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### BCOR 2200Chapter 6

Interest Rates and Bond Valuation

Chapter 6 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

Key Concepts and Skills:

- Know the important bond features and bond types
- Understand how to calculate bond prices
- Understand why prices fluctuate
- Understand bond ratings
- Understand the impact of inflation on interest rates
- Understand the Term Structure of Interest Rates and the determinants of bond yields
- Understand the determinants of bond yields

6.1 Bonds and Bond Valuation

First some terminology:

A 30 year $1,000 bond pays $100per year (Annual payments) and repays the $1,000 in thirty years. The market requires a 9% return on loans to this company.

- COUPON = $100
- COUPON RATE = Coupon/Face = $100/$1,000= 10%
- COUPON PERIOD = Annual
- FACE VALUE or PAR VALUE= $1,000
- TIME (or TERM) TO MATURITY = 30 years
- YIELD TO MATURITY (YTM) = required rate = 9%

A Bond is Defined (or Identified) by:

- Issuer (the borrower)
- A Corporation (corps): GE, GM…
- A Municipality (Munis): Boulder, New York State, E470 Public Highway Authority…
- The Federal Government (Govies): Issued to finance the deficit and debt
- Maturity (when the loan is repaid)
- Usually a date (but in this class we’ll use a time to maturity)
- A ten year bond is a 2024 but we’ll call it a ten year.
- Structure (how the money is repaid)
- Coupon, Zero, Amortizing

A Bond is Defined (or Identified) by:

- Coupon Rate (if applicable)
- 7% Semi-Annual, 10% Annual, or “Zero-Coupon”
- Bond Features
- Callable, Convertible... (we’ll talk about this later)
- Rating
- Rating agencies (S&P, Moodys, Fitch…) issue public rating.
- BBB or higher are Investment Grade
- BB or lower are Non-Investment Grade(junk)

Example: GE 5.125 of 2028 (“five and an eighth of 28”)

- GE corp bond. Matures in 2028. Pays a 5.125% S-A coupon. AAA rated

Calculate the price of the bond:

- 30year $1,000 Face or Par Value, 10% annual coupon and 10%required rate (so YTM = 10%)

Two Components:

- Thirty payments of $100, discounted at 10%:

PV = 100/(1.1)+ 100/(1.1)2 + 100/(1.1)3 + … + 100/(1.1)30 = $942.69

N = 30 I/YR = 10 PMT = 100 PV = -942.69

- One payment of $1,000 in thirty years

PV = 1,000/(1.1)30 = $57.31

N = 30 I/YR = 10 FV = 1000 PV = -57.31

Or both at the same time:

Total Price: = $942.69 + $57.31 = $1,000

N = 30 I/YR = 10 PMT = 100 FV = 1000 PV = -1000

Equal to the Face Value or Par Value

Calculate the price of the bond:

Same bond with different YTMs:

- Why might the bond’s YTM change?
- The issuer’s credit rating has changed
- All rates have changed

30 year $1,000 Face or Par Value, 10% annual coupon

8% required rate (YTM = 8%)

N = 30 I/YR = 8 PMT = 100 FV = 1000 PV = -1225

- Price = $1,225 > $1,000
- Great than the Face Value or Par Value (Premium)

30 year $1,000 Face or Par Value, 10% annual coupon

12% required rate (YTM = 12%)

N = 30 I/YR = 12 PMT = 100 FV = 1000 PV = -839

- Price = $839 < $1,000
- Less than the Face Value or Par Value (Discount)

- A 20 year, $1,000 face-value bond pays an 7% annual coupon and has a discount rate of 9%.
- Calculate the bond’s price
- Is it priced at par, a discount or a premium?
- $217 (Discount)
- $234 (Discount)
- $817 (Discount)
- $1,000 (Par)
- $1,196 (Premium)

- 20 year, $1,000 face-value bond pays an 7% annual coupon and has a discount rate of 9%:
- PMT = $1,000 x 0.07 = $70

N = 20 I/YR = 9% PMT = 70 FV = 1,000 PV = -817

The price is $817.

