 10 years at 9%, is
$10,000(0.4224) = $4,224. Therefore, the firm could invest $4,224 in Treasury bonds
and expect to be able to meet its obligation 10 years hence.1
Suppose, however, that interest rates change from the current 9% rate immedi-
ately after the firm has bought the Treasury bonds. How will this affect the situation?
The answer is, “It all depends.” If rates fall, then the value of the bonds in the port-
folio will rise, but this benefit will be offset to a greater or lesser degree by a decline
in the rate at which the coupon payment of 0.09($4,224) = $380.16 can be reinvested.
The reverse would hold if interest rates rose above 9%. Here are some examples (for
simplicity, we assume annual coupons).
1. The firm buys $4,224 of 9%, 10-year maturity bonds; rates fall to 7% immediately
after the purchase and remain at that level:
Portfolio value at
the end of 10 years
¼
Future value of
10 interest payments
of $380:16 each
compounded at 7%
þ Maturity
value
¼ $5; 252 þ $4; 224
¼ $9; 476
Therefore, the firm cannot meet its $10,000 obligation, and it must contribute
additional funds.
1For the sake of simplicity, we assume that the firm can buy a fraction of a bond.
1
2. The firm buys $4,224 of 9%, 40-year bonds; rates fall to 7% immediately after the
purchase and remain at that level:
Portfolio value at
the end of 10 years
¼ $5; 252 þ
Value of
30-year
9% bonds
when rd ¼ 7%
¼ $5; 252 þ $5; 272
¼ $10; 524
In this situation, the firm has excess capital at the end of the 10-year period.
3. The firm buys $4,224 of 9%, 10-year bonds; rates rise to 12% immediately after the
purchase and remain at that level:
Portfolio value at
the end of 10 years
¼
Future value of
10 interest payments
of $380:16 each
compounded at 12%
þ Maturity
value
¼ $6; 671 þ $4; 224
¼ $10; 895
This situation also produces a funding surplus.
4. The firm buys $4,224 of 9%, 40-year bonds; rates rise to 12% immediately after the
purchase and remain at that level:
Portfolio value at
the end of 10 years
¼ $6; 671 þ
Value of
30-year
9% bonds
when rd ¼ 12%
¼ $6; 671 þ $3; 203
¼ $9; 874
This time, a shortfall occurs.
Here are some generalizations drawn from the examples.
1. If interest rates fall and the portfolio is invested in relatively short-term bonds,
then the reinvestment rate penalty exceeds the capital gains and so a net short-
fall occurs. However, if the portfolio had been invested in relatively long-term
bonds, then a drop in rates would produce capital gains that would more than
offset the shortfall caused by low reinvestment rates.
2. If interest rates rise and the portfolio is invested in relatively short-term bonds,
then gains from high reinvestment rates will more than offset capital losses,
and the final portfolio value will exceed the required amount. However, if the
portfolio had been invested in long-term bonds, then capital losses would more
than offset reinvestment gains, and a net shortfall would result.
If a company has many cash obligations expected in the future, then the
complexity of estimating the effects of interest rate changes is obviously exac-
erbated. Still, methods have been devised to help deal with the risks associated
with changing interest rates. Several methods are discussed in the following
sections.
2 Web Extension 5C: A Closer Look at Bond Risk: Duration
5.2 IMMUNIZATION
Bond portfolios can be immunized against interest rate and reinvestment rate risk,
much as people can be immunized against the flu. In brief, immunization involves
selecting bonds with coupons and maturities such that the benefits or losses from
changes in reinvestment rates are exactly offset by losses or gains in the prices of
the bonds. In other words, if a bond’s reinvestment rate risk exactly matches its inter-
est rate price risk, then the bond is immunized against the adverse effects of changes
in interest rates.
To see what is involved, refer back to our example of a firm that buys $4,224 of
9% Treasury bonds to meet a $10,000 obligation 10 years hence. In the example, we
see that if the firm buys bonds with a 10-year maturity and interest rates remain con-
stant, then the obligation can be met exactly. However, if interest rates fall from 9%
to 7%, then a shortfall will occur because the coupons received will be reinvested at a
rate of 7% rather than the 9% reinvestment rate required to reach the $10,000 tar-
get. But suppose the firm had bought 40-year rather than 10-year bonds. A decline in
interest rates would still have the same effect on the compounded coupon payments,
but now the firm would hold 9% coupon, 30-year bonds in a 7% market 10 years
hence, so the bonds would have a value greater than par. In this case, the bonds
would have risen by more than enough to offset the shortfall in compounded interest.
Can we buy bonds with a maturity somewhere between 10 and 40 years such that the
net effect of changes in reinvested cash flows and changes in bond values at year 10
will always be positive? The answer is “yes,” as we explain next.
5.3 DURATION
The key to immunizing a portfolio is to buy bonds that have a duration equal to the
years until the funds will be needed. Duration cannot be defined in simple terms like
maturity, but it can be thought of as the weighted average maturity of all the cash
flows (coupon payments plus maturity value) provided by a bond, and it is exception-
ally useful to help manage the risk inherent in a bond portfolio. The duration for-
mula and an example of the calculation are provided below, but first we present
some additional points about duration.
1. Duration is similar to the concept of discounted payback in capital budgeting in
the sense that, the longer the duration, the longer funds are tied up in the bond.
