Bond Valuation and Interest Rates
The Basic Structure of Bonds
Bond Valuation
Bond Yields
Current Yield, Capital Gain (loss) Yield, &Total Yield
Yield to Maturity, Yield to Call, Holding Period Yield
Realized Rate of Return
Relationship between interest rate and bond price
Interest Rate Determinants
Bond Ratings
Other Types of Bonds/Debt Instruments
1
The Basic Structure of Bonds
• In the broadest sense, a bond is any debt instrument that
promises a fixed income stream to the holder
• Bonds usually have the following characteristics:
• A fixed face or par value, paid to the holder at maturity
• A fixed coupon, which specifies the interest payable over the life of
the bond
• A fixed maturity date
• Fixed income securities are often classified according to maturity:
• Bills or paper have maturities less than one year
• Notes have maturities between one and seven years
• Bonds have maturities greater than seven years
• Bonds may be either bearer bonds or registered bonds
2
The Basic Structure of Bonds
• The market price of a bond is the present value of the
payments promised by the bond
0
1
2
3
t
|---------------|------------------|------------------|---|---|---|----|
C
C
C
C
C C
C
+F
• The bond indenture is the contract between issuer and
holder, which specifies:
•
•
•
•
•
Details regarding payment terms
Collateral
Positive or negative covenants
Par value or face value (usually in increments of $1,000)
Bond pricing, usually shown as the price per $100 of par value
which is equal to a percentage of the bond’s face value
3
The Basic Structure of Bonds
• Term to maturity is the time remaining to the maturity date
• Coupon rate is the annual percentage interest paid on the bond’s
face value
• Multiply the coupon rate by the bond’s face value to calculate coupon,
and divide by two if the coupon is paid semi-annually
• Example: A $1,000 bond with an 8% coupon rate will have an $80
annual coupon or a $40 semi-annual coupon
• Security and protective provisions:
• Mortgage bonds are secured by real assets
• Debentures are either unsecured or secured, with a floating charge
over the firm’s assets
• Collateral trust bonds are secured by pledged financial assets, such as
common stock, other bonds or Treasury bills
• Equipment trust certificates are secured by pledged equipment, such
as railway rolling stock
4
The Basic Structure of Bonds
• Covenants are another type of protective provision
• Positive covenants specify actions the firm agrees to do, such as supply
periodic financial statements and maintain certain ratios
• Negative covenants specify actions the firm agrees to avoid, such as
restrictions on the size of its debt or acquiring or disposing of assets
• Call features allow the issuer to redeem or pay off the bond prior to
maturity, usually at a premium
• Retractable bonds allow the holder to extend bonds back to the
issuer before maturity
• Extendible bonds allow the holder to extend the bond’s maturity
• Sinking funds are 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 predetermined conversion price
5
Bond Market Reporting
CANADA
Canada
Coupon
6.375
Mat. Date
Dec 31/20
The Government
of Canada issued
this bond
The bond pays
an annual
coupon rate of
6.375%
The bond
matures on
December 31,
2020
Bid $
107.05
Yld%
5.00
The bond’s
quoted annual
yield to
maturity is 5%
The bond is selling
at 107.05% of the
face value of
$1,000
Level Coupon Bonds
• Make periodic coupon payments in addition to the
maturity value
• The payments are equal each period. Therefore, the
bond is just a combination of an annuity and a
terminal (maturity) value.
• Coupon payments are typically semiannual.
7
Level-Coupon Bonds
Information needed to value level-coupon bonds:
– Coupon payment dates and time to maturity (t)
– Coupon payment (C) per period and Face value (F)
– Discount rate
0
1
2
3
t
|-------------------------|-------------------------|-------------------------|---|---|---|---|-------|
C
C
C C C C C
C+F
Value of a Level-coupon bond
= PV of coupon payment annuity + PV of face value
𝐶𝐶
1
𝐹𝐹
𝑃𝑃𝑃𝑃 = 1 −
+
= 𝐶𝐶
𝑟𝑟
(1 + 𝑟𝑟)𝑡𝑡
(1 + 𝑟𝑟)𝑡𝑡
1−
1
1 + 𝑟𝑟
𝑟𝑟
𝑡𝑡
𝐹𝐹
+
1 + 𝑟𝑟
𝑡𝑡
8
LEVEL COUPON BOND
QUESTION 1
On January 1, 2014 CC Rated Inc. (CCI) issued
a 10-year bond with a 14% coupon payable
annually. The required rate of return on this
type of bond at the time of issue was 14%.
a. Determine the price of the bond at the
time of issue.
