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### Interest Rates and Bond Valuation

Chapter 7

Key Concepts and Skills

- Know the important bond features and bond types
- Understand bond values and why they fluctuate
- Understand bond ratings and what they mean
- Understand the impact of inflation on interest rates
- Understand the term structure of interest rates and the determinants of bond yields

Chapter Outline

- Bonds and Bond Valuation
- More on Bond Features
- Bond Ratings
- Some Different Types of Bonds
- Bond Markets
- Inflation and Interest Rates
- Determinants of Bond Yields

Bond Definitions

- Bond
- Par value (face value)
- Coupon rate
- fixed rate
- floating rate
- zero coupon
- Coupon payment
- Maturity date
- Yield or Yield to maturity
- indenture
- call provision
- sinking fund
- convertible bonds
- bonds w/ warrants - options that permit holder to buy stock for stated price
- income bonds; debenture (unsecured) vs. mortgage bond
- indexed bonds - e.g., Treasury Inflation indexed bonds

Key Features of a bond:

1. Par value: Face amount; paid at maturity. Assume $1000.

2. Coupon interest rate: Stated interest rate. Multiply by par to get $ of interest. Generally fixed.

example - of Louisiana bearer bond - paid $181.25 semi-annually (how much per year)??

par value = $5000; coupon rate = ??

3. Maturity: Years until bond must be repaid. Declines.

4. Issue date: Date when bond was issued.

Present Value of Cash Flows as Rates Change

- Bond Value = PV of coupons + PV of par
- Bond Value = PV annuity + PV of lump sum
- Remember, as interest rates increase the PV’s decrease
- So, as interest rates increase, bond prices decrease and vice versa

Valuing a Discount Bond with Annual Coupons

- Consider a bond with a coupon rate of 10% and coupons paid annually. The par value is $1000 and the bond has 5 years to maturity. The yield to maturity is 11%. What is the value of the bond?
- Using the formula:
- B = PV of annuity + PV of lump sum
- B = 100[1 – 1/(1.11)5] / .11 + 1000 / (1.11)5
- B = 369.59 + 593.45 = 963.04
- Using the calculator:
- N = 5; I/Y = 11; PMT = 100; FV = 1000
- CPT PV = -963.04

Valuing a Premium Bond with Annual Coupons

- Suppose you are looking at a bond that has a 10% annual coupon and a face value of $1000. There are 20 years to maturity and the yield to maturity is 8%. What is the price of this bond?
- Using the formula:
- B = PV of annuity + PV of lump sum
- B = 100[1 – 1/(1.08)20] / .08 + 1000 / (1.08)20
- B = 981.81 + 214.55 = 1196.36
- Using the calculator:
- N = 20; I/Y = 8; PMT = 100; FV = 1000
- CPT PV = -1196.36

Bond Prices: Relationship Between Couponand Yield

- If YTM = coupon rate, then par value = bond price
- If YTM > coupon rate, then par value > bond price
- Why?
- Selling at a discount, called a discount bond
- If YTM < coupon rate, then par value < bond price
- Why?
- Selling at a premium, called a premium bond

What’s the value of a 10-year, 10% coupon bond if r = 10%?(here C = $100 & F = $1000)

0

1

2

10

10%

. . .

B = ?

100 + 1000

100

100

= $90.91 + . . . + $38.55 + $385.54

= $1,000.

The bond consists of a 10-yr, 10%

annuity of $100/yr plus a $1,000

lump sum at t = 10:

PV annuity

PV maturity value

Value of bond

= $614.46

= 385.54

= $1000.00

INPUTS 10 10 100 1,000

N I/YR PV PMT FV

-1,000

INPUTS

OUTPUT

12

What would happen if expected inflation rose by 3%, causing r = 13%?

INPUTS

10 13 100 1000

N I/YR PV PMT FV

-837.21

OUTPUT

When r rises, above the coupon rate, the bond’s value falls below par, so it sells at a discount.

