OBJECTIVES


To show how to value contracts and securities that promise a stream of cash flows that is known with certainty.
To understand how bond prices and yields change over time.
CONTENIS
8.1
Using present value Formulas to Value Known Cash Flows
8.2
The Basic Building Blocks: Pure Discount Bonds
8.3
Coupon Bonds, Current Yield, and Yield to Maturity
8.4
Reading Bond Listings
8.5
Why Yields for the Same Maturity May Differ
8.5
The Behavior of Bond Prices over Time
Chapter 7 shows that the essence of the valuation process is to estimate an asset’s market value using
information about the prices of comparable assets, making adjustments for differences. A valuation model is a
quantitative method used to infer an asset's value (the output of the model) from market information about the prices of
other assets and market interest rates (the inputs to the model).
In this chapter, we examine the valuation of fixed-income securities and other contracts promising a stream of
known future cash payments. Examples are fixed- income securities like bonds and contracts such as mortgages and
pension annuities. These securities and contracts are important to households because they represent major sources of
income and sources of financing for housing and other consumer durables. They are also important to firms and
governments, primarily as sources of financing.
Having a method to value such contracts is important for at least two reasons. First, the parties to the contracts
need to have an agreed-upon valuation procedure in setting the terms of the contracts at the outset. Second,
fixed-income securities are often sold before they mature. Because the market factors determining their value-namely,
interest rates-change over time, both buyers and sellers have to reevaluate them each time they are traded.
Section 8.1 presents a basic valuation model that uses a discounted cash flow formula with a single discount rate
to estimate the value of a stream of promised future cash flows. Section 8.2 shows how to modify such a model to take
account of the fact that generally the yield curve is not flat (i.e., that interest rates vary with maturity).Sections 8.3-85
explain the main features of bonds in the real world and discuss how these features affect the prices and yields of bonds.
Section 8.6 explores how changes in interest rates over time affect the market prices of bonds.
8.1 USING PRESENT VALUE FORMULAS TO VALUE KNOWN CASH FLOWS
Chapter 4 shows that in a world with a single risk--free interest rate, computing the present value of any stream
of future cash flows is relatively uncomplicated. It involves applying a discounted cash now formula using the
risk-free interest rate as the discount rate.
For example, suppose you buy a fixed-income security that promises to pay $100 each year for the next three
years. How much is this three-year annuity worth if you know that the appropriate discount rate is 6% per year? As
shown in chapter, the answer-$267.30-can be found easily using a financial calculator, a table of present value factors,
or by applying the algebraic formula for the present valr4f an annuity.
Recall that the formula for the present value of an ordinary annuity of an ordinary period for n periods at an
interest rate of i is:
pv 
1  (1  i)  n
i
On a financial calculator, we would enter the values for n, i, and PMT, and compute the PV:
n
3
i
6
PV
?
FV
0
PMT
100
Result
PV=267.30
Now suppose that an hour after you buy the security, the risk-free interest rises from 6% to 7% per year, and you
want to sell. How much can you get for it?
The level of market interest rates has changed, but the promised future cash flows from your security have not.
In order for an investor to earn 7% per year on your security, its price has to drop. How much? The answer is that it
must fall to the point at which its price equals the present va111e of the promised cash flows discounted at7% per
year:
n
3
i
7
PV
?
FV
0
PMT
100
Result
PV=262.43
At a price of $262.43, a fixed-income security that promises to pay $100 each year for the next three years offers
its purchaser a rate of return of 7% per year. Thus, the price of any existing fixed-income security falls when market
interest rates rise because investors will only be willing to buy them if they offer a competitive yield.
Thus, a rise of 1% in the interest rate causes a drop of $4.87in the market value of your security. Similarly, a fall
in interest rates causes a rise in its market value.
This illustrates a basic principle in valuing known cash flows:
A change in market interest rates causes a change in the opposite direction in the market values of all existing contracts
promising fixed payments in the future.
Because interest rate changes arc not predictable, it follows that the prices of fixed- income securities arc
uncertain up to the time they mature.
In practice, valuation of known cash flows is not as simple as we just described because in practice you do not
usually know which discount rate to use in the present value formula. As shown in chapter 2, market interest rates arc
not the same for all maturities. We reproduce as Figure 8.1 the graph showing the yield curve for U.S. Treasury bonds.