9
QUESTION 1 a
14 15
16
17
18
19
20
21
22
23
24
|----|-------|--------------|-------------|--------------|-------------|-------------|--------------|-------------|-------------|
$140 140
140
140
140
140
140
140
140
140
+1000
1
𝐹𝐹
1 + 𝑟𝑟 𝑡𝑡
+
𝑃𝑃𝑃𝑃 = 𝐶𝐶
𝑟𝑟
1 + 𝑟𝑟
= 730.26 + 269.74 = $1000
1−
𝑡𝑡
= 140
1−
1
1 + 0.14
0.14
10
+
1000
1 + 0.14
10
10
b.
Suppose the required rate of return increased to 20% after one year and
stayed at that level for the remaining life of the bond.
1. Determine the bond price on January 1, 2015 and on January 1, 2016
January 1, 2015
Now it is a 9 year bond
𝑃𝑃2015 = 140
January 1, 2016
1
1.20
0.20
1−
9
1000
+
= $758.14
1.20 9
Now it is an 8 year bond
𝑃𝑃2016 = 140
1
1.20
0.20
1−
8
r = 20%
r = 20%
1000
+
= $769.77
1.20 8
11
Current, Capital Gain, & Total Yield
b2.
If you purchase the bond on January 1, 2015 and sold on
January 1, 2016, what will be your current (interest) yield,
capital gain yield, and total yield?
Current (Interest)Yield = C/P15 = $140/758.14 =
Capital Gain Yield = (P16 – P15)/P15 = (769.77-758.14)/758.14 =
Total Yield
0.1847
0.0153
0.2000
12
Current, Capital Gain, & Total Yield
b3.
If you purchase the bond on January 1, 2014 and sold on
January 1, 2015, what will be your current (interest) yield,
capital gain yield, and total yield?
Current (Interest)Yield = C/P14= $140/1000 =
Capital Loss Yield = (P15– P14)/P14 = (758.14-1000)/1000 =
Total Yield
0.1400
-0.2419
-0.1019
13
c.
Suppose the required rate of return declined to 8% after one year and
stayed at that level for the remaining life of the bond.
1. Determine the bond price on January 1, 2015 and on January 1, 2016
January 1, 2015
Now it is a 9 year bond
1
1.08
0.08
1−
𝑃𝑃2015 = 140
January 1, 2016
1000
+
= $1374.81
1.08 9
9
Now it is an 8 year bond
𝑃𝑃2016 = 140
1
1.08
0.08
1−
r = 8%
8
+
r = 8%
1000
= $1344.80
1.08 8
14
Current, Capital Gain, & Total Yield
c2.
If you purchase the bond on January 1, 2015 and sold on
January 1, 2016, what will be your current (interest) yield,
capital gain yield, and total yield?
Current (Interest)Yield = C/P15 = $140/1374.81 =
0.1018
Capital Loss Yield = (P16 – P15)/P15 = (1344.80-1374.81)/1374.81 = -0.0218
Total Yield
0.0800
15
Current, Capital Gain, & Total Yield
c3.
If you purchase the bond on January 1, 2014 and sold on
January 1, 2015, what will be your current (interest) yield,
capital gain yield, and total yield?
Current (Interest)Yield = C/P14 = $140/1000 =
Capital Gain Yield = (P15– P14)/P14 = (1374.81-1000)/1000 =
Total Yield
0.1400
0.3748
0.5148
16
Bond Yields
• The YIELD TO MATURITY (YTM) is the discount rate used for bond
valuation
• YTM is the yield an investor would earn if:
• She purchases the bond at the current market price
• She holds the bond to maturity
• She reinvests all of the coupons paid by the bond at the YTM
• YTM is, therefore, the bond’s internal rate of return (IRR)
• YTM is, also, the discount rate that causes the present value of
the bond’s future cash flows to equal its current price
𝑃𝑃0 = 𝐶𝐶
1−
1
𝐹𝐹
(1 + 𝑌𝑌𝑌𝑌𝑌𝑌)𝑡𝑡
+
𝑌𝑌𝑌𝑌𝑌𝑌
(1 + 𝑌𝑌𝑌𝑌𝑌𝑌)𝑡𝑡
17
d.
Suppose on January 1, 2015 CCI’s bonds were selling for $907.87. What will
be the Yield to Maturity (YTM)?
