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5-1

INTEREST RATES AND BOND VALUAT ION

Chapter 5

5-2

KEY CONCEPTS AND SKILLS

• Know the important bond features and bond types

• Comprehend bond values (prices) and why they fluctuate

• Compute bond values and fluctuations• Appreciate bond ratings, their meaning, and

relationship to bond terms and value• Understand the impact of inflation on interest

rates• Grasp the term structure of interest rates and

the determinants of bond yields

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5-3

CHAPTER OUTLINE

5.1 Bonds and Bond Valuation5.2 More on Bond Features5.3 Bond Ratings5.4 Some Different Types of Bonds5.5 Bond Markets5.6 Inflation and Interest Rates5.7 Determinants of Bond Yields

5-4

5.1 BONDS AND BOND VALUATION

• A bond is a legally binding agreement be-tween a borrower and a lender that specifies:• Par (face) value • Coupon rate (which deter-

mines coupon payment) • Maturity date• Bondholders have a required rate of return on

the bond• Do not confuse the coupon rate with the re-

quired rate of return in the market• The yield to maturity (YTM) is the expected

rate of return on the bond• Almost always, YTM = the required return

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BOND VALUATION

• Primary Principle:• (Fundamental) Value of any financial security = Present

value of that security’s expected future cash flows • Bond value is, therefore, determined by the pres-

ent value of the coupon payments plus the present value of the par, or face, value

• Present values (i.e., bond values) are inversely related to market interest rates, which them-selves affect bondholders’ required rates of return.

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CONCEPTUAL CASH FLOW OF A 10 YEAR BOND

• Xanth Co. has issued a 10 year bond with an 8% annual coupon. The cash flows from the bond would be paid as follows:

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THE BOND-PRICING EQUATION

Notice that:• The first term is the present value of the coupon payments

(an annuity)• The second term is the present value of the face-value

payment

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FREQUENCY OF COUPON PAYMENTS

• Bond terms dictate the frequency of coupon payments• The coupon rate is expressed in annual terms• If the rate is expressed annually and the payments

are more frequent, calculation of bond value requires:• Dividing the annual coupon payment by the number of

compounding periods per year to arrive at the value of each periodic coupon payment (C);

• Dividing the annual required rate of return by the number of compounding periods per year to arrive at the desired peri-odic required return (or desired periodic yield) (i.e., r);

• Multiplying the remaining years of the bond’s life by the number of compounding periods per year to arrive at the remaining number of coupon payments (T).

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BOND EXAMPLE• On 1-1-16, consider a US government bond with

a 63/8% coupon that expires in Dec. 2020.• The Par Value of the bond is $1,000.• Coupon payments are made semi-annually (June

30 and Dec. 31 for this particular bond).• Coupon payment is $31.875: 63/8% x 1000 / 2• Using Jan. 1, 2010 as today, the timeline show-

ing the sizes and timing of cash flows is this:

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BOND EXAMPLE

• On Jan. 1, 2016, the required yield is 5%/year.• The size and timing of the cash flows are:

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BOND EXAMPLE: CALCULATOR

PMT

I/Y

FV

PV

N

PV

31.875 =

2.5

1,000

– 1,060.17

10

1,000×0.06375

2

Find the present value (as of January 1, 2016), of a 6 3/8% coupon bond with semi-annual payments, and a maturity date of Decem-ber 2020 if the current required return is 5%.

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BOND EXAMPLE• Now assume that the required yield is 11%. • For this new req’d return, calculate the bond’s

price.

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REQUIRED RETURN & BOND VALUE

800

1000

1100

1200

1300

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

Discount Rate(Required Ret.)

Bon

d V

alu

e

6 3/8%

When the required return < coupon rate, the bond trades at a premium.

When the required ret. = coupon rate, the bond trades at par.

When the required ret. > coupon rate, the bond trades at a discount.

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BOND CONCEPTS

• Bond prices and market interest rates (i.e., required returns) move in opposite directions.

• When required return > coupon rate, price < face (or par) value

• The bond is a ‘discount’ bond

• When required return = coupon rate, price = face value

• The bond sells at par value

• When required return < coupon rate, price > face value

• The bond is a ‘premium’ bond

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INTEREST-RATE RISK

• Price Risk• Change in price due to changes in market interest

rates, which affect bondholders’ required returns• Long-term bonds have more price risk than short-

term bonds• Low-coupon-rate bonds have more price risk than

high-coupon-rate bonds

• Reinvestment Rate Risk• Uncertainty concerning rates at which cash flows

can be reinvested• Short-term bonds have more reinvestment rate risk

than long-term bonds• High-coupon-rate bonds have more reinvestment

rate risk than low-coupon-rate bonds

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MATURITY AND BOND PRICE VOLATILITY

C

Consider two otherwise identical bonds.

