Module 3
Bond and Pricing Assignment
a. Bond Prices
U.S. Treasury bills (commonly known as T-bills) are the most straightforward
type of bond. Each T-bill represents a promise by the U.S. government to pay $100 on
a fixed future date. There are no coupon payments, which is why T-bills are known as
zerocoupon bonds. They are also called pure discount bonds (or just discount bonds),
because the price is less than their face value—they sell at a discount. This isn’t a
discount in the sense of a markdown at a clothing store, however.
When a $100 face value Treasury bill (T-bill) sells for $96, the $4 difference
between the purchase price and the face value represents the interest earned by the
investor. This difference is the payment to the lender, compensating them for
providing funds to the government for the duration of the T-bill's term.
To elaborate, T-bills are short-term debt securities issued by the U.S. Treasury,
typically with maturities of one year or less. They are sold at a discount to their face
value, meaning investors purchase them for less than their nominal value. At maturity,
the U.S. Treasury pays the investor the full face value of the T-bill. The difference
between the purchase price and the face value is the interest income, which reflects
the return on the investment.
In this example, the investor purchases the T-bill for $96. When the T-bill
matures, the government repays the full face value of $100. The investor's return on
investment is the $4 difference between the purchase price and the face value. This $4
represents the interest earned over the life of the T-bill, providing a clear and
straightforward method for calculating the yield.
To understand the yield on this investment, we can calculate the percentage
return based on the initial investment. The yield is determined by dividing the interest
earned by the purchase price and then annualizing this return if necessary. This yield
represents the annualized return if the T-bill matures in one year. For T-bills with
shorter maturities, such as three or six months, the yield calculation would be adjusted
to reflect the shorter time frame.
Investors are attracted to T-bills for several reasons. First, they are considered
one of the safest investments available because they are backed by the full faith and
credit of the U.S. government. The low risk associated with T-bills makes them an
attractive option for conservative investors seeking to preserve capital while earning a
modest return. Second, T-bills are highly liquid, meaning they can be easily bought
and sold in the secondary market without significant price fluctuations. This liquidity
provides investors with flexibility and quick access to cash if needed.
Additionally, T-bills are exempt from state and local income taxes, although
they are subject to federal income tax. This tax advantage can be particularly
appealing to investors in high-tax states, as it effectively increases the after-tax return
on the investment.
The interest earned on T-bills, as illustrated by the $4 difference in our
example, is an essential component of their overall appeal. This interest payment
compensates investors for the opportunity cost of tying up their funds in the T-bill
rather than using them for other investments or consumption. It also provides a
benchmark for other short-term interest rates in the financial markets, influencing
rates on savings accounts, money market funds, and other fixed-income securities.
The simplicity and transparency of T-bill pricing and yield calculations make
them an excellent educational tool for understanding basic principles of fixed-income
investing. By examining the relationship between the purchase price, face value, and
interest earned, investors can gain insights into how debt securities generate returns
and how to compare different investment options based on yield and risk.
In summary, when a $100 face value T-bill sells for $96, the $4 difference is
the interest earned by the investor. This interest compensates the lender for providing
funds to the government and represents the return on investment for the T-bill. The
yield calculation provides a straightforward way to assess the annualized return,
highlighting the appeal of T-bills as a safe, liquid, and tax-advantaged investment
option. Understanding these concepts is fundamental for making informed decisions
in the fixed-income market and managing a diversified investment portfolio.
Because a Treasury bill makes a single payment on a future date, its price is
just the present value of that payment: Price of $100 face value zero-coupon bond =
$100 (1 + i)n (1) where i is the interest rate expressed in decimal form and n is the
time until the payment is made, measured in the same time units as the interest rate.
Suppose the annual interest rate is 5 percent. What is the price of a one-year T-bill? To
figure out the answer, take the present value formula, set i at 0.05 and n at 1, and then
compute the price: Price of one-year Treasury bill = $100 (1 + 0.05) = $95.24 The
U.S. Treasury doesn’t issue T-bills with a maturity of more than one year; sixmonth T-
bills are much more common. At an annual interest rate of 5 percent, what is the price
of such a zero-coupon bond? We can use the present-value formula, again, but this
time we have to be careful. Recall that we need to measure i and n in the same time
units. Because i is the interest rate for one year, we need to measure n in years, and
because six months is half a year: Price of a six-month Treasury bill = $100 (1 +
0.05)1/2 = $97.59.
As you can see, the price of a six-month Treasury bill is higher than that of a
one-year T-bill. The shorter the time until the payment is made, the more we are
willing to pay for it now. If we go on to compute the price of a three-month T-bill,
setting n at 0.25 (one-fourth of a year), we find the answer is $99.02. Equation (1)
shows that for a zero-coupon bond, the relationship between the price and the interest
rate is the same as the one we saw in our discussion of present value. When the price
moves, the interest rate moves in the opposite direction. Thus, we can compute the
interest rate from the price using the present-value formula. For example, if the price
of a one-year T-bill is $95, then the interest rate is i = ($100/$95) − 1 = 0.0526, or
5.26 percent.
Conventional home mortgages and car loans are called fixed-payment loans
because they promise a fixed number of equal payments at regular intervals. These
loans are amortized, meaning that the borrower pays off the principal along with the
interest over the life of the loan. Each payment includes both interest and a portion of
the principal. Pricing these sorts of loans is straightforward using the present-value
formula: The value of the loan today is the present value of all the payments.
If we assume that the annual interest rate is denoted by iii (measured as a
decimal), and that the loan specifies nnn payments, we can delve into the mechanics
of how the loan is structured and repaid over time. This involves understanding how
the interest rate and the number of payments interact to determine the payment
amounts, the total interest paid over the life of the loan, and the remaining balance at
any point in time.
For instance, consider a loan where the annual interest rate iii is applied to
calculate periodic payments over a specified number of periods nnn. This structure is
typical in various financial products, including mortgages, car loans, and personal
loans. These loans are often repaid in equal periodic installments, which consist of
both interest and principal components.