It is tempting to think that the interest rate corresponding to a three-year maturity can be applied as the correct
discount rate to use in valuing the three-year annuity in our example. But that would not be correct. The correct
procedure for using the information contained in the yield curve to value other streams of known cash payments is
more complicated; that is the subject of the next few sections.
8.2 THE BASIC BUILDING BLOCKS: PURE DISCOUNT BONDS
In valuing contracts promising a stream of known cash flows, the place to start is a listing of the market prices of
pure discount bonds (also called zero-coupon bonds). These are bonds that promise a single payment of cash at some
date in the future called the maturity date.
Pure discount bonds are the basic building blocks for valuing al1 contracts promising streams of known cash
flows. This is because we can always decompose any contract-no matter how complicated its pattern of certain future
cash flows into its component cash flows, value each one separately, and then add them up.
The promised cash payment on a pure discount bond is called its face value or par value .The interest earned by
investors on pure discount bonds is the differ between the price paid for the bond and the face value received at the
maturity date. Thus, for a pure discount bond with a face value of $1,000maturing in one year and a purchase price of
$950, the interest earned is the $50 difference between the $1,OOO face value and the $950 purchase price.
The yield (interest rate) on a pure discount bond is the annualized rate of return to investors who buy it and hold
it until it matures. For a pure discount bond with a one-year maturity such as the one in our example, we get
Yield on 1-Year Pure Discount Bond = (Face Value – Price) / Price
= ($1000-$950) / $950
=0.0526 or 5.26%
If, however, the bond has a maturity different from one year, we would use the present value formula to find its
annualized yield. Thus, suppose that we observe two-year pure discount bond with a face value of $1,OOO and a price
of $880. We would compute the annualized yield on this bond as the discount rate that makes its face value equal to its
price-On a financial calculator, we would enter the values for n, PV,FV, and compute i.
n
2
i
?
PV
-880
FV
1,000
PMT
0
Result
i=6.60%
Return to the valuation of the security of section 8.1 that promises to pay $100 each year for the next three years.
Suppose that we observe the set of pure discount bond prices in Table8.1. Following standard practice, the bond prices
are quoted as a fraction of their face value.
There are two alternative procedures that we can use to arrive at a correct value for the security. The first
procedure uses the prices in the second column of Table 8.1, and the second procedure uses the yields in the last
column. Procedure 1 multiplies each of the three promised cash payments by its corresponding per-dollar price and
then adds them up:
Present Value of First Year's Cash Flow =$100 × 0.95=$95.00
Present Value of Second Year's Cash Flow =$100 × 0.88=$88.OO
Present Value of Third Year's Cash Flow =$100 × 0.80=$80.00
Total Present Value =$263
The resulting estimate of the security's value is $263.
Procedure 2 gets the same result by discounting each Year's promised cash payment at the yield corresponding to
that maturity:
Present Value of First Year's Cash Flow =$100/1.0526=$9500
Present Value of Second Year's Cash Flow=$100/1.06602=$88.00
Present Value of Third Year's Cash Flow=$100/107723=$80.00
Total Present Value =$263
Note, however, that it would be a mistake to discount all three cash flows using the same three-year yield of
7.72% per year listed in the last row of Table 8.1. If we do so, we get a value of $259, which is $4 too low:
n
3
i
7.72%
PV
?
FV
0
PMT
100
Result
PV=$259
Is there a single discount rate that we can use to discount all three of the payments the way we did in section 8.1
to get a value of $263 for the security? The answer is yes: That single discount rate is 6.88% per year. To verify this,
substitute 6.88% for i in the formula for the present value of an annuity or in the calculator:
n
3
i
6.88%
PV
?
FV
0
PMT
100
Result
PV=$263
The problem is that the 6.88% per-year discount rate appropriate for valuing the three-year annuity is not one of
the rates listed anywhere in Table 8.1. We derived it from our knowledge that the value of the security has to be $263.
In other words, we solved the present value equation to find i:
n
3
i
?
PV
-263
FV
0
PMT
100
Result
i=6.88%
But it was that value (i.e., $263) that we were trying to estimate in the first place. Therefore, we have no direct
way to find the value of the three-year annuity using a single discount rate with the bond price information available to
us in Table 8.1.
We can summarize the main conclusion from this section as follows: When the yield curve is not flat (i.e., when
observed yields are not the same for all maturities), the correct procedure for valuing a contract or a security promising
a stream of known cash payments is to discount each of the payments at the rate corresponding to a pure discount bond
of its maturity and then add the resulting individual payment values.