The answer is C.

Important thing about a bond’s price:

- If YTM = Coupon Rate Price = Par

N = 30 I/YR = 10 PMT = 100 FV = 1000 PV = -1000

- If YTM > Coupon Rate Price < Par
- Priced at a “discount”

N = 30 I/YR = 12 PMT = 100 FV = 1000 PV = -839

- If YTM < Coupon Rate Price > Par
- Priced at a “premium”

N = 30 I/YR = 8 PMT = 100 FV = 1000 PV = -1225

Why?

- A 20 year, $1,000 face-value bond pays an 5.50% annual coupon and is priced at par.
- Calculate the yield to maturity.
- 3.50%
- 5.50%
- 7.50%
- 9.50%
- Not enough information to calculate YTM

- A 20 year, $1,000 face-value bond pays an 5.50% annual coupon and is priced at par.
- If the price equals par, then the YTM equals the coupon rate.

The Answer is B.

Bonds are Generally Issued at Par

- By tradition, a bond is issued at par
- This means the YTM equals the Coupon Rate
- A company that wants to borrow money, goes to the market to determine what rate of return lenders (bond buyers) require on the company’s bonds
- Call this is the Required Return
- Or the Required Yield
- Or the Yield to Maturity (YTM)
- The company then sets the coupon rate of the bond equal to the YTM
- And then the bond is issued at par.

Changes in Price as Time Changes

Back to the $1,000, 30 year, 10% coupon, 10% YTM bond

N = 30 I/YR = 10 PMT = $100 FV = $1,000 PV = -1,000

Now one year has passed. Recalculate the Bonds value:

N = 29 I/YR = 10 PMT = $100 FV = $1,000 PV = -1,000

So still $1,000 since Coupon Rate = YTM!

Examine two components:

- Twenty-nine $100 payments discounted at 10% (No FV)

N = 29 I/YR = 10 PMT = $100 PV = -936.96

- One $1,000 payment in twenty-nine years (No PMTs)

N = 29 I/YR = 10 FV = $1,000 PV = -63.04

29 yrs Total Value = $936.96+ $63.04 = $1,000

30 yrs Total Value = $942.69 + $57.31 = $1,000

One fewer annuity payment but $1,000 FV received one year sooner.

Changes in Price as the YTM Changes

Back to the 30 years to maturity. But required rate is now 11%.

Why?

N = 30 I/YR = 11PMT = $100 FV = $1,000 PV = -913.06

So less than $1,000 since denominator (the rate) has increased

Examine two components:

- Thirty $100 payments discounted at 11%

N = 30 I/YR = 11PMT = $100 PV = -869.38

- One $1,000 payment in thirty years

N = 30 I/YR = 11FV = $1,000 PV = -43.68

- 11% YTM Total Value = $869.38+ $43.68 = $913.06
- 10% YTM Total Value = $942.69 + $57.31 = $1,000

Why is price lower?

Changes in Price as the YTM Changes

- Call this Interest Rate Risk
- It is the Volatility of the bond’s value caused by changes in rates

The Price function:

PV = C/(1 + r)1 + C/(1 + r)2 + C/(1 + r)3 + … + (C + FV)/(1 + r)t

C = Coupon (Fixed)

FV = Face Value (Fixed)

t = Time to Maturity (slow and can see that one coming)

r = YTM or the discount rate

- So what will change the price of the bond by tomorrow?
- Changes in the YTM (aka changes in Rates)!
- So bond holders incur Interest Rate Risk

Determinants of Interest Rate Risk

- What causes Interest Rate Risk?
- Or what characteristics of a bond increase Interest Rate Risk?
- We will see there are three factors that increase Interest Rate Risk:
- Interest rate risk Increases as

Time to MaturityINCREASES

- Longer Maturity, higher interest rate risk
- Interest rate risk Increases as the

Coupon RateDECREASES

- Lower coupon, higher interest rate risk
- Interest rate risk Increases as the

Starting YTMDECREASES

- Lower YTM, higher interest rate risk

Time to Maturity

- The longer the term to maturity (ceteris paribus), the greater the Interest Rate Risk
- The % change in price increases as maturity increases.

