Bond Valuation Capital Budgetingand Corporate...
Transcript of Bond Valuation Capital Budgetingand Corporate...
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Bond Valuation
Capital Budgeting and CorporateObjectives
Professor Ron Kaniel
Simon School of Business
University of Rochester
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Bond Valuation An Overview
Introduction to bonds and bond markets» What are they? Some examples
Zero coupon bonds» Valuation» Interest rate sensitivity
Coupon bonds» Valuation» Interest rate sensitivity
The term structure of interest rates
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What is a Bond?
A bond is a security that obligates the issuer to make specified interest and principal payments to the holder on specified dates.
» Coupon rate
» Face value (or par)
» Maturity (or term)
Bonds are also called fixed income securities.
Bonds differ in several respects:
» Repayment type
» Issuer
» Maturity
» Security
» Priority in case of default
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Repayment Schemes
Pure Discount or Zero-Coupon Bonds» Pay no coupons prior to maturity.» Pay the bond’s face value at maturity.
Coupon Bonds» Pay a stated coupon at periodic intervals prior to maturity.» Pay the bond’s face value at maturity.
Floating-Rate Bonds» Pay a variable coupon, reset periodically to a reference rate. » Pay the bond’s face value at maturity.
Perpetual Bonds (Consols)» No maturity date.» Pay a stated coupon at periodic intervals.
Annuity or Self-Amortizing Bonds » Pay a regular fixed amount each payment period.» Principal repaid over time rather than at maturity.
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Types of Bonds: Issuers
Bonds Issuer
Government Bonds US Treasury, Government Agencies
Mortgage-Backed Securities Government agencies (FNMA etc)
Municipal Bonds State and local government
Corporate Bonds Corporations
Asset-Back Securities Corporations
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U.S. Government Bonds
Treasury Bills» No coupons (zero coupon security)
» Face value paid at maturity
» Maturities up to one year
Treasury Notes» Coupons paid semiannually
» Face value paid at maturity
» Maturities from 2-10 years
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U.S. Government Bonds (Cont.)
Treasury Bonds» Coupons paid semiannually» Face value paid at maturity» Maturities over 10 years» The 30-year bond is called the long bond.
Treasury Strips» Zero-coupon bond» Created by “stripping” the coupons and principal from Treasury
bonds and notes.
No default risk. Considered to be risk free. Exempt from state and local taxes. Sold regularly through a network of primary dealers. Traded regularly in the over-the-counter market.
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Agency and Municipal Bonds
Agency bonds: mortgage-backed bonds» Bonds issued by U.S. Government agencies that are backed by a
pool of home mortgages.» Self-amortizing bonds. (mostly monthly payments)» Maturities up to 30 years.» Prepayment risk.
Municipal bonds» Maturities from one month to 40 years.» Usually exempt from federal, state, and local taxes.» Generally two types:
– Revenue bonds– General Obligation bonds
» Riskier than U.S. Government bonds.
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Corporate Bonds
Bonds issued by corporations» Bonds vs. Debentures
» Fixed-rate versus floating-rate bonds.
» Investment-grade vs. Below investment-grade bonds.
» Additional features:– call provisions
– convertible bonds
– puttable bonds
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Seniority of Corporate Bonds
In case of default, different classes of bonds have different claim priority on the assets of a corporation.
Secured Bonds (Asset-Backed)» Secured by real property.
» Ownership of the property reverts to the bondholders upon default.
Debentures» Same priority as general creditors.
» Have priority over stockholders, but subordinate to secured debt.
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Bond Ratings
Moody’s S&P Quality of Issue
Aaa AAA Highest quality. Very small risk of default.
Aa AA High quality. Small risk of default.
A A High-Medium quality. Strong attributes, but potentially vulnerable.
Baa BBB Medium quality. Currently adequate, but potentially unreliable.
Ba BB Some speculative element. Long-run prospects questionable.
B B Able to pay currently, but at risk of default in the future.
Caa CCC Poor quality. Clear danger of default.
Ca CC High speculative quality. May be in default.
C C Lowest rated. Poor prospects of repayment.
D - In default.
