Booth Cleary 2nd Edition Chapter 6 - Bond Valuation and Interest Rates
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Transcript of Booth Cleary 2nd Edition Chapter 6 - Bond Valuation and Interest Rates
Chapter 6 – Bond Valuation and Interest Rates
INTRODUCTION TO CORPORATE FINANCE
Second Edition
Laurence Booth • W. Sean Cleary
Prepared by
Ken Hartviksen
Lecture Agenda
1. Basic Structure of Bonds1. Basic Structure of Bonds
2. Bond Valuation 2. Bond Valuation
3. Bond Yields3. Bond Yields
4. Interest Rate Determinants4. Interest Rate Determinants
5. Other Debt Instruments5. Other Debt Instruments
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Learning Objectives
The basic features of different types of bonds How to value bonds given an appropriate discount rate How to determine the discount rate or yield given the
market value of a bond How market interest rates or yields affect bond investors How bond prices change over time The factors (both domestic and global) that affect interest
rates
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Basic Structure of Bonds
The Basic Structure of Bonds
What is a bond? In its broadest sense, a bond is any debt
instrument that promises a fixed income stream to the holder
Fixed income securities are often classified according to maturity, as follows:
Less than one year – Bills or “Paper” 1 year < Maturity < 7 years – Notes < 7 years – Bonds
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
The Basic Structure of Bonds
A typical bond has the following characteristics: A fixed face or par value, paid to the holder of the
bond, at maturity A fixed coupon, which specifies the interest payable
over the life of the bond Coupons are usually paid either annually or semi-
annually A fixed maturity date
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Bonds may be either: Bearer bonds Registered bonds
Bond indenture - the contract between the issuer of the bond and the investors who hold it
The market price of a bond is equal to the present value of the payments promised by the bond
(See the basic pattern of cash flows from a traditional bond on the next slide)
The Basic Structure of Bonds
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
The Basic Structure of BondsCash Flow Pattern for a Traditional Coupon-Paying Bond
0 1 2 3 … n
I I I I I
F
0 1 2 3 … n
I I I I I
F
FIGURE 6-1
I = interest payments, and F = principal repayment
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Cash Flow Pattern of a Bond
The Purchase Price or Market Price of a bond is simply the present value of the cash inflows, discounted at the bond’s yield-to-maturity
0 2 3 4 n1
Coupon Coupon Coupon Coupon Coupon +Face Value
Purchase Price
Cash Inflows to the Investor
Cash Outflows to the Investor
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Bond indenture is the contract between the issuer and the holder. It specifies:
Details regarding payment terms Collateral Positive and negative covenants Par or face value (usually increments of $1,000) Bond pricing – usually shown as the price per $100
of par value, which is equal to the percentage of the bond’s face value
The Basic Structure of Bonds
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Term-to-maturity – the time remaining to the bond’s maturity date
Coupon rate – the annual percentage interest paid on the bond’s face value; to calculate the dollar value of the annual coupon, multiply the coupon rate by the face value
If the coupon is paid twice a year, divide the annual coupon by two
Example: A $1,000 bond with an 8% coupon rate will have an $80 coupon if paid annually or a $40 coupon if paid semi-annually
The Basic Structure of Bonds
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Security and Protective Provisions
Mortgage bonds – secured by real assets Debentures – either unsecured or secured with
a floating charge over the firm’s assets Collateral trust bonds – secured by a pledge of
financial assets, such as common stock, other bonds or treasury bills
Equipment trust certificates – secured by a pledge of equipment, such as railway rolling stock
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Security and Protective Provisions
Covenants Positive covenants – things the firm agrees to do
Supply periodic financial statements Maintain certain ratios
Negative covenants – things the firm agrees not to do Restricts the amount of debt the firm can take on Prevents the firm from acquiring or disposing of
assets
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
More Bond Features
Call feature – allows the issuer to redeem or pay off the bond prior to maturity, usually at a premium
Retractable bonds – allows the holder to sell the bonds back to the issuer before maturity
Extendible bonds – allows the holder to extend the maturity of the bond
Sinking funds – funds set aside by the issuer to ensure the firm is able to redeem the bond at maturity
Convertible bonds – can be converted into common stock at a pre-determined conversion price
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Bond Valuation
Bond Valuation
The value of a bond is a function of: Par value Term to maturity Coupon rate Investor’s required rate of return (discount rate is
also known as the bond’s yield to maturity)
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Bond ValueGeneral Formula
)k(
Fk
)k(IB
nbb
nb
1
111
1
[ 6-1]
Where:I = interest (or coupon ) paymentskb = the bond discount rate (or market rate)n = the term to maturityF = Face (or par) value of the bond
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Bond Valuation: Example
What is the market price of a ten-year, $1,000 bond with a 5% coupon, if the bond’s yield-to-maturity is 6%?
