CHAPTER 6INTEREST RATES AND BOND VALUATION

CHAPTER 6 QUIZ

CHAPTER ORGANIZATION

6.1 Bonds and Bond Valuation - When a corporation (or government) wishes to borrow money from the public on a long-term basis, it usually does so by issuing, or selling, debt securities that are generically called bonds.

Bond Features and Prices

o Face value, par value, maturity value: The principal amount of the bond that is repaid at maturity. The par value remains constant throughout the life of the bond. The par value of most bonds is assumed to be $1,000 unless otherwise stated.

o Coupon rate: This is the stated rate of interest the bond pays each period, normally annually or semi-annually. The coupon rate remains constant throughout the life of the bond (assuming a fixed-rate bond).

o Coupon or coupon interest payment: The dollar ($) amount of interest the bond pays each period, which equals the par value x coupon rate. The par value of the bond as well as the coupon rate on the bond remains constant throughout the bond’s life; therefore, the coupon interest payment also remains constant.

o Maturity date: The date on which the bond ceases to earn interest. On this date, the last interest payment will be made, and the face value of the bond will be repaid. When valuing a bond, we are interested in the “time to maturity,” the length of time between the date the bond is purchased and its maturity date

o Yield to maturity, required return, market rate: The percentage rate of return paid on a bond, note, or other fixed income security if the investor buys and holds it to its maturity date. The calculation for YTM is based on the coupon rate, length of time to maturity, and market price. It assumes that coupon interest paid over the life of the bond will be reinvested at the same rate. The required return on the bond can change at any time over the life of the bond.

Bond Values and Yields - FinSim

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The cash flows from a bond are the coupons and the face value. The value of a bond (market price) is the present value of the expected cash flows discounted at the market rate of interest.

Find the value of the Xanth bond assuming a yield to maturity (market rate) of 8%:

Bond value = PV of coupons + PV of face valueBond value = PV of an annuity + PV of a lump sum

Bond value = + , where

C = coupon paid each periodr = market rate (required return) per periodt = number of periodsF = face (par) value of the bond

Bond value = +

Bond value = $80(6.710081399) + $1,000(.463193488)Bond value = $536.81 + $463.19 = $1,000.00

The bond is selling at par. This will occur any time the market rate equals the bond’s coupon rate.

t r Bond Value C FN I/YR PV PMT FV10 8 -1,000 80 1,000

Find the value of the Xanth bond assuming a yield to maturity (market rate) of 9.5%:

Bond value = +

Bond value = $80( ) + $1,000( )Bond value = $ + $ = $

The bond is selling at a ______ . This will occur any time the market rate ____ the bond’s coupon rate.

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t r Bond Value C FN I/YR PV PMT FV10 9.5 80 1,000

Find the value of the Xanth bond assuming a yield to maturity (market rate) of 6%:

Bond value = +

Bond value = $80( ) + $1,000( )Bond value = $ + $ = $

The bond is selling at a ______ . This will occur any time the market rate ____ the bond’s coupon rate.

t r Bond Value C FN I/YR PV PMT FV10 6 80 1,000

Note: Bond prices and interest rates move in opposite directions, i.e., they are inversely related. This is intuitive given that the price of a bond is simply the present value of its cash flows, and present values move opposite to changes in the discount rate.

In practice, bonds issued in the United States usually make coupon payments twice a year. Bond yields are quoted like APRs; the quoted rate is equal to the actual rate per period multiplied by the number of periods.

Find the value of the Xanth bond assuming the coupon is paid semi-annually and the market rate (yield to maturity) is 6%*:

Bond value = +

Bond value = +

Bond value = +

Bond value = $40( ) + $1,000( )Bond value = $ + $ = $

t r Bond Value C FN I/YR PV PMT FV20 3 40 1,000

* To calculate the effective yield on this bond, note that 3 percent every six months is equivalent to:

Effective annual rate = (1 + .03)2 - 1 = 1.0609 – 1 = .0609 = 6.09%Prepared by Jim Keys 3

Brittany Co. issued 15-year bonds one year ago at a coupon rate of 8.50 percent. The bonds make semiannual payments. If the YTM on these bonds is 7.90 percent, what is the current bond price?

