Bond Bo nd Price ri ces s and and Yield ields s Chapter 12
Bond Characteristics
Face or par value Coupon rate Coupon rate= annual coupon payment ($)/Face Value Zero coupon bond
Indenture A contract between the issuer and the bond holder Specifies: coupons rate, par value, maturity, and bond provisions.
Bond Characteristics
Face or par value Coupon rate Coupon rate= annual coupon payment ($)/Face Value Zero coupon bond
Indenture A contract between the issuer and the bond holder Specifies: coupons rate, par value, maturity, and bond provisions.
Provisions of Bonds
Secured or unsecured Call provision Convertible provision Retractable and extendible bonds
Known Known as putable putable bonds bonds in the US
Floating vs. fixed rate bond Sinking funds
Principal and Interest Payments for a 4% Real Return Bond
Bond Pricing T
P B =
C t
∑ (1 + r )t + t =1
PB = Ct = T = r =
ParValue
(1 + r )T
price of the bond interest or coupon payments number of periods to maturity the appropriate semi-annual discount rate
Bond Quotation (Nov. 17, 2011) Issuer Name
Symbol
Rating Coupon Maturity Moody's/S&P/ Fitch
MORGAN STANLEY
MS.MHU
5.500% Jul 2021
US CENTRAL CREDIT UNION
USFD.GB 1.900% Oct 2012
JPMORGAN CHASE & CO
JPM.SEM 4.350% Aug 2021
AMERICAN EXPRESS CREDIT CORP
AXP.NX
JPMORGAN CHASE & CO
JPM.SCH 3.150% Jul 2016
CITIGROUP FUNDING
Low
Last
Change Yield %
93.946 89.884 90.394
-1.752
6.873
Aaa/AA+/--
101.563 101.484 101.484
-0.079
0.268
Aa3/A+/--
101.518 98.758 98.891
-0.545
4.491
2.800% Sep 2016 A2/BBB+/A+ 102.646 100.043 100.088
-1.234
2.780
99.300
-0.033
3.314
C.HTB
1.875% Oct 2012 Aaa/AA+/AAA 101.620 101.517 101.620
0.450
0.112
CITIGROUP
C.ALY
4.500% Jan 2022
--/A/--
99.814 96.364 99.550
2.520
4.554
GENERAL ELECTRIC CAPITAL CORP
GE.HDS
5.250% Oct 2012
Aa2/AA+/--
104.115 103.291 103.386
-0.289
1.493
BHP BILLITON FINANCE (USA)
BHP.GH
4.800% Apr 2013
A1/A+/A+
106.063 105.793 105.846
-0.026
0.600
ALLY FINANCIAL
GMAC.IRI 2.200% Dec 2012 Aaa/AA+/AAA 102.280 101.346 101.591
-0.552
0.715
Fannie Mae Issues Rate
Matur ity
A2/A/--
High
Aa3/A+/AA-
100.900
99.203
Federal Provincial Corporate
Bid
Ask ed
Yield
6.13
3-12
101:31
102:00
...
4.88 5.25
5-12 8-12
102:13 103:14
102:14 103:15
0.04 0.29
4.38 4.38
9-12 3-13
103:14 105:14
103:15 105:15
0.16 0.26
Coupon
Maturity Date
Bid $
Yield %
Bell Canada
4.850
06/30/2014
106.56
2.25
Bell Canada
3.600
12/02/2015
103.71
2.62
Bell Canada
4.400
03/16/2018
106.45
3.26
Bell Canada
5.000
02/15/2017
109.66
2.99
Bell Canada
6.100
03/16/2035
108.02
5.49
Bank of Montreal
4.650
03/14/2013
104.08
1.52
Bank of Montreal
4.870
04/22/2015
107.88
2.45
Problem 12.1 A bond with semi-annual coupon payment: • Annual coupon: 80 • Par value: 1000 • Years to maturity: 30 years • Yield to maturity (annual rate): 10% What is the price of the bond?
Bond Prices and Interest Rates
Prices and market interest rates have an inverse relationship
When interest rates get very high the value of the bond will be very low When rates approach zero, the value of the bond approaches the sum of the cash flows
Bond Prices at Different Interest Rates (8% Coupon Bond, Coupons Paid Semiannually)
Bond Valuation Between Coupon Payments
When a bond is sold between coupon payments, part of the next coupon payment (CP) belongs to the sellers. This is called accrued interest. Entire Coupon Period
Most Recent CP Interest earned by Seller
Next CP
Interest to be earned by buyer
In order to value a bond between coupon payment dates, we need to incorporate the fractional coupon period into our bond value model.
