Chapter 10 Bond Prices and Yields. 10-2 Straight Bond Obligates the issuer of the bond to pay the...
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Transcript of Chapter 10 Bond Prices and Yields. 10-2 Straight Bond Obligates the issuer of the bond to pay the...
Chapter 10
Bond Prices and Yields
10-2
Straight Bond • Obligates the issuer of the bond to pay the
holder of the bond:• A fixed sum of money (principal, par value, or face
value) at the bond’s maturity• Constant, periodic interest payments (coupons)
during the life of the bond (Sometimes)
• Special features may be attached• Convertible bonds• Callable bonds• Putable bonds
10-3
Straight Bond Basics
• $1,000 face value• Semiannual coupon payments
Price Bond
Coupon Annual YieldCurrent (10.2)
ValuePar
Coupon Annual Rate Coupon (10.1)
10-4
Straight Bonds• Suppose a straight bond pays a semiannual
coupon of $45 and is currently priced at $960.• What is the coupon rate?• What is the current yield?
%.
%.
3759$960
2 $45 YieldCurrent (10.2)
009$1,000
2 $45 Rate Coupon (10.1)
10-5
Straight Bond Prices & Yield to Maturity• Bond Price:
• Present value of the bond’s coupon payments
• + Present value of the bond’s face value
• Yield to maturity (YTM):• The discount rate that equates today’s
bond price with the present value of the future cash flows of the bond
10-6
Bond Pricing Formula
MM YTM
FV
YTMYTM
C22
2121
11 Price Bond (10.3)
Where:
C = Annual coupon payment
FV = Face value
M = Maturity in years
YTM = Yield to maturity
PV of coupons PV of FV
10-7
Straight Bond Prices Calculator Solution
MM YTM
FV
YTMYTM
C22
2121
11 Price Bond (10.3)
Where:
C = Annual coupon payment
FV = Face value
M = Maturity in years
YTM = Yield to maturity
N = 2M
I/Y = YTM/2
PMT = C/2
FV = 1000
CPT PV
10-8
Straight Bond Prices
41457
2081
11
08
60 Coupons ofPV
24.$
..
MM YTM
FV
YTMYTM
C22
2121
11 Price Bond (10.3)
PV of coupons PV of FV
12390
2081
1000 FV ofPV
24.$
.
For a straight bond with 12 years to maturity, a coupon rate of 6% and a YTM of 8%, what is the current price?
Price = $457.41 + $390.12 = $847.53
10-9
Calculating a Straight Bond Price Using Excel
• Excel function to price straight bonds:
=PRICE(“Today”,“Maturity”,Coupon Rate,YTM,100,2,3)
• Enter “Today” and “Maturity” in quotes, using mm/dd/yyyy format.• Enter the Coupon Rate and the YTM as a decimal.• The "100" tells Excel to us $100 as the par value.• The "2" tells Excel to use semi-annual coupons.• The "3" tells Excel to use an actual day count with 365 days per year.
Note: Excel returns a price per $100 face.
10-10
Spreadsheet Analysis
10-11
Par, Premium and Discount Bonds
Par bonds: Price = par valueYTM = coupon rate
Premium bonds: Price > par value
YTM < coupon rate
The longer the term to maturity, the greater the premium over par
Discount bonds: Price < par value
YTM > coupon rate
The longer the term to maturity, the greater the discount from
par
10-12
Premium Bond Price
42677
2061
11
06
80 Coupons ofPV
24.$
..
93491
2031
1000 FV ofPV
24.$
.
12 years to maturity
8% coupon rate, paid semiannually
YTM = 6%
Price = $457.41 + $390.12 = $1,169.36
10-13
Discount Bonds
15906
2081
1000
2081
11
08
60 yr)(6 Price Bond
1212.$
...
Consider two straight bonds with a coupon rate of 6% and a YTM of 8%.
If one bond matures in 6 years and one in 12, what are their current prices?
53847
2081
1000
2081
11
08
60 yr)(12Price Bond
2442.$
...
10-14
Premium Bonds
361691
2061
1000
2061
11
06
80 yr)(12 Price Bond
2424.,$
...
Consider two straight bonds with a coupon rate of 8% and a YTM of 6%.
If one bond matures in 6 years and one in 12, what are their current prices?
540991
2061
1000
2061
11
06
80 yr)(6 Price Bond
1212.,$
...