The % change in price for the 30 year bond is 10.27%

Starting YTM

- The Lower the Starting YTM, The higher the interest rate risk
- The % change in price increases as the starting YTM decreases

The % change in price when YTM starts at 5% is 15.20%

Starting YTM

- Compare the SLOPE of the line at the Red, Blue and Green dots.
- The slope is steepest at the Red dot
- Steeper slope means greater a change in price (Rise) for given change in rates (Run)
- Interest Rate Risk is measured by the slope

Figure 6.2 from the text

Table and Chart of the Price-Yield Relationship:

Red is a 30 Year Bond

Blue is a 1 Year Bond

(The 30 Year Bond has a steeper slope)

Fig 6.2

- Compare the slope of the RED LINE at 5% and 20% -
- It is much steeper at 5% (greater slope)
- Steeper means greater change in Price (Rise) for given change in rates (Run)
- Interest Rate Risk is measured by the slope

Coupon Rate

- The lower the coupon rate, the greater the Interest Rate Risk
- The % change in price is increases as the coupon rate decreases

The % change in price for the 5% coupon bond is 11.43%

Interest Rate Risk Factors Recap:

- Interest rate risk Increases as

the Time to Maturity INCREASES

- Longer Maturity, higher interest rate risk
- Interest rate risk Increases as

the Starting YTMDECREASES

- Lower YTM, higher interest rate risk
- Interest rate risk Increases as

the Coupon RateDECREASES

- Lower coupon, higher interest rate risk

- Consider Bonds 1 and 2:

Bond 1: 10 year bond, 10% annual coupon, 10% YTM

Bond 2: 20 year bond, 10% annual coupon, 10% YTM

- Consider Bonds 3 and 4:

Bond 3: 10 year bond, 5% annual coupon, 5% YTM

Bond 4: 10 year bond, 10% annual coupon, 5% YTM

Which of the following is true about interest rate risk?

- Bond 1 > Bond 2 and Bond 3 > Bond 4
- Bond 2 > Bond 1 and Bond 3 > Bond 4
- Bond 1 > Bond 2 and Bond 4 > Bond 3
- Bond 2 > Bond 1 and Bond 4 > Bond 3

Bond 1: 10 year bond, 10% annual coupon, 10% YTM

Bond 2: 20 year bond, 10% annual coupon, 10% YTM

Longer Maturity Greater Interest Rate Risk 2 > 1

Bond 3: 10 year bond, 5% annual coupon, 5% YTM

Bond 4: 10 year bond, 10% annual coupon, 5% YTM

Lower Coupon Greater Interest Rate Risk 3 > 4

Which of the following is true about interest rate risk?

- Bond 1 > Bond 2 and Bond 3 > Bond 4
- Bond 2 > Bond 1 and Bond 3 > Bond 4
- Bond 1 > Bond 2 and Bond 4 > Bond 3
- Bond 2 > Bond 1 and Bond 4 > Bond 3

Is Interest Rate Risk Bad?

- Not Necessarily!
- Interest Rate Risk is:

Change in the price for a change in yield

- If RATES go up, bond PRICES go down
- A big down-change in price is bad
- If RATES go down, bond PRICES go up
- A big up-change in price is good!
- So if you think rates will decrease…

then more interest rate risk is good!

Semi-Annual Coupon Bonds

There are special “rules” for bonds that pay Semi-Annual coupons:

- Example: 21 year, 5.125% S-A bond. YTM = 5.88%

Calculate the price:

- Coupon Rate = 5.125% means 0.05125/2 = 0.025625

NOT 0.03 or 0.026 or 0.0256 or…

Use FULL PRECISSION!!

