OBJECTIVES
To show how to value contracts and securities that promise a stream of cashflows that is known with certainty.
To understand how bond prices and yields change over time.
CONTENIS
8.1Using present value Formulas to Value Known CashFlows
8.2The Basic Building Blocks:Pure Discount Bonds
8.3Coupon Bonds,Current Yield,and Yield to Maturity
8.4Reading Bond Listings
8.5 Why Yields for the Same Maturity May Differ
8.5The 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.Avaluation model is a quantitativemethod used to infer an asset's value (the output ofthe 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 procedurein 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 8.1presents 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 8.2shows how to modify such a model to take account of thefactthat generallythe yieldcurve isnot flat (i.e.,that interestratesvarywith maturity).Sections8.3-85explain the main features ofbonds in the realworld 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
Chapter4shows 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. Itinvolves applying adiscounted 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 $100each year for the next three years.How much is this three-year annuity worth if you know that the appropriate discount rate is6%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:
On a financial calculator,we would enter the values for n, i, and PMT, and compute the PV:
n / i / PV / FV / PMT / Result3 / 6 / ? / 0 / 100 / PV=267.30
Now suppose that an hour after you buy the security,the risk-free interest rises from 6%to7%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 yearon 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 / i / PV / FV / PMT / Result3 / 7 / ? / 0 / 100 / PV=262.43
At a price of $262.43,a fixed-income security that promises to pay $100each 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 willonly 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 valueof 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:
Achange 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 discountrate to use in thepresent value formula.As shown in chapter 2,market interest rates arc not the same for all maturities.We reproduce as Figure 8.1the 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 ofthe 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 listingofthemarket prices ofpurediscountbonds(alsocalledzero-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 valueor 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 $50difference betweenthe $1,OOOface value and the $950purchase price.
The yield(interest rate)on a pure discount bond is the annualized rate ofreturn 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,OOOand 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 / i / PV / FV / PMT / Result2 / ? / -880 / 1,000 / 0 / i=6.60%
Return to the valuation of the security of section 8.1that promises to pay $100 each year for the next three years.Suppose that we observe the set of pure discount bond prices inTable8.1.Following standard practice,the bond prices are quoted as afraction 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 ofTable8.1, and the second procedure uses the yields in the last column.Procedure 1multiplies 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 ofThird Year's Cash Flow =$100 ×0.80=$80.00
Total Present Value =$263
The resulting estimate of the security's value is $263.
Procedure 2gets the same result by discounting each Year's promised cashpayment at the yield corresponding tothat maturity:
PresentValue ofFirstYear's Cash Flow =$100/1.0526=$9500
PresentValue ofSecondYear's CashFlow=$100/1.06602=$88.00
Present Value of ThirdYear's CashFlow=$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 ofTable 8.1.If we do so,we get a value of $259,which is$4too low:
n / i / PV / FV / PMT / Result3 / 7.72% / ? / 0 / 100 / 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 8.1to get a value of $263for the security? The answer is yes: That single discount rate is6.88%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 / i / PV / FV / PMT / Result3 / 6.88% / ? / 0 / 100 / 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 / i / PV / FV / PMT / Result3 / ? / -263 / 0 / 100 / 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 annuityusing 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.
8.3COUPON BONDS, CURRENT YEEID, AND YIELD TOMATURITY
Acoupon 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 mostbonds hadcoupons attached tothem thatinvestorswould tearoffandpresent 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 $1,000that makes, annual coupon payments at a coupon rate of 10%obligates the issuer topay 0.lOX$1,000=$100every 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 inFigure8.2.We see that the stream of promised cash flows has an annuity component(a fixed per pt amount)of$100peryear and a“balloon”or“bullet”payment of$1,OOOat maturity.
The $100annual 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,OOOface 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 19years 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,itis 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--$47.62less 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,OOOat 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 thanjust 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:
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 / i / PV / FV / PMT / Result2 / ? / -1,100 / 1,000 / 100 / 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 maturingtwo 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,OOOface value,not the $950price 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,OOOat maturity. Using the financial calculator to compute thebond's yield to maturity,we find:
n / i / PV / FV / PMT / Result2 / ? / -950 / 1,000 / 40 / 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
8.3.1Beware 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.