Understanding these calculations helps borrowers plan their finances by
knowing how much they will need to pay periodically, how much total interest they
will pay over the life of the loan, and how the balance will decrease over time.
Additionally, this knowledge is crucial for comparing different loan offers.
Borrowers can use the calculated periodic payment amounts, total interest, and other
loan terms to evaluate which loan best meets their financial needs and goals. For
instance, a lower interest rate or a longer repayment period might reduce the monthly
payment amount, making the loan more affordable on a month-to-month basis, but
could also result in higher total interest paid over the life of the loan.
Lenders, on the other hand, use these calculations to determine appropriate
interest rates and repayment schedules based on the risk profile of the borrower and
market conditions. By understanding the relationship between the interest rate,
number of payments, and the periodic payment amount, lenders can structure loans
that are both profitable and attractive to borrowers.
In summary, assuming an annual interest rate iii (measured as a decimal) and a
specified number of payments nnn allows for the detailed calculation of periodic
payments and the overall financial impact of a loan. This involves converting the
annual interest rate to a periodic rate, applying amortization formulas to determine
payment amounts, and understanding how these payments break down into interest
and principal components over the life of the loan. Such calculations are fundamental
for both borrowers and lenders in making informed financial decisions.
The right side of this equation has two parts. The first part, in brackets, looks
just like the fixed-payment loan—and it is, with the important exception that it
represents only the interest. The second part, on the far right, looks just like a zero-
coupon bond, and it is. It represents the value of the promise to repay the principal at
maturity. Another type of bond offers only periodic payments. That is, the borrower
pays only interest, never repaying the principal. These loans, called consols or
perpetuities, are like coupon bonds whose payments last forever.
Governments are uniquely positioned in the financial world because they are
the only borrowers that can credibly promise to make payments indefinitely. This
credibility stems from their ability to levy taxes, print money, and their general
perpetual existence, all of which ensure that they can continue to meet their financial
obligations over the long term. This characteristic is why government-issued
perpetual bonds, known as consols, exist. Consols are bonds with no maturity date
that pay a fixed annual interest payment forever. Investors who purchase consols
receive interest payments in perpetuity, making these bonds a unique and interesting
financial instrument.
In contrast, privately issued consols are virtually non-existent. This is
primarily because private entities, such as corporations, do not have the same level of
permanence or financial backing as governments. Corporations can go out of
business, face financial distress, or undergo significant changes that could impair their
ability to make indefinite payments. The lack of perpetual certainty and the risk
associated with corporate financial stability make it impractical for private entities to
issue consols.
However, there have been instances where corporations have issued very long-
term bonds, sometimes referred to as "century bonds," which have maturities of 100
years. These bonds are not perpetual like consols, but they are issued with an
exceptionally long time horizon, reflecting a high level of confidence in the
company's long-term prospects. Century bonds are rare and typically issued by large,
stable corporations with a strong credit rating. They appeal to a specific segment of
investors who are willing to lock in their investment for an extended period in
exchange for the potential stability and interest payments offered by such long-term
securities.
The issuance of century bonds by corporations underscores the challenges and
limitations private entities face when attempting to promise payments over extremely
long periods. Even the most stable and well-established companies must contend with
uncertainties and risks that can affect their ability to honor such long-term
commitments. Economic cycles, market conditions, technological advancements, and
changes in consumer preferences can all impact a corporation's financial health over a
century.
In addition to century bonds, some corporations have also issued other forms
of long-term debt, such as bonds with maturities of 30, 40, or 50 years. These bonds,
while not as extreme as century bonds, still represent a significant commitment and
require careful consideration by both issuers and investors. Issuers must ensure they
have the financial strength and strategic vision to meet their obligations over the long
term, while investors must evaluate the credit risk and potential returns associated
with these long-dated securities.
The difference between government consols and corporate century bonds
highlights the unique financial dynamics of public versus private entities.
Governments, with their sovereign powers and perpetual existence, can credibly
commit to making indefinite payments, making consols a feasible and attractive
financial instrument for certain investors. Corporations, on the other hand, must
navigate a more complex and uncertain landscape, which limits their ability to issue
perpetual bonds and necessitates careful management of long-term debt.
For investors, government consols offer a unique opportunity to secure a
stable, long-term income stream with minimal credit risk, given the backing of a
sovereign entity. These instruments can play a valuable role in a diversified
investment portfolio, particularly for those seeking reliable, fixed-income returns over
an indefinite period. Corporate century bonds, while riskier than government consols,
can also provide attractive returns and serve as a hedge against inflation and interest
rate fluctuations over the long term.
In summary, the ability of governments to credibly promise perpetual
payments underpins the existence of consols, a unique and valuable financial
instrument. In contrast, the inherent uncertainties and risks faced by private entities
make the issuance of consols impractical, leading to the rare issuance of century
bonds by only the most stable and confident corporations. Understanding the
distinctions between these types of long-term debt securities is essential for both
issuers and investors in navigating the complexities of financial markets and making
informed investment decisions.
The British government retired the last of its consols, originally issued in the
18th century, in 2015. The U.S. government sold consols once in 1900. The bonds had
a special provision allowing the Treasury to buy them back starting in 1930. The
Treasury bought back all the consols, so you would not be able to find one today. You
won’t be surprised to learn that the price of a consol is the present value of all the
future interest payments.
The fact that the number of payments is infinite indeed complicates things
when it comes to valuing financial instruments. However, despite this complexity, we
can derive a straightforward formula for the price of a consol, which is a type of bond
that makes a coupon payment every year indefinitely. The value of a consol is
essentially the present value of an infinite series of fixed annual coupon payments.
To understand this, let's start by considering the nature of the payments.
Suppose the consol makes an annual coupon payment CCC. Because these payments
are to be made forever, the consol creates a perpetuity. A perpetuity is a type of
annuity that continues for an infinite number of periods. The present value of a
perpetuity can be calculated if we know the amount of the payment and the discount
rate, which is the interest rate used to discount future payments to the present value.