What would happen if inflation fell, and r declined to 7%?

INPUTS

10 7 100 1,000

N I/YR PV PMT FV

-1,210.71

OUTPUT

When r falls, below the coupon rate,

the bond’s value rises above par,

so it sells at a premium.

Value of price of a security or loan

- PV of stream of contractual CF discounted at the market rate of return
- Fixed coupon rates establish the periodic CF
- Price varies to provide buyer the market rate of return
- if yield falls, price must rise so that new purchases of the bonds will result in the lower yield (when r declined to 7%, price of bond rose to 1,210.71)
- if yields rise, buyers want to pay a lower price (which is the only way that they can obtain the higher yield on the outstanding bond) (when r rose to 13%, price of bond fell to 837.21)

s

The bond was issued 20 years ago and now has 10 years to maturity. What would happen to its value over time if the required rate of return remained at 10%, or at 13%, or at 7%?

1,372

r =7%.

1,211

r =10%.

F

1,000

r =13%.

837

775

30 25 20 15 10 5 0

Years remaining to Maturity

At maturity, the value of any bond must equal its par value.

- The value of a premium bond would decrease to $1,000.
- The value of a discount bond would increase to $1,000.
- A par bond stays at $1,000 if r remains constant.

Example 7.1

- Find present values based on the payment period
- How many coupon payments are there?
- What is the semiannual coupon payment?
- What is the semiannual yield?
- B = 70[1 – 1/(1.08)14] / .08 + 1000 / (1.08)14 = 917.56
- Or PMT = 70; N = 14; I/Y = 8; FV = 1000; CPT PV = -917.56

Interest Rate Risk

- Price Risk
- Change in price due to changes in interest rates
- Long-term bonds have more price risk than short-term 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

What’s interest rate (or price) risk? Does a 1-yr or 10-yr 10% bond have more risk?

Interest rate risk: Rising r causes bond’s price to fall. Longer maturity bond is more sensitive to r change than shorter maturity bond

Computing Yield-to-maturity

- Yield-to-maturity is the rate implied by the current bond price
- Finding the YTM requires trial and error if you do not have a financial calculator and is similar to the process for finding r with an annuity
- If you have a financial calculator, enter N, PV, PMT and FV, remembering the sign convention (PMT and FV need to have the same sign, PV the opposite sign)

YTM with Annual Coupons

- Consider a bond with a 10% annual coupon rate, 15 years to maturity and a par value of $1000. The current price is $928.09.
- Will the yield be more or less than 10%?
- N = 15; PV = -928.09; FV = 1000; PMT = 100
- CPT I/Y = 11%

YTM with Semiannual Coupons

- Suppose a bond with a 10% coupon rate and semiannual coupons, has a face value of $1000, 20 years to maturity and is selling for $1197.93.
- Is the YTM more or less than 10%?
- What is the semiannual coupon payment?
- How many periods are there?
- N = 40; PV = -1197.93; PMT = 50; FV = 1000; CPT I/Y = 4% (Is this the YTM?)
- YTM = 4%*2 = 8%

Table 7.1

- Finding the value of a bond
- Bond Value = C x {1 – [1/(1 + r)t] }/ r + F/(1 + r) t
- Where
- C= Coupon paid each period
- r = Rate per period
- t = Number of periods
- F = Face value (aka “par value” or “maturity value” of bond
- Finding the yield on a bond
- Given a bond value, coupon, time to maturity, & face value, it is possible to find the implicit discount rate, or yield to maturity, by trial and error only. To do this, try different discount rates until the calculated bond value equals the given value (or let a financial calculator do it for you). Remember that increasing the rate decreases the bond value.