10
10
1 1
1
1 1.06 1,00050
0.06 1.06
$926.40
n
bn
b b
k FB I
k k
Calculator Approach:1,000 FV50PMT10NI/Y 6CPT PV 926.40
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
ExampleExample
Factors Affecting Bond PricesBond Price-Yield Curve
Market Yield (%)
FIGURE 6-2
Price ($)
When interest rates increase, bond prices fallWhen interest rates increase, bond prices fall
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
The relationship between the coupon rate and the bond’s yield-to-maturity (YTM) determines if the bond will sell at a premium, at a discount, or at par
If Then Bond Sells at a:
Coupon < YTM Market < Face Discount
Coupon = YTM Market = Face Par
Coupon > YTM Market > Face Premium
Factors Affecting Bond Prices
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Bond Valuation: Semi-Annual Coupons
So far, we have assumed that all bonds have annual pay coupons. While this is true for many Eurobonds, it is not true for most domestic bond issues, which have coupons that are paid semi-annually
To adjust for semi-annual coupons, we must make three changes:
Size of the coupon payment (divide by 2) Number of periods (multiply by 2) Yield-to-maturity (divide by 2)
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Bond Valuation: Semi-Annual Coupons
For example, suppose you want to value a five-year, $10,000 Government of Canada bond with a 4% coupon, paid twice a year, given a YTM of 6%.
2
2
2 5
2 5
1 12
212 2
.061 1
400 10,00020.062 .06
12 2
$9,146.98
n
b
nb b
x
x
kI F
Bk k
Calculator Approach:10,000 FV400 ÷ 2 = PMT5 x 2 = N6 ÷ 2 = I/YCPT PV 926.40
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Factors Affecting Bond Prices
There are three factors that affect the price volatility of a bond
Yield to maturity Time to maturity Size of coupon
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Factors Affecting Bond Prices
Yield to maturity Bond prices go down when the YTM goes up Bond prices go up when the YTM goes down
Look at the graph on the next slide. It shows how the price of a 25 year, 10% coupon bond changes as the bond’s YTM varies from 1% to 30%
Note that the graph is not linear – instead it is said to be convex to the origin
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Factors Affecting Bond Prices Price and Yield: 25 Year Bond, 10% Coupon
Price/Yield Relationship
0
50
100
150
200
250
300
350
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Percent YTM
Pri
ce p
er $
100
of F
ace
Val
ue
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
The convexity of the price/YTM graph reveals two important insights:
The price rise due to a fall in YTM is greater than the price decline due to a rise in YTM, given an identical change in the YTM
For a given change in YTM, bond prices will change more when interest rates are low than when they are high
Factors Affecting Bond Prices Bond Convexity
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Factors Affecting Bond Prices
Time to maturity Long bonds have greater price volatility than
short bonds The longer the bond, the longer the period for
which the cash flows are fixed
Size of coupon Low coupon bonds have greater price volatility
than high coupon bonds High coupons act like a stabilizing device, since a
greater proportion of the bond’s total cash flows occur closer to today & are therefore less affected by a change in YTM
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Interest Rate Risk & Duration
The sensitivity of bond prices to changes in interest rates is a measure of the bond’s interest rate risk
A bond’s interest rate risk is affected by: Yield to maturity Term to maturity Size of coupon
These three factors are all captured in one number called duration
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Duration
Duration is a measure of interest rate risk The higher the duration, the more sensitive the
bond is to changes in interest rates A bond’s duration will be higher if its:
YTM is lower Term to maturity is longer Coupon is lower
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Bond Quotations
Issuer Coupon Maturity Price Yield
Canada 5.500 2009-Jun-01 103.79 4.16
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Cash Versus Quoted Prices
The quoted price is the price reported by the media
The cash price is the price paid by an investor The cash price includes both the quoted price
plus any interest that has accrued since the last coupon payment date
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Cash Versus Quoted Price: Example
Assume you want to purchase a $1,000 bond with a 5% coupon, paid semi-annually. Today is July 15th. The last coupon was paid June 30th. If the quoted price is $902, how much is the cash price?