Let iii represent the annual interest rate (or discount rate) expressed as a
decimal. The key to valuing a consol is recognizing that each coupon payment is a
future cash flow that must be discounted back to its present value. The present value
of the consol is the sum of the present values of all future coupon payments.
This formula shows that the price of a consol is simply the annual coupon
payment divided by the annual interest rate. This relationship highlights an important
feature of consols: their price is inversely related to the interest rate. If the interest rate
increases, the present value of the future coupon payments decreases, leading to a
lower price for the consol. Conversely, if the interest rate decreases, the present value
of the future coupon payments increases, resulting in a higher price for the consol.
The simplicity of this formula belies the complexity behind the concept of
valuing an infinite series of payments. The ability to distill such an infinite series into
a manageable formula is a testament to the power of mathematical finance. This
formula is useful not only for valuing consols but also for understanding the broader
concept of perpetuities in finance.
In the real world, consols are quite rare, and they are mostly issued by
governments. The British government, for instance, issued consols in the past, which
were used to fund various expenditures, including wars and public projects. These
consols paid a fixed interest payment annually and were seen as a stable investment,
particularly during times of economic stability.
For investors, the primary appeal of consols lies in their predictable and
perpetual income stream. Because the payments continue indefinitely, consols can be
a valuable component of a diversified investment portfolio, providing steady income
regardless of market fluctuations. However, the infinite nature of the payments also
means that consols are highly sensitive to changes in interest rates. Investors need to
be aware of this sensitivity when considering consols as part of their investment
strategy.
In summary, while the infinite number of payments associated with consols
adds complexity to their valuation, the derived formula P=CiP = \frac{C}{i}P=iC
provides a straightforward method for determining their price. This formula captures
the essence of consols as perpetuities and underscores the inverse relationship
between their price and the prevailing interest rate. Understanding this relationship is
crucial for both issuers and investors in managing and evaluating long-term financial
commitments.
The price of a consol is determined by a straightforward formula: it equals the
annual coupon payment divided by the interest rate. This relationship allows investors
to easily calculate the value of a consol based on prevailing interest rates. To illustrate
this concept, consider a consol that promises to pay $1 per year forever.
Thus, with an interest rate of 5 percent, a consol that makes an annual coupon
payment of $1 will sell for $20. This calculation highlights the inverse relationship
between interest rates and the price of consols. As the interest rate rises, the price of
the consol falls because the present value of future coupon payments decreases. At an
interest rate of 4 percent, the same consol with an annual coupon payment of $1
would sell for $25. This increase in price reflects the higher present value of the
perpetual $1 payments when discounted at a lower interest rate. The inverse
relationship between interest rates and consol prices becomes evident: as the interest
rate decreases, the price of the consol increases.
At an interest rate of 6 percent, the consol's price drops to approximately
$16.67. This decrease reflects the lower present value of the future coupon payments
when discounted at a higher rate. The same inverse relationship is at play: higher
interest rates lead to lower consol prices.
The relationship between interest rates and consol prices is a fundamental
principle in financial markets. It affects not only consols but also other types of fixed-
income securities. For example, the prices of traditional bonds with fixed coupon
payments and finite maturities also move inversely with interest rates. When interest
rates rise, the present value of future cash flows from these bonds decreases, leading
to lower bond prices. Conversely, when interest rates fall, the present value of future
cash flows increases, resulting in higher bond prices.
Investors need to be acutely aware of this inverse relationship when managing
their investment portfolios. For those holding fixed-income securities, changes in
interest rates can have significant implications for the market value of their
investments. Rising interest rates can lead to capital losses on existing bond holdings,
while falling interest rates can result in capital gains. This sensitivity to interest rate
changes is known as "interest rate risk."
Interest rate risk is particularly relevant for consols because their payments
extend indefinitely. The infinite nature of consols means their prices are especially
sensitive to changes in the discount rate. Even small fluctuations in interest rates can
have a pronounced impact on the price of a consol. Therefore, investors in consols
must carefully monitor interest rate trends and consider the potential effects on their
investments.
Central banks, such as the Federal Reserve in the United States, play a crucial
role in setting interest rates through monetary policy. By adjusting the federal funds
rate, central banks influence short-term interest rates and, indirectly, long-term
interest rates. Investors closely watch central bank actions and statements for clues
about future interest rate movements, as these can significantly affect the prices of
fixed-income securities.
In addition to interest rate risk, investors in consols should also consider
inflation risk. Inflation erodes the purchasing power of fixed payments over time,
reducing the real value of the income received from consols. If inflation rises, the real
yield on consols declines, making them less attractive to investors. This inflation risk
is another factor that can influence consol prices and investor behavior.
In summary, the price of a consol is inversely related to the interest rate. When
the interest rate is 5 percent, a consol with an annual payment of $1 will sell for $20.
If the interest rate drops to 4 percent, the price rises to $25. This inverse relationship
highlights the sensitivity of consol prices to changes in interest rates. Understanding
this dynamic is crucial for investors in fixed-income securities, as it affects the market
value of their investments and their exposure to interest rate and inflation risks. By
monitoring interest rate trends and central bank policies, investors can make informed
decisions and manage their portfolios effectively.
b. Bond Yields
Now that we know how to calculate a bond price given the interest rate, we
need to move in the other direction and calculate the interest rate, or the return to an
investor, implicit in the bond’s price. Doing so means combining information about
the promised payments with the price to obtain what is called the yield—a measure of
the cost of borrowing and the reward for lending. When people talk about bonds they
use the terms yield and interest rate interchangeably, so we will too.
The most useful measure of the return on holding a bond is called the yield to
maturity, or the yield bondholders receive if they hold the bond to its maturity when
the final principal payment is made. Take a $100 face value 5 percent coupon bond
with one year to maturity. At maturity, the owner of this bond receives a coupon
payment of $5 plus a principal payment of $100.
The fact that the return on a bond depends on the price you pay for it really
isn’t that mysterious. If you pay $95 for a $100 face value bond, for example, you will
receive both the interest payments and the increase in value from $95 to $100. This
rise in value, referred to as a capital gain, is part of the return on your investment. So
when the price of the bond is below the face value, the return is above the coupon
rate.
