Finance 100 Prof. Michael R. Roberts docx

» Maturity or term » Face value or par: Notional amount used to compute interest payments » Coupon rate: Determines the amount of each coupon payment, expressed as an » Priority in case

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z Introduction to bonds and bond markets

z Zero coupon bonds

» Bond Prices Over Time

» Yield Curve Revisited

» Interest rate sensitivity – Duration & Immunization

z Forward Rates

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What is a Bond and What are its Features?

z A bond is a security that obligates the issuer to make interest and principal

payments to the holder on specified dates.

» Maturity (or term)

» Face value (or par): Notional amount used to compute interest payments

» Coupon rate: Determines the amount of each coupon payment, expressed as an

» Priority in case of default

Coupon Rate Face Value Number of Coupon Payments per Year

Repayment Schemes

z Bonds with a balloon (or bullet) payment

» Pure discount or zero-coupon bonds

– Pay no coupons prior to maturity.

» Coupon bonds

– Pay a stated coupon at periodic intervals prior to maturity.

» Floating-rate bonds

– Pay a variable coupon, reset periodically to a reference rate

z Bonds without a balloon payment

» Perpetual bonds

– Pay a stated coupon at periodic intervals.

» Annuity or self-amortizing bonds

– Pay a regular fixed amount each payment period.

– Principal repaid over time rather than at maturity.

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Who Issues Bonds?

z US Government (Treasuries)

» T-bills: 4,13,16-week maturity, zero coupon bonds

» T-notes: 2,3,5,10 year, semi-annual coupon bonds

» T-bonds: 20 & 30-year, semi-annual coupon bonds

» TIPS: 5,10,20-year, semi-annual coupon bond, principal π-adjusted

» Strips: Wide-ranging maturity, zero-coupon bond, IB-structured

z Foreign Governments

z Municipalities

» Maturities from one month to 40 years, semiannual coupons

» Exempt from federal taxes (sometimes state and local as well).

» Generally two types: Revenue bonds vs General Obligation bonds

» Riskier than government bonds (e.g., Orange County)

Who Issues Bonds? (Cont.)

z Agencies:

» E.g Government National Mortgage Association (Ginnie Mae),

Student Loan Marketing Association (Sallie Mae)

» Most issues are mortgage-backed, pass-through securities.

» Typically 30-year, monthly paying annuities mirroring underlying

securities

» Prepayment risk.

z Corporations

» 4 types: notes, debentures, mortgage, asset-backed

» ~30 year maturity, semi-annual coupon set to price at par

» Additional features/provisions:

– Callable: right to retire all bonds on (or after) call date, for call price

– convertible bonds

– putable bonds

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Bond Ratings

Aaa AAA Highest quality Very small risk of default

Aa AA High quality Small risk of default

A A High-Medium quality Strong attributes, but potentially

B B Able to pay currently, but at risk of default in the future

Caa CCC Poor quality Clear danger of default

Ca CC High speculative quality May be in default

C C Lowest rated Poor prospects of repayment

The US Bond Market – Flows

Amount ($bil.) Source: Flow of Funds Data 2005-2007

132.3 104.4

94.5 Consumer Credit

1417.5

53.6 195 307.3

2005

1397.1

213.4 177.3 183.7

2006

1053.2

314.1 214.6 237.5

2007

Mortgages

Corporate Municipal U.S Gov.

Debt Instrument

Dollar volume of bonds traded daily is 10 times that of equity markets!

Outstanding investment-grade dollar denominated debt is about $8.3 trillion (e.g.,

treasuries, agencies, corporate and MBSs

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Zero Coupon Bonds (a.k.a Pure Discount Bonds)

z Notation Reminder:

» V n = B n = Market price of the bond in period n

» F = Face value

» R= Annual percentage rate

» m = compounding periods (annual Æ m = 1, semiannual Æ m = 2,…)

» i = Effective periodic interest rate; i=R/m

» T = Maturity (in years)

» N = Number of compounding periods; N = T*m

z Value a 5 year, U.S Treasury strip with face value of $1,000

The APR is 7.5% with quarterly compounding?

» Approach 1: Using R (APR) and i (effective periodic rate)

» Approach 2: Using r (EAR)

» Approach 3: Using r (periodic discount rate)

?

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Yield to Maturity

z The Yield to Maturity (YTM) is the one discount rate that

sets the present value of the promised bond payments equal to

the current market price of the bond

» Doesn’t this sound vaguely familiar…

z Example: Zero-Coupon Bond

» But this is just the IRR since

⎛ ⎞

Yields for Different Maturities

z Note: bonds of different maturities have different YTMs

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Spot Rates, Term Structure, Yield Curve

z A spot rate is the interest rate on a T-year loan that is to be made today

» r 1=5% indicates that the current rate for a one-year loan today is 5%.

» r 2=6% indicates that the current rate for a two-year loan today is 6%.

» Etc.