Solution: The cash price is equal to: Quoted price of $902 Plus 15 days of interest
15902 1,000 0.05
365
902 2.05
$904.05
Cash price = Quoted Price+ Accrued Interest
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Bond Yields
Bond Yields
Yield-to-maturity (YTM) – the discount rate used to evaluate bonds
The yield earned by a bond investor who: Purchases the bond at the current market price Held the bond to maturity Reinvested all of the coupons at the YTM
Is the bond’s Internal Rate of Return (IRR)
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Bond Yield to Maturity
The yield to maturity is that discount rate that causes the sum of the present value of promised cash flows to equal the current bond price.
YTM)(
FYTM
YTM)(IB
n
n
1
111
1
[ 6-2]
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Solving for YTM
To solve for YTM, solve for YTM in the following formula:
Problem: can’t solve for YTM algebraically; therefore, must either use a financial calculator, spreadsheet, trial and error, or approximation formula.
1 1
1
n
n
YTM FB I
YTM YTM
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Solving for YTM
Example: What is the YTM on a 10 year, 5% coupon bond (annual pay coupons) that is selling for $980?
10
10
1 1
1
1 1 1,000980 50
1
5.26%
n
n
YTM FB I
YTM YTM
YTM
YTM YTM
YTM
Financial Calculator1,000 FV980 +/- PV50 PMT51 NI/Y 5.26%
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Solving for YTM: Semi-annual Coupons
When solving for YTM with a semi-annual pay coupon, the yield obtained must be multiplied by two to obtain the annual YTM
Example: What is the YTM for a 20 year, $1,000 bond with a 6% coupon, paid semi-annually, given a current market price of $1,030?
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Solving for YTM: Semi-annual Coupons
40
40
1 1
1
1 1 1,0001,030 30
1
2.87 2 5.74%
n
n
YTM FB I
YTM YTM
YTM
YTM YTM
YTM x
Financial Calculator1,000 FV1,030 +/- PV30 PMT40 NI/Y 2.87 x
2 =
5.746%
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
The Approximation Formula
Where
F = Face Value = Par Value = $1,000
B = Bond Price
I = the semi annual coupon interest
N = number of semi-annual periods left to maturity
1YTM) annual-semi (1YTM
YTM annual-semi 2YTM2
nB-F
Maturity toYield annual-Semi
2
BF
I
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Example
Find the yield-to-maturity of a 5 year 6% coupon bond that is currently priced at $850. (Always assume the coupon interest is paid semi-annually.)
Therefore there is coupon interest of $30 paid semi-annually
There are 10 semi-annual periods left until maturity
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Solution
The actual answer is 9.87%...so of course, the approximation approach only gives us an approximate answer…but that is just fine for tests and exams.
The actual answer is 9.87%...so of course, the approximation approach only gives us an approximate answer…but that is just fine for tests and exams.
%97.91)0486.1(1YTM) annual-semi (1YTM
9.3%0.0927320.0486YTM annual-semi 2YTM
0486.0925$
30$15$
2850,1$
30$10
850$000,1$
2
nB-F
Maturity toYield annual-Semi
22
BF
I
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
The Logic of the EquationApproximation Formula for YTM
The numerator simply represents the average semi-annual returns on the investment; it is made up of two components: The first component is the average capital gain (if it is a
discount bond) or capital loss (if it is a premium priced bond) per semi-annual period.
The second component is the semi-annual coupon interest received.
The denominator represents the average price of the bond. Therefore the formula is basically, average semi-annual
return on average investment. Of course, we annualize the semi-annual return so that we
can compare this return to other returns on other investments for comparison purposes.
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Yield to Call
If a bond has a call feature, the issuer can call the bond prior to its stated maturity
To calculate the yield to call, replace the maturity date with the first call date
Yield to Call
The yield to call is that discount rate that causes the present value of all promised cash flows including the call price (CP) to equal the current bond price.