When the price of a bond exceeds its face value, commonly known as trading
at a premium, several financial implications arise for bondholders. Firstly, this
scenario typically results in a situation where the bondholder incurs a capital loss if
they were to sell the bond before maturity, as they would receive less than the price
they paid. This capital loss stems from the fact that the bond's market price is higher
than its face value, which is the amount the issuer promises to pay back at maturity.
Moreover, when a bond trades at a premium, its yield to maturity (YTM) tends
to be lower than its coupon rate. The coupon rate is the fixed annual interest rate paid
by the issuer to the bondholder based on the bond's face value. As the bond price
increases above its face value, the effective yield to maturity decreases because the
bondholder pays more to purchase the bond but still receives the fixed coupon
payments based on the lower face value. This results in a lower overall return on
investment compared to the coupon rate, reflecting the premium paid upfront for the
bond.
Furthermore, the dynamics of bond pricing at a premium are influenced by
market interest rates. If market interest rates decline after the bond is issued, investors
may bid up the price of the bond above its face value to capture the higher fixed
coupon payments relative to current market rates. This scenario highlights how bond
prices and yields interact inversely: as prices rise, yields fall, and vice versa.
In summary, when a bond's price exceeds its face value, bondholders face the
prospect of a capital loss upon sale and a yield to maturity that is lower than the
coupon rate due to the premium paid for the bond. These dynamics underscore the
complex interplay between bond prices, yields, and market interest rates in fixed-
income investing.
Putting all this together, we see the relationship between the current yield and
the coupon rate. Again, it comes from the fact that current yield moves in the opposite
direction from the price: it falls when the bond’s price goes up and rises when the
price goes down. So when the price equals the face value of the bond, theMcurrent
yield and coupon rate are equal. When the price rises above the face value, the current
yield falls below the coupon rate. And when the price falls below the face value, the
current yield rises above the coupon rate.
When a bond trades at a price lower than its face value, typically referred to as
trading at a discount, it introduces several key financial dynamics for bondholders.
Firstly, in such scenarios, bondholders have the potential to realize a capital gain if
they were to sell the bond before maturity. This gain arises because the market price at
which the bond can be sold is higher than the lower face value at which it was
purchased, resulting in a profit for the investor.
Moreover, the relationship between bond prices, coupon rates, and yields
shifts in favor of the investor when bonds are priced at a discount. The current yield,
which is calculated by dividing the annual coupon payment by the current market
price of the bond, increases when the bond is priced below face value. This is because
the coupon payments remain fixed based on the face value, while the investor pays
less to acquire the bond in the market, thereby increasing the yield relative to the
lower purchase price.
Similarly, the yield to maturity (YTM) also rises above the bond's coupon rate
when the bond trades at a discount. YTM reflects the total return an investor can
expect to earn if they hold the bond until maturity, accounting for both the periodic
coupon payments and the capital gain or loss upon maturity. When the bond is priced
below its face value, the YTM increases because the investor's total return is enhanced
by the capital gain realized when the bond matures at its higher face value. This
higher YTM compensates investors for the discount at which they purchased the
bond, effectively offering a higher yield than the fixed coupon rate alone.
Furthermore, market dynamics play a crucial role in the pricing of discounted
bonds. If market interest rates rise after a bond is issued, causing its market price to
fall below face value, investors may find the bond attractive due to its higher current
yield and YTM relative to new issuances offering lower coupon rates. This
phenomenon illustrates how discounted bonds can present opportunities for investors
seeking higher yields in a rising interest rate environment.
In summary, when a bond's price is below its face value, bondholders stand to
potentially realize a capital gain upon sale and benefit from both a higher current yield
and yield to maturity compared to the fixed coupon rate. These dynamics underscore
the attractiveness of discounted bonds in providing enhanced returns for investors
willing to capitalize on market pricing disparities in the fixed-income securities
market.
When discussing bond yields, it's crucial to understand the nuances between
different types of yields: current yield, yield to maturity (YTM), and coupon rate. The
current yield represents the annual income generated by a bond as a percentage of its
current market price. It's a straightforward calculation but doesn't consider capital
gains or losses if the bond is bought at a price different from its face value.
On the other hand, yield to maturity takes into account not only the annual
coupon payments but also any capital gains or losses that occur if the bond is held
until maturity. This yield metric provides a more comprehensive picture of the total
return an investor can expect if the bond is held until maturity. Importantly, when the
bond price is above its face value (trading at a premium), the yield to maturity will be
lower than the current yield because the investor is paying more than the face value,
resulting in reduced capital gains over time.
Conversely, if the bond price is below its face value (trading at a discount), the
yield to maturity will be higher than the current yield because the investor stands to
gain from price appreciation as the bond approaches maturity. Thus, the relationship
between current yield, yield to maturity, and coupon rate varies based on market
conditions and the price of the bond relative to its face value.
Investors often use these metrics to assess the attractiveness of bonds within
their investment portfolios, weighing the trade-offs between current income and
potential capital gains or losses. Understanding these concepts helps investors make
informed decisions based on their risk tolerance, investment objectives, and market
expectations.
We have emphasized that if you buy a bond whose yield to maturity deviates
from the coupon rate, the price will not be the face value. Similarly, the return from
holding a bond need not be the coupon rate. For example, if you pay $95 for a
oneyear 6Mpercent coupon bond, one year later you will get both the $6 coupon
payment and the $5 difference between the purchase price and the $100 face value at
maturity. But this example is really too simple, because it assumes that the investor
holds the bond to maturity. Most holders of long-term bonds plan to sell them well
before they mature.
The concept of holding period return (HPR) versus yield to maturity (YTM)
highlights an important aspect of bond investing: the potential variability in returns
based on changes in bond prices over time. When an investor buys a bond and sells it
before its maturity, the actual return realized, known as the holding period return, can
differ significantly from the yield to maturity initially calculated at the time of
purchase.