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The U.S. Bond Market:
Source: U.S. Federal Reserve. Table L.4, Credit Market Debt, All Sectors, by Instrument,
Amount ($bil.), July , 2019
Debt Instrument 2019 Q1
Treasury securities 18,248
Agency backed securities 9,146
Municipal securities 3,816
Corporate and foreign bonds 13,503
Mortgages 15,527
Consumer Credit 4,000
Corporate equities 48,029
Mutual fund shares 16,180
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Bond Valuation: Zero Coupon Bonds
B = Market price of the Bond of bondF = Face valueR = Annual percentage ratem = compounding period (annual m = 1, semiannual m = 2,…)i = Effective periodic interest rate; i=R/mT = Maturity (in years)N = Number of compounding periods; N = T*m
Two cash flows to purchaser of bond:» -B at time 0» F at time T
What is the price of a bond?Use present value formula:
Ni
FB
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Valuing Zero Coupon Bonds:An Example
Value a 5 year, U.S. Treasury strip with face value of $1,000. The APR is R=7.5% with annual compounding? What about quarterly compounding?
What is the APR on a U.S. Treasury strip that pays $1,000 in exactly 7 years and is currently selling for $591.11 under annual compounding? Semi-annual compounding?
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Interest Rate Sensitivity:Zero Coupon Bonds
Consider the following 1, 2 and 10-year zero-coupon bonds, all with» face value of F=$1,000
» APR of R=10%, compounded annually.
We obtain the following table for increases and decreases of the interest rate by 1%:
Bond prices move up if interest rates drop, decrease if interest rates rise
Interest Rate Bond 1 Bond 2 Bond 31-Year 2-Year 10-Year
9.0% $917.43 $841.68 $422.4110.0% $909.09 $826.45 $385.5411.0% $900.90 $811.62 $352.18
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Bond Prices and Interest Rates
$0
$200
$400
$600
$800
$1,000
$1,200
0.0% 5.0% 10.0% 15.0% 20.0% 25.0%
1-Year
2-Year
10-Year
Bond prices are inversely related to IR
Longer term bonds are more sensitive to IR changes than short term bonds
The lower the IR, the more sensitive the price.
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Measuring Interest Rate SensitivityZero Coupon Bonds
We would like to measure the interest rate sensitivity of a bond or a portfolio of bonds.» How much do bond prices change if interest rates change by a small
amount?» Why is this important?
Use “Dollar value of a one basis point decrease” (DV01):» Basis point (bp): 1/100 of one percentage point =0.01%=0.0001» Calculate DV01:
– Method 1: Difference of moving one basis point down:DV01= B(R-0.01%)-B(R).
– Method 2: Difference of moving 1/2bp down minus 1/2pb up:DV01=B(R-0.005%) -B(R+0.005%).
– Method 3: Use calculus:
0001.001
R
BDV
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Computing DV01: An Example
Reconsider the 1, 2 and 10- year bonds discussed before:
Method 3:
11 10.1
1*10.0$*0001.0
10.1
000,1$0001.0
TT
TTR
B
Interest Rate Bond 1 Bond 2 Bond 31-Year 2-Year 10-Year
9.990% $909.1736 $826.5966 $385.89409.995% $909.1322 $826.5214 $385.7186
10.000% $909.0909 $826.4463 $385.543310.005% $909.0496 $826.3712 $385.3681
Method 1 $0.082652 $0.150283 $0.350669Method 2 $0.082645 $0.150263 $0.350494Method 3 $0.082645 $0.150263 $0.350494
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DV01: A Graphical Approach
DV01 estimates the change in the Price-Interest rate curve using a linear approximation.
higher slope implies greater sensitivity
10-Year
$0.00
$200.00
$400.00
$600.00
$800.00
$1,000.00
$1,200.00
Interest Rate
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Valuing Coupon BondsExample 1: Amortization Bonds
Consider Amortization Bond» T=2
» m=2
» C=$2,000 c = C/m = $2,000/2 = $1,000
» R=10% i = R/m = 10%/2 = 5%
How can we value this security?» Brute force discounting
» Similar to another security we already know how to value?
