Chapter 2
Bond Prices and Yields
FIXED-INCOME SECURITIES
Outline
• Bond Pricing• Time-Value of Money• Present Value Formula• Interest Rates• Frequency• Continuous Compounding• Coupon Rate • Current Yield• Yield-to-Maturity• Bank Discount Rate• Forward Rates
Bond Pricing
• Bond pricing is a 2 steps process– Step 1: find the cash-flows the bondholder is entitled to– Step 2: find the bond price as the discounted value of the cash-flows
• Step 1 - Example– Government of Canada bond issued in the domestic market pays one-half
of its coupon rate times its principal value every six months up to and including the maturity date
– Thus, a bond with an 8% coupon and $5,000 face value maturing on December 1, 2005 will make future coupon payments of 4% of principal value every 6 months
– That is $200 on each June 1 and December 1 between the purchase date and the maturity date
Bond Pricing
• Step 2 is discounting
T
tt
t
r
FP
10 )1(
• Does it make sense to discount all cash-flows with same discount rate?
• Notion of the term structure of interest rates – see next chapter
• Rationale behind discounting: time value of money
Time-Value of Money
• Would you prefer to receive $1 now or $1 in a year from now?
• Chances are that you would go for money now• First, you might have a consumption need sooner
rather than later. – That shouldn’t matter: that’s what fixed-income markets are for. – You may as well borrow today against this future income, and consume
now. (Individual need vs. Generalized preference)
• In the presence of money market, the only reason why one would prefer receiving $1 as opposed to $1 in a year from now is because of time-value of money
Present Value Formula
• If you receive $1 today– Invest it in the money market (say buy a one-year T-Bill) – Obtain some interest r on it– Better off as long as r strictly positive: 1+r>1 iff r>0
• How much is worth a piece of paper (contract, bond) promising $1 in 1 year?
– Since you are not willing to exchange $1 now for $1 in a year from now, it must be that the present value of $1 in a year from now is less than $1
– Now, how much exactly is worth this $1 received in a year from now?– Would you be willing to pay 90, 80, 20, 10 cents to acquire this dollar paid
in a year from now?
• Answer is 1/(1+r) : the exact amount of money that allows you to get $1 in 1 year
Interest Rates
• Specifying the rate is not enough• One should also specify
– Maturity– Frequency of interest payments– Date of interest rates payment (beginning or end of periods)
• Basic formula– After 1 period, capital is C1= C0 (1+ r )– After n period, capital is Cn = C0(1+ r )n
– Interests : I = Cn - C0
• Example– Invest $10,000 for 3 years at 6% with annual compounding– Obtain $11,910 = 10,000 x (1+ .06)3 at the end of the 3 years– Interests: $1,910
Frequency
• Watch out for – Time-basis (rates are usually expressed on an annual basis) – Compounding frequency
• Examples– Invest $100 at a 6% two-year annual rate with semi-annual compounding
• 100 x (1+ 3%) after 6 months• 100 x (1+ 3%)2 after 1 year• 100 x (1+ 3%)3 after 1.5 year• 100 x (1+ 3%)4 after 2 years
– Invest $100 at a 6% one-year annual rate with monthly compounding• 100 x (1+ 6/12%) after 1 month• 100 x (1+ 6/12%)2 after 2 months• …. 100 x (1+ 6/12%)12 = $106.1678 after 1 year• Equivalent to 6.1678% annual rate with annual compounding
• The effective equivalent annual (i.e., compounded once a year) rate ra is defined as the solution to
• More generally– Amount x invested at the interest rate r– Expressed in an annual basis – Compounded n times per year – For T years– Grows to the amount
Frequency
1nT
rx
n
1 (1 )nT
a Trx x r
n
11
na
n
rr
or
• The equivalent annual rate of a 6% continuously compounded interest rate is e6% –1 = 6.1837%
• Very convenient: present value of X is Xe-rT
• One may of course easily obtain the effective equivalent annual ra
Continuous Compounding
• What happens if we compound infinitely often, i.e. we use continuous compounding?