Bond Pricing Theorems

- Bonds of similar risk (and maturity) will be priced to yield about the same return, regardless of the coupon rate
- If you know the price of one bond, you can estimate its YTM and use that to find the price of the second bond
- This is a useful concept that can be transferred to valuing assets other than bonds

summary

- Bond prices are inversely related to bond yields
- the price volatility of a longer-maturity security is larger than that of a shorter maturity security, holding the coupon rate constant
- the price volatility of a low couponsecurity is larger than that of a high coupon security, holding maturity constant (extremely low coupon is zero, which has most price volatility, as you will see in advanced finance course)
- Interest rate risk occurs when realized return doesn’t = YTM
- why does this happen? how can it be avoided?

Bond Prices with a Spreadsheet

- There is a specific formula for finding bond prices on a spreadsheet
- PRICE (Settlement, Maturity, Rate, Yld, Redemption, Frequency, Basis)
- YIELD (Settlement, Maturity, Rate, Pr, Redemption, Frequency, Basis)
- Settlement and maturity need to be actual dates
- The redemption and Pr need to given as % of par value
- Click on the Excel icon for an example

Debt

Not an ownership interest

Creditors do not have voting rights

Interest is considered a cost of doing business and is tax deductible

Creditors have legal recourse if interest or principal payments 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 not a liability of the firm and stockholders have no legal recourse if dividends are not paid

An all equity firm can not go bankrupt

Differences Between Debt and EquityThe Bond Indenture

- Contract between the company and the bondholders and includes
- The basic terms of the bonds
- The total amount of bonds issued
- A description of property used as security, if applicable
- Sinking fund provisions
- Call provisions
- Details of protective covenants

Bond Classifications

- Registered vs. Bearer Forms
- Security
- Collateral – secured by financial securities
- Mortgage – secured by real property, normally land or buildings
- Debentures – unsecured
- Notes – unsecured debt with original maturity less than or equal to 10 years
- Seniority
- Senior vs. subordinated

Bond Characteristics and Required Returns

- The coupon rate depends on the risk characteristics of the bond when issued
- Which bonds will have the higher coupon, all else equal?
- Secured debt versus a debenture
- Subordinated debenture versus senior debt
- A bond with a sinking fund versus one without
- A callable bond versus a non-callable bond
- Issuer can refund if rates decline. That helps the issuer but hurts the investor.
- Therefore, borrowers are willing to pay more, and lenders require more, on callable bonds.
- Most corporate & many muni. bonds have a deferred call and a declining call premium. (older US government bonds had call provision)

Bond Ratings – Investment Quality

- High Grade
- Moody’s Aaa and S&P AAA – capacity to pay is extremely strong
- Moody’s Aa and S&P AA – capacity to pay 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 adequate, adverse conditions will have more impact on the firm’s ability to pay

Bond Ratings - Speculative

- Low Grade
- Moody’s Ba, B, Caa and Ca
- S&P BB, B, CCC, CC
- Considered speculative with respect to capacity to pay. The “B” ratings are the lowest degree of speculation.
- Very Low Grade
- Moody’s C and S&P C – income bonds with no interest being paid
- Moody’s D and S&P D – in default with principal and interest in arrears

Bond Ratings Provide One Measure of Default Risk (below Baa/BBB – junk)

What factors affect default risk and bond ratings?

- Financial performance
- Debt ratio
- TIE, FCC ratios
- Current ratios
- Provisions in the bond contract
- Secured vs. unsecured debt
- Senior vs. subordinated debt
- Guarantee provisions
- Sinking fund provisions
- Debt maturity
- Other factors
- Earnings stability
- Regulatory environment
- Potential product liability
- Accounting policies

Government Bonds

- Treasury Securities
- Federal government debt
- T-bills – pure discount bonds with original maturity of one year or less
- 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

Example 7.3

- A taxable bond has a yield of 8% and a municipal bond has a yield of 6%
- If you are in a 40% tax bracket, which bond do you prefer?
- 8%(1 - .4) = 4.8%
- The after-tax return on the corporate bond is 4.8%, compared to a 6% return on the municipal
- At what tax rate would you be indifferent between the two bonds?
- 8%(1 – T) = 6%
- T = 25%