8.3 COUPON BONDS, CURRENT YEEID, AND YIELD TO MATURITY
A coupon bond obligates the issuer to make periodic payments of interest--called coupon payments-to the
bondholder for the life of the bond, and then to pay the face value of the bond when the bond matures (i.e., when the
last payment comes due). The periodic payments of interest are called coupons because at one time most bonds had
coupons attached to them that investors would tear off and present to the bond issuer for payment.
The coupon rate of the bond is the interest rate applied to the face value to compute the coupon payment. Thus,
a bond with a face value of $1,000 that makes, annual coupon payments at a coupon rate of 10% obligates the issuer to
pay 0.lOX$1,000=$100 every year. If the bond's maturity is six years, then at the end of six years, the issuer pays the
last coupon of $100and the face value of $1,O00.
The cash flows from this coupon bond arc displayed in Figure 8.2. We see that the stream of promised cash
flows has an annuity component (a fixed per pt amount) of $100 per year and a “balloon” or “bullet” payment of
$1,OOO at maturity.
The $100 annual coupon payment is fixed at the time the bond is issued and remains constant until the bond's
maturity date. On the date the bond is issued, it usually has a price (equal to its face value) of $1,000.
The relation between prices and yie1ds on coupon bonds is more complicated than for pure discount bonds. As
we will see, when the prices of coupon bonds are different from their face value, the meaning of the term yield is itself
ambiguous.
Coupon bonds with a market price equal to their face value are called par bonds. When a coupon bond's market
price equals its face value, its yield is the same as its coupon rate. For example, consider a bond maturing in one year
that pays an annual coupon at a rate of 10% of its $1,OOO face value. This bond will pay its holder $1100a year from
now--a coupon payment of $100and the face value of $1,000. Thus, if the current price of our 10% coupon bond is
$1,000, its yield is 10%.
Bond Pricing Principle 1: Par Bonds
If a bond's price equals its face value, then its yield equals its coupon rate
Often the price of a coupon bond and its face value arc not the same. This situation would occur, for instance, if
the level of interest rates in the economy falls after the bond is issued. So, for example, suppose that our one--year 10%
coupon bond was originally issued as a 20--year--maturity bond 19 years ago. At that time, the yield curve was flat
at10% per year. Now the bond has one year remaining before it matures, and the interest rate on one-year bonds is5%
per year.
Although the 10%coupon bond was issued at par ($1,000), its market price will now be $1,047.62. Because the
bond's price is now higher than its face value, it is called a premium bond.
What is its yield?
There are two different yields that we can compute. The first is called the current yield, the annual coupon
divided by the bond's price:
Current Yield = Coupon / Price
= $100 / $1,047.62
=9.55%
The current yield overstates the true yield on the premium bond because it ignores the fact that at maturity you
will receive only $1,000--$47.62 less than you paid for the bond.
To take account of the fact that a bond's face value and its price may differ, we compute a different yield called
the yield to maturity. The yield to maturity is defined as the discount rate that makes the present value of the bond's
stream of promised cash payments equal to its price.
The yield to maturity takes account of all of the cash payments you will receive from purchasing the bond,
including the face value of $1,OOO at maturity. In our example, because the bond is maturing in one year, it is easy to
compute the yield to maturity.
Yield to Maturity = (Coupon + Face value – Price) / Price
Yield to Maturity = ($100 + $1,000-$1,047.62) / $1,047.62
= 5%
Thus, we see that if you used the current yield of 9.55% as a guide to what you would be earning if you bought
the bond, you would be seriously misled.
When the maturity of a coupon bond is greater than a year, the calculation of its yield to maturity is more
complicated than just shown. For example, suppose that you are considering buying a two-year 10% coupon bond with
a face va1ue of $1,000 and a current price of $1,100. What is its yield?
Its current yield is 9.09%:
Current Yield = Coupon / Price
= $100 / $1,100
= 9.09%
But as in the case of the one-year premium bond, the current yield ignores the fact that at maturity, you will
receive less than the $1,100that you paid-me yield to maturity when bond maturity is greater than one year is the
discount rate the makes the present value of the stream of cash payments equal to the bond's price:
n
PV  
i 1
PMT
FV

t
(1  i ) (1  i ) n
Where n is the number of annual payment periods until the bond's maturity, i is the annual yield to maturity, PMT is the
coupon payment, and FV is the face value of the bond received at maturity.