YTC)(
CPYTC
YTC)(IB
n
n
1
111
1
[ 6-3]
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Solving for YTC: Semi-Annual Coupons
Financial Calculator1,050 FV1,030 +/- PV30 PMT10 NI/Y 3.081 x
2 =
6.16%
YTC on a 20-year 6 percent bond that is callable in five years at a call price of $1,050. The bond pays semi-annual coupons and is selling for $1,030.
%16.62%081.3
%081.3
1
050,1$11
130$030,1$
1
111
1
10
10
YTC
annuallysemiYTC
YTC)(YTCYTC)(
YTC)(
CPYTC
YTC)(IB
n
n
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Current Yield
The current yield is the yield on the bond’s current market price provided by the annual coupon
It is not a true measure of the return to the bondholder because it does not consider potential capital gain or capital losses based on the relationship between the purchase price of the bond and it’s par value.
B
interestAnnualCY [ 6-4]
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Current YieldExample
The current yield is the yield on the bond’s current market price provided by the annual coupon
Example: If a bond has a 5.5% annual pay coupon and the current market price of the bond is $1,050, the current yield is:
55
1,050
5.24%
Annual CouponCurrent Yield =
Current Market Price
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Interest Rate Determinants
Interest Rate Determinants
Interest is the “price” of money Basis points – 1/100 of 1% Interest rates go:
Up – when the demand for loanable funds rises Down – when the demand for loanable funds falls
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Risk-free Interest Rate
Usually use the yield on short federal government treasury bills as a proxy for the risk-free rate (RF)
The risk-free rate is comprised of two components: Real rate – compensation for deferring consumption Expected inflation – compensation for the expected loss in
purchasing power
(See Figure 6-3 to see rates of inflation and yields on long Canada bonds since 1961)
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Inflation and Yields over Time
FIGURE 6-3
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Fisher Equation
If we call the risk-free rate the nominal rate, then the relationship between the real rate, the nominal rate and expected inflation is usually referred to as the Fisher Equation (after Irving Fisher)
inflation Expectedrate RealRF [ 6-5]
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Fisher Equation
When inflation is low, can safely use the approximation formula:
When inflation is high, use the exact form of the Fisher Equation:
Nominal RealR = R + Expected Inflation
1 1 1Nominal RealR = R Expected Inflation
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Fisher Equation Example
If the real rate is 3% and the nominal rate is 5.5%, what is the approximate expected future inflation rate?
5.5 3
2.5%
Nominal RealR = R + Expected Inflation
Expected Inflation
Expected Inflation
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Global Influences on Interest Rates
Canadian domestic interest rates are heavily influenced by global interest rates
Interest rate parity (IRP) theory states that FX forward rates will be established that equalize the yield an investor can earn, whether investing domestically or in a foreign jurisdiction
A country with high inflation and high interest rates will have a depreciating currency
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Term Structure of Interest Rates
Is that set of rates (YTM) for a given risk-class of debt securities (for example, Government of Canada Bonds) at a given point in time.
When plotted on a graph, the line is called a Yield Curve
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Term Structure of Interest Rates
The Yield Curve is the graph created by putting term to maturity on the X axis, YTM on the Y axis and then plotting the yield at each maturity.
The four typical shapes of yield curves: Upward sloping (the most common shape) Downward sloping Flat Humped
(See Figure 6-4 for Yield curves that existed at various times in Canada)
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Historical Yield Curves1990, 1994, 1998, 2004
Pe
rce
nt
Term Left to Maturity
16
14
12
10
8
6
4
2
0
1 mth 3 mths 6 mths 1 yr 2yrs 5 yrs 7 yrs 10 yrs 30 yrs
FIGURE 6-4
1990 1994 1998 2004
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Theories of the Term Structure
Three theories are used to explain the shape of the term structure
Liquidity preference theory Investors must be paid a “liquidity premium” to hold
less liquid, long-term debt Expectations theory
The long rate is the average of expected future short interest rates
Market segmentation theory Distinct markets exist for securities of different
maturities
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Term Structure of Interest RatesRisk Premiums
More risky bonds (i.e.. BBB rated Corporate Bonds) will have their own yield curve and it will plot at higher YTM at every term to maturity because of the default risk that BBBs carry
The difference between the YTM on a 10-year BBB corporate bond and a 10-year Government of Canada bond is called a yield spread and represents a default-risk premium investors demand for investing in more risky securities.