Coupon payment = 0.025625 x $1,000 = $25.625

- YTM = 5.88% means 0.0588/2 = 0.0294

NOT 0.03 or 0.029 (or 3 or 2.9 entered into the calculator)

- Number of Semi-Annual Periods = 21 x 2 = 42

N = 42 I/YR = 2.94 PMT = 25.625 FV = 1000 PV = ?

(Greater or less than $1,000?)

PV = -909.62

- A 15 year, $1,000 face-value bond pays an 8.25%

SEMI-ANNUAL coupon and has a YTM of 10%.

- Calculate the bond’s price
- Note: Both the Coupon Rate and the YTM are quoted at twice the periodic rate (Always! Always! Always!)
- $850
- $865
- $950
- $1,000
- $1,200

- A 15 year, $1,000 face-value bond pays an 8.25%

SEMI-ANNUAL coupon and has a YTM of 10%.

- Calculate the bond’s price
- Note: Both the Coupon Rate and the YTM are quoted at twice the periodic rate (Always! Always! Always!)

N = 15 x 2 = 30

I/YR = 10/2 = 5

PMT = (0.0825/2)1,000 = 41.25

FV = 1,000

PV = -865 (so the answer is B)

Debt

Not an ownership interest

No voting rights

Interest is 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 to elect the board of directors and on other issues

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

Dividends are not a liability of the firm until declared. Stockholders have no legal recourse if dividends are not declared

An all-equity firm cannot go bankrupt (although it can go “out of business”)

6.2 More on Corporate Bond FeaturesDifferences between Debt and Equity:A Bond is Formally called a Bond Indenture

It is a contract between the issuer and the lender

Contract Terms Includes:

- The Basic Terms of the Bonds
- Interest rate, Payment Structure, Repayment Schedule (or Amortization)…
- Total Amount of Bonds Issued
- Assets used to collateralize the bond
- Debentures – unsecured
- Collateralized – secured by financial securities
- Mortgage bonds – secured by real property (land or buildings)
- Notes – unsecured debt with original maturity less than 10 years
- Seniority
- Which bonds gets paid first?
- “Senior Debt” gets paid first
- “Subordinate Debt” get paid next

Some More Terms

More Terms of a Bond Contract:

- Sinking Fund Provisions
- Money set aside to repay the bond is called a sinking provision
- How much money is set aside and when
- Call Provisions
- Price and time when the company can “buy back” or “call” the bond (if it wants)
- Conversion Provisions
- Price and time when the bond holders can convert the bond to shares of stock (if they want)
- Details of Restrictive Covenants
- Must maintain a D/E ratio below a certain level
- May not sell certain assets (the company can’t sell its trucks)
- May not issue new debt senior to this issue
- Must limit dividends (payments to stockholders) Why?

6.3 Bond Ratings

- Bonds that are sold to the public are rated by one of the major credit-rating agencies (S&P, Moody’s, Fitch…)
- The agencies are private companies that are paid by the issuer to produce ratings
- Many investment funds (mutual funds, insurance companies, endowments…) will not buy unrated debt
- Unrated or bonds rated BB or below are called…

Non-Investment Grade or “Junk Bonds”

- Some low quality bond issuers will buy “bond insurance”
- The issuer pays 6% to the bond owners and 5% to the insurance company
- The net of 11% is less than the issuer would have paid without the insurance
- If not insured, the issuer probably would not have been able to sell bonds at all

6.4 Types of bonds

Bonds are Defined by:

- The type of issuer
- Government, Municipality, Corporation
- How is the interest paid?
- Is there a coupon? Annual? Semi-Annual?
- Is the coupon rate fixed or does it float?
- Do you buy it at a discount today and receive all the interest (along with principal) at the end? (Zero-Coupon)
- How is the principal paid?
- All at the end? (Bullet Maturity)
- A little each period? (Self-Amortizing)

Government Bonds

Called Treasury Securities (Treasury Site)