The yield to maturity on a multiperiod coupon bond can be computed eau on most financial calculators by
entering the bond's maturity as n, its price as PV (with a negative sign), its face value as FV, its coupon as PMT, and
computing i.
n
2
i
?
PV
-1,100
FV
1,000
PMT
100
Result
i=4.65%
Thus, the yield to maturity on this two-year premium bond is considerably less than its current yield.
These examples illustrate a general principle about the relation between bond prices and yields:
Bond Pricing Principle 2: Premium Bonds
If a coupon bond has a price higher than its face value, its yield to maturity is less than its current yield, which is in turn less than
its coupon rate.
For a premium bond:
Yield to Maturity <Current Yield <Coupon Rate
Now let us consider a bond with a 4% coupon rate maturing two years. Suppose that its price is $950. Because
the price is below the face value of the bond, we call it a discount bond. (Note it is not a pure discount bond because it
does pr coupon.)
What is its yield? As in the previous case of a premium bond, we can compute two different yield--the current
yield and the yield to maturity.
Current Yield = Coupon / Price
= $40 / $950
= 4.21%
The current yield understates the true yield in the case of the discount bond because it ignores the fact that at
maturity you will receive more than you paid for the bond. When the discount bond matures, you receive the $1,OOO
face value, not the $950 price that you paid for it.
The yield to maturity takes account of a1l of the cash payments you will receive from purchasing the bond,
including the face value of $1,OOO at maturity. Using the financial calculator to compute the bond's yield to maturity,
we find:
n
2
i
?
PV
-950
FV
1,000
PMT
40
Result
i=6.76%
Thus, the yield to maturity on this discount bond is greater than its current yield.
Bond Pricing Principle 3: Discount Bonds
If a coupon bond has a price lower than its face value, its yield to maturity is greater than its current yield, which is in turn greater
than its coupon rate
For discount bonds:
Yield to Maturity >Current Yield >Coupon Rate
8.3.1 Beware of “High-Yield” U.S. Treasury Bond Funds
In the past, some investment companies that invest exclusively in U.S. Treasury bonds have advertised yields that
appear much higher than the interest rates on other known investments of the same maturity. The yields that they are
advertising are current yields, and the bonds that they are investing in are premium bonds that have relatively high
coupon rates. Thus, according to Bond Pricing Principle 2, the actual return you will earn is expected to be
considerably less than the advertised current yield.
Suppose that you have $10,OOO to invest for one year-You are deciding between putting your money m a
one-year, government-insured, bank CD offering an interest rate of 5% and investing in the shares of a U.S. Treasury
bond fund that holds one-year bonds with a coupon rate of 8%.Ihe bonds held by the fund are selling at a premium over
their face value: For every$10,OOO of face value that you will receive at maturity a year from now, you must pay
$10285.71 now. The current yield on the fund is $800/$10,285.71 or 7.78%,and this is the yield that the fund is
advertising. If the fund charges a 1% annual fee for its services, what rate of return will you actually earn?
If there were no fees at all for investing in the fund, your rate of return for the year would be 5%, precisely the same
rate of return as on the bank CD. This is because investing your $109000 in the fund will achieve the same return as
buying an 8% coupon bond with a face value of $10,000 for a price of $10285.71:
Rate of Return = (Coupon + Face – Price) / Price
= ($800 + $10,000 - $10,285.71) / $10,286
= 5%
Because you have to pay the fund a fee equal to 1%of your $10,000, your rate of return will be only 4% rather
than the 5%you can earn on the bank CD.
8.6 READING BOND LISTINGS
8.7
The prices of bonds are published in a variety of places. For investors and analysts who need the most
up-to-the-minute price data, the best sources are on-line information services that feed the information electronically
to computer terminals. For those who do not need data that are quite SO UP-to-date, the daily financial press provides
bond listings.
Table 8.2 shows that the asked price for a Treasury strip maturing in May 2000 was 89 and 19/32(8959375)
cents per dollar of face va111e and for one maturing in May 2027,17 and 31/32 (or 17.96875 cents per dollar of face
value).
To interpret the prices, we must understand several conventions:
1. Type in the second column tells the original source of the strip: ci is coupon interest, bp is principal from a
Treasury bond, and np is principal from a Treasury note. Bonds have original maturities of more than 10 years;
notes have original 2 maturities of 10 years or less.