Spreads will increase when pessimism increases in the economy
Spreads will narrow during times of economic expansion (confidence)
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Yield Curves for Different Risk ClassesRisk Premiums (Yield Spreads)
Pe
rce
nt
Term Left to Maturity
16
14
12
10
8
6
4
2
0
1 mth 3 mths 6 mths 1 yr 2yrs 5 yrs 7 yrs 10 yrs 30 yrs
BBB Corporates Government of Canada Bonds
Yield Spread
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Risk Premiums
The YTM on a corporate bond is comprised of:
The maturity yield differential is explained by the term structure
Spread is the additional yield due to default risk
Spread aldifferenti yieldMaturity -/ RFkb YTM[ 6-6]
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Debt Ratings
All publicly traded bonds are assigned a “risk rating” by a rating agency, such as Dominion Bond Rating Service (DBRS), Standard & Poors (S&P), Moodys, Fitch, etc.
Bonds are categorized as Investment grade – top four rating categories (AAA,
AA, A & BBB) Junk or high yield – everything below investment
grade (BB, B, CCC, CC, D, Suspended)
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Why Do Bonds Have Different Yields?
Default risk – the higher the default risk, the higher the required YTM
Liquidity – the less liquid the bond, the higher the required YTM
Call features – increase required YTM Extendible feature – reduce required YTM Retractable feature – reduce required YTM
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Other Types of Bonds/Debt Instruments
Treasury Bills
Treasury bills are short-term obligations of government with an initial term to maturity of one year or less
Issued at a discount and mature at face value The difference between the issue price and the face
value is treated as interest income To calculate the price of a T-bill, use the following
formula
1T Bill
FP
nBEY
B
Where:P = market price of the T BillF = face value of the T BillBEY = the bond equivalent yieldn = the number of days until maturityB = the annual basis (365 days in Canada)
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Treasury Bills: Example
What is the price of a $1,000,000 Canadian T bill with 80 days to maturity and a BEY of 4.5%?
1
1,000,00080
1 .045365
$990,233.32
T Bill
FP
nBEY
B
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Solving for Yield on a T Bill
To solve for the yield on a T bill, rearrange the previous formula and solve for BEY.
Example: What is the yield on a $100,000 T bill with 180 days to maturity and a market price of $98,200?
100,000 98,200 365
98,200 180
3.72%
F P BBEY
P n
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Zero Coupon Bonds
A zero coupon bond is a bond issued at a discount that matures at par or face value
A zero coupon bond has no reinvestment rate risk, since there are no coupons to be reinvested
To calculate the price of a zero coupon bond, solve for the PV of the face amount
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Zero Coupon Bonds
Example: What is the market price of a $50,000 zero coupon bond with 25 years to maturity that is currently yielding 6%?
25
F
1
50,000
1.06
$11,649.93
n
b
Bk
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Floating Rate & Real Return Bonds
Floating rate bonds have a coupon that floats with some reference rate, such as the yield on T bills
Because the coupon floats, the market price will typically be close to the bond’s face value
Real return bonds are issued by the Government of Canada to protect investors against unexpected inflation
Each period, the face value of the bond is grossed up by the inflation rate. The coupon is then paid on the grossed up face value.
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Canada Savings Bonds
A Canada Savings Bond (CSB) is a special type of bond issued by the Government of Canada
It is issued in two forms: Regular interest – interest is paid annually Compound interest – interest compounds over the
life of the bond
CSBs are redeemable at any chartered bank in Canada at their face value
There is no secondary market for CSBs
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Summary and Conclusions
In this chapter you have learned: About the nature of bonds as an investment How to value a bond using discounted cash flow
concepts About the determinants of interest rates and
theories used to explain the term structure of interest rates
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
Interesting Web Links
1. Bond Characteristics
2. Bond Valuation
3. Bond Yields
4. Bond Ratings and Agencies
Bond Basics Bond Basics Valuation Valuation YieldsYields DeterminantsDeterminants Other DebtOther Debt
End of Chapter Problems
Chapter 6 – Bond Valuation and Interest Rates
Question 6 - 1
Which of the following statements concerning bonds is incorrect?