- T-Bills
- Pure discount bonds
- Original maturity of one year or less
- T-Notes
- Coupon debt
- Original maturity between one and ten years
- T-Bonds
- Coupon debt
- Original maturity greater than ten years
- TIPS
- Treasury Inflation Protection Securities
- The Principaladjusts with CPI
- Pays a fixed rate on an adjusted principal amount
- TIPS Rates & Terms; TIPS in Depth

Municipal Bonds

- Issued by state and local governments and Authorities, School Districts, Special Districts…
- It is how they borrow money to build bridges, schools, airports, buy open space…
- Interest earned by the bond owners is tax-exempt at the federal level

Example:

A corporate bond pays 8% and a Muni bond pays 5%.

- If your marginal tax bracket is 30%, which bond to you prefer?

You keep 1 - 0.30 of the corporate coupon

(1-0.30)(0.08) = 0.056 = 5.6% so the 8% corporate bond is better

- If your marginal tax bracket is 40%, which bond to you prefer?

(1-0.40)(0.08) = 0.048 = 4.8% so the 5% muni bond is better

- At what tax bracket are you indifferent between the bonds?

(1 - T)(0.08) = 0.05 T = 1 - (0.05/0.08) = 0.375 = 37.5%

More on Munis

Either General Obligationor Revenue Bonds

- GO bonds are backed by the “full faith and credit” of the issuer
- All a city’s or state’s taxes and revenue sources are committed to paying the coupons and principal
- Revenue bonds (Revs) are backed by revenue from a specific project
- E470 Tolls
- DIA Landing and gate fees

Floating Rate Bonds (Floaters)

- Coupons set at an “Index rate” plus a spread
- Index rates: T-bill rate, T-bonds rate, LIBOR
- If the coupon is T-bills + 200 bps
- And the T-bill rate is 4.75%
- Then the Floater pays 6.75%

Price of a Floater is usually close to Par

- A company’s default risk premium is 200 bps above T-bills
- So coupon rate is set to T-bills + 200
- Then the Coupon Rate = Discount Rate (a.k.a. YTM)
- If Coupon Rate = YTM Price = Par
- As TVM changes, T-bill rate changes and coupon rate changes
- But coupon rate and discount rate are still equal Price = Par
- Unless default risk changes.
- Say default risk increase to 300 bps
- Then Coupon Rate (T-bill + 200) < YTM (T-Bill + 300) Price < Par

LIBOR

London Inter-Bank Offered Rate

- The average rate charged by London banks for off-shore dollar loans
- Off-shore dollars are called “eurodollars”
- So LIBOR is also called the eurodollar rate
- Eurodollars are not the same as euros
- Why London bank rates and not US bank rates?
- London banks are less regulated
- US banks charge the Fed Funds rate “plus”
- LIBOR is determined more by “natural” supply and demand
- But is also affected by the Fed Funds rate

6.5 Bond Markets

- Dealers at securities firms stand ready to buy and sell certain bonds
- They are linked electronically and by phone
- This is called an Over-the-Counter market (OTC)
- Distinguish this from a big room full of traders called an exchange
- There are an extremely large number of bond issues
- Multiple bonds from each company
- Generally illiquidity (low volume) in any single bond issue
- Most recent treasury at each maturity is an exception
- Only one Federal government
- The most recent treasury at each maturity is called the “on-the-run” treasury

Read about bond quotes on pages 187 to 191

There is a HW question on this.