» Replication
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0 1 2 3 4Buy Coupon Bond -$3,545.95 $1,000.00 $1,000.00 $1,000.00 $1,000.00
Buy 6-Month Zero -$952.38Buy 1-Year Zero -$907.03Buy 1.5-Year Zero -$863.84Buy 2-Year Zero -$822.70Portfolio -$3,545.95
Valuing Coupon BondsExample 1: Amortization Bonds
Compare with a portfolio of zero coupon bonds:
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A First Look at Arbitrage
Reconsider amortization bond; suppose bond trades at $3,500 (as opposed to computed price of $3,545.95)» Can we make a profit without any risk?
– What is the strategy?
– What is the profit?
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A First Look at Arbitrage
Reconsider amortization bond; suppose bond trades at $3,500 (as opposed to computed price of $3,545.95)
» Can make risk less profit
– Buy low: buy amortization bond
– Sell high: Sell portfolio of zero coupon bonds
– riskless profit of $45.95
– no riskless profit if price is correct
0 1 2 3 4Buy Coupon Bond -$3,500.00 $1,000.00 $1,000.00 $1,000.00 $1,000.00
Sell 6-Month Zero $952.38 -$1,000.00 $0.00 $0.00 $0.00Sell 1-Year Zero $907.03 $0.00 -$1,000.00 $0.00 $0.00Sell 1.5-Year Zero $863.84 $0.00 $0.00 -$1,000.00 $0.00Sell 2-Year Zero $822.70 $0.00 $0.00 $0.00 -$1,000.00Portfolio $3,545.95 -$1,000.00 -$1,000.00 -$1,000.00 -$1,000.00Net Cash Flow $45.95 $0.00 $0.00 $0.00 $0.00
Time Period
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What is the market price of a U.S. Treasury bond that has a coupon rate of 9%, a face value of $1,000 and matures exactly 10 years from today if the interest rate is 10% compounded semiannually?
0 6 12 18 24 ... 120 Months
45 45 45 45 1045
Valuation of Coupon Bonds:Example 2: Straight Bonds
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What is the market price of a bond that has an annual coupon C, face value F and matures exactly T years from today if the required rate of return is R, with m-periodic compounding?
» Coupon payment is: c = C/m
» Effective periodic interest rate is: i = R/m
» number of periods N = Tm
0 1 2 3 4 ... … N
c c c c … … c+F
Valuing Coupon BondsThe General Formula
NN i
F
ii
c
ZeroAnnuityB
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The Concept of a “Yield to Maturity”
So far we have valued bonds by using a given interest rate, then discounted all payments to the bond.
Prices are usually given from trade prices» need to infer interest rate that has been used
Definition: The yield to maturity is that interest rate that equates the present discounted value of all future payments to bondholders to the market price:
Algebraic:
NN myield
F
myieldmyield
cB
/1/1
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/
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Yield to MaturityA Graphical Interpretation
$0.00
$500.00
$1,000.00
$1,500.00
$2,000.00
$2,500.00
0% 2% 4% 6% 8% 10%
12%
14%
16%
18%
20%
22%
24%
Consider a U.S. Treasury bond that has a coupon rate of 10%, a face value of $1,000 and matures exactly 10 years from now. » Market price of $1,500, implies a yield of 3.91% (semi-annual
compounding); for B=$1,000 we obviously find R=10%.
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Interest Rate Sensitivity:Coupon Bonds
Coupon bonds can be represented as portfolios of zero-coupon bonds» Implication for price sensitivity
Consider purchasing the US Treasury bond discussed earlier (10 year, 9% coupon, $1,000 face)» Suppose immediately thereafter interest rates fall to 8%,
compounded semiannually.
» Suppose immediately thereafter interest rate rises to 12% compounded semiannually.
» Suppose the interest rate equals 9%, compounded semiannually.
What are the pricing implications of these scenarios?
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Recall the general formula:
What is the price of the bond if the APR is 8% compounded semiannually?
Similarly:
If R=12%: B=$ 827.95
If R= 9%: B=$1,000.00
Implication of Interest Rate Changes on Coupon Bond Prices
NN i
F
ii
cB
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Relationship Between Coupon Bond Prices and Interest Rates
Bond prices are inversely related to interest rates (or yields).