A. They involve blended payments of principal and interest.B. They have a fixed maturity date at which time the issuer
repays the full principal amount.C. Bondholders are paid a series of fixed periodic amounts
before the maturity date.D. The bond indenture is a legal document, specifying
payment requirements and so on.Solution: A
Mortgages have “blended” payments including interest and principal payment, not bonds.
End-of-Chapter Problems
Question 6 - 2
Which of the following statements is incorrect?A. Callable bonds give the bond issuer an option to call the
bond at a predetermined price.B. All debentures are secured bonds.C. Extendible bonds allow bondholders to extend the
maturity date.D. Convertible bonds give the bondholders an option to
convert into common shares at a predetermined conversion ratio.
Solution: B
Debentures are generally unsecured.
End-of-Chapter Problems
Question 6 - 3
Determine the price of a five-year 7 percent annual coupon bond when the market rate is 8 percent. The face value is $100.
A. $100
B. $102.50
C. $96.01
D. $104.10
Solution: C
Or, by financial calculator:
N = 5; I/Y= 8; PMT = 7; FV = 100; CPT PV = –96.01
$96.0168.058327.9490
)08.1(
1100
08.)08.1(
11
75
5
B
End-of-Chapter Problems
Question 6 - 4
Which of the following bond prices is most sensitive to market rate changes? The par value = $100 for all.
A. 5-year, 5 percent coupon rate, yield = 5.5 percent
B. 3-year, 8 percent coupon rate, yield = 5.6 percent
C. 7.5-year, 4.5 percent coupon rate, yield = 5.5 percent
D. 10-year, 4.5 percent coupon rate, yield = 5.5 percent
Solution: DAll else being equal, interest rate risk is positively related to term to
maturity, but negatively related to coupon rate and market yields. The bond in choice D has the lowest yield, lowest coupon rate and the longest term to maturity relative to other bonds. Therefore it has the highest interest rate risk.
End-of-Chapter Problems
Question 6 - 5
Determine the yield to maturity on a six-year, 7 percent, semi-annual-pay bond, which is now priced at $993. Use a financial calculator.
A. 7.05 percentB. 6.98 percentC. 3.57 percentD. 7.15 percent
Solution: DCoupon = ($1,000)(7%)/2=$35, N=6×2=12, FV=1,000, PV= –993
Using a financial calculator, N=12, PV= –993, PMT=35, FV=1,000 CPT I/Y=3.5727Therefore, YTM=3.5727%×2=7.15%
End-of-Chapter Problems
Question 6 - 6
Which of the following statements is correct?A. Current yield is the ratio of annual coupon payment divided by
the par value.
B. When the coupon rate is higher than the market rate, the bond is priced at discount.
C. When the market rate is higher than the coupon rate, the bond is priced at premium.
D. If a bond is at discount, the coupon rate < current yield < YTM.
Solution: DCurrent yield is the ratio of annual coupon divided by the current market price. When coupon rate is higher than the market rate, bond is at premium and when the market rate is higher, bond is at discount.
End-of-Chapter Problems
Question 6 - 7
According to interest rate parity (IRP) theory,A. differences in interest rates across countries cannot be totally offset by
expected changes in exchange rates.B. forward exchange rates may be locked in today to eliminate foreign
exchange risk and ensure investors can profit from moving capital to countries with higher interest rates.
C. the inflation differentials between countries affect both interest rates and currency exchange rates.
D. the country with a higher inflation rate will see its currency appreciate against another country with a lower inflation rate.
Solution: CDifference in interest rates across countries can be offset by expected changes in exchange rates. If a country has a higher expected inflation rate, its currency will depreciate.
IRP states that forward exchange rates locked-in today to eliminate foreign exchange risk ensure investors earn the same amount no matter where they invest.
End-of-Chapter Problems
Question 6 - 8
Which statement is incorrect?A. The liquidity preference theory states that investors prefer
short-term debt.
B. According to the expectations theory, a downward-sloping yield curve implies that interest rates are expected to decline in the future.
C. The risk premium in the bond yield reflects default risk, liquidity risk, and issue-specific features.
D. A debt rating of AAA is a worse rating than BB for S&P.
Solution: DAAA is rated higher than BB.
End-of-Chapter Problems
Question 6 - 9
Calculate the quoted price of a 182-day Canadian T-bill that has a face value of $10,000 and a quoted yield of 5.5 percent.