To find the price of this bond, we need to realize that the maturity of the bond is 14 years. The bond was issued one year ago, with 15 years to maturity, so there are 14 years left on the bond. Also, the coupons are semiannual, so we need to use the semiannual interest rate and the number of semiannual periods. The price of the bond is:

Price = $42.50(PVIFA3.95%,28) + $1,000(PVIF3.95%,28) = $1,050.28

t r Bond Value C FN I/YR PV PMT FV28 3.95 -1,050.28 42.50 1,000

Interest Rate Price Risk - The risk that arises for bond owners from fluctuating interest rates is called interest rate risk. How much interest rate risk a bond has depends on how sensitive its price is to interest rate changes.

1. All other things being equal, the longer the time to maturity, the greater the interest rate risk.

The reason that longer-term bonds have greater interest rate sensitivity is that a large portion of a bond's value comes from the $1,000 face amount. The present value of this amount isn't greatly affected by a small change in interest rates if the amount is to be received in one year. Even a small change in the interest rate, however, once it is compounded for 30 years, can have a significant effect on the present

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value. As a result, the present value of the face amount will be much more volatile with a longer-term bond.

2. All other things being equal, the lower the coupon rate, the greater the interest rate risk.

The reason that bonds with lower coupons have greater interest rate risk is easy to understand. As we discussed earlier, the value of a bond depends on the present value of its coupons and the present value of the face amount. If two bonds with different coupon rates have the same maturity, then the value of the one with the lower coupon is proportionately more dependent on the face amount to be received at maturity. As a result, all other things being equal, its value will fluctuate more as interest rates change. Put another way, the bond with the higher coupon has a larger cash flow early in its life, so its value is less sensitive to changes in the discount rate.

Interest Rate Reinvestment Risk - The YTM calculation assumes that the investor reinvests all coupons received from a bond at a rate equal to the computed YTM on that bond, thereby earning interest on interest over the life of the bond at the computed YTM. In effect, this calculation assumes that the reinvestment rate is the yield to maturity. If the investor spends the coupons, or reinvests them at a rate different from the assumed reinvestment rate, the realized yield that will actually be earned at the termination of the investment in the bond will differ from the promised YTM. And, in fact, coupons almost always will be reinvested at rates higher or lower than the computed YTM, resulting in a realized yield that differs from the promised yield. This gives rise to reinvestment rate risk.

1. Holding everything else constant, the longer the maturity of a bond, the greater the reinvestment risk.

2. Holding everything else constant, the higher the coupon rate, the greater the dependence of the total dollar return from the bond on the reinvestment of the coupon payments.

Finding the Yield to Maturity: More Trial and Error - The yield to maturity (YTM) of a bond is the compound average annual expected rate of return if the bond is purchased at its current market price and held to maturity. The YTM also assumes that the interest payments are reinvested for the life of the bond at the same yield. The YTM is the internal rate of return (IRR) of the bond.

A corporate bond is currently selling for $863.07, has 20 years left to maturity, and a coupon rate of 7.5%. If you purchase the bond today at the listed price and hold until maturity, what is your yield to maturity?

Trial and error - Set up the bond pricing equation with the given values and solve for the YTM (r):

Bond value = +

$863.07 = +

$863.07 = $75(PVIFA r,20) + $1,000(PVIF r,20)

At this point, you can look up (using the interest tables) or compute the PVIFA and PVIF factors for your estimate of “r” and plug them into the equation above. You have found the yield to maturity when your estimate causes the right-hand side of the equation to equal the left-hand side.

Alternatively, the approximation formula given below will give you a fairly close estimate unless the bond is selling at a steep discount or premium:

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Of course, the best alternative is to use a spreadsheet or a financial calculator to solve:

t YTM (r) Bond Value C FN I/YR PV PMT FV20 9.00 -863.07 75 1,000

Current yield - The current yield is simply the annual interest payment divided by the current market price of the bond ( $C / $Bond price ).

For the bond shown above, the current yield = $75 / $863.07 = .086899 = 8.69%

The total rate of return on the bond (YTM) = Current yield + Capital gains (loss) yield

For the bond shown above:

.09 = .0869 + Capital gains yield

Capital gains yield = .09 - .0869 = .0031 = .31%

Yield to call – The average rate of return earned on a bond if it is held until the first call date.

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6.2 More on Bond Features

Is it Debt or Equity? - In general, debt securities are characterized by the following attributes:

Creditors (or lenders or bondholders) generally have no voting rights. Payment of interest on debt is a tax-deductible business expense. Unpaid debt is a liability, so default subjects the firm to legal action by its creditors.

It is sometimes difficult to tell whether a hybrid security is debt or equity. The distinction is important for many reasons, not the least of which is that (a) the IRS takes a keen interest in the firm’s financing expenses in order to be sure that nondeductible expenses are not deducted, and (b) investors are concerned with the strength of their claims on firm cash flows.

