MGT 326 Ch 6: Bonds (bdh) 1 Bonds Definition: A bond is a type of annuity A bond is a long or short term debt instrument (a loan) issued by corporations.

MGT 326 Ch 6: Bonds (bdh)

1

BondsDefinition: A bond is a type of annuityA bond is a long or short term debt instrument (a loan) issued by corporations and municipal, state and federal agencies.A bond is a contract; it’s an IOU

When a corporation or government agency issues bonds (request a loan, borrows money), it is said to be issuing debtWhen you buy a bond, you become a creditor to the issuing agency which, in turn, becomes a debtor to youBonds are issued in uniform denominations (i.e. $1,000, $10,000, etc.); most corporate and gov’t bonds are for $1,000Bonds are traded (bought and sold); mostly OTCBonds are the most predominate method for financing business and government projectsBonds are a fundamental investment class

Some Terms: Principal, Face Value, Maturity Value, and Par Value: The amount of money the firm borrows and promises to repay at some future date, usually at the maturity dateCoupon Interest Rate: rcpn , the stated annual rate of interest paid on a bond. Coupon Payment: The specified number of dollars of interest paid each period. Bonds most commonly pay semiannual interest. It’s called a “coupon” because……..

Original Maturity: The number of years (or months) to maturity at the time the bond is issued. Also referred to as the “term”. Some bonds have been issued with 100 year original maturity

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More Terms:Maturity Date: A specified date on which the par value of a bond must be repaid. “Maturity” or “term”: The time left until a bond matures. What do you call a 10-year original maturity bond that was issued 8 years ago and thus has only 2 years left to maturity? Answer: a 2-year bond

Coupon Interest RateThis is the cost of debt to the issuer (borrower) A coupon bond pays periodic interestA zero-coupon bond pays interest only once (simple interest) (more on this later); applies mainly to short-term bondsFor both types of bonds, the principle is repaid at maturity

Coupon PaymentExample: a $10k face value bond has a coupon rate of 5.0000% per annum, paid annually. How much is the coupon payment?

CPN(Cpn) = Face Value(rcoupon/m) = $10k(0.05/1) = $500Example: a $10k face value bond has a coupon rate of 5% per annum, paid semiannually. How much is the coupon payment?

CPN(Cpn) = Face Value(rcoupon/m) = $10k(0.05/2) = $250


3

Bond Ratings:Bonds from companies with similar risk characteristics are grouped together into categories as shown below

Moody's Aaa Aa A Baa Ba B Caa CS & P, Fitch AAA AA A BBB BB B CCC D

Junk BondsHigh Quality Investment Grade Substandard Speculative

Risk / rd HigherLower

Each of the rating categories above will have a different term structure of interest rates (see next slide) Standard & Poors, Moody’s and Fitch are the main firms that provide these bond ratingsEach category has sub-ratings:

→Moody’s: 1, 2, 3→S&P: -, no symbol, +→Fitch: a, aa, aa3

Each rating category is also referred to as a “bond market” (i.e. all AAA bonds of the same maturity are traded in the AAA “bond market”)The cost of debt for a particular term in a particular bond market is called the “market interest rate”All bonds in the same bond market and having the same maturity have the same market interest rate (“Law of One Price” from Ch 3)The current market interest rate is also called the “spot rate”A firm’s bond rating can change over time and are influenced by::

Financial health/strength of the issuing company Mortgage (collateral) provisionsSeniority of the debtRestrictive covenantsSinking fund or deferred callLitigation possibilitiesRegulation effects on issuer

Higher risk, higher rd


4

Issuer Name Symbol CPN Rate MaturityRating

Moody’s/S&P/Fitch

High Low Last Change Yield%

COOPERATIEVE CENTRALE RAIFFEIS

RABO.BA 3.375% Jan 2017 Aaa /AA / 103.3640 101.3540 102.6430 0.295000 2.7939

TRANSOCEAN INC

RIG.HG 6.375% Dec 2021Baa3

/BBB- /BBB- 113.9220 113.2830 113.7560 0.478000 4.6140

WELLS FARGO & CO NEW

WFC.GDP 5.250% Oct 2012 A2 /A+ /AA- 103.1500 99.8200 103.1405 -0.025417 N/A

WACHOVIA CORP GLOBAL MTN

WFC.PO 5.500% May 2013 A2 /A+ /AA- 105.5970 102.1100 105.4100 0.008000 N/A

GOLDMAN SACHS GROUP INC

GS.AOK 5.250% Jul 2021 A1 /A- /A 101.4000 98.5150 98.7680 -0.332000 5.4180

EXELON GENERATION CO LLC

EXC.DE 4.000% Oct 2020A3 /BBB /BBB+

106.8730 103.7190 105.8500 0.208000 3.1958

JPMORGAN CHASE & CO

JPM.SEM 4.350% Aug 2021 Aa3 /A /AA- 104.1096 101.7500 102.7450 -1.238490 3.9989

MORGAN STANLEY

MS.MHU 5.500% Jul 2021 A2 /A- /A 102.5000 98.0000 98.18800 -2.482000 5.7510

BANK AMER CORP

BAC.BP 5.625% Jul 2020 Baa1 /A- /A 103.4000 99.5480 102.3490 -0.096500 5.2725

MORGAN STANLEY

MS.HHP 6.000% May 2014 A2 /A- /A 105.8000 104.5500 105.5080 -0.407000 3.3995

WSJ Bond Yields (as of 22 Feb 2012)