Zero-Coupon Bonds

- Make no periodic interest payments (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, or deep discount bonds
- Treasury Bills and principal only Treasury strips are good examples of zeroes

Floating Rate Bonds

- Coupon rate floats depending on some index value
- Examples – adjustable rate mortgages and inflation-linked Treasuries
- There is less price risk with floating rate bonds
- The coupon floats, so it is less likely to differ substantially from the yield-to-maturity
- Coupons may have a “collar” – the rate cannot go above a specified “ceiling” or below a specified “floor”

Other Bond Types

- Disaster bonds
- Income bonds
- Convertible bonds
- Put bond
- There are many other types of provisions that can be added to a bond and many bonds have several provisions – it is important to recognize how these provisions affect required returns

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 municipal issues
- Treasury securities are an exception

Work the Web Example

- Bond quotes are available online
- One good site is Bonds Online
- Click on the web surfer to go to the site
- Follow the bond search, corporate links
- Choose a company, enter it under Express Search Issue and see what you can find!

Bond Quotations

- Highlighted quote in Figure 7.3
- ATT 6s09 6.4 177 93 7/8 + ¼
- What company are we looking at?
- What is the coupon rate? If the bond has a $1000 face value, what is the coupon payment each year?
- When does the bond mature?
- What is the current yield? How is it computed?
- How many bonds trade that day?
- What is the quoted price?
- How much did the price change from the previous day?

Definitions:

Current yield = Annual coupon pmt.

Current price

Cap gains yield = Change in price

Beginning price

Coupon rate = Annual coupon payment

Par

Exp. total = YTM = Expected + Exp. Capital

return Curr. yld. gains yield

Treasury Quotations

- Highlighted quote in Figure 7.4
- 8 Nov 21 125:05 125:11 -46 5.86
- What is the coupon rate on the bond?
- When does the bond mature?
- What is the bid price? What does this mean?
- What is the ask price? What does this mean?
- How much did the price change from the previous day?
- What is the yield based on the ask price?

Treasury 2 year Note Auction-WSJ 8/28/03 p. C-11

- AUCTION RESULTS
- Here are results of Wednesday's Treasury auction of two-year notes. All bids are awarded at a single price at the market-clearing yield. Rates are determined by the difference between that price and the face value.
- Applications $43,141,607,000
- Accepted bids $25,000,062,000
- Bids at market-clearing yield accepted .89%
- Accepted noncompetitively $836,794,000
- foreign noncompetitively $0
- Auction price (Rate) 99.922 (2.040%)
- Interest rate (this is the "coupon rate") 2.00%
- CUSIP number 912828BJ8
- The notes are dated Aug. 31, 2003, and mature Aug. 31, 2005. until issued, security trades WI - when issued; settles after issue; can be sold “short”

Treasury 2 year Note Auction-WSJ 2/27/03 p. C-15

- AUCTION RESULTS
- Here are results of Wednesday's Treasury auction of two-year notes. All bids are awarded at a single price at the market-clearing yield. Rates are determined by the difference between that price and the face value.
- Applications $53,881,152,000
- Accepted bids $27,000,074,000
- Bids at market-clearing yield accepted 90.26%
- Accepted noncompetitively $819,852,000
- foreign noncompetitively $0
- Auction price (Rate) 99.853 (1.575%)
- Interest rate (this is the "coupon rate") 1.50%
- CUSIP number 912828AV2
- The notes are dated Feb. 28, 2003, and mature Feb. 28, 2005. until issued, security trades WI - when issued; settles after issue; can be sold “short”

Treasury 5 yr. Note Auction-WSJ 2/12/03 p. C-13

- Five-Year Notes
- Here are the results of the Treasury auction of five-year notes. All bids are awarded at a single price at the market-clearing yield. Rates are determined by the difference between that price and the face value.
- Applications $34,162,483,000
- Accepted bids $24,000,032,000
- Accepted at low price 71.96%
- Accepted noncompetitively $237,378,000
- " foreign noncompetitively $30,000,000
- Auction price (Rate) 99.866 (3.029%)
- Interest rate 3% CUSIP number 912828AT7
- The notes are dated Feb. 15, 2003, & mature Feb. 15, 2008.