10-15
M
Premium
1,000
Discount
12 6 0
CR>YTM 8%>6%
CR<YTM 6%<8%
YTM = CR
Bond Value ($) vs Years to Maturity
10-16
Premium and Discount Bonds• In general, when the coupon rate and YTM are
held constant:
For premium bonds: the longer the term to maturity, the greater the premium over par value.
For discount bonds: the longer the term to maturity, the greater the discount from par value.
10-17
Relationships among Yield Measures
For premium bonds: coupon rate > current yield > YTM
For discount bonds:coupon rate < current yield < YTM
For par value bonds:coupon rate = current yield = YTM
10-18
A Note on Bond Quotations
• If you buy a bond between coupon dates:• You will receive the next coupon payment • You might have to pay taxes on it• You must compensate the seller for any
accrued interest.
10-19
A Note on Bond Quotations• Clean Price = Flat Price
• Bond quoting convention ignores accrued interest.
• Clean price = a quoted price net of accrued interest
• Dirty Price = Full Price = Invoice Price• The price the buyer actually pays • Includes accrued interest added to the clean
price.
10-20
Clean vs. Dirty PricesExample
• Today is April 1. Suppose you want to buy a bond with a 8% annual coupon payable on January 1 and July 1.
• The bond is currently quoted at $1,020
• The Clean price = the quoted price = $1,020• The Dirty or Invoice price = $1,020 plus
(3mo/6mo)*$40 = $1,040
10-21
Calculating Yields
MM YTM
FV
YTMYTM
C22
2121
11 Price Bond
• Trial and error
• Calculator
• Spreadsheet
10-22
Calculating YieldsTrial & Error
A 5% bond with 12 years to maturity is priced at 90% of par ($900).
Selling at a discount YTM > 5%
Try 6% --- price = $915.32 too high
Try 6.5% --- price = $876.34 too low
Try 6.25% --- price = $895.56 a little low
Actual = 6.1933%
10-23
Calculating YieldsCalculatorA 5% bond with 12 years to maturity is priced at
90% of par ($900).
N = 24
PV = -900
PMT = 25
FV = 1000
CPT I/Y = 3.0966 x 2 = 6.1933%
10-24
Calculating YieldsSpreadsheet5% bond with 12 years to maturity,priced at 90% of par
=YIELD(“Now”,”Maturity”,Coupon, Price,100,2,3)
“Now” = “06/01/2008”
“Maturity” = “06/01/2020”
Coupon = .05
Price = 90 (entered as a % of par)
100 redemption value as a % of face value
“2” semiannual coupon payments
“3” actual day count (365)
=YIELD(“06/01/2008”,”06/01/2020”,0.05,90,100,2,3) = 0.06193276
10-25
Spreadsheet Analysis
10-26
Callable Bonds• Gives the issuer the option to:
• Buy back the bond • At a specified call price • Anytime after an initial call protection
period.
• Most bonds are callable Yield-to-call may be more relevant
10-27
Yield to Call
TT YTC
CP
YTCYTC
C22
2121
11 Price Bond Callable
Where:
C = constant annual coupon
CP = Call price of bond
T = Time in years to earliest call date
YTC = Yield to call
10-28
Yield to Call
• Suppose a 5% bond, priced at 104% of par with 12 years to maturity is callable in 2 years with a $20 call premium. What is its yield to call?
N = 4 # periods to first call date
PV = -1040
PMT = 25
FV = 1,020 Face value + call premium
CPT I/Y = 1.9368 x 2 = 3.874%
Remember: resulting rate = 6 month rate
10-29
Interest Rate Risk
• Interest Rate Risk = possibility that changes in interest rates will result in losses in the bond’s value
• Realized Yield = yield actually earned or “realized” on a bond
• Realized yield is almost never exactly equal to the yield to maturity, or promised yield
10-30
Interest Rate Risk and Maturity
10-31
Malkiel’s Theorems1. Bond prices and bond yields move in
opposite directions.
2. For a given change in a bond’s YTM, the longer the term to maturity, the greater the magnitude of the change in the bond’s price.
3. For a given change in a bond’s YTM, the size of the change in the bond’s price increases at a diminishing rate as the bond’s term ot maturity lengthens.
10-32
Malkiel’s Theorems
4. For a given change in a bond’s YTM, the absolute magnitude of the resulting change in the bond’s price is inversely related to the bond’s coupon rate.
5. For a given absolute change in a bond’s YTM, the magnitude of the price increase caused by a decrease in yield is greater than the price decrease caused by an increase in yield.
10-33
Bond Prices and Yields
10-34
Duration• Duration measure the sensitivity of a bond
price to changes in bond yields.