AA Bonds

Maturity (yrs) rd

1 4.25%

2 4.56%

3 5.02%

4 5.54%

5 6.13%

10 7.17%

20 8.93%

B Bonds

Maturity (yrs) rd

1 5.75%

2 6.06%

3 6.52%

4 7.04%

5 7.63%

10 8.67%

20 10.43%

Example: Interest Rate Term Structure for Different Bond Ratings

Bonds in different rating category but with the same maturity will have a different rd. Why?

http://cxa.marketwatch.com/finra/BondCenter/BondDetail.aspx?ID=MjE2ODZDQUQyIA==

http://cxa.marketwatch.com/finra/BondCenter/BondDetail.aspx?ID=ODkzODMwQkI0IA==


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The Market Interest Rate (rd) is the Required ROR for all bonds of the same rating and maturity

Main Types of Bonds (By Issuers):

Treasury Bonds: issued by the U.S. Government; also known as “treasuries” or “government bonds”; three types (by maturity):

T-bills: orig. maturity of 1 year or lessTreasury Notes: 1 > orig. maturity < 10 yearsTreasury Bonds: 10 > maturity; the maximum maturity currently available for new bonds is 30 years

Corporate Bonds: issued by private firmsMunicipal Bonds:

issued by state and local govt’s; called “munis”; interest earned on most munis is tax deductible

Foreign Bonds: Subject to all risk/reward characteristics of domestic bondsAdditional risk due to currency exchange rate fluctuation

Bond Issue(s) or Series

Companies don’t issue one bond at a timeThey issue several million $ worth at a time with each bond usually having a $1,000 Face/Maturity ValueExample: Intel issues $50m worth of bonds on 1 Sep 2004, each bond has a Maturity Value of $1,000; that equates to 50,000 bondsThis single grouping of bonds is called a bond “issue” or “series”All bonds in a series have the same bond rating and YTM (more on YTM later)


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Bond Retail Price: Zero-Coupon Bonds

Recall that a zero-coupon bond pays interest only at maturity (simple interest), it does not pay periodic interestthis is true for some types of zero-coupon bonds

In reality most zero-coupon bonds pay no interestthey are, instead, sold at a discount; they are issued as Original Issue Discount (OID) bondsthe cost of borrowing money using a zero-coupon bond is reflected by the issue price, which is significantly lower than par valueAt issue, the coupon rate is usually equal to rd

Bonds are valuated using the TVM principles and techniques we learned in Ch 4

Example 1: A firm with a BB bond rating wants to issue a 1-year $1,000 zero-coupon bond. This will be an OID bond What is the price of the bond at issue if rd for 1-year BB bonds is 3.0000%?

0 1

FV = $1000

VB = ?

rd = 3%

N=1, I/YR=3, FV=1000; PV (VB) = $970.87


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Bond Valuation: Coupon BondsCoupon Bonds can be modeled as simple present value problemsExample: Consider a $1,000 face value bond that pays annual interest and has a maturity of 5 years. Draw the Cash Flow diagram.

The Ch 4 version cash flow diagram for a bond looks like this:

Bond Buyer’s (Holder, Lender) Perspective

Bond Seller’s (Issuer, Borrower) Perspective

Coupon (Interest) Payments (CPN or INT)

Fair Market Valueor VB or Vd = PV

PV = ?

0 1 2 3 4 5

Note: In Ch 4, the interest payments were implied

In this chapter, the bond cash flow diagram looks like this:

0 1 2 3 4 5

The interest payments are implied, but we will treat them as if they were specified; thus the bond looks an annuity (in arrears)We will treat and analyze bonds as though they are annuities but they really aren’t (there’s only 1 payment each period, not 2 as is the case for an annuity as described in Ch 4)It is necessary to do this in order to do more complicated analysis on bonds which is beyond the scope of this course

Face Value (M) ≠ Future ValueCoupon (Interest) Payments (CPN or INT)