A bond sells at par only if its interest rate equals the coupon rate
A bond sells at a premium if its coupon rate is above the interest rate.
A bond sells at a discount if its coupon rate is below the interest rate.
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Interest Rate Sensitivity of Coupon Bonds
Consider two bonds with 10% annual coupons with maturities of 5 years and 10 years.
The APR is 8%
What are the responses to a .01% (1bp) interest rate change?
Does the sensitivity of a coupon bond always increase with the term to maturity?
Yield 5-Year Bond $ Change % Change 10-Year Bond $ Change % Change7.995% $1,080.06 $0.21019 0.0195% $1,134.57 $0.36585 0.0323%8.000% $1,079.85 $1,134.208.005% $1,079.64 -$0.21013 -0.0195% $1,133.84 -$0.36569 -0.0322%DV01 $0.42032 $0.73154
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$0.00
$500.00
$1,000.00
$1,500.00
$2,000.00
$2,500.00
0%
2%
4%
6%
8%
10
%
12
%
14
%
16
%
18
%
20
%
22
%
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%
Interest Rate (R)
Pri
ce
(P
)
5-Year Bond
10-Year Bond
Bond Prices and Interest Rates
Longer term bonds are more sensitive to changes in interest rates than shorter term bonds, in general.
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Consider the following two bonds:» Both have a maturity of 5 years» Both have yield of 8%» First has 6% coupon, other has 10% coupon, compounded
annually. Then, what are the price sensitivities of these bonds, measured by
DV01 as for zero coupon bonds?
Why do we get different answers for two bonds with the same yield and same maturity?
Bond Yields and Prices
Yield 6%-Bond $ Change % change 10%-Bond $ Change % change
7.995% $920.33 $0.1891 $1,080.06 $0.2102
8.000% $920.15 $1,079.85
8.005% $919.96 ($0.1891) $1,079.64 ($0.2101)
0.0411% 0.0389%
DV01 $0.3782 $0.4203
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Maturity and Price Risk
Zero coupon bonds have well-defined relationship between maturity and interest rate sensitivity:
Coupon bonds can have different sensitivities for the same maturity» DV01 now depends on maturity and coupon
Need concept of “average maturity” of coupon bond:» Duration
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Duration
Duration is a weighted average term to maturity where the weights are relative size of the contemporaneous cash flow.
Duration is a unitless number that quantifies the percentage change in a bond’s price for a 1 percentage change in the interest rate.
B
FPVN
TB
NcPV
NT
B
cPVT
B
cPVTDuration )()()2(
2
)1(
1
1
1
BB R BDuration
RR BR
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Duration (cont.)
The duration of a bond is less than its time to maturity (except forzero coupon bonds).
The duration of the bond decreases the greater the coupon rate.This is because more weight (present value weight) is being givento the coupon payments.
As market interest rate increases, the duration of the bonddecreases. This is a direct result of discounting. Discounting at ahigher rate means lower weight on payments in the far future.Hence, the weighting of the cash flows will be more heavilyplaced on the early cash flows -- decreasing the duration.
Modified Duration = Duration / (1+yield)
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Spot Rates
A spot rate is a rate agreed upon today, for a loan that is to be made today» r1=5% indicates that the current rate for a one-year loan is 5%.
» r2=6% indicates that the current rate for a two-year loan is 6%.
» Etc.
The term structure of interest rates is the series of spot rates r1, r2, r3,…» We can build using STRIPS or coupon bond yields.
» Explanations of the term structure.
Term Structure, July 15, 2019
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Term Structure, August, 2000
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Term Structure, May, 2007
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Term Structure of Interest RatesSource: National Economic Trends (St. Louis Fed)
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Summary
Bonds can be valued by discounting their future cash flows
Bond prices change inversely with yield Price response of bond to interest rates depends on term
to maturity.» Works well for zero-coupon bond, but not for coupon bonds
Measure interest rate sensitivity using ‘DV01’ and duration.
The term structure implies terms for future borrowing:» Forward rates» Compare with expected future spot rates