2. The ask price is the price at which dealers in Treasury bonds are willing to sell and the bid price is the price at
which they are willing to buy. Therefore, asked price always exceeds the bid price. The difference is, in effect, the
dealer’s commission. Ask Yld, in the last column is the yield to maturity computed using the asked price. It
assumes semiannual compounding.
3. The price quotations are cents per $1 of face value.
4. The numbers after the colon mean 32nds and not moths of a cent. Thus,97:11 means 97and (or $0.9734375),not
$0.9711.
Table 8.2shows that the ask price for a Treasury strip maturing in February 1996 was 97and 11/32 (97.34375)
cents per dollar of face value and for one maturing in February 2004, 57and 5/32 (or57.15625cents per dollar of face
value).
Table 8.3 is a partial listing of the prices of U.S. Treasury Bonds taken from The Wall Street Journal. It differs
from the previous listing in that it displays each bond's coupon rate in the first column. The letter n that appears after
the maturity date indicates that the bond is a U.S. Treasury note, meaning that it had an original maturity of less than
10years.
8.5 WHY YIELDS FOR THE SAME MATURITY MAY DIFFER
Often we observe that two U.S. Treasury bonds with the same maturity have different yields to maturity. Is this a
violation of the Law of One Price? The answer is no. In fact, for bonds with different coupon rates, the Law of One
Price implies that, unless the yield curve is flat, bonds of the same maturity will have different yields to maturity.
8.5.1 The Effect of the Coupon Rate
For example, consider two different two-year coupon bonds-one with a coupon rate of 5% and the other with a
coupon rate of 10%.Suppose the current market prices and yields of one-and two-year pure discount bonds are as
follows:
Maturity
1 year
2 year
Price per $1 of Face Value
$0.961538
$0.889996
Yield(per year)
4%
6%
According to the Law of One Price, the first-year cash flows from each coupon bond must have a per-dollar
price of $0.961538, and the second-year cash flows must have a per-dollar price of $0.889996.Therefore,the market
prices of the two different coupon bonds should be:
For the 5% coupon bond:
0.961538 × $50+0.889996 × $1,050=$98257
For the 10% coupon bond:
0.961538 × $100+0.889996 × $1,100=$1,07515
Now let us compute the yields to maturity on each of the coupon bonds must correspond to these market
prices-Using the financial calculator we find:
For the 5% coupon bond:
n
2
i
?
PV
-982.57
FV
1,000
PMT
50
Result
i=5.9500%
n
2
i
?
PV
-1075.15
FV
1,000
PMT
100
Result
i=5.9064%
For the 10% coupon bond:
Thus, we see that in order to obey the Law of One Price, the two bonds must have different yields to maturity. As a
general principle:
When the yield curve is not flat, bonds of the same maturity with different coupon rates have different yields to maturity.
8.5.2 The Effect of Default Risk and Taxes
At times, one will encounter examples of bonds with the same coupon rate and maturity selling at different prices.
These differences occur because of the other way seemingly identical securities differ.
Bonds offering the same future stream of promised payments can differ in a number of ways, but the two most
important are default risk and taxability. To illustrate, consider a bond promising to pay $1,OOO a year from now.
Suppose that one-year U.S. Treasury rate is 6% per year. If the bond is completely free of default risk, its price would,
therefore, be $1,000/1.06=$94340.But if it is subject to some default risk (i.e., that what is promised may not be
paid),no matter how slight, its price will be less than $943.40,and its yield will be higher than 6% per year.
The taxability of bonds can vary according to the issuer or type of bond, and this fact will certainly influence
their price. For example, interest earned on bonds issued by state and local governments in the United States is exempt
from federal income taxes. Other things equal, this feature makes them more attractive to taxpaying investors and will
cause their prices to be higher (and their yields lower) than otherwise comparable bonds.
8.5.3 Other Effects on Bond Yields
There are many other features that may differentiate seemingly identical fixed- income securities and, therefore,
cause their prices to differ. Check your intuition about the effect of the following two bond features. In each case
consider whether the inclusion of the feature should increase or decrease the price of an otherwise identical bond (i.e.,
one which offers the same stream of promised cash flows) that does not have the feature:
1. Callability. This feature gives the issuer of the bond the right to redeem it before the final maturity date. A
bond that has this feature is a callable bond.
2. Convertibility. This feature gives the holder of a bond issued by a corporation the right to convert the bond into
a prespecified number of shares of common stock. A bond that has this feature is a convertible bond.