Fair Market Valueor VB or Vd = PV

0 1 2 3 4 5

Face Value (M) ≠ Future Value

M = Face Value = $1,000 ≠ Future Value


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m: the number of compounding periods/payments per year T: the number of years left until maturityn: the number of years or months or days, etc. (number of compounding periods/payments) until the bond matures; n = m x TM: Face Value; the principle; the Par ValueCoupon Rate(rcoupon): This is the interest the borrower promises to pay the lender. It is an APR.Coupon Payment (CPN): This is the interest payment (periodic or simple) CPN = Face Value(rCPN/m)rd: The current market interest rate

it is the current cost of money for all bonds of the same maturity and risk category (bond rating)it is the current ROR for all bonds of the same maturity and risk category thus it is an opportunity cost/best investment opportunity WRT risk, i.e. it is the Opportunity Cost of Capital (from Ch’s 3 & 5)since our investment will grow at this rate we must also use it as the discount rate to compute PV

Things that will not change during the life of a bond:→M (the face value)→the Coupon Rate (promised interest rate)→the Coupon Payment (promised interest payment)→these are contractual obligations

Bond Valuation (continued)


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Finding the Retail Price a Bond

We want to know how to determine the appropriate current market price of bonds so we know how much to pay for or charge for that bondThis is the current “market price” or “retail price” or “settlement price” of a bondSymbols: VB or Vd

Basic Approach: Find Present Value of an Annuity

Basic Valuation Equation: VB = CPN/(1 + rd /m)1 + CPN/(1 + rd/m)2 …….+ CPN/(1 + rd/m)n-1 + CPN/(1 + rd/m)n + M/(1 + rd /m)n

Note: There’s an additional term to account for the maturity value (face value, redemption value)

rd changes all the time due to factors we’ve preciously discussedSince rd changes all the time, a bond’s Fair Market Value changes all the time Why?Answer: rd is the rate at which we will discount all future cash flows to find PV. If the discount rate changes, PV must change.

0

CPNVB

n1 2 n - 1

CPN CPN CPN

M +


10

Finding the Retail Price a Bond (continued)When the bond is issued, the Cpn Rate is generally equal to the market interest rate (rd) at that time in order to sell the bond at par value

Example 1: Two years ago Jamaica Jim’s Cruise Lines (BB bond rating) issued $100m worth of $1,000 bonds with an original maturity of 4 years and annual coupon payments. The bonds have a CPN Rate of 6.0000% since that was the cost of debt (market interest rate, rd) for a 4-year loan for all BB rated firms at the time. The current market interest rate (rd) for BB bonds with 2 year maturity is 5.0000%. (i.e. the current cost of a 2-year loan for all BB rated firms is 5.0000%). What is the current retail price (VB) of these bonds? (The coupon due at t = 2 was paid yesterday)

CPN = $60.00

M = Face Value = $1000

0 1 2

PV =VB = $1000.00

Cash Flow Diagram When the Bond Was Issued:Cpn Rate = rcpn = 6%; Current Mkt Interest Rate = rd = 6%Cpn Pmt = M(rd/m) =$1000(0.06/1) = $60.00

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CPN = $60.00

M = Face Value = $1000

0 1 2

PV =VB = ?

Cash Flow Diagram Current Situation:Cpn Rate = rcpn = 6%Current Mkt Interest Rate = rd = 5%Cpn Pmt = M(rcpn/m) =$1000(0.06/1) = $60.00


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Finding the Retail Price a Bond (continued)Example 1: (continued)

Numerical Solution:1) Find CPN: CPN = Face Value(rcpn/m) = $1k(0.06/1) = $602) Find VB:VB = CPN/(1 + rd /m)1 + CPN/(1 + rd /m)2…+ CPN/(1 + rd /m)n + M(1 + rd /m)n

= 60/(1 + 0.05)1 + 60/(1 + 0.05)2 + 1000/(1 + 0.05)2 = 60/(1.05) + 1060/(1.05)2

= 60/1.05 + 1060/1.1025 = 57.1429 + 961.4512 = $1,018.59

Why do we discount the cash flows using rd?Why is the current market price greater than the face value?Financial Function Solution:1) Find CPN: CPN = Face Value(rcpn/m) = $1k(0.06/1) = $602) Clear your calculator: [2nd, CLR TVM]3) Set/ensure 1 payment per year [2nd, P/Y, 1, ENTER]4) Set payment timing to end of year: [2nd, BGN, 2nd, SET] (Note: the “BGN” should NOT appear in the display)5) Enter parameters:

Enter N [2, N]Enter discount rate (rd) [5, I/Y]Enter CPN payment [60, PMT]Enter Face Value [1000, FV] Note: the FV(Future Value) of two payments of $60 over two years is not $1,000! When working with bonds, the FV key on the TI BA II PLUS can be interpreted as meaning “Face Value”Find VB [CPT,PV] and voila! PV(VB) = $1.018.59