𝑃𝑃15 = 907.87 = 140
1
1 + 𝑌𝑌𝑌𝑌𝑌𝑌
𝑌𝑌𝑌𝑌𝑌𝑌
1−
9
+
1000
1 + 𝑌𝑌𝑌𝑌𝑌𝑌
9
Since bond is selling at a discount YTM or r > coupon rate
Trial and Error
Try YTM = 18%
P2005 = $827.92
Try YTM = 16%
$907.87 = 140
∴ 𝑌𝑌𝑌𝑌𝑌𝑌 = 16%
1
1.16
0.16
1−
9
+
1000
= $907.87
1.16 9
Formula to get approximate YTM
𝑌𝑌𝑌𝑌𝑌𝑌 ≃
𝐶𝐶 + 𝐹𝐹 − 𝑃𝑃0 /𝑡𝑡 140 + 1000 − 907.87 /9
=
= 0.1575
𝐹𝐹 + 𝑃𝑃0 /2
1000 + 907.87 /2
18
Bond Concepts
1.
Bond prices and market interest rates move in opposite
directions.
2.
When coupon rate = YTM, price = par value.
When coupon rate > YTM, price > par value (premium
bond)
When coupon rate < YTM, price < par value (discount
bond)
19
Bond Concepts
3.
A bond with longer maturity has higher relative (%) price
change than one with shorter maturity when interest rate
(YTM) changes. All other features are identical.
Example:
Current
Market Interest
0%
5
10
15
20
Current Market Value
1 Year, 10% Bond
12 Year, 10% Bond
$1,100.00
1,047.62
1,000.00
956.52
916.67
$2,200.00
1,443.16
1,000.00
728.97
556.08
20
Bond Value
Maturity and Bond Price Volatility
Consider two otherwise identical bonds.
The long-maturity bond will have much more volatility
with respect to changes in the discount rate
Par
Short Maturity Bond
C
Discount Rate
Long Maturity Bond
Bond Concepts
4.
A lower coupon bond has a higher relative price change
than a higher coupon bond when YTM changes. All other
features are identical.
22
Bond Value
Coupon Rate and Bond Price Volatility
Consider two otherwise identical bonds.
The low-coupon bond will have much more volatility with
respect to changes in the discount rate
High Coupon Bond
Low Coupon Bond
Discount Rate
Semiannual Coupon Payments
e1.
Determine the bond price on January 1, 2015, if the
required rate of return is 16% compounded
semiannually .
𝑃𝑃15
𝑃𝑃15
1
0.16 9×2
1+
140
1000
2
=
+
0.16
2
0.16 9×2
1+
2
2
1
1−
1000
1.08 18
+
= 70
= $906.28
0.08
1.08 18
1−
24
e2.
Determine YTM if the bond is selling for $1101.06 on
January 1, 2016. What will be the effective yield?
𝐶𝐶
𝐹𝐹 − 𝑃𝑃
+
𝑌𝑌𝑌𝑌𝑌𝑌 2
70 + 1000 − 1101.06 /16
𝑡𝑡 × 2
≃
=
𝐹𝐹 + 𝑃𝑃
1000 + 1101.06 /2
2
2
= 0.0606
∴ 𝑌𝑌𝑌𝑌𝑌𝑌 ≃ 0.0606 × 2 = 0.1212
Try YTM = 12%
$1101.06 = 70
1−
1
1.06
0.06
= $1101.06
∴ 𝑌𝑌𝑌𝑌𝑌𝑌 = 0.06 × 2 = 0.12
16
+
1000
1.06 16
and Effective Yield = (1.06)2 -1 =0.1236 = 12.36%
25
Yield to Call (YTC or rC)
• If a bond has a call feature, the issuer can call (or force the
investor to sell the bond back to it) before the maturity stated in
the bond indenture
• Callable bonds are initially protected from call for a period of a
few years (e.g., five, seven or ten), after which the issuer may call
the bond
• Replace the maturity date with the first call date, and the face
value with the call price (CP).
t = periods until company can call the bond
CP = Call Price, the company must pay to call the bond
𝑃𝑃0 = 𝐶𝐶
1−
1
𝐶𝐶𝐶𝐶
(1 + 𝑟𝑟𝐶𝐶 )𝑡𝑡
+
𝑟𝑟𝑐𝑐
(1 + 𝑟𝑟𝐶𝐶 )𝑡𝑡
26
e3.
Determine the Yield to Call, if bond is callable after 4 years at a call
price of $1,052.23.