Two bonds with the same duration, but not necessarily the same maturity, will have approximately the same price sensitivity to a (small) change in bond yields.
2YTM1
YTM in ChangeDurationPrice Bond %
10-35
Macaulay Duration
% Bond Price ≈ -Duration x 2YTM1
ΔYTM
A bond has a Macaulay Duration = 10 years, its yield increases from 7% to 7.5%.
How much will its price change?
Duration = 10
Change in YTM = .075-.070 = .005
YTM/2 = .035
%Price ≈ -10 x (.005/1.035) = -4.83%
(10.5)
10-36
Modified Duration• Some analysts prefer a variation of Macaulay’s
Duration, known as Modified Duration.
• The relationship between percentage changes in bond prices and changes in bond yields is approximately:
2YTM
1
DurationMacaulayDurationModified
YTM in Change Duration Modified- Price Bond in Change Pct.
10-37
Modified Duration
2YTM1
Duration Macaulay Duration Modified
(10.7) %Bond Price ≈ -Modified Duration x YTM
(10.6)
A bond has a Macaulay duration of 9.2 years and a YTM of 7%. What is it’s modified duration?
Modified duration = 9.2/1.035 = 8.89 years
10-38
Modified Duration
2YTM1
Duration Macaulay Duration Modified
(10.7) %Bond Price ≈ -Modified Duration x YTM
(10.6)
A bond has a Modified duration of 7.2 years and a YTM of 7%. If the yield increases to 7.5%, what happens to the price?
%Price ≈ -7.2 x .005 = -3.6%
10-39
Calculating Macaulay’s Duration
• Macaulay’s duration values stated in years• Often called a bond’s effective maturity
• For a zero-coupon bond:Duration = maturity
• For a coupon bond:Duration = a weighted average of individual maturities of all the bond’s separate cash flows, where the weights are proportionate to the present values of each cash flow.
10-40
Calculating Macaulay Duration
M22YTM1
11
YTM
2YTM1 duration bond ValuePar(10.8)
Suppose a par value bond has 12 years to maturity and an 8% coupon. What is its duration?
years10.472.081
11
.08
2.081 duration bond ValuePar
24
10-41
General Macaulay Duration Formula
12YTM1CPRYTM
YTM)M(CPR2YTM1
YTM
2YTM1Duration 2M
(10.9)
Where:
CPR = Constant annual coupon rate
M = Bond maturity in years
YTM = Yield to maturity assuming semiannual coupons
10-42
Calculating Macaulay’s Duration
• In general, for a bond paying constant semiannual coupons, the formula for Macaulay’s Duration is:
• In the formula, C is the annual coupon rate, M is the bond maturity (in years), and YTM is the yield to maturity, assuming semiannual coupons.
12YTM1CYTM
YTMCM2YTM1
YTM2
YTM1Duration
2M
10-43
Using the General Macaulay Duration Formula
• What is the modified duration for a bond that matures in 15 years, has a coupon rate of 5% and a yield to maturity of 6.5%?
• Steps:1. Calculate Macaulay duration using 10.9
2. Convert to Modified duration using 10.6
10-44
Using the General Macaulay Duration FormulaBond matures in 15 yearsCoupon rate = 5% Yield to maturity = 6.5%
1. Calculate Macaulay duration using 10.9
years10.34 Duration
12.0651.05.065
.065)15(.052.0651
.065
2.0651Duration 30
12YTM1CPRYTM
YTM)M(CPR2YTM1
YTM
2YTM1Duration 2M
10-45
Using the General Macaulay Duration Formula
Bond matures in 15 yearsCoupon rate = 5% Yield to maturity = 6.5%Macaulay Duration = 10.34 years
2. Convert to Modified duration using 10.6
2YTM1
Duration Macaulay Duration Modified
years01102.0651
10.34 Duration Modified .
10-46
Calculating Duration Using Excel
• Macaulay Duration -- DURATION function• Modified Duration -- MDURATION function
=DURATION(“Today”,“Maturity”,Coupon Rate,YTM,2,3)
10-47
Calculating Macaulay and Modified Duration
10-48
Duration Properties: All Else Equal
1. The longer a bond’s maturity, the longer its duration.
2. A bond’s duration increases at a decreasing rate as maturity lengthens.
3. The higher a bond’s coupon, the shorter is its duration.
4. A higher yield to maturity implies a shorter duration, and a lower yield to maturity implies a longer duration.
10-49
Properties of Duration
10-50
Bond Risk Measures based on Duration
• Dollar Value of an 01: (10.10)
≈ Modified Duration x Bond Price x 0.0001
= Value of a basis point change
• Yield Value of a 32nd: (10.11)
≈ 01 an of ValueDollar32
1
In both cases, the bond price is per $100 face value.