Treasury 10 year Note Auction-WSJ 2/13/03 p. C?

- AUCTION RESULTS-10 year Treasury notes
- Here are results of Wednesday's Treasury auction of 10-year notes. All bids are awarded at a single price at the market-clearing yield. Rates are determined by the difference between that price and the face value.
- Applications $33,261,722,000
- Accepted bids $18,000,044,000
- Bids at market-clearing yield accepted 51.28%
- Accepted noncompetitively $128,291,000
- " foreign noncompetitively $50,000,000
- Auction price (Rate) 99.304 (3.960%)
- Interest rate (“coupon” rate) 3.875%
- CUSIP number 912828AU4
- The notes are dated Feb. 15 & mature Feb. 15, 2013.
- http://www.publicdebt.treas.gov/of/ofaucrt.htm

Inflation and Interest Rates

- Real rate of interest – change in purchasing power
- Nominal rate of interest – quoted rate of interest, change in purchasing power and inflation
- The ex ante nominal rate of interest includes our desired real rate of return plus an adjustment for expected inflation

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
- r = real rate
- h = expected inflation rate
- Approximation
- R = r + h

Example 7.6

- If we require a 10% real return and we expect inflation to be 8%, what is the nominal rate?
- R = (1.1)(1.08) – 1 = .188 = 18.8%
- Approximation: R = 10% + 8% = 18%
- Because the real return and expected inflation are relatively high, there is significant difference between the actual Fisher Effect and the approximation.

Term Structure of Interest Rates

- Term structure is the relationship between time to maturity and yields, 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
- Normal – upward-sloping, long-term yields are higher than short-term yields
- Inverted – downward-sloping, long-term yields are lower than short-term yields

What factors can explain the shape of this yield curve?

- This constructed yield curve is upward sloping.
- This is due to increasing expected inflation and an increasing maturity risk premium.

Factors Affecting Required Return

- Default risk premium – remember bond ratings
- Taxability premium – remember municipal versus taxable
- Liquidity premium – bonds that have more frequent trading will generally have lower required returns
- Anything else that affects the risk of the cash flows to the bondholders, will affect the required returns

Interest rate determinants

- real risk free rate of interest (r) is interest rate that would exist on a riskless security if no inflation were expected; - rate of interest that would exist on short term Treasury securities in an inflation-free world
- quoted or nominal risk free rate
- (RRF) = r + h, where h is Inflation premium h = (avg. inflation rate expected over life of security) & compensates investors for expected loss of purchasing power.

Interest rate determinants

- DRP - Default risk premium compensates investors for risk that a borrower will default & not pay interest or principal - credit risk premium
- LP - liquidity premium - added to real rate for securities that are not liquid (liquid security can be sold & quickly converted to cash at a fair market value)
- MRP-maturity risk premium - added to longer-term securities to compensate investors for interest rate risk. longer term securities are more price sensitive to interest rate changes than are short-term securities.

r

= real risk-free rate.

T-Bond rate if no inflation;

1% to 4%.

R

= any nominal rate.

RRF

= Rate on T-securities.

define the terms IP, DRP, LP, MRPR = r + h + DRP + LP + MRP

Here:

r = Real risk-free rate

h = Inflation premium

DRP = Default risk premium

LP = Liquidity premium

MRP = Maturity risk premium

What kind of relationship exists between the Treasury yield curve and the yield curves for corporate issues?

- Corporate yield curves are higher than that of the Treasury bond. However, corporate yield curves are not neces-sarily parallel to the Treasury curve.
- The spread between a corporate yield curve and the Treasury curve widens as the corporate bond rating decreases.