Holding period return is calculated as the total return an investor earns from
holding an investment for a specific period, taking into account any capital gains or
losses and the income generated from coupon payments. Unlike yield to maturity,
which assumes the bond is held until maturity and thus reflects the average annualized
return if all coupon payments are reinvested at the YTM rate, HPR is influenced by
changes in bond prices in the secondary market.
The primary reason HPR may diverge from YTM is fluctuations in bond
prices due to changes in interest rates, credit risk perceptions, market demand, and
other economic factors. If interest rates rise after a bond is purchased, its market price
typically decreases, potentially resulting in a capital loss if the bond is sold before
maturity. Conversely, falling interest rates can lead to an increase in bond prices and
capital gains upon sale.
Furthermore, market conditions and investor sentiment can influence the
attractiveness of a bond relative to its YTM. For instance, if a bond's market price
rises significantly due to increased demand or improved credit ratings, selling the
bond before maturity can result in a higher HPR than initially anticipated based on
YTM calculations. Conversely, adverse market conditions or credit downgrades may
lower the bond's market price, reducing the HPR compared to YTM expectations.
Investors often monitor market conditions and adjust their investment
strategies based on expectations of interest rate movements and economic trends to
optimize their bond investment returns. Understanding the relationship between HPR
and YTM helps investors assess the potential risks and rewards of holding bonds until
maturity versus selling them earlier in response to market conditions.
In summary, while yield to maturity provides a standardized measure of
expected return assuming bonds are held until maturity, holding period return reflects
the actual return realized from buying and selling bonds before maturity, accounting
for changes in market prices and reinvestment opportunities. This distinction
underscores the dynamic nature of bond investing and the importance of monitoring
market conditions to maximize investment returns.
Take an example in which you pay $100 for a 10-year, 6Mpercent coupon bond
with a face value of $100. You intend to hold the bond for one year. That is, you are
going to buy a 10-year bond and then a year later, you’ll sell a nine-year bond. What
is your return from holding this bond? If the interest rate doesn’t change (that is, it
stays at 6Mpercent) your return will be $6/$100 = 0.06, or 6 percent. But if the interest
rate changes, calculating your return becomes more complicated. Say that over the
year you hold the bond, the interest rate falls from 6 to 5 percent. That is, the yield to
maturity falls to 5 percent. Using equation (3), we can figure out that you have bought
a 10-year bond for $100 and sold a 9-year bond for $107.11. What is your one-year
holding period return on the initial $100 investment?
When evaluating the holding period return (HPR) of a bond investment, it's
essential to delve into its components, which include both the coupon payments
received and any capital gains or losses realized from changes in the bond's market
price during the holding period.
The first component of HPR is the coupon payment. This fixed periodic
payment represents the interest income paid by the bond issuer to the bondholder
based on the bond's face value and coupon rate. For example, if a bond has a face
value of $1,000 and a coupon rate of 6%, the investor would receive $60 annually
($1,000 * 6%) or $30 semi-annually, depending on the payment frequency. These
coupon payments provide a steady stream of income to investors throughout the
bond's term.
The second component of HPR is the capital gain or loss realized when the
bond is sold before maturity. This gain or loss is determined by the difference between
the price at which the investor purchased the bond initially and the price at which it is
sold in the secondary market. If the bond's market price has increased since purchase,
selling it would result in a capital gain. Conversely, if the market price has decreased,
a capital loss would be incurred upon sale.
For instance, suppose an investor purchases a bond at $950 and sells it later at
$957. This transaction would result in a capital gain of $7 ($957 - $950). When added
to the total coupon payments received during the holding period, the holding period
return would consist of both the $6 coupon payment and the $7 capital gain, totaling
$13.
It's important to note that the actual holding period return can vary depending
on factors such as the timing of coupon payments, changes in interest rates affecting
bond prices, and market liquidity conditions. Investors often compare the realized
HPR with the yield to maturity (YTM) calculated at the time of purchase to assess the
performance of their bond investments relative to initial expectations.
Moreover, understanding the components of HPR allows investors to make
informed decisions regarding bond investments, considering factors such as risk
tolerance, income objectives, and market conditions. By monitoring coupon
payments, capital gains or losses, and overall market trends, investors can
strategically manage their bond portfolios to optimize returns while managing risks
associated with interest rate fluctuations and economic uncertainties.
In summary, the holding period return of a bond investment encompasses both
coupon payments and capital gains or losses realized from changes in market prices
during the holding period. This dual-component approach provides investors with a
comprehensive view of their investment performance and aids in decision-making
regarding bond portfolio management and asset allocation strategies.
c. The Bond Market and the Determination of Interest Rates
Now that we understand the relationship between bond prices and various
measures of interest rates, we need to figure out how bond prices are determined and
why they change. The best way to do that is to look at bond supply, bond demand, and
equilibrium prices in the bond market. Once we understand how the bond market
determines bond prices, we can figure out why the prices change. To keep the analysis
simple, we need to make a few choices about how to proceed. First, we’ll restrict the
discussion to the quantity of bonds outstanding, called the stock of bonds. (We could
look at what causes the changes in the quantity of bonds outstanding— the flow—but
that would complicate matters.) Second, we are going to talk about bond prices rather
than interest rates. Because a bond’s price, together with its various characteristics,
determines its yield, it really doesn’t matter whether we talk about yields (interest
rates) or bond prices. Once we know the price, we know the yield. Finally, we’re
going to consider the market for a one-year zero-coupon bond (one that makes no
coupon payments) with a face value of $100.
If we assume the investor is planning to purchase a one-year bond and hold it
to maturity—the investor has a one-year investment horizon—then the holding period
return equals the bond’s yield to maturity, and both are determined directlyM from the
price. The present-value formula shows that the relationship between the price and the
yield on such a bond is simply P = $100/(1 + i), so i =M[($100 – P)/P]. For example, if
a bond sells for $95, then the yield is i = $5/$95 = 0.0526, or 5.26 percent.