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0 1 2rCpn = 6%

Beginning Balance: $0.00 $0.00 $60.00 Interest Earned: $0.00 $60.00 $63.60 Repay Principle: $1,000.00 Ending Balance: $0.00 $60.00 $1,123.60

Future Value

= $60 + $60 x 0.06

M = Face Value

Finding the Retail Price a Bond (continued)

Example 1 (cont)Here’s an accounting explanation of what happens at each time period

More Bond Terms:Premium Bond: a bond whose current market value is greater than its par value (rd < Cpn Rate)Discount Bond: a bond whose current market value is less than its par value (rd > Cpn Rate)When a bond is first issued it is referred to as a new issueOnce a bond has been on the market a while, it’s referred to as a seasoned issue or an outstanding bond


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Finding the Retail Price a Bond (continued)

Example 2: (other than 1 compounding period per year): What is the retail price (VB) of a 3-year, $10,000 Face Value bond that has a coupon rate of 6.0000% p.a. and has semiannual coupon payments? The current market interest rate (rd) for this bond is 7.0000%.

0 4 5 61 2 31 2 3 yrs

$10,000CPN?Coupon Rate = 6%rd = 7%rperiodic = ?

VB = ?

m = 2T = 3 yearsn = m x T = 6


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Yield to Maturity (YTM)Definition: The average rate of return earned on a particular bond if it is bought at its current price and held to maturity.The ROR or “Internal ROR” based on its current price It is the discount rate that will cause the sum of the PVs of all future cash flows to equal the current market price of the bond.

VB = CPN/(1 + YTM /m)1 + CPN/(1 + YTM /m)2……+ CPN/(1 + YTM /m)n-1

+ CPN/(1 + YTM /m)n + M(1 + YTM /m)n


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Yield to Maturity (YTM) (continued)

Example: (finding YTM): The current retail price of a 4-year, $10,000 AA bond issued by GM paying an annual coupon rate of 5.0000% p.a. is $9,913.89. What is the yield to maturity (YTM) of this bond?

Approach: Given all the normal bond parameters (i.e. face value, coupon rate and maturity) and the current price of the bond, find YTM (this is like solving for r of an annuity)

0 1 2 3 4

M = $10,000

Price = 9,913.89

rd = YTM = ?rcpn = 5%

CPN

Financial Calculator Solution: 1) Find CPN: CPN = Face Value(rcpn/m) = $10k(0.05/1) = $5002) Enter parameters:Set payments per year [2nd, P/Y, 1]Enter number of periods [ 4, N]Enter Price [-9913.89, PV] Enter CPN payment [500, PMT] Enter Face Value [10000, FV]Find YTM [CPT,I/Y] and voila! YTM (rd) = 5.2442%

Interpretation: If you buy this bond today it will earn 5.2442% p.a. until it matures.This bond has an Internal ROR of 5.2442%The price is based on the current market interest rate (rd) so….. YTM should be equal to the current market interest rate (rd)


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0 1 2 3 4

M = $1,000

Price = $973.23

rd = YTM = ?rcpn = 6.25%

CPN

m =2T= 2n = m x T = 2 x 2 = 4

Yield to Maturity (YTM) (continued)

Example: (finding YTM w/ other-than-annual cpn payments): The current FMV of a $1,000 BB bond issued by Home Depot paying a semi-annual coupon rate of 6.2500% APR with 2 years left to maturity is $973.23. What is the yield to maturity of this bond?


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Why would we want to know a bond’s YTM?Answer: to determine if the bond is selling at an appropriate market priceEach bond issue/series from a particular firm has a particular YTMAll BB bonds of the same maturity will have the same market interest rate and thus the prices of all these bonds should be equal or very close to each other (Law of One Price)Thus the YTMs for all BB rated bonds of similar maturity will be equal to the market interest rate (rd)However, the price of a particular firm’s bonds is a result of market perception of that firm’s financial healthIf a bond’s reported YTM is higher than that of other bonds in the same bond market it means the bond is selling for less than FMV

What Happens When rd = Cpn Rate?When the market interest rate (rd) equals the coupon interest rate (rcpn) the market value of a bond equals its face value (VB = M)

Example: What is the current fair market value (VB) of a 2-year, $1,000 Face Value bond that has a coupon rate of 6% APR and has annual coupon payments? The current market interest rate (rd) for this bond is 6%. (rcpn = rd). Answer: $1,000

VB = CPN/(1 + rd)1 + CPN/(1 + rd)2…+ CPN/(1 + rd)n + M/(1 + rd)n

= $1,000(0.06)/(1 + 0.06)1 + $1,000(0.06)/(1 + 0.06)2 + $1,000/(1 + 0.06)2 = $60/1.06 + $60/1.1236 + $1,000/1.1236 = $56.6038 + $53.3998 + $899.9964 = $1,000When the fair market value of a bond equals its face value, the bond is said to be “trading at par” (recall: face value = par value)When the bond is first issued, the Cpn Rate is generally = to rd in order to sell the bond at or very close to par value