AAA-Rated

Hypothetical Treasury and Corporate Yield CurvesInterest

Rate (%)

BB- Continental

Airlines (Junk)

15

10

Treasury

Yield Curve

6.0%

5.9%

5

5.2%

Years to

Maturity

0

0

1

5

10

15

20

How does the volume of corporate bond issues compare to that of Treasury securities?

Gross U.S. Treasury Issuance (in blue)

Investment Grade Corporate Bond Issuance (in red)

600

450

300

150

Billions of dollars

‘95 ‘96 ‘97 ‘98 ‘99

In late 1990s, the volume of investment grade corporate bond issues has overtaken Treasury issues because US government had a surplus & did not need to issue so much debt.

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 important?
- How does inflation affect interest rates?
- What is the term structure of interest rates?
- What factors determine the required return on bonds?

Current events - WSJ- p. C1 & Cred. Markets Col.

- Traders look for indication of whether FOMC (Federal Open Market Committee, the policy-making arm of the Fed, will act when it meets.
- http://www.federalreserve.gov/fomc/should have minutes from the last FOMC meeting.
- When is next FOMC meeting?
- Who is Greenspan?
- What was yield on 10 year T-note on Thurs.? - see Credit Markets column or Treasury Bills, Notes & Bonds report on Friday
- What was Fed Funds rate on Thursday - see Money Rates in WSJ on Friday
- What was close of S&P 500, DJIA, NASDAQ on Thursday, see first page of section C of WSJ on Friday
- what was market reaction to most recently released economic data?

Treasury Bills

- Characteristics
- Sold on discount basis
- Maturities up to one year
- Multiple denominations up to $1 million
- Pricing Treasury Bills
- Treasury bills are priced on a bank discount rate basis, a traditional yield calculation.
- The bank discount rate, rd, is:

Treasury Bills

- The Wall Journal lists T-Bill yields on a bond equivalent basis where the discounted price is the denominator and 365 days is used as the annualizer.
- The effective annual yield assuming compounding twice a year is:Effective Yield = [(Face Value/Price)365/D -1] x 100%

zero coupon securities

- in Treasury bonds, notes & Bills column, we saw 2 “zero coupon” security types:
- U.S. Treasury strips
- Treasury Bills
- B = F/ ( 1 = I/r)2N
- this is just finding price of single payment out 2N periods ( N years; 2N six-month periods) Like a perpetuity
- in Financial Markets and Instruments we price T-bills - not a Fin 325 topic

Bond price volatility- zero coupon

- What is the volatility of a 30-year zero coupon bond if market rates change from 5% to 6%
- B0 = 1000/ (1.05)30 = 231.38
- B1 = 1000/ (1.06)30 = 174.11
- % D B = (174.11 - 231.38)/231.38 = - 32.9%
- conclusion: lower coupon has more volatility ( - 32.9 % vs. -13.8% for 5% coupon)
- _ __________________________________________________
- if assume 60-semi annual compounding periods with INT = r = 2.5%, B0 = 1000/ (1.025)60= 227.28
- or B1 = 1000/ (1.03)60 = 169.73
- % D B = (169.73 - 227.28)/227.28 = - 33.9%

Other Bonds see WSJ 2/27/03 p. B4

- Bond Market DataMajor Bond Indexes: See statistics on indexes tracking Treasurys, U.S. corporate-debt issues, mortgage-backed securities and more, updated at the end of the most recent session. http://online.wsj.com/documents/majbondi.htm
- New York Exchange Bonds p. C15
- Government Agency p. C11
- Mortgage-Backed Securities, CMOs p. B4
- “Tombstones” p. C5
- High Yield Bonds - p. B4
- Tax-Exempt Bonds - p. B4
- Futures prices p. C14
- Options prices p. C11
- stock prices pp. C3-C10
- Foreign Exchange p.C16
- Yield comparisons p. C15

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