6.6 Interest Rates and Inflation

- Inflation decreases purchasing power
- Nominal return is amount paid
- Real Return factors out inflation
- Measures increase in purchasing power
- Let Nominal Return = 10%, Inflation = 3%
- $100(1.10) = $110 only buys (1.10/1.03) -1 = 6.80% more stuff
- Not 10% more stuff

Interest Rates and Inflation

Before the year:

- Given Ex Ante or ExpectedInflation (h)
- Calculate the Required Nominal Return

After the year:

- Given Ex Post or MeasuredInflation (i)
- Calculate the realized real return

Ex-Ante (or before you hold the bond)

Use EXPECTED INFLATION = h

Example:

- You will hold a bond for one year.
- The Required real return is 5%
- ExpectedInflationis 3%
- Calculate the Required Nominal Return on a bond

R = Nominal Return

r = Required Real Return = 5%

h = expected inflation = 3%

R = (1 + r)(1 + h) - 1

R = (1.05)(1.03) -1 = 8.15%

- You require a real return of 10% on a bond.
- You expect inflation to be 5%
- Calculate the nominal return on the bond.
- 5.0%
- 5.5%
- 10.5%
- 15.0%
- 15.5%

r = required return = 10%

h = expected inflation = 5%

R = Nominal Return

= (1 + r)(1 + h) - 1

= (1.10)(1.05) -1 = 15.5%

The Answer is E

What is the ESTIMATED Nominal Return?

10% + 5% = 15%

Ex-Post (or after you have held the bond)

Use REALIZED or MEASURED INFLAITON= i

Example:

- The nominal return was 8.15%
- Measured Inflationwas 2%
- Calculate the Realized Real Return

R = Nominal Return = 8.15%

r = Required Real Return

i = measured inflation = 2%

r = (1 + R)/(1 + i) - 1

r = (1.0815)/(1.02) – 1 = 6.03% > 5%

- You have earned 12% nominal return on a bond.
- Over the period you held the bond, measured inflation was 4%
- Calculate the realizedREAL return on the bond
- 4.00%
- 7.69%
- 8.00%
- 8.69%
- 12.00%

R = 12% nominal on a bond.

i = 4%

r = (1 + R)/(1 + i) - 1

= (1.12)/(1.04) – 1 = 7.69%

The answer is B.

What is the ESTIMATED realized real return?

12% - 4% = 8%

6.7 The Determinants of Bond Yields (YTM)

The Term Structure of Interest Rates

- Relationship between Rates and the Term to Maturity
- TheYield Curve refers to yields of on-the-run US Treasury bonds
- No default risk premium
- No (or very) little liquidity premium
- US Treasury yield curve site (treasury.gov)
- Dynamic Yield Curve
- Graphical depiction of US Treasury Rates is called
- TheYield Curve
- Lets look at the current Yield Curve
- and some historic Yield Curves

Does the yield curve slope up or down?

Upward (usually): Short rates are lower than long rates

Downward or Inverted (rare): Short rates are higher than Long rates

Three components of government rates:

- Required Real Rate of return
- A premium for Expected Inflation
- Usually increasing in maturity
- But what if it isn’t? People might expect Deflation
- Deflation is lower prices over time
- So required nominal rates decrease over time (maybe)
- A premium for increased Interest Rate Risk
- Longer Maturity, more interest rate risk (See slide #26)

Upward-Sloping Yield Curve

- Shown with constant real rate
- Not necessarily the case
- Increasing interest rate risk premium
- Increasing inflation premium

Some Notes on Bond Terminology

YTM vs. Pure Discount Rates

- The YTM for a 20 year T-bond is an “average” rate for all 40 bond payments (20 semi-annual coupons)
- Each payment has it’s own Pure Discount rate
- Averaged together (sort of) they are the YTM for the whole bond
- The Yield Curverefers to YTMs (which are averaged rates)
- The Term Structure usually refers to Pure Discount rates

YTM is sometimes called the “Promised” yield

- You only earn the YTM if:
- The bond is held to maturity
- Each coupon is reinvested at the same YTM rate
- Think of YTM as
- A rough indication of what the bond pays
- A way to price the bond.
- Pricing convention: The price equals the PV of the CFs discounted by the YTM

What’s Next:

- We just covered how companies borrow money
- Companies issue BONDS
- Next we’ll talk about how sell ownership stakes
- Companies issue STOCKS
- So we’ll talk about stocks
- How they work
- How they are valued

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