How are bond prices (and bond yields) determined? Not surprisingly, by
supply and demand. Some investors are supplying bonds, while others are demanding
them. The bond supply curve is the relationship between the price and the quantity of
bonds people are willing to sell, all other things being equal. The higher theM price of a
bond, the larger the quantity supplied will be for two reasons. From investors’ point of
view, the higher the price, the more tempting it is to sell a bond they currently hold.
From the point of view of companies seeking finance for new projects, the higher the
price at which they can sell bonds, the better. Taking our example of a $100 one-year
zero-coupon bond, the quantity supplied will be higher at $95 per bond than it will be
at $90 per bond, all other things being equal. This means that the bond supply curve
slopes upward.
The bond demand curve is the relationship between the price and quantity of
bonds that investors demand, all other things being equal. As the price falls, the
reward for holding the bond rises, so the demand goes up. That is, the lower the price
potential bondholders must pay for a fixed-dollar payment on a future date, the more
likely they are to buy a bond. Again, think of the zero-coupon bond promising to pay
$100 in one year. That bond will attract more demand at $90 than it will at $95 per
bond, all other things being equal. Thus, the bond demand curve slopes downward.
Because the price of bonds is inversely related to the yield, the demand curve implies
that the higher the demand for bonds, the higher the yield.
As is always the case with supply and demand analysis, we need to explain
how the market adjusts when the price deviates from the price that equates supply and
demand—point P0. Let’s look briefly at the two possibilities: Either the price is too
high or the price is too low. If bond prices start out above the equilibrium point,
somewhere greater than P0, quantity supplied will exceed quantity demanded. That is,
excess supply means that suppliers cannot sell the bonds they want to at the current
price. To make the sale, they will start cutting the price. The excess supply will put
downward pressure on the price until supply equals demand.
When the price is below the equilibrium point, quantity demanded will
exceedM quantity supplied. Those people who wish to buy bonds cannot get all
theyMwant at the prevailing price. Their reaction is to start bidding up the price. Excess
demand continues to put upward pressure on the price until the market reaches
equilibrium. So far, so good. But to really understand how bond prices (and bond
yields) change over time, we need to learn what determines the location of the supply
and demand curves. Over time they shift around, leading to changes in the
equilibrium prices. As we discuss the causes of such shifts in the following section,
make sure you remember the distinction between moving along a supply or demand
curve and shifting a curve. When the quantity demanded or quantity supplied changes
because of a change in the price, it produces a movement along the curve. But when
the quantity demanded or supplied at a given price changes, it shifts the entire curve.
More important, in the bond market, a shift in either the supply or the demand curve
changes the price of bonds, so it changes the yield as well.
The government’s need to issue bonds affects the supply of bonds out there.
Both changes in tax policy and adjustments in spending can affect a government’s
need to borrow. Regardless of the reason, any increase in the government’s borrowing
needs increases the quantity of bonds outstanding, shifting the bond supply curve to
the right. The result is an increase in quantity of the bonds supplied at every price.
Because the demand curve stays where it is (remember, we’re holding everything else
constant), the increase in supply drives the price down. The added supply of U.S.
government bonds has reduced prices, raising interest rates.
During business cycle expansions, when general business conditions are good,
investment opportunities abound, prompting firms to increase their borrowing. As the
amount of debt in the economy rises, the quantity of bonds outstanding with a given
risk goes up. So as business conditions improve, the bond supply curve shifts to the
right, forcing bond prices down and interest rates up. This connection between general
business conditions and the supply of bonds also helps explain how weak economic
growth can lead to rising bond prices and lower interest rates for bonds with
unchanged risk.
Bond issuers care about the real cost of borrowing—the cost of the loan taking
inflation into account. At a given nominal interest rate, higher expected inflation
means a lower real interest rate. And at a lower real interest rate, fewer real resources
are required to make the payments promised by a bond. So when expected inflation
rises, the cost of borrowing falls and the desire to borrow at every nominal interest
rate rises. Higher expected inflation increases the bond supply, reducing bond prices
and raising the nominal interest rate.
. Before moving on to shifts in the demand for bonds, we should mention that
there is one other factor that shifts the bond supply: changes in corporate taxation.
Because such changes in the tax code require government legislation, they don’t occur
very often. But when they do, they can affect the economywide supply of bonds.
Corporations pay taxes on their profits, just as individuals pay taxes on their income,
so they are concerned with after-tax profits. Governments often create special tax
subsidies that make corporate investments less costly. These tax incentives increase
the supply of bonds because they raise the after-tax profitability of investing in new
equipment purchased with funds raised from selling bonds. Like the other three
factors we have considered, government tax incentives increase bond supply, shift the
supply curve to the right, and lower the price of bonds.
The more rapidly the economy grows, the wealthier individuals become. As
their wealth increases, they increase their investment in stocks, bonds, real estate, and
art. Thus, increases in wealth shift the demand for bonds to the right, raising bond
prices and lowering yields. This is what happens in a business cycle expansion. In a
recession, as wealth falls, the demand for bonds falls with it, lowering bond prices and
raising interest rates.
Changes in expected inflation alter investors’ willingness to purchase bonds
promising fixed-dollar payments. A decline in expected inflation means that the
payments promised by the bond’s issuer have a higher value than borrowers originally
thought, so the bond will become more attractive. This fall in expected inflation shifts
the bond demand curve to the right, increasing demand at each price and lowering the
yield. In short, the higher real return on the bond increases the willingness of would-
be lenders to buy it at any given price. Note that the decline in expected inflation has
reduced the nominal interest rate that investors require in order to make a loan.
An investor’s desire to hold any particular financial instrument depends on
how its return compares to those of alternative instruments. Bonds are no different. If
the expected return on bonds rises relative to the return on alternative investments, the
quantity of bonds demanded at every price will rise, shifting the bond demand curve
to the right. This leads us to conclude that bond prices are connected to the stock
market. Investors see bonds as an alternative to stocks, so when the stock market
outlook worsens, they shift their portfolios into bonds, increasing demand, driving
bond prices up and interest rates down. Similarly, when interest rates are expected to
change in the future, bond prices adjust immediately. Recall that the holding period
return on a bond depends on the coupon payment plus the capital gain or loss. When
interest rates fall, bond prices rise, creating a capital gain. Whenever interest rates are
expected to fall, then bond prices are expected to rise, creating an expectation of a
capital gain. This makes bonds more attractive. Knowing that bonds are a good
investment, investors increase their demand immediately, driving bond prices up. So
an increase in the expected return on a bond, relative to the return on alternatives,
shifts bond demand to the right.