A. $9,733.07
B. $9,478.67
C. $97.3307
D. $94.7867
Solution: C
Because it is quoted on a basis of $100, therefore quoted price is $97.3307.
07.733,9$02742466.1
000,10
)365182
055.01(
000,10
P
End-of-Chapter Problems
Question 6 - 10
Which of the following statements is false?A. Zero coupon bonds are deep-discount bonds.
B. Zero coupon bonds are often created when cash flows are stripped from traditional bonds.
C. Floating rate bonds provide protection against decreasing interest rates.
D. There are two forms of return available for Canadian Savings Bond buyers.
Solution: C
Floating rate bonds provide protection against increasing interest rates compared to fixed rate bonds.
End-of-Chapter Problems
Question 6 - 11
State the relationship between the market rates and bond prices.
Solution:
When market interest rates increase, prices of bonds decrease. When market interest rates decrease, prices of bonds increase.
End-of-Chapter Problems
Question 6 - 12
Find the price of a bond with FV = $1,000, a coupon rate of 6 percent (paid semi-annually), and three years to maturity when
A. kb = 7 percent.B. kb = 6 percent.C. kb = 5 percent.
Solution:
Using a financial calculator:A. N = 6; I/Y= 3.5; PMT = 30; FV = 1,000; CPT PV = –973.36B. N = 6; I/Y= 3; PMT = 30; FV = 1,000; CPT PV = –1,000C. N = 6; I/Y= 2.5; PMT = 30; FV = 1,000; CPT PV = –1,027.54Clearly, the price of bonds is inversely related to the market yield.
End-of-Chapter Problems
Question 6 - 13
At maturity, each of the following zero-coupon bonds (pure discount bonds) will be worth $1,000. For each bond, find the missing quantity in the table below.
Solution:
Using a financial calculator:A. N = 30; PMT = 0; FV = 1,000; PV = - 400; CPT I/Y = 3.10%B. I/Y = 8; PMT = 0; FV = 1,000; PV = - 400; CPT N = 11.9 yearsC. N = 10; I/Y = 12; PMT = 0; FV = 1,000; CPT PV = $321.97
Price Maturity (years)
Yield to Maturity
A. $400 30
B. $400 8%
C. 10 12%
End-of-Chapter Problems
Question 6 - 14
Suppose that a government of Canada 9 percent annual-pay bond that matures in two years has a yield to maturity of 9.80 percent. If inflation is expected to be 3 percent per year over the next two years, what coupon rate would you expect to find on a Real Return Bond that is otherwise identical?
Solution:
Using the approximate relationship: 9.80% = Real Rate + 3%, so the Real Rate = 6.8%.
Using the exact relationship:Real Rate = (1+0.098)/(1+0.03) – 1 = 6.60%
End-of-Chapter Problems
Question 6 - 15
Suppose the inflation rate in Canada, as measured by the CPI, has been averaging 2.5 percent in recent years. The most recent Bank of Canada announcement indicates that it expects 4 percent inflation over the next year. If the real rate of return on Canadian T-bills is 1.75 percent, what is the nominal risk-free rate?
Solution:
The Fisher relationship uses expected inflation figures, not the actual rate of inflation experienced in the past. Therefore:
RF = 1.75% + 4% = 5.75%
Using the exact relationship:
RF = (1+0.0175)(1.04)–1=5.82%
End-of-Chapter Problems
Question 6 - 16
The following values are the spread for corporate bond yields:
A. One-year T-bills are trading with a YTM of 6 percent. What yield would you expect to find on A rated corporate bonds maturing in one year?
B. Five-year government bonds have a maturity yield differential of 50 basis points. What yield would you expect to observe on non-investment grade (BB rated) corporate bonds with a five-year maturity?
Solution: A. We can assume that T-bills (short-term, federal government bonds) have the highest possible credit rating,
“AAA”, because they have virtually no default risk. We expect lower rated (riskier) bonds to have a higher yield. For “A” rated bonds, we should expect kb = 6% + 0.45% = 6.45%
B. Adding the maturity yield and the “spread” for this bond rating, we find: kb = 6% + 0.50% + 1.10% = 7.60%
Bond Rating Spread over AAA
AA 30 basis points
A 45 basis points
BBB 70 basis points
BB 110 basis points
End-of-Chapter Problems
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