Your intuition should tell you that any feature that makes the bond more attractive to the issuer will lower its
price, and any feature that makes it more attractive to the bondholders will raise its price. Thus, caliability will cause a
bond to have a lower price (and a higher yield to maturity).On the other hand, convertibility will cause a bond to have a
higher price and a lower yield to maturity.
8.6 THE BEHAVIOR OF BOND PRICES OVER TIME
In this section we examine how bond prices change over time as a result of the passage of time and changes in
interest rates.
8.6.1 The Effect of the Passage of Time
If the yield curve were flat and interest rates did not change, any default-free discount bond's price would rise
with the passage of time, and any premium bond's price would fall. This is because eventually bonds mature, and their
price must equal their face value at maturity. We should, therefore, expect the prices of discount bonds and premium
bonds to move toward their face value as they approach maturity. This implied price pattern is illustrated for the case of
20-year pure discount bonds in Figure 8.3.
Let us illustrate the calculation assuming the face value of the bond is $1,OOO and the yield remains constant at
6% per year. Initially the bond has a maturity of 20 years and its price is:
n
2
i
6%
PV
?
FV
1,000
PMT
0
Result
PV=$311.80
After one year goes by, the bond has a remaining maturity of 19 years and its price is:
n
19
i
6%
PV
?
FV
1,000
PMT
0
Result
PV=330.51
The proportional change in price is, therefore, exactly equal to the 6% per year yield on the bond:
Proportional Change in Price = ($330.51- $311.80) / $311.80
= 6%
8.6.2 Interest-Rate Risk
Normally, we think of buying U.S-Treasury bonds as a conservative investment policy because there is no risk of
default involved. However, an economic environment of changing interest rates can produce big gains or losses for
investors in long-term bonds.
Figure 8.4 illustrates the sensitivity of long-term bond prices to interest rate. It shows the magnitude of the
changes that would occur in the prices of 30-year pure discount bonds and 30-year 8% coupon par bonds if the level of
interest rates moved to a value different from 8% immediately after the bonds are purchased. Each curve in Figure 8.4
corresponds to a different bond. Along the ordinate we measure the ratio of the bond's price computed using the
indicated interest rate to its price computed at a discount rate of 8%.
For example, at an interest rate of 8%per year, the price of a 30-year 8% coupon bond with a face value of
$1,OOO would be $1,000, whereas at an interest rate of 9% per year its price is $89726. The ratio of its price at a 9%
interest rate to its price at an 8% interest rate, therefore, 89726/1,000=0.89726. We can, therefore, say that if the level
of interest rates were to rise from 8% to 9%, the price of the par bond would fall by roughly 10%.
The Figure shows the magnitude of the changes that would occur in the prices of 30-year pure discount bonds
and 30-year 8% coupon par bonds if the level of interest rates moved to a value different from 8% immediately after
the bonds are purchased. The ordinate measures the ratio of the bond's price computed at the indicated interest rate to
its price computed at a discount rate of 8%. Thus, at an interest rate of 8%, the price ratios for both bonds are 1.
On the other hand, the price of a 30-year pure discount bond with a face value of $1,000 is $99.38 at an interest
rate of 8% per year and $75.37 at an interest rate of 9%. The ratio of its price at a 9% interest rate to its price at an 8%
interest rate is, therefore, 7537/99.38=0.7684. We can, therefore, say that if the level of interest rates were to rise from
8% to 9%, the price of the pure discount bond would fall by roughly 23%.
Note in Figure 8.4 that the curve corresponding to the pure discount bond is steeper than the par bond's curve.
This greater steepness reflects its greater interest-rate sensitivity.
Summary
A change in market interest rates causes a change in the opposite direction in the market va1ues of all existing
contracts promising fixed payments in the future.
The market prices of $1 to be received at every possible date in the future the basic building blocks for valuing all
other streams of known cash flows. These prices are inferred from the observed market prices of traded bonds and then
applied to other streams of known cash flows to value them.
An equivalent valuation can be carried out by applying a discounted cash flow now formula with a different
discount rate for each future time period.
Differences in the prices of fixed-income securities of a given maturity arise from differences in coupon rates,
default risk, tax treatment, culpability, convertibility, and other features.
Over time the prices of bonds converge toward their face value .Before maturity, however, bond prices can
fluctuate a great deal as a result of changes in market interest rates.