Long-Term Debt: The Basics – Promises made by the issuing firm to pay principal when due andto make timely interest payments on the unpaid balance (notes, debentures, bonds).

Public issues – offered to the general public Private placement – directly placed with a lender or group of lenders

The Indenture - written agreement between issuer and creditors detailing terms of borrowing. (Also deed of trust.) The indenture includes the following provisions:

Bond termso Registered form – ownership is recorded, payment made directly to ownero Bearer form – payment is made to holder (bearer) of bond

The total face amount of bonds issued A description of any property used as security

o Collateral – strictly speaking, pledged securitieso Mortgage securities – secured by mortgage on real propertyo Debenture – an unsecured debt with 10 or more years to maturityo Note – a debenture with 10 years or less maturityo Seniority – order of precedence of claimso Subordinated debenture – of lower priority than senior debt

The repayment arrangementso Sinking fund – an account managed by the bond trustee for early redemption

Any call provisionso Call provision – allows company to “call” or repurchase part or all of issueo Call premium – amount by which the call price exceeds the par valueo Deferred call – firm cannot call bonds for a designated periodo Call protected – the description of a bond during the period it can’t be called

Any protective covenantso Protective covenants – indenture conditions that limit the actions of firmso Negative covenant – “thou shalt not” sell major assets, etc.o Positive covenant – “thou shalt” keep working capital at or above $X, etc.

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* “Make whole” call feature

6.3 Bond Ratings - Firms frequently pay to have their debt rated. The two leading bond-rating firms are Moody's and Standard and Poor's (S&P). The debt ratings are an assessment of the creditworthiness of the corporate issuer. The definitions of creditworthiness used by Moody's and S&P are based on how likely the firm is to default and the protection creditors have in the event of a default. It is important to recognize that bond ratings are concerned only with the possibility of default; they do not measure interest rate risk.

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6.4 Some Different Types of Bonds

Government Bonds - Long-term debt instruments issued by a governmental entity. Treasury bonds are bonds issued by a federal government; a state or local government issues municipal bonds. In the US, Treasuries are exempt from state taxation and “munis” are exempt from federal taxation.

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Zero Coupon Bonds - Zero coupon bonds are bonds that are offered at deep discounts because there are no periodic coupon payments. Although, no cash interest is paid, firms deduct the implicit interest while holders report it as income. Interest expense equals the periodic change in the amortized value of the bond.

The U.S. Treasury created the STRIPS program in February 1985: Separate Trading of Registered Interest and Principal of Securities (STRIPS). Treasury strips are created when a coupon-bearing Treasury issue is purchased, placed in escrow, and the coupon payments are “stripped away” from the principal portion. Each component is then sold separately to investors with different objectives: the coupon portion is purchased by those desirous of safe current income, while the principal portion is purchased by those with cash needs in the future. (The latter portion is, in essence, a synthetically created zero-coupon bond.) Merrill Lynch was the first to offer these instruments, calling them “TIGRs” (Treasury Investment Growth Receipts), soon to be followed Salomon Brothers’ CATs (Certificates of Accrual of Treasury securities).

Valuation of a zero-coupon bond: Zero-coupon bond price =

You are offered a bond that will pay no interest but will return the par value of $1,000 twenty years from now. If your required return for this bond is 7.35%, what are you willing to pay?

Zero-coupon bond price = = = $1,000(.2420800635) = $242.08

Yield to maturity on a zero-coupon bond: YTMzero =

You purchased a zero-coupon bond that matures in 25 years for $165.00. What YTM do you expect to earn on this bond assuming you hold it until maturity?

YTMzero = = = (6.060606061).04 – 1 = .074733 = 7.47%

Floating-Rate Bonds - coupon payments adjust periodically according to an index.

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o put provision – holder can sell back to issuer at paro collar - coupon rate has a floor and a ceiling

Other Types of Bondso Income bonds – coupon is paid if income is sufficiento Convertible bonds – can be traded for a fixed number of shares of stocko Put bonds – shareholders can redeem for par at their discretion

6.5 Bond Markets

How Bonds are Bought and Sold - Most transactions are OTC (over-the-counter), therefore, the bond market is not transparent. Daily bond trading volume exceeds stock trading volume, but trading in individual issues tends to be very thin.