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Finding the No Arbitrage Price / Theoretical Value / Fair Market Value of a Bond

The “Retail Price” of a bond as we discussed earlier is determined using one discount rate (rd or YTM) to discount all future cash flows to t = 0 and then adding them upThe “No Arbitrage Price” of a bond is found the same way (by discounting all future cash flows to t = 0 and adding them up). However, a different discount rate is used for each future cash flow

Recall the topic of yield curves from Ch 5

8

6

4

2

0

Inte

rest

Rat

e (%

)

1 5 10 20

Maturity

Note that the market interest rate is different at each year of maturityThis means that at each different future point in time when the coupon is paid, the opportunity cost of capital will be differentTo accurately determine the No Arbitrage Price / Theoretical Value / Fair Market Value, we must discount each future coupon payment at the appropriate opportunity cost of capital


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Finding the No Arbitrage Price / Theoretical Value / Fair Market Value of a Bond (continued)

Example: Consider a four-year U.S. Treasury $1,000 face value note paying 3.0000% p.a. coupon. What is the no arbitrage price of this bond? (The most recent coupon payment was yesterday).

The current U.S Treasuries yield curve is:5

4

3

2

1

0

Inte

rest

Rat

e (%

)

Maturity0 1 2 3 4

2.5400%

3.5500%

4.0600%4.3700%

1) Find CPN: CPN = Face Value(rcpn/m) = $1k(0.03/1) = $302) Find VB:VB = CPN1/(1 + rd,1 /m)1 + CPN2/(1 + rd,2 /m)2 + CPN3/(1 + rd,3 /m)3 + + CPN4/(1 + rd,4 /m)4 + M(1 + rd,4 /m)4

= 30/(1 + 0.0254/1)1 + 30/(1 + 0.0355/1)2 + 30/(1 + 0.0406/1)3 + 30/(1 + 0.0437/1)4 + 1,000/(1 + 0.0437/1)4 = 30/(1.0254) + 30/(1.0723) + 30/(1.1268) + 30/(1.1866) + 1000/(1.1866) = 29.2569 + 27.9783 + 26.6238 + 25.2824 + 842.7471 = $951.89 = No Arbitrage Price / Theoretical Value / Fair Market Value


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Finding the No Arbitrage Price / Theoretical Value / Fair Market Value of a Bond (continued)

Example (continued): Consider a four-year U.S. Treasury $1,000 face value note paying 3.0000% p.a. coupon. It’s YTM is 3.1500%. What is the retail price of this bond? (The most recent coupon payment was yesterday).

1) Find CPN: CPN = Face Value(rcpn/m) = $1k(0.03/1) = $302) Find VB(Using Calculator Financial Functions:N=4, I/Y= 3.15, PMT=30, FV=1000; CPT PV, VB = $994.44

Point:Licensed bond brokers/traders (bond market insiders) sell bonds to each other at the No Arbitrage Price / Theoretical Value / Fair Market Value (i.e. at $951.89)Bond brokers sell bonds to you and me (bond market outsiders) at the retail price plus what ever extra profit current market conditions will allow (i.e. at least $994.44)


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Relationship Between rd and VB:When rd goes down (over time), VB goes up [rd↓, VB↑]When rd goes up (over time), VB goes down [rd↑, VB↓]

Example: A $1,000 bond issued by Home Depot paying a semi-annual coupon rate of 6.2500% APR with 2 years left to maturity has a current rd of 7.7200%. a) What is the VB of this bond? Answer: $973.23b) What is the VB of this bond if rd fell to 4.5000%?

Answer: $1,033.12c) What is the VB of this bond if rd rose to 8.5000%?

Answer: $959.40

rd (YTM)

$880.00

$900.00

$920.00

$940.00

$960.00

$980.00

$1,000.00

$1,020.00

$1,040.00

$1,060.00

$1,080.00

3.00% 3.50% 4.00% 4.50% 5.00% 5.50% 6.00% 6.50% 7.00% 7.50% 8.00% 8.50% 9.00%

@ rd = 6.25%, this bond trades at par (VB = $1,000)

Val

ue

@ rd > 6.25%, VB < $1,000@ rd < 6.25%, VB > $1,000

Cpn Rate = 6.25%


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What happens to the Price (Value) of a bond over time?Example: A $1,000 face value bond with an annual coupon rate of 5.0000% p.a., paying annual coupon payments and a maturity of 5 years is issued at t = 0. What is its value at t = 1 if it’s trading at par?

M = $1,000CPN

VB1 = ?