The long-maturity bond will have much more volatility with respect to changes

in the discount rate.

Discount Rate(Req’d Ret.)

Bon

d V

alu

e

ParShort-Maturity Bond

Long-Maturity Bond

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COUPON RATES AND BOND PRICES

Consider two otherwise identical bonds.

The low-coupon bond will have more volatility with respect to changes in the

discount rate.

Discount Rate(Req’d Ret.)

Bon

d V

alu

e

High-Coupon Bond

Low-Coupon Bond

Par

C

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COMPUTING YIELD TO MATURITY

• Mechanically, yield to maturity (YTM) is the ratethat sets the present value of a bond’s future cash flows equal to the bond’s current price

• YTM is implied by the bond’s price and cash flows• YTM can be thought of as the return expected by

the bondholders (i.e., as the expected return)• Finding YTM requires trial and error if you do not

not have a financial calculator, an Excel sheet, or mult.-choice answers from which to choose.

• If you have a financial calculator, enter PV, PMT, N, and FV, remembering the sign convention (PMT and FV need to have the same sign, PV the oppo-site sign). Then, compute R (or I/Y).

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YTM WITH ANNUAL COUPONS

• Consider a bond with a 10% annual coupon rate, 15 years to maturity, and a par value of $1,000. Current price is $928.09. Calculate YTM. Poss-ible answers are: 9%, 10%, 11%, 12%, 13%.

• Financial calculator:• N = 15; PV = –928.09; FV = 1,000; PMT = 100• CPT I/Y = 11%

• Math:• 100 / 0.13 ∙ ( 1 – 1/1.1315 ) + 1000 / 1.1315 = 806.12• 100 / 0.12 ∙ ( 1 – 1/1.1215 ) + 1000 / 1.1215 = 863.78• 100 / 0.11 ∙ ( 1 – 1/1.1115 ) + 1000 / 1.1115 = 928.09

11% is the yield to maturity

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YTM WITH SEMIANNUAL COUPONS

• Consider a bond with 10% coupon rate, semi-annual coupons, $1000 face value, and maturity in 20 years. Price = $1,197.93. Calculate YTM.• Will the YTM be more or less than 10%?• What’s the semi-annual coupon pmt.? ( 10% / 2 ) x $1K• How many periods (half-years) are there? 20 x 2• So, N = 40; PV = -1,197.93; PMT = 50; FV = 1,00

CPT I/Y = 4% (Is this the Annual YTM?)• YTM = 4% (per half-year) * 2 = 8% per year

• Math … supposing 8%/yr. is one mult.-ch. answer.• Convert the potential answer to a semi-annual rate: 4%• Then do the usual math & confirm the answer:• 50 / 0.04 ∙ ( 1 – 1/1.0440 ) + 1000 / 1.0440 = 1197.93

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CURRENT YIELD VS. YIELD TO MATURITY

• Current yield = annual coupon / price• Yield to maturity = current yield + capital gains

yield• Example: 10% coupon bond, with semi-annual

coupons, face value of 1,000, 20 years to maturity, $1,197.93 price• Current yield = $100 / 1197.93 = .0835 = 8.35%• Price in 1 yr., assuming no change in required return,

= 1,193.68 … calculated using 19 years to maturity• Capital gain yield = (1193.68 – 1197.93) / 1197.93 =

–0.0035, or –0.35%• YTM = 8.35% – 0.35% = 8.00%, same as the YTM

that we computed earlier for this same bond

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BOND PRICING THEOREMS

• Bonds of same risk (and maturity) should be priced to yield about the same return, regardless of the coupon rate. Call these Bonds A and B.

• If you know the price of Bond A, you can calcu-late A’s YTM. In turn, this YTM (i.e., the expected return on A) is a reasonable required rate of return for B, given that A & B have the same risk.

• Next, use this rate as the required return to cal-culate PV of B’s cash flows and, hence, B’s price.

• This is a useful concept that can be transferred to valuing assets other than bonds.

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BOND PRICING WITH A SPREADSHEET

• Excel offers formulas for finding prices and YTMs.• PRICE(Settlement,Maturity,Rate,Yld,Redemp-

tion,Frequency,Basis)• YIELD(Settlement,Maturity,Rate,Pr,Redemp-

tion,Frequency,Basis)• Settlement & maturity need to be actual dates• Redemption & Pr are entered as % of par value