Bond Price Reporting

o Bid price – the price a dealer is willing to pay for a securityo Asked price – the price a dealer is willing to accept for a securityo Bid-Ask spread – the difference between the bid price and the asked price

Treasury prices are quoted in 32nds. For example, a quote of 101:16 means that the bond is selling at 101.50 percent of its face value. Therefore, if the face value of the bond is $1,000, its price would be $1,015.

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Locate the Treasury issue in Figure 6.3 maturing in February 2015. What is its coupon rate? What is its bid price? What was the previous day's asked price?The bond listed as 2015 Feb 15 is the one we seek. Its coupon rate is 11.25 percent of face value. The bid price is 148:31, or 148.96875 percent of face value. The ask price is 149:00, which is up by 28 ticks from the previous day. This means that the ask price on the previous day was equal to 149 – (28/32) = 148.125 = 148:04.

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Bond Price Quotes and Accrued Interest (from www.tvmcalcs.com)It is important to understand that bond prices are quoted by dealers without the accrued interest. So, if you get a quote of $950 to purchase a bond, then you will pay $950 plus however much interest has accrued to the seller of the bond since the last coupon payment. That is, the invoiced price is the quoted price plus accrued interest. There are three terms that you should understand:

Accrued interest is the interest that has been earned, but not yet been paid by the bond issuer, since the last coupon payment. Note that interest accrues equally on every day during the period. That is, it does not compound. So, halfway through the period, you will have accrued exactly one-half of the period's interest payment. It works the same way for any other fraction of a payment period.

The clean price is the price of the bond excluding the accrued interest. This is also known as the quoted price.

The dirty price is the total price of the bond, including accrued interest. This is the amount that you would actually pay (or receive) if you purchase (or sell) the bond. The dirty price is simply the clean price plus the accrued interest.

6.6 Inflation and Interest Rates

Real versus Nominal Rates – Real rates have been adjusted for inflation, nominal rates have not.

The Fisher Effect is a theoretical relationship between nominal returns, real returns and the expected inflation rate. Let R be the nominal rate, r the real rate, and h the expected inflation rate; then,

(1 + R) = (1 + r)(1 + h), and R = (r + h) + (r x h)

A reasonable approximation, when expected inflation is relatively low, is R r + h.

A definition whereby the real rate can be found by deflating the nominal rate by the inflation rate: r = [(1 + R) / (1 + h)] – 1.

6.7 Determinants of Bond Yields

The Term Structure of Interest Rates - The relationship between short- and long-term interest rates is known as the term structure of interest rates. The relationship between nominal interest rates on default-free, pure discount securities and time to maturity; that is, the pure time value of money.

What determines the shape of the term structure? There are three basic components. The first two are the real rate of interest and the rate of inflation. The real rate of interest is the compensation investors demand for forgoing the use of their money. You can think of it as the pure time value of money after adjusting for the effects of inflation. The real rate of interest is the basic component underlying every interest rate, regardless of the time to maturity. When the real rate is high, all interest rates will tend to be higher, and vice versa. Thus, the real rate doesn't really determine the shape of the term structure; instead, it mostly influences the overall level of interest rates.

In contrast, the prospect of future inflation very strongly influences the shape of the term structure. Investors thinking about loaning money for various lengths of time recognize that future inflation erodes the value of the dollars that will be returned. As a result, investors demand compensation for this loss in the form of higher nominal rates. This extra compensation is called the inflation premium.

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The third, and last, component of the term structure has to do with interest rate risk. Longer-term bonds have much greater risk of loss resulting from changes in interest rates than do shorter-term bonds. Investors recognize this risk, and they demand extra compensation in the form of higher rates for bearing it. This extra compensation is called the interest rate risk premium. The longer the term to maturity, the greater is the interest rate risk, so the interest rate risk premium increases with maturity.

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Bond Yields and the Yield Curve: Putting It All Together

Treasury yield curve – plot of yields on Treasury notes and bonds relative to maturityDefault risk premium – the portion of a nominal rate that represents compensation for the possibility of

defaultTaxability premium – the portion of a nominal rate that represents compensation for unfavorable tax statusLiquidity premium – the portion of a nominal rate that represents compensation for lack of liquidity

If we combine all of the things we have discussed regarding bond yields, we find that bond yields represent the combined effect of no fewer than six things. The first is the real rate of interest. On top of the real rate are five premiums representing compensation for (1) expected future inflation, (2) interest rate risk, (3) default risk, (4) taxability, and (5) lack of liquidity. As a result, determining the appropriate yield on a bond requires careful analysis of each of these effects.

“Living” yield curve

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