On May 13, 2002, a headline in The Wall Street Journal read “Japan Gets Irate
at Having Its Risk Compared to Botswana.” What’s going on here? Japan is the third-
largest economy in the world, with a population of more than 125 million and a GDP
approaching $5 trillion. Botswana is a landlocked country in southern Africa with a
population of 2 million people and a GDP of about $17 billion. The problem was that
investors had two reasons to question Japan’s budget outlook. First, the fiscal deficit
was a very high 7Mpercent of GDP. Second, over the next few decades, the Japanese
government would have to find a way to meet its promises to make pension payments
to the growing number of retirees. Together these created the perception that Japan’s
bonds were risky, which meant that investors would be less interested in holding
them. Remember that investors require compensation for risk, which means that when
a bond becomes more or less risky, the demand for the bond changes. The less risky
the bond, the higher the price investors are willing to pay for it, all other things being
equal. From this we can conclude that if a bond becomes less risky relative to
alternative investments, the demand for the bond shifts to the right. The reason Japan
was irate was because the price of its bonds would be lower, so its borrowing costs
would be higher.
During the financial crisis in the fall of 1998, for example, the bonds issued by
emerging market governments in Latin America and eastern Europe became virtually
impossible to sell. The same thing happened to some U.S. mortgage-related bonds
during the crisis of 2007–2009 and to the bonds of some euro-area countries
beginning in 2010. For all practical purposes, the market for them disappeared. When
a buyer could be found, prices were severely depressed. Who wants to buy a bond that
is difficult to sell? Liquidity matters. The less liquid a bond is, the lower the demand
for it, and the lower the price. So when a bond becomes more liquid relative to
alternatives, the demand curve shifts to the right.
Before we continue, let’s look again at how bond prices and interest rates
move in response to changes in expected inflation and a change in general business
conditions. Recall that expected inflation affects both bond supply and bond demand.
An increase in expected inflation reduces the real cost of borrowing, shifting bond
supply to the right. But at the same time, this increase in expected inflation lowers the
real return on lending, shifting bond demand to the left. These two effects reinforce
each other, lowering the price of the bond and raising the interest rate.
d. Why Bonds Are Risky
How can bonds be risky? They are promises to make fixed payments on future
dates. Where is the risk in that? The fact is that the return an investor receives for
holding a bond is far from riskless. Bondholders face three major risks. Default risk is
the chance that the bond’s issuer may fail to make the promised payment. Inflation
risk means an investor can’t be sure of what the real value of the payments will be,
even if they are made. And interest rate risk arises from a bondholder’s investment
horizon, which may be shorter than the maturity of the bond. If, for example, the
interest rate were to rise between the time the bond is purchased and the time it is
sold, the investor would suffer a capital loss.
There is no guarantee that a bond issuer will make the promised payments.
While we usually ignore default risk in thinking about U.S. Treasury bonds, we
cannot do so when discussing bonds issued by many other governments or by private
corporations. When corporations or governments fail to meet their payments, what
happens to the price of their bonds? To figure out the answer, let’s list all the
possibilities and payoffs that might occur, along with their probabilities. We can then
calculate the expected value of the promised payments, from which we can compute
the bond’s price and yield. Suppose, for example, that the one-year risk-free interest
rate is 5 percent.
Flim.com, an innovative Internet firm that aims to launch its proprietary e-
cash brand known as "Flam," has ventured into the bond market by issuing one-year
bonds. These bonds come with a 5 percent coupon rate and a face value of $100. This
financial instrument represents a promise by Flim.com to pay bondholders $105 at the
end of one year, which includes the $100 face value plus the $5 coupon payment.
To understand the price of this bond, we need to consider several factors,
including the bond's yield to maturity, the prevailing market interest rates, and the
credit risk associated with Flim.com. The bond price calculation essentially discounts
the future cash flows—the face value and the coupon payment—back to their present
value.
Assuming the market interest rate or yield for similar one-year bonds is
denoted as rrr, we can calculate the present value of the bond’s cash flows. If the
market interest rate matches the bond's coupon rate (5%), the bond would typically be
priced at its face value because the return from the coupon payments aligns with the
return demanded by the market.
These examples illustrate how the bond’s price varies inversely with changes
in market interest rates. A key consideration for potential investors in Flim.com’s
bonds is the perceived credit risk associated with the company. As Flim.com is an
internet firm launching a new e-cash product, the perceived risk might be higher due
to the speculative nature of their business venture. Higher perceived risk could lead
investors to demand a higher yield, which would further lower the bond's price.
Investors would also evaluate Flim.com’s financial health, market position,
and potential for success with its e-cash brand, Flam. If investors believe in the
company’s growth prospects and financial stability, they might be more willing to
accept a lower yield, thereby supporting a higher bond price. Conversely, if investors
are skeptical about the company's future or concerned about its ability to meet its
financial obligations, they would demand a higher yield, which would decrease the
bond's price.
In summary, while the initial pricing of Flim.com’s one-year, 5 percent coupon
bond with a $100 face value would theoretically be at par value ($100) if market
interest rates align with the coupon rate, the actual price in the market could vary
based on prevailing interest rates and investor perceptions of risk. By analyzing these
factors, investors can make informed decisions about the attractiveness and value of
Flim.com’s bond offering.
Converting the decimal to a percentage, we get an interest rate of 16.67
percent. 8 Because the default-risk premium is the promised yield to maturity minus
the risk-free rate, it is 16.67 percent − 5 percent = 11.67 percent. In calculating the
default-risk premium on Flim.com’s bond, we computed the expected value of
holding the bond—the yield at which the bond is a fair bet. But we know that risk-
averse investors require some compensation for bearing risk. The more risk, the
greater the compensation they demand. Only a risk-neutral investor would be willing
to pay $90 for this bond. Any risk premium will drive the price down below $90 and
push the yield to maturity above 16.67 percent. This example shows that the higher
the default risk, the higher the probability that the bondholders will not receive the
promised payments.