𝐶𝐶𝐶𝐶 − 𝑃𝑃0
𝐶𝐶
+
𝑌𝑌𝑌𝑌𝑌𝑌 2
70 + 1052.23 − 1000 /8
𝑡𝑡 × 2
≃
=
= 0.07458
𝐶𝐶𝐶𝐶 + 𝑃𝑃0
2
1052.23 + 1000 /2
2
∴ 𝑌𝑌𝑌𝑌𝑌𝑌 ≃ 0.07458 × 2 = 0.14916 = 14.916%
Try YTC = 15%
$1,000 = 70
= $1,000
∴ 𝑌𝑌𝑌𝑌𝑌𝑌 = 15%
1
1.075
0.075
1−
8
1052.23
+
1.075 8
and Effective YTC or rc = (1.075)2 -1 = 15.56%
27
Holding Period Yield (HPY)
e4. CCI’s bonds which you bought on January 1, 2014, you sold them on January 1, 2019 after
holding them for 5 years. The required rate of return on January 1, 2019 was 16%
compounded semiannually. Determine HPY.
𝑃𝑃2019 = 70
1−
1
1 + 0.08
0.08
10
+
1000
= $932.90
1.08 10
𝐻𝐻𝐻𝐻𝐻𝐻 70 + 932.90 − 1000 /10
≃
= 0.06549
2
932.90 + 1000 /2
∴ 𝐻𝐻𝐻𝐻𝐻𝐻 ≃ 0.06549 × 2 = 0.13098
Try HPY = 13%
$1000 = 70
1−
1
1.065
0.065
10
+
932.90
1.065 10
= $1000
∴ 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = (1.065)2 − 1 = 0.134225
= 13.4225%
28
e5.
Annual Realized Rate of Return
Right after you bought the bond on January 1, 2014, interest rate
declined to 8% compounded semiannually, and remained at this
level till maturity. Determine the annual realized rate of return, if you
keep the bond for 4 years and earn 8% compounded semiannually
on coupon reinvestments.
𝑃𝑃2018 = 70
= $1,281.55
1−
1
1.04
0.04
FV of Coupons on January 1, 2018
70
12
+
1000
1.04 12
1.04 8 − 1
= $645.00
0.04
Wealth on January 1, 2018 = $1,281.55 + 645 = $1,926.55
Annual Realized Rate of Return
1926.55
1000
1
4
− 1 = 0.1781 = 17.81%
29
Cash Versus Quoted Prices
• The quoted price is the price reported by the media
• The cash price is the price paid by an investor, and includes both
the quoted price plus any interest that has accrued since the last
coupon payment date.
e6. How much in total (cash price) you have to pay, if you buy
the bond on May 1, 2015. Assume that the required rate of
return is 16% compounded semiannually.
30
Cash Versus Quoted Price
𝑃𝑃𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽,2015
1
1−
1.08
= $70
0.08
1000
= $908.78
+
17
1.08
17
𝑃𝑃𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽,2015 + 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 = $908.78 + 70 = $978.78
𝑃𝑃𝑀𝑀𝑀𝑀𝑀𝑀𝑀,2015 =
978.78
1.08
1
3
= $953.99
31
Interest Rate Determinants
• Interest is the “price” of borrowed money
• Interest is measured in basis points, or 1/100th of 1% (e.g., 250
basis points is 2.5%)
• Interest rates rise when the demand for loanable funds rises
• Interest rates fall when the demand for loanable funds falls
The Risk-Free Interest Rate
• The risk-free rate is an abstract concept, and usually the yield
on short-term government treasury bills is used as a proxy for
practical purposes
• The risk-free rate is comprised of two components
• The real rate, which is compensation for deferring consumption
• The expected inflation rate, which is compensation for the expected
loss of purchasing power over the term of the short-term T-bill
32
Inflation and Government Bond Yields
• The following figure shows relationship between inflation
rates and interest rates
33
Interest Rate Term Structure
• The term structure of interest rates is the set of rates (YTM) for all maturities
of a given risk-class of debt securities (e.g., Government of Canada bonds) at a
given point in time
• When plotted on a graph the result is called a yield curve
• There are three typical shapes for the yield curve: upward sloping or normal
(1994), downward sloping or inverted (1990) and flat (1998)
34
Risk Premiums and Yield Spreads
• More risky bonds (i.e., BBB-rated corporate bonds) will have their
own yield curve and it will plot at higher YTM at every maturity
than government bonds because of the additional default risk
that BBBs carry
• The yield spread is the difference between the YTM on a BBBrated corporate bond and a Government of Canada bond of the