Cpn Rate = 5%rd = 5%

0 1 2 53 4

VB1 = $1,000 (rd = Cpn Rate, it’s trading at par)

What is the bond’s value at t = 1 if market interest rates have dropped to 2.8000%? Answer: VB1 = $1,082.17What is the value of the bond with only 3 years remaining until maturity (t=2) if the market rate is 2.8000%? Answer: VB2 = $1,062.47What is the value of the bond with only 1 year remaining until maturity (t=4) if the market rate is 2.8000% ? Answer: VB4 = $1,021.40

As time goes on, the fair market value of a bond approaches (converges) to its par value (M, FV, etc)As a bond approaches maturity, value fluctuation due to changes in rd decreases

the value of a premium bond would decrease towards $1,000.the value of a discount bond would increase towards $1,000.

At maturity, the value of any bond must equal its par value (after the last coupon payment is paid)


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What happens to the Value of a bond over time? (continued)

As time goes on, the fair market value of a bond approaches (converges) to its par value (M, FV, etc)

Adjusting Bond FMV by Prorating the Upcoming Coupon Pmt

Notice that the lines for the coupon bonds are not smoothThis is because the bond value drops by exactly the value of the coupon payment just after the coupon is paidThen the value of the bond slowly rises until the next coupon is paid. This is because we have to adjust FMV for the upcoming coupon paymentThis was not done in previous examples because we calculated VB just after the most recent coupon was paid


24

Adjusting Bond FMV By Prorating the Upcoming Coupon Pmt (continued)Consider the bond from Example 1 again: Jamaica Jim’s Cruise Lines (BB bond rating) issued $100m worth of $1,000 bonds with an original maturity of 4 years and annual coupon payments. The bonds have a CPN Rate of 6.0000% since that was the cost of debt (market interest rate, rd) for a 4-year loan for all BB rated firms at the time. The current market interest rate (rd) is 5.0000%. What is the value of the bond?

$60.00$1000.00

0 1 2PV =VB = ?

43

VB = CPN/(1 + rd /m)1 + CPN/(1 + rd /m)2…+ CPN/(1 + rd /m)n + M(1 + rd /m)n

= 60/(1+0.05)1 + 60/(1+0.05)2 + 60/(1+0.05)3 + 60/(1+0.05)4 + 1000/(1+0.05)4

= 60/(1.05) + 1060/(1.05)2 + 60/(1.05)3 + 60/(1.05)4 + 1000/(1.05)4

= 60/1.05 + 60/1.1025 + 60/1.1576 + 60/1.2155 + 1000/1.2155 = 57.1429 + 54.4218 + 51.8303 + 49.3621 + 822.7025 = $1,035.46 (Note: since rd = 5% and rcpn = 6%, the bonds won’t sell at par)

$60.00 $60.00 $60.00

Now exactly two years go by. The coupon due at year 2 will be paid today. What is the value of the bond is rd = 5%?

$60.00$1000.00

0 1 2PV =VB = ?

$60.00$60.00

VB = 60 + 60/(1+0.05)1 + 60/(1+0.05)2 + 1000/(1+0.05)2

= 60 + 60/(1.05) + 1060/(1.05)2 + 1000/(1.05)2

= 60 + 60/1.05 + 60/1.1025 + 1000/1.1025 = 60 + 57.1429 + 54.4218 + 907.0295 = $1,078.59

What is the value of the bond the day after the coupon at year 2 was paid if rd = 5%?

$60.00$1000.00

0 1 2PV =VB = $1,018.59 (as previously computed on p. 11)

$60.00


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Now exactly one more year goes by. The coupon due at year 3 will be paid today. What is the value of the bond is rd = 5%?

$60.00$1000.00

0 1PV =VB = ?

$60.00

VB = 60 + 60/(1+0.05)1 + 1000/(1+0.05)1

= 60 + 60/(1.05) + 1000/(1.05)1

= 60 + 60/1.05 + 1000/1.05 = 60 + 57.1429 + 952.3810 = $1,069.52

What is the value of the bond the day after the coupon at year 3 was paid if rd = 5%?

$60.00$1000.00

0 1PV =VB = ?

VB = 60/(1+0.05)1 + 1000/(1+0.05)1

= 60/(1.05) + 1000/(1.05)1

= 60/1.05 + 1000/1.05 = 57.1429 + 952.3810 = $1,009.52

A plot of the VB’s looks like this:

This is why the line in the diagram on p. 23 is jagged

$1,009.52

$1,069.52

$1,018.59

$1,078.59

Adjusting Bond FMV By Prorating the Upcoming Coupon Pmt (continued)


26

Computing the Retail Price of a Bond Between Coupon Payments (The “Clean Price”)

Example: A $1,000 face value Smithfields Foods bond matures 1 Jun 2012. It has a 7.0000% coupon rate and pays coupons semiannually. Its YTM as of 1 Oct 2010 was 6.1970%. What was this bond’s FMV as of 1 Oct 2010?