10-51
Calculating Bond Risk MesuresBond matures in 15 years M = 15Coupon rate = 5% C = $50Yield to maturity = 6.5% YTM = .065
Macaulay Duration = 10.34 yearsModified Duration = 10.01 years
• First we have to find the price of the bond:
64857
20651
1000
20651
11
065
50 Price Bond
2121
11 Price Bond (10.3)
3030
22
.$...
YTM
FV
YTMYTM
CMM
10-52
Bond Risk Measures based on Duration
• Dollar Value of an 01: (10.10)
≈ Modified Duration x Bond Price x 0.0001
≈ 10.01 x 85.764 x 0.0001 = 0.0859
Bond matures in 15 years M = 15Coupon rate = 5% C = $50Yield to maturity = 6.5% YTM = .065Macaulay Duration = 10.34 yearsModified Duration = 10.01 yearsPrice = 85.764
10-53
Bond Risk Measures based on Duration
• Yield Value of a 32nd: (10.11)
≈ 364.00859.032
1
01 an of ValueDollar32
1
Bond matures in 15 years M = 15Coupon rate = 5% C = $50Yield to maturity = 6.5% YTM = .065Macaulay Duration = 10.34 yearsModified Duration = 10.01 yearsPrice = 85.764
10-54
Dedicated Portfolios
• Bond portfolio created to prepare for a future cash payment, e.g. pension funds
• Target Date = date the payment is due
10-55
Reinvestment Risk & Price Risk
• Reinvestment Rate Risk:• Uncertainty about the value of the portfolio on the
target date• Stems from the need to reinvest bond coupons at
yields not known in advance
• Price Risk:• Risk that bond prices will decrease• Arises in dedicated portfolios when the target date
value of a bond is not known with certainty
10-56
Price Risk vs. Reinvestment Rate Risk For a Dedicated Portfolio• Interest rate increases have two effects:
in interest rates decrease bond prices, but
in interest rates increase the future value of reinvested coupons
• Interest rate decreases have two effects: in interest rates increase bond prices, but
Decreases in interest rates decrease the future value of reinvested coupons
10-57
Immunization
• Immunization = constructing a dedicated portfolio that minimizes uncertainty surrounding the target date value • Engineer a portfolio so that price risk and
reinvestment rate risk offset each other (just about entirely).
• Duration matching = matching the duration of the portfolio to its target date
10-58
Immunization by Duration Matching
10-59
Dynamic Immunization
• Periodic rebalancing of a dedicated bond portfolio for the purpose of maintaining a duration that matches the target maturity date• Advantage = reinvestment risk greatly
reduced• Drawback = each rebalancing incurs
management and transaction costs
10-60
Constructing a Dedicated Portfolio
• Suppose a Pension Fund estimates it will need to pay out about $50 million in 4 years. The fund decides to buy coupon bonds paying 6%, maturing in 4 years and selling at par.
• Assuming interest rates do not change over the next four years, how much should the fund invest to have $50 million in 4 years?
10-61
Constructing a Dedicated Portfolio
2M
M2M2
2)YTM(1
Required ValueFuture ValueFace
value Face12YTM1YTM
C)2YTM1(P
10.12
10.13
Future Value required = $50 million
Time = 4 years (M=8)
Par value bonds paying 6%
462,470,39$)03.(1
0$50,000,00 ValueFace
8
10-62
Constructing a Dedicated Portfolio6-month Value at End
Year Period Payment of Year 51 1 $1,184,113.86 1,456,310.69$
2 $1,184,113.86 1,413,893.87$ 2 3 $1,184,113.86 1,372,712.50$
4 $1,184,113.86 1,332,730.58$ 3 5 $1,184,113.86 1,293,913.19$
6 $1,184,113.86 1,256,226.39$ 4 7 $1,184,113.86 1,219,637.28$
8 $1,184,113.86 1,184,113.86$
Future Value of Coupons 10,529,538.36$ Face Value 39,470,462.00$
50,000,000.36$
10-63
Useful Internet Sites
• www.sifma.org (check out the bonds section)• www.jamesbaker.com (a practical view of bond
portfolio management)• www.bondsonline.com (bond basics and current
market data)• www.investinginbonds.com (bond basics and
current market data) • www.bloomberg.com (for information on
government bonds)