» Spot rate = YTM on default-free zero bonds.

z The term structure of interest rates is the series of spot rates r 1 , r 2 , r 3,…

relating interest rates to investment term

z The yield curve is just a plot of the term structure: interest rates against

investment term (or maturity)

» Zero-Coupon Yield Curve: built from zero-coupon bond yields (STRIPS)

» Coupon Yield Curve: built from coupon bond yields (Treasuries)

» Corporate Yield Curve: built from corporate bond yields of similar risk (i.e.,

credit rating)

Term Structure of Risk-Free U.S Interest

Rates, January 2004, 2005, and 2006

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Using the Yield Curve

z We should discount each cash flow by its appropriate discount

rate, governed by the timing of the cash flow

today (Use the term structure from January 2004)

z Generally speaking, we must use the appropriate discount rate

for each cash flow:

z All of our valuation formulas (e.g., perpetuity, annuity)

assume a flat term structure.

» I.e., there is only one discount rate for cash flows received at any point

C PV

r g

=

−

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Interest Rate Sensitivity Zero Coupon Bonds

z Why do zero-coupon bond prices change? Interest rates

change!

z The price of a zero-coupon bond maturing in one year from

today with face value $100 and an APR of 10% is:

bond, the interest rate increase to 15% What is the price of the

i

= +

Characterizing the Price Rate Sensitivity

of Zero Coupon Bonds

z Consider the following 1, 2 and 10-year zero-coupon bonds, all with

3 Longer term bonds are more sensitive to IR changes than short term bonds

4 The lower the IR, the more sensitive the price.

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Quantifying the Interest Rate Sensitivity

of Zero Coupon Bonds – DV01

z What’s the natural thing to do? Compute the derivative

» If we change the interest rate by a little (e.g., 0.0001 or 1 basis point) than

multiplying this number by the derivative should tell me how much the price

will change, all else equal (i.e., DV01 = Dollar Value of 1 Basis Point)

z Alternatively, we can just compute the prices at two different interest rates

and look at the difference: B 0 (i) – B 0 (i+0.0001)

2

2 0

i V

z Consider an amortization bond maturing in two years with

semiannual payments of $1,000 Assume that the APR is 10%

with semiannual compounding

z How can we value this security?

1 Brute force discounting

2 Recognize the stream of cash flows as an annuity

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Replication

z Can we construct the same cash flows as our amortization

bond using other securities?

A First Look at Arbitrage

z What if the bond is selling for $3,500 in the market?

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Valuation of Straight Coupon Bond

Example

z What is the market price of a U.S Treasury bond that has a

coupon rate of 9%, a face value of $1,000 and matures

exactly 10 years from today if the interest rate is 10%

Present Value = Current Price = ?

Valuation of Straight Coupon Bond

General Formula

z What is the market price of a bond that has an annual coupon

C, face value F and matures exactly T years from today if the

required rate of return is R, with m-periodic compounding?

» Coupon payment is: c = C/m

» Effective periodic interest rate is: i = R/m

i

F i

i c

Zero Annuity

V

1 )

1 ( 1

0

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Relationship Between Coupon Bond Prices

and Interest Rates

z Bond prices are inversely related to interest rates (or yields).

z A bond sells at par only if its interest rate equals the coupon

rate

» Most bonds set the coupon rate at origination to sell at par

z A bond sells at a premium if its coupon rate is above the

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YTM and Bond

Price Fluctuations

Over Time

Yield to Maturity Coupon Bonds

z Recall: The Yield to Maturity is the one discount rate that sets the

present value of the promised bond payments equal to the current market

price of the bond

z Prices are usually given from trade prices

» need to infer interest rate that has been used

» This is not the annualized yield, which equals yield* = ( 1 + yield / m) m-1

z Typically must solve using a computer

» E.g., IRR function in excel or your calculator since:

m yield

F m

yield m

yield

c B

/ 1

1 1

−

=

m yield

F m

yield m

yield

c B

/ 1

1 1

−

=

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The Yield Curve Revisited

» Often referred to as “the yield curve”

» Same idea as the zero-coupon yield curve except we use the

yields from coupon paying bonds, as opposed to

zero-coupon bonds

– Treasury notes and bonds are semi-annual coupon paying bonds

» We often use On-the-Run Bonds to estimate the yields

– On-the-Run Bonds are the most recently issued bonds

Interest Rate Sensitivity

Duration

z The Duration of a security is the percent sensitivity of the

price to a small parallel shift in the level of interest rates

» A small uniform change dy across maturities might by 1 basis point.

» Duration gives the proportionate decline in value associated with a rise

in yield

» Negative sign is to cancel negative first derivative

z Alternatively, given a duration DB of a security with price B, a

uniform change in the level of interest rates brings about a

change in value of

1

B dB Duration D

B dy

= = −

B

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Duration of a Coupon Bond

which we can rearrange

1 1 Time in Years "Weight" on until n payment n payment

N

n n

z Compute the duration of a two-year, semi-annual, 10%

coupon, par bond, with face value of $100

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