The seller is entitled to accrued interest on the next coupon paymentThe Hard Way:Step 1: Find CPN: CPN = Face Value(rcpn/m) = $1k(0.07/2) = $35Step 2: Discount all future cash flows back to 1 Dec 2010 $35 + [P/Y=2, N=3, I/Y=6.1970, PMT=35, FV=1000; CPT PV] PV = $35 + $1,011.3354 = $1,046.3354Step 3: Discount this value back to 1 Oct 2010 a) Compute n: n = 61/183 = 0.3333 b) Find PV: N=0.3333, I/Y=6.1970, FV=1,046.3354; CPT PV, PV = $1,035.7465Step 3: Find the accrued interest (prorated cpn pmt) owed to the seller Accrued interest: $35(122 / (122 + 61)) = $35(122 / 183) = $23.3333Step 4: Compute VB (Settlement Value) VB = $1,035.7465 - $23.3333 = $1,012.41 = “Clean Price”

Summary:The “Dirty Price” is the retail price computed just after a coupon is issued (i.e. Example 1 on p. 11)Clean Price = Dirty Price – Accrued Interest on upcoming coupon

1 Dec ’10$35.00

1 Oct ‘10PV =VB = ?

1 Jun ‘101 Jun ’11

$35.001 Dec ’11

$35.001 Jun ’12

$35.00 + $1000.00

122 days + 61 days = 183 days



27

Computing the Value of a Bond Between Coupon Payments (The “Clean Price”)Example (repeated): A $1,000 face value Smithfields Foods bond matures 1 Jun 2012. It has a 7.0000% coupon rate and pays coupons semiannually. Its YTM as of 1 Oct 2010 was 6.1970%. What was this bond’s FMV as of 1 Oct 2010?

The Easy Way: Use the Bond Worksheet1) Access Bond Worksheet: [2nd, BOND]2) Clear the worksheet [2nd, CLR WORK]3) Enter parameters:

Enter settlement date SDT (PV date)[10.01.10, ENTER↓]Enter coupon rate (not coupon payment) CPN [7, ↓]Enter redemption (maturity) date RDT [6.01.12, ENTER]Enter YLD (yield to maturity) {press down arrow until YLD appears} [6.197, ENTER, ↓]Find VB (redemption value as a % of face value); {press down arrow unit PR = 0.000000 appears} CPT, PRI(VB) = 101.2413%

multiply PRI by 10 → VB = $1,012.41 = “Clean Price”

Point: Very few bonds get traded right after a coupon payment Most bonds get traded somewhere between coupon payments; therefore the retail price must be computed as the clean price

1 Dec ’10$35.00

1 Oct ‘10PV =VB = ?

1 Jun ‘101 Jun ’11

$35.001 Dec ’11

$35.001 Jun ’12

$35.00 + $1000.00

122 days + 61 days = 183 days



28

Computing the Value of a Bond Between Coupon PaymentsExample: A $5,000 face value Diamond Jim’s Corporation bond matures 1 Sep 2013. It has a 6.2500% coupon rate and pays coupons semiannually. Its YTM as of 1 Oct 2010 was 5.8250%. What was this bond’s FMV as of 1 Oct 2010?

Use the Bond Worksheet


29

Risk of Long-term Bonds vs Short-term Bonds

30 25 20 15 10 5 0Years to Maturity

Value Fluctuation Due to Changes in rd

M Val

ue

Dec

reas

ing r

dIn

crea

sin

g r

d

rd > Coupon Rate

rd < Coupon Rate

As time goes on, the fair market value of a bond approaches (converges) to its par value (M, FV, etc)The price range of a 30-year bond is much greater than that of shorter maturity bondsThe prices of bonds with longer remaining maturity are much more influenced by a change in rd than the price of bonds with

shorter remaining maturity Why?Price volatility of long maturity bonds is greater than that of short maturity bondsLonger term bonds are more sensitive to change in rd than shorter term bondsLong maturity bonds are riskier than shorter maturity bonds; this is reflected by the higher rd

Pric

e R

an

ge


30

1-yr Bond

20-yr BondFV = $1,000rcoupon = 7.50%Annual Coupon

rd = rcoupon thus VB = FV

Investment Implications of Bond Sensitivity to a change in rd

When market interest rates are falling, it’s good to have an inventory of long-term bonds to sell:

when rd decreases, bond values risesince long-term bond prices are more sensitive (react more) to changes in rd, the profits from selling them will be greater than for short term bonds

When market interest rates bottom out and start to rise, it’s better to deal in (buy, hold, sell) short-term bonds:

when rd increases, bond prices fallshorter-term bonds are less sensitive to changes in rd and will have lower int. rate (price) and reinvestment risk

Can we foresee what interest rates might be?