• Click on the Excel icon for an example.• Alternately, PV(Rate,Nper,Pmt,FV) works • pretty decently, too, for Price• Additionally, RATE(Nper,PV,FV) works

pretty decently, too, for Yield to Maturity

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5.2 MORE ON BOND FEATURES

• There are two kinds of securities issued by corporations:• Equity – Represents Ownership Inter-

est• Debt – Represents Short- or Long-Term

Borrowing by the Firm

• Bonds are classified as Debt

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DEBT VERSUS EQUITY

• Debt• Not an ownership interest• Creditors do not have

voting rights• Interest paid on debt is

considered a cost of do-ing business and is tax-deductible on a firm’s in-come statement

• Creditors have legal re-course if interest or prin-cipal pmts. are missed

• Excess debt can lead to financial distress and bankruptcy

• Equity• Ownership interest• Common stockholders

vote for the board of directors and other issues

• Dividends are not considered a cost of doing business and are not tax-deductible

• Dividends are neither a liability nor an obligation of the firm; stockholders have no legal recourse if dividends are not paid

• An all-equity firm cannot go bankrupt

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THE BOND INDENTURE

• Indenture ≡ A contract between the issuing com-pany and the bondholders that includes:• The basic terms of the bonds

(Face Value, Coupon Rate, Maturity Date, Fre-quency of Coupon Pmts.)

• The total amount of bonds issued• A description of property used as security, if

applicable• Sinking fund provisions (if applicable)• Call provisions (if applicable)• Details of protective debt covenants

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SAMPLE BOND FEATURES

• Features of a recent CSX bond issue demonstrate the range of items covered in the bond inden-ture:

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BOND CHARACTERISTICS

• Security• Collateral – secured by financial securities• Mortgage – secured by real property, normally

land or buildings• Debentures ≡ unsecured debt• Notes – unsecured debt with original maturity

less than 10 years• Seniority• A firm will often have multiple different bonds

outstanding at once• In many cases, a hierarchy exists among bonds

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BOND CHARACTERISTICS (CONT.)

• Sinking Funds• Funds into which firm makes payments (as sec-

urity) – these funds are used (by a trustee) to reduce a firm’s overall bond obligation period-

ically, across time

• Call Provisions (if applicable)• Deferred Call – date after which a callable bond

can be called• Call Premium – difference between the stated

call price (per bond) and the actual price of the bond

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BOND CHARACTERISTICS (CONT.)

• Protective Covenants• Rules and restrictions written into the indenture• Place constraints on the firm• Often written in terms of ratios• For example, “a firm cannot issue additional

long-term debt” might be expressed as “the debt/equity ratio cannot exceed, say, 0.6”

• See page 141 of textbook for representative examples of debt covenants

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REQUIRED YIELDS

• The coupon rate depends on the risk characteris-tics of the bond when issued.

• Which bonds will have the higher coupon rate, all else equal?

• Secured debt versus a debenture (unsecured debt)• Subordinated debenture versus senior debt• A bond with a sinking fund versus one without• A callable bond versus a non-callable bond• A bond with protective covenants versus a bond without

any such covenants

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5.3 BOND RATINGS –INVESTMENT QUALITY

• High-Grade• Moody’s Aaa and Standard & Poors (S&P) AAA:

Capacity to make payments is extremely strong• Moody’s Aa and S&P AA:

Capacity to make payments is very strong

• Medium-Grade• Moody’s A and S&P A: Capacity to pay is strong,

but more susceptible to changes in circumstances• Moody’s Baa and S&P BBB: Capacity to pay is ade-

quate, adverse conditions will have more impact on the firm’s ability to pay

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BOND RATINGS - SPECULATIVE

• Low-Grade• Moody’s Ba and B; S&P BB and B

Considered speculative investments, with respect to capacity to pay.

• Very-Low-Grade• Moody’s C and S&P C and D• Considered to have highly uncertain repay-

ment and, in many cases, already in default with principal and interest in arrears.

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5.4 SOME DIFFERENT TYPES OF BONDS

• There are many types of bonds• Some common bonds include:• Government Bonds• Federal• State and Municipal

• Zero-Coupon Bonds (a.k.a., Pure-Discount Bonds)

• Floating-Rate Bonds

• Each is discussed below

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GOVERNMENT BONDS

• Treasury Securities• Federal government debt• Treasury bills (T-bills) – pure discount bonds

with original maturity less than one year • T-notes – coupon debt with original maturity

between one and ten years• T-bonds – coupon debt with original maturity

greater than ten years

• Municipal Securities• Debt of state and local governments• Varying degrees of default risk, rated similar to

corporate debt• Interest received is tax-exempt at the federal

level

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AFTER-TAX YIELDS

• Example: A taxable (corporate) bond’s YTM is 8.0% & a municipal bond’s yield is 6.0%. If you are in a 40% tax brack-et, which bond do you prefer?• 8.0% ∙ (1 – 40%) = 4.8%• The after-tax return on the corp. bond is