• The amount of money obtained per dollar invested after T years is
lim 1nT
rT
n
rx xe
n
(1 ) 1rT a T a re r r e
Bond Prices
• Bond price
T
tTt rr
F
r
FP
1 1
11
)1(
T
tt
t
r
FP
1 )1(
TT
tTTt r
N
rr
cN
r
N
r
cNP
11
11
1)1(1
• Shortcut when cash-flows are all identical (Exercise: can you prove it?)
• Coupon bond
Bond Prices - Example
• Example– Consider a bond with 5% coupon rate– 10 year maturity – $1,000 face value– All discount rates equal to 6%
• Present value
10
10 10 101
50 1,000 50 1 1,0001 $926.3991
(1 6%) 6%1 6% 1 6% 1 6%i
i
P
• We could have guessed that price was below par– You do not want to pay the full price for a bond paying 5% when interest rates are at
6%
• What happens if rates decrease to 5%?– Price = $1,000
Perpetuity
• When the bond has infinite maturity (consol bond)
r
cN
r
N
rr
cNP
TTT
11
11
000,2$05.
100P
• Example– How much money should you be willing to pay to buy a contract offering
$100 per year for perpetuity? – Assume the discount rate is 5% – The answer is
– Perpetuities are issued by the British government (consol bonds)
Coupon Rate and Current Yield
• Coupon rate is the stated interest rate on a security– It is referred to as an annual percentage of face value– It is usually paid twice a year – It is called the coupon rate because bearer bonds carry coupons for
interest payments – It is only used to obtain the cash-flows
• Current yield gives you a first idea of the return on a bond
P
cNyc
%78.7900
70cy
• Example– A $1,000 bond has a coupon rate of 7 percent– If you buy the bond for $900, your actual current yield is
Yield to Maturity (YTM)
• Definition: It is the interest rate that makes the present value of the bond’s payments equal to its price. (Memorize it!)
• It is the solution to (T is number of periods)
T
tt
t
YTM
FP
1 )1(• YTM is the IRR of cash-flows delivered by bonds
– YTM may easily be computed by trial-and-error– YTM is typically a semi-annual rate because coupons usually paid semi-
annually– Each cash-flow is discounted using the same rate– Implicitly assume that the yield curve is flat at a point in time– It is a complex average of pure discount rates (see below)
BEY versus EAY
• Bond equivalent yield (BEY): obtained using simple interest to annualize the semi-annual YTM (street convention): y = 2 YTM
• One can always turn a bond yield into an effective annual yield (EAY), i.e., an interest rate expressed on a yearly basis with annual compounding
• Example – What is the effective annual yield of a bond with a 5.5% annual YTM– Answer is
%5756.512
%5.51
2
ar
One Last Complication
• What happens if we don’t have integer # of periods?• Example
– Consider the US T-Bond with coupon 4.625% and maturity date 05/15/2006, quoted price is 101.739641 on 01/07/2002
– What is the YTM and EAY?
• Solution– There are 128 calendar days between 01/07/2002 and the next coupon
date (05/15/2002)
%353.4)21(
3125.102
)21(
2%625.4
739641.1018181
128
8
0 181128
YTMYTMYTMt
t
• EAY is %40.412
%353.41
2
Quoted Bond Prices - Screen
Source: Wall Street Journal
Government Bonds & Notes Friday, October 16, 1998
Rate Maturity Mo/Yr
Bid Asked Chg Ask Yield
7 1/2 Feb 05n 116:30 117:02 -5 4.38
6 1/2 May 05n 112:02 112:06 -3 4.35
8 1/4 May 00-05 105:22 105:24 -1 4.42
12 May 05 142:10 142:16 -5 4.47
5 1/2 Aug 28 108:10 108:11 -8 4.96
Quoted Bond Prices - Paper
Quoted Bond Prices
• Bonds are– Sold in denominations of $1,000 par value
– Quoted as a percentage of par value
• Prices– Integer number + n/32ths (Treasury bonds) or + n/8ths (corporate bonds)
– Example: 112:06 = 112 6/32 = 112.1875%
– Change -5: closing bid price went down by 5/32%
• Ask yield– YTM based on ask price (APR basis:1/2 year x 2)
– Not compounded (Bond Equivalent Yield as opposed to Effective Annual Yield)
Examples
• Example– Consider a $1,000 face value 2-year bond with 8% coupon
– Current price is 103:23
– What is the yield to maturity of this bond?