same maturity and it represents the default risk premium
investors demand for investing in the more risky corporate bond
• Yield spreads widen during recessions and narrow during times of
economic expansion
35
Risk Premiums and Yield Spreads
• The following equation can be used to determine the YTM on a
corporate bond:
𝑌𝑌𝑌𝑌𝑌𝑌 = 𝑅𝑅𝑅𝑅 ± Maturity yield differential + Spread
• RF is the risk-free rate
• The maturity yield differential is the extra yield required for
taking on a longer maturity
• The spread is the additional yield required for default risk
36
Debt Ratings
• Debt ratings – rating agencies, such as the Dominion Bond Rating
Service (DBRS), Standard & Poors (S&P), and Moody’s assign all
publicly traded bonds a risk rating
37
•
•
•
•
•
Risk, Liquidity, and Bond Features
Determine YTM
The greater the default risk, the higher the required YTM
The less liquid the bond, the higher the required YTM
Call features generally increase the required YTM
Extendable bonds generally have lower required YTMs
Retractable bonds generally have lower required YTMs
38
Other Types of Bonds and Debt Instruments:
Treasury Bills
• Treasury bills are short-term obligations of the government with an
initial term to maturity of one year or less
• Issued at a discount to face value with face value being paid at maturity
• The difference between the discounted issue price and the face value is
treated as interest income
• The following equation can be used to value a Treasury bill:
•
•
•
•
𝑃𝑃T−bill =
𝐹𝐹
1 + 𝑟𝑟𝐵𝐵𝐵𝐵𝐵𝐵
P = the market price of the T-bill
F = the face value of the T-bill
rBEY = the bond equivalent yield
t = the number of days until maturity
𝑡𝑡
365
39
Other Types of Bonds and Debt Instruments:
Treasury Bills
• Example: What is the price of a $1 million Canadian treasury bill with
80 days until maturity and a bond-equivalent yield of 4.5%?
𝑃𝑃T−bill =
𝐹𝐹
1 + 𝑟𝑟𝐵𝐵𝐵𝐵𝐵𝐵
𝑡𝑡
365
=
$1,000,000
= $990,233.32
80
1 + 0.045
365
• Example: What is the yield on a $100,000 Treasury bill with 180 days
until maturity and a market price of $98,200?
𝑟𝑟𝐵𝐵𝐵𝐵𝐵𝐵
𝐹𝐹 − 𝑃𝑃 365
$100,000 − $98,200 365
=
=
= 3.72%
𝑃𝑃
𝑡𝑡
$98,200
180
40
Other Types of Bonds and Debt Instruments:
Zero Coupon Bonds
• Zero coupon bonds are bonds issued at a discount which pay no
coupons and mature at par or face value
• Since no coupons are paid, there is no reinvestment rate risk
• The price of a zero-coupon bond is the present value of the face
value of the bond:
𝑃𝑃0 =
𝐹𝐹
(1 + 𝑟𝑟)𝑡𝑡
41
Zero Coupon Bond: Example
Find the value of a 30-year zero-coupon bond
with a $1,000 par value and a YTM of 6%.
0
$0
$0
1
2
⋯
$0
$1,000
29
30
$0
$1,000
⋯
2
30
0
1
29
𝐹𝐹
$1,000
𝑃𝑃0 =
=
= $174.11
(1 + 𝑟𝑟)𝑡𝑡 (1.06)30
42
Consols
• Not all bonds have a final maturity.
• British consols pay a set amount (i.e., coupon) every
period forever.
• These are examples of a perpetuity.
𝐶𝐶
𝑃𝑃𝑃𝑃 =
𝑅𝑅
43
CONSOL Example
A Consol bond pays $30 coupon per year.
Determine the price if the required rate of return is
6%.
PV= C/r = $30/0.06 = $500
44
Other Types of Bonds and Debt Instruments
Floating Rate, Real Return, and CSBs
Floating rate bonds have coupon rates that float with some reference rate,
such as the yield on Treasury bills
• Since the coupon rate floats, or is variable, 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 real return bond is grossed up by the
inflation rate. The coupon is then paid on the grossed up face value.
Canada Savings Bonds (CSBs) are issued by the Government of Canada as
either regular interest bonds (interest paid annually) or compound
interest bonds (interest compounds over the life of the bond)
• There is no secondary market for Canada Savings Bonds; instead, they
are redeemable at any chartered bank in Canada at their face value
45