Note: Coupon pmts are not prorated

Risk of Long-term Bonds vs Short-term Bonds


31

Other Risks Associated With Bonds

Interest Rate Risk (Bond Price Risk): This is the uncertainty concerning the future value of a bond due to changes in rd

Reinvestment Risk: The risk that income from a bond portfolio will vary because cash flows have to be reinvested at current market ratesWhen a bond matures, the owner may not be able to reinvest the face value of that bond at a rate at least a favorable as the one currently paid by that bond.

These two types of risk are what is really being compensated by the Maturity Risk Premium (MRP)

Point: Shorter term bonds have less exposure to these types of risk than longer term bonds. That’s why MRP is smaller for short-term bonds than for long-term bondsA possible decision criteria for selecting bond maturities


32

Realized Capital Gains Yield

This is how much you earn if/when you sell a bondIt is capital gain(loss) due to changes in rd It’s a percentageIt’s like computing Rate of Return (ROR)

Example: At the beginning of the year, a $1,000 bond paying 8.25% APR semiannually bought for $1,048 (FMV). At the end of the year, this bond was sold for $1,059 (FMV). What is the capital gains yield on this bond?

Capital Gains Yield = (VB1 – VB0)/VB0 = (End Value - Begin. Value)/ Begin. Value = ($1,059 - $1,048)/$1,048 = 0.010496 = 1.0496%

Realized Total Yield

This is the total return from interest and capital gains when you sell a bondUsually expressed as an annual rateRealized Total Return = EAR of Cpn Rate + Realized Cap. Gains Yield(Important Assumption: interest payments are reinvested at the coupon rate)


33

Realized Total Yield (continued)

Example: One year ago you bought a AA rated $1,000 face value bond which pays 8.2500% coupon rate semiannually and has an 8 year term. You purchased the bond immediately after it issued its most recent coupon and at that time, the market interest rate for all AA rated bonds of 8 year maturity was 8.6000%. You sold this bond today, immediately after it paid its most recent coupon at par. What is the total return on this bond?

1) Find EAR Cpn Rate : [2nd, ICONV, 8.25, ENTER, ↓, ↓, 2, ENTER, ↓, ↓, CPT] = 8.4202% p.a.2) Find Realized Capital Gains Yield: a) (VB1 – VB0)/VB0

i) Find VB0 (the beginning value) T=8, m=2; n=T x m = 8 x 2 = 16 Cpn = FV(rcpn/m) = $1,000(0.0825/2) = $41.25 P/Y=2, N=16, I/Y=8.6, PMT=41.25, FV=1000; CPT PV: PV(VB0) = $980.0525ii) Find VB1 (the ending value) Since the bond sold at par,

VB1 = $1,000iii) (VB1 – VB0)/VB0 = ($1,000 - $980.0525) / $980.0525 = 2.0354% p.a.

3) Find Realized Total Yield: 8.4202% + 2.0354% = 10.4556% p.a.


34

Types of Bond (From Ch 14)

Mortgage Bonds: A bond with some fixed asset presented as collateral

higher claim priority than unsecured bondsusually have lower interest rates

Debentures: A long-term bond that is not secured by a mortgage on a specific property

higher interest rates than secured bonds Why?Subordinated Debentures: A bond that is specifically designated to be of lower claim priority than other bonds

Bonds with Provisions (Indentures, Restrictions):

Convertible bonds: A bond that can be converted to a fixed number of stock shares specified at time of issue (it’s like a stock option)

have a lower coupon rate than non-convertible bonds Why?offers the holder a chance at capital gains through stock

Warrants: Similar to a convertible bond but the price of the stock is specified at time of issue, not the number of sharesoffer a chance at capital gains if stock price at time of conversion is higher than the specified stock price

Income Bond: pays interest only if the issuing firm has sufficient income to pay the interest

A real safe deal from the issuer’s stand point; this type of bond can’t bankrupt a firmThese type bonds pay high relative interest (otherwise, who would buy them?)


35

Putable Bonds: a bond that can be redeemed for cash at the bond holder’s option

pay lower interest rates Why?when would it be a good idea to redeem this bond?

Callable Bond: a bond that has a “call provision” which allows the issuer to redeem the bond for cash at the issuer’s option prior to the stated maturity date

pay higher interest rates. Why?when would it be a good idea to redeem this bond?Call premium: Compensation paid to the holder by the issuer if/when the bond is “called”

MGT 326 Ch 6: Bonds (bdh) 1 Bonds Definition: A bond is a type of annuity A bond is a long or short term debt instrument (a loan) issued by corporations.

Documents

bond ratings

year bond coupon

zerocoupon bond

face value bond

coupon rate

bonds bdh

particular bond market

aaa bond market