• To answer that question– First note that 103:23 means 103 + (23/32)%=103.72%
– And obtain the following equation
432 )21(
040,1
)21(
40
)21(
40
)21(
402.037,1
yyyy
– With solution y/2 = 3% or y = 6%
Accrued Interest
• The quoted price (or market price) of a bond is usually its clean price, that is its gross price (or dirty or full price) minus the accrued interest
• Example– An investor buys on 12/10/01 a given amount of the US Treasury bond with
coupon 3.5% and maturity 11/15/2006– The current market price is 96.15625– The accrued interest period is equal to 26 days; this is the number of calendar
days between the settlement date (12/11/2001) and the last coupon payment date (11/15/2001)
– Hence the accrued interest is equal to the coupon payment (1.75) times 26 divided by the number of calendar days between the next coupon payment date (05/15/2002) and the last coupon payment date (11/15/2001)
– In this case, the accrued interest is equal to $1.75x(26/181) = $0.25138– The investor will pay 96.40763 = 96.15625 + 0.25138 for this bond
– where P is price of T-Bill– n is # of days until maturity
• Example: 90 days T-Bill, P = $9,800
Bank Discount Rate (T-Bills)
• Bank discount rate is the quoted rate on T-Bills
%890
360
000,10
800,9000,10
BDr
• Can’t compare T-bill directly to bond– 360 vs 365 days – Return is figured on par vs. price paid
10,000
10,000 / 360 10,000
10,000 360
10,000
BD
BD
P Discount
P r n
Pr
n
Bond Equivalent Yield
• Adjust the bank discounted rate to make it comparable 10,000 10,000 365
1365
BEY
BEY
PP r
n P nr
%28.890
365
800,9
800,9000,10
BEYr
• Example: same as before
BDBEY rP
r 365
360
000,10
• BDR versus BEY
• (exercise: Show it!)
Spot Zero-Coupon (or Discount) Rate
• Spot Zero-Coupon (or Discount) Rate is the annualized rate on a pure discount bond
– where B(0,t) is the market price at date 0 of a bond paying off $1 at date t– See Chapter 4 for how to extract implicit spot rates from bond prices
• General pricing formula
tBR t
t
,0)1(
1
,0
01 10,
0,(1 )
T Tt
ttt tt
FP F B t
R
Bond Par Yield
• Recall that a par bond is a bond with a coupon identical to its yield to maturity
• The bond's price is therefore equal to its principal• Then we define the par yield c(n) so that a n-year
maturity fixed bond paying annually a coupon rate of c(n) with a $100 face value quotes at par
• Typically, the par yield curve is used to determine the coupon level of a bond issued at par
n
ii
i
nn
nn
n
ii
i
R
Rnc
RR
nc
1 ,0
,0
,01 ,0
)1(1
)1(1
1
)()1(
100
)1(
)(100100
Forward Rates
• One may represent the term structure of interest rates as set of implicit forward rates
• Consider two choices for a 2-year horizon:– Choice A: Buy 2-year zero
– Choice B: Buy 1-year zero and rollover for 1 year
• What yield from year 1 to year 2 will make you indifferent between the two choices?
)1(
)1(1
1,0
22,0
1,1 R
RF
Forward Rates (continued)
• They are ‘implicit’ in the term structure• Rates that explain the relationship between spot
rates of different maturity• Example:
– Suppose the one year spot rate is 4% and the eighteen month spot rate is 4.5%
%51.5;)1)(04.1()045.1(
)1)(1()1(
6,122/1
6,122/3
2/16,121,0
2/318,0
mmmm
mmm
FF
FRR
Recap: Taxonomy of Rates
• Coupon Rate • Current Yield• Yield to Maturity • Zero-Coupon Rate• Bond Par Yield • Forward Rate
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