2. To immunize a bond portfolio, buy bonds that have a duration equal to the
number of years until the funds will be needed. In our example, the firm should
buy bonds with a duration of 10 years.
3. Duration is a measure of bond volatility. The percentage change in the value of a
bond (or bond portfolio) will be approximately equal to the bond’s duration mul-
tiplied by the percentage-point change in interest rates.2 Therefore, a 2-
percentage-point increase in interest rates will lower the value of a bond with a
10-year duration by about 20%, but the value of a 5-year duration bond will fall
by only 10%. As this example shows, if Bond A has twice the duration of Bond B
then Bond A also has twice the volatility of Bond B. A corporate treasurer (or any
other investor) who is concerned about declines in the market value of his or her
2Actually, duration measures the negative of the percentage change in bond price. Duration is a measure
of the bond price’s elasticity with respect to the interest rate, but this elasticity measure is valid only for
small changes in the interest rate.
Web Extension 5C: A Closer Look at Bond Risk: Duration 3
portfolio should buy bonds with low durations. (This is important even if the in-
vestor buys a bond mutual fund.)
4. The duration of a zero coupon bond is equal to its maturity, but the duration of
any coupon bond is less than its maturity. (Remember, the duration is a weighted
average maturity of the cash flows, the only cash flow from a zero occurs at ma-
turity, and coupon bonds have cash flows prior to maturity.) And with all else
held constant, the higher the coupon rate the shorter the duration, because a
high-coupon bond provides significant early cash flows even if it has a long
maturity.
Duration is calculated using this formula:
Duration ¼
∑
N
t¼1
tðCFtÞ
ð1 þ rdÞt
∑
N
t¼1
CFt
ð1 þ rdÞt
¼
∑
N
t¼1
tðCFtÞ
ð1 þ rdÞt
VB
(5C-1)
Here rd is the required return on the bond, N is the bond’s years to maturity, t is
the year each cash flow occurs, and CFt is the cash flow in year t (CFt = INT for t <
N and CFt = INT + M for t = N, where INT is the interest payment and M is the
principal payment). Notice that the denominator of Equation 5C-1 is simply the
value of the bond, VB.
To simplify calculations in Excel, we define the present value of cash flow t (PV of
CFt ) as
PV of CFt ¼
CFt
ð1 þ rdÞt
We can rewrite Equation 5C-1 as
Duration ¼
∑
N
t¼1
tðPVofCFtÞ
VB
(5C-2)
To illustrate the duration calculation, consider a 20-year, 9% annual coupon
bond bought at its par value of $1,000. It provides cash flows of $90 per year
for 19 years plus $1,090 in the 20th year. To calculate duration, we used an Excel
model as shown in Figure 5C-1; see the worksheet Web 5C in the file Ch05 Tool
Kit.xls for details.
Column 1 of Figure 5C-1 gives the year each cash flow occurs, Column 2 gives the
cash flows, Column 3 shows the PV of each cash flow, and Column 4 shows the product
of t and the PV of each cash flow. The sum the PVs in Column 3 is the value of
the bond, VB. The duration is equal to the value of the bond divided by the sum of
Column 4.
This 20-year bond’s 9.95 duration is close to that of our illustrative firm’s 10-year
liability. So, if the firm bought a portfolio of these 20-year bonds and then reinvested
4 Web Extension 5C: A Closer Look at Bond Risk: Duration
the coupons as they came in, then the accumulated interest payments, plus the
value of the bond after 10 years, would be close to or exceed $10,000 regardless of
whether interest rates rose, fell, or remained constant at 9%. See Ch05 Tool Kit.xls
for an example. There is also an Excel function for duration; the Tool Kit also de-
monstrates its use.
Unfortunately, other complications arise. Our simple example looked at a single
interest rate change that occurred immediately after funding. In reality, interest rates
change every day, which causes bonds’ durations to change; this, in turn, requires
that bond portfolios be rebalanced periodically to remain immunized.
FIGURE 5C-1 Duration
7
A B C D E
Inputs
Years to maturity = 20
9.00%
$90.0
$1,080
9.00%
t
(1)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
$90
$90
$90
$90
$90
$90
$90
$90
$90
$90
$90
$90
$90
$90
$90
$90
$90
$90
$90
$1,090 $194.49
VB =
Duration = Sum of t(PV of CFt)/VB =
$1,000,00
Sum of
t(PV of CFt) =
$17.50
$19.08
$20.80
$22.67
$24.71
$26.93
$29.36
$32.00
$34.88
$38.02
$41.44
$45.17
$49.23
$53.66
$58.49
$63.76
$69.50
$75.75
$82.57 82.57
151.50
208.49
255.03
292.47
321.98
344.63
361.34
372.95
380.17
383.66
383.98
381.63
377.05
370.63
362.69
353.54
343.43
332.58
3,889.79
$9,990.11
CFt
(2)
PV of CFt
(3)
t(PV of CFt)
(4)
Coupon rate =
Annual payment =
Par value = FV =
Going rate, r =
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39 9.95
resource
See the worksheet Web
5C in Ch05 Tool Kit.xls on
the textbook’s Web site.
Web Extension 5C: A Closer Look at Bond Risk: Duration 5