Risk is a fundamental factor in bond investing that significantly influences an
investor's decision-making process. It impacts both the expected value of the bond
and the price an investor is willing to pay for it. One of the primary types of risk that
investors consider is default risk, which is the possibility that the bond issuer will fail
to make the promised interest payments or repay the principal amount at maturity.
When a bond has a higher default risk, the expected value of the promised
payments decreases. This is because there is an increased likelihood that the issuer
will not be able to meet its financial obligations. As a result, investors perceive such
bonds as less attractive compared to those with lower default risks. To compensate for
this higher risk, investors demand a higher yield on these bonds. The yield, in this
context, is the effective rate of return an investor expects to earn from the bond, taking
into account both the coupon payments and any potential capital gains or losses.
The relationship between risk and yield is direct: the higher the default risk,
the higher the yield investors will require. This higher yield serves as a premium for
taking on additional risk. For instance, if two bonds have similar characteristics, such
as maturity and coupon rate, but one bond is issued by a company with a lower credit
rating (indicating higher default risk), investors will demand a higher yield on the
riskier bond. This demand for a higher yield results in a lower price for the bond
because yield and price are inversely related in bond markets.
Moreover, investors use various tools and metrics to assess the default risk of
bonds. Credit rating agencies such as Moody's, Standard & Poor's, and Fitch provide
credit ratings that evaluate the creditworthiness of bond issuers. Bonds rated lower by
these agencies are considered to have higher default risk and, consequently, trade at
higher yields. Investors rely on these ratings to make informed decisions about the
risk-return trade-off of their bond investments.
In addition to default risk, other types of risks can also affect the expected
value and yield of a bond. Interest rate risk, for example, arises from fluctuations in
market interest rates that can lead to changes in bond prices. When interest rates rise,
bond prices typically fall, and vice versa. Investors must consider these risks when
evaluating the overall risk profile of a bond investment.
Furthermore, market risk, liquidity risk, and inflation risk also play a role in
determining the price and yield of bonds. Market risk refers to the potential for bond
prices to be affected by broader economic and financial market conditions. Liquidity
risk pertains to the ease with which a bond can be bought or sold without significantly
affecting its price. Inflation risk involves the potential erosion of purchasing power
due to rising inflation rates, which can reduce the real return on bond investments.
In summary, risk plays a crucial role in bond investing by affecting the
expected value of promised payments, bond prices, and yields. Higher default risk
leads to lower bond prices and higher yields as investors seek compensation for taking
on additional risk. Understanding the interplay between various types of risks and
their impact on bond investments is essential for making informed investment
decisions and optimizing the risk-return profile of a bond portfolio.
With few exceptions, bonds promise to make fixed-dollar payments. That is, a
$100 face value, one-year bond at 5 percent is a promise to make a $105 payment in
one year. If this promise is free of default risk, the bondholder can be sure of receiving
the $105 payment. Still, there is a risk of inflation. Remember that what you care
about is the purchasing power of the money, not the number of dollars. In other
words, bondholders are interested in the real interest rate, not just the nominal interest
rate. And they don’t know what the inflation rate will be. Let’s look at an example that
shows how inflation risk affects the interest rate. To begin with, think about the
interest rate as having three components: the real interest rate, expected inflation, and
a compensation for inflation risk. Suppose the real interest rate is 3 percent but we are
unsure what the inflation rate will be. It could be either 1Mpercent or 3 percent with
equal probability. Expected inflation is 2 percent, with a standard deviation of 1.0
percent. This means the nominal interest rate should equal the 3 percent real interest
rate plus the 2 percent expected inflation plus the compensation for inflation risk. The
greater the inflation risk, the larger the compensation for it will be.
In Cases II and III, expected inflation is the same (2 percent) but the standard
deviation is lower because we are more certain that the inflation rate will be close to
its expected value. That is, Case III is less risky than Case II, which is less risky than
CaseMI. Because risk requires compensation, we would expect the interest rate to be
highest in Case I and lowest in Case III. While we may not see this distinction much
in the United States or Europe, where inflation is stable, emerging market countries
can go through periods when increases in inflation risk substantially drive up nominal
interest rates.
To explain interest rate risk, we’ll focus on a U.S. Treasury bond and assume
that it is free of default risk and that we know how much inflation there will be, so
there also is no inflation risk. Interest rate risk arises from the fact that investors don’t
know the holding period return of a long-term bond. Remember that when interest
rates change, bond prices move; the longer the term of the bond, the larger the price
change for a given change in the interest rate. Now think about what happens if you
have a short investment horizon. If you buy a long-term bond, you will need to sell
the bondMbefore it matures, so you have to worry about what will happen if the interest
rate changes.
Whenever there is a mismatch between your investment horizon and a bond’s
maturity, there is interest rate risk. Because the prices of long-term bonds can change
dramatically, this can be an important source of risk. For example, on March 18,
2019, the 8.75 percent coupon Treasury bond that matures on August 15, 2020, traded
at a price of $108.58 (per $100 face value). When it was originally issued as a 30-year
bond in August 1990, its price was $98.747.9 An investor who bought the bond when
it was originally issued and sold it nearly 29 years later on March 18, 2019, earned a
capital gain of nearly 10Mpercent. By comparison, an investor who purchased the 2.75
percent coupon 30-year Treasury bond when it was issued on November 15, 2012, at
$100.69 and sold it less than 7 years later at its market price of $95.91 on March 18,
2019, suffered a capital loss of nearly 5 percent. The lesson is that any move in
interest rates changes the price of a bond. For investors with holding periods shorter
than the maturity of the bond, the potential for a change in interest rates creates risk.
The more likely interest rates are to change during the bondholder’s investment
horizon, the larger the risk of holding a bond.