4.8%, compared to a 6.0% “after-tax” return on the muni’ bond

• At what tax rate would you be indifferent be-tween the two bonds?• 8.0% ∙ (1 – T) = 6.0%• Solve for T = 25%

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ZERO-COUPON BONDS

• Make no periodic interest payments (i.e., have a coupon rate = 0%)

• The entire yield to maturity comes from the difference between the purchase price and the par value

• Cannot sell for more than par value• Sometimes called “zeroes”, “deep-discount

bonds”, or “original-issue discount bonds” (OIDs or OID bonds)

• Treasury Bills and principal-only Treasury STRIPS (or “strips”) are good examples of zeroes

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PURE-DISCOUNT BONDS

Information needed for valuing pure discount bonds:• Time to maturity (T) = Maturity date -

today’s date• Face value (F)• Discount rate (r)

Present value of a pure-discount bond’s single cash flow:

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PURE DISCOUNT BONDS: EXAMPLE

Find the value of a 30-year zero-coupon bond with a $1,000 par value and a YTM of 6%.

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FLOATING-RATE BONDS

• On this type of bond, the coupon rate floats depending on some index value

• Examples – adjustable-rate mortgage bonds & inflation-linked Treasury bills, notes, bonds

• There is less price risk with floating-rate bonds.• The coupon floats, so it is less likely to diff-

er substantially from required return.• Coupons may have a “collar” – the rate can-

not go above a specified “ceiling” or below a specified “floor.”

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5.5 BOND MARKETS

• Primarily over-the-counter transactions with dealers connected electronically

• Extremely large number of bond issues, but generally low daily volume in single

issues• Makes getting up-to-date prices difficult,

particularly on small-company or muni-cipal issues

• Treasury securities are an exception

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TREASURY QUOTATIONS (Fig. 5.4)

11/15/39 4.375 130.1250 130.2031 –1.4063 2.779

What is the coupon rate on the bond?

When does the bond mature?

What is the bid price?

What is the ask price?

How much did the price change from the previous day?

What is the current yield based on the ask price?

43/8% per yr.

2039

1301/8% of $1000, or $1301.150

130.2031% of $1K: $1302.031

Down by 1.4063% of $1K, or $14.063

2.779%

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5.6 INFLATION AND INTEREST RATES

• Real rate of interest – change in purchasing power• Nominal rate of interest – quoted (or stated) rate of

interest, includes change in purchasing power andinflation

• The ex ante nominal rate of interest includes our desired real rate of return plus an adjustment for expected inflation

• Fisher notes that investors realize inflation reduces purchasing power and insist on high returns during inflationary times:• “A rise in the rate of inflation causes the nominal rate to

rise just enough so that the real rate of interest is unaffected.”

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THE FISHER EFFECT

• The Fisher Effect defines the relationship between real rates, nominal rates, and inflation.

• (1 + R) = (1 + r)∙(1 + h), where• R = nominal rate, or stated rate on an investment• r = real rate• h = expected inflation rate

• Approximation• R ≈ r + h

(1+R) = (1+r) ∙ (1+h)1+R = 1+ r + h + r∙h1+R ≈ 1+ r + h

R ≈ r + h• Another approximation

• r ≈ R – h

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THE FISHER EFFECT: EXAMPLE

• If an investment states an 18.0% nominal rate and we expect inflation to be 8.0%, what is the underlying real rate of return?

• (1 + R) = (1 + r)∙(1 + h)• 1.180 = (1 + r)∙(1.080)• Solve for r = 0.0926 , or 9.26%

• Approximation: r ≈ 18.0% – 8.0% = 10.0%• Because the expected inflation is relatively high,

there is a significant difference between the actual Fisher Effect and the approximation

….. nearly 3/4 percent!

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5.7 DETERMINANTS OF BOND YIELDS

• Term structure (of interest rates) is the relation-ship between yields and time to maturity, all

else equal.• It is important to recognize that we pull out the

effect of default risk, different coupons, etc.• Yield curve – graphical representation of the term

structure (see Figure 5.6)• Normal – upward-sloping, long-term yields

are higher than short-term yields• Inverted – downward-sloping, long-term

yields are lower than short-term yields

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SAMPLE YIELD CURVE (Fig. 5.7)

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FACTORS AFFECTING REQUIRED RETURN

• Default-risk premium – remember bond ratings• Taxability premium – remember municipal versus

taxable• Liquidity premium – bonds that have more fre-

quent trading will generally have lower re-quired returns

• Anything else that affects the risk of the cash flows to the bondholders will affect the

required returns.

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QUICK QUIZ

• How do you find the value of a bond, and why do bond prices change?

• What is a bond indenture, and what are some of the important features?

• What are bond ratings, and why are they im-portant?

• How does inflation affect interest rates?• What is the term structure of interest rates?• What factors determine the required return

on bonds?