Bond Valuation Between Coupon Payment •
The fractional period is W =
•
# days between settlement and next coupon # days in the coupon period
And the valuation equation becomes T
P B = ∑ t =1
• •
C t t −1+W
(1+ y)
+
Par T −1+W
(1+ y)
The above price is called invoice price (or dirty pri ce, full price) which is the actual price paid when buying the bond. Bond prices are quoted without the accrued interest. This is often referred to as the clean price (or just price). To determine the clean price, we must compute the accrued interest and subtract this from the dirty price. Conversely, if we know clean price, we need to add accrued interest to determine the actual price we need to pay.
Problem 12.2
Consider A 3-year bond, with $1,000 par value, 6% semiannual coupon bond, YTM=12%. Suppose that the maturity of this bond is January 15, 2004, and you are valuing the bond for settlement on April 20, 2001. The next coupon is due July 15, 2001, and you will assume a 30E/360 day count convention. What is the dirty price, accrued interest and clean price of the bond on the settlement date?
Yield to Maturity
YTM: interest rate that makes the present value of the bond’s payments equal to its price. It is the promised rate of return based on the current market price if
Bond is held to maturity Coupons are reinvested at the same rate
Solve the bond price formula for y T
PB =
∑ t =1
T Ct + ParValue T t (1+ y) (1+ y)
Problem 12.3 Yield to Maturity Consider a 10-year bond, with an 7% annual coupon rate, making payments semiannually. The bond’s par value is $1,000, and its quoted market price is $950. Find the bond’s yield to maturity. What is the yield to maturity (semi-annual rate)?
Yield Measures
Bond Equivalent Yield (quoted yield): 3.86% x 2 = 7.72% Effective Annual Yield: (1+0.0386)2 − 1 = 7.88% Current Yield (=Annual Interest/Market Price): $70/$950 = 7.37% Yield to Call: Yield calculated using call price and # of periods to call
For premium bonds
Coupon rate > Current yield > YTM
For discount bonds, relationships are reversed Coupon rate < Current yield < YTM
Callable Bonds The company (issuer) has an option to buy back (call) the entire bond issue at the call price. The issuer has an incentive to call when the bond price exceeds the call price. An equivalent argument: when interest rates fall, the issuer can refinance its debt at a lower rate. The price of a callable bond is capped – a rational bondholder will never agree to pay for the bond a price higher than the call price
The Inverse Relationship between Bond Prices and Yields for a Callable Bond
Yield to Call • There is always a chance that the bond will be called before maturity • Yield to call may be more relevant to bondholders than yield to maturity (especially if market price is close to the call price) • Solve the following formula for yC T C
P=∑ t =1
C t
(1 + y c )
t
+
Call Price (1 + y c )
T c
where yc = Yield To Call and Tc = expected time of call
Problem 12.4: Yield to Call • Consider a 10-year callable bond, with an 8% coupon rate, making payments semiannually. The bond’s par value is $1,000, and its current quoted market price is $1,150 • The bond indenture indicates the call schedule. Find the bond’s yield to first call
Period (years)
Call price
6-7
1,100
7-8
1, 050
8-10
1, 000
Problem 12.5
Two bonds have identical times to maturity and coupon rates. One is callable at 105, the other at 110. Which should have the higher yield to maturity?
Realized Yield versus YTM
Reinvestment Assumptions Holding Period Return
Changes in rates affects returns Reinvestment of coupon payments Change in price of the bond
Growth of Invested Funds
Problem 12.6 Bonds of Zello Corporation with a par value of $1,000 sell for $960, mature in five years, and have a 7 percent annual coupon rate paid semiannually. a) Calculate Current yield I. II. Yield to maturity III. Realized compound yield for an investors with a threeyear holding period and a re-investment rate of 6 percent over the period; at the end of three years the 7 percent coupon bonds with two years remaining will sell to yield 7 percent b) Discuss one major shortcoming for each of the three yield measures.
Price Paths of Coupon Bonds Price Premium bond
1,000
Discount bond
0
Maturity date
Time
Taxation on Bond Investment
Ordinary income component: Assuming yield remains constant, constan t, discount bond prices rise over time and premium bond prices decline over time. For bonds originally issued at a discount/premium, the price appreciation/depreci appreciation/depreciation ation (based on unchanged unchanged yield) yield) is taxed as ordinary income/loss.
Capital Gain Component:
Price changes stemming from yield changes are taxed as capital gains/lo gains/loss ss if the the bond is is sold. sold.
Problem 12.7
Consider a 30-year bond with 4% coupon rate (paid annually), issued at an 8% YTM. Calculate the tax payment, if sold one year later, when YTM=7%. Income tax rate is 36%.
Problem 12.8 Assume you have a one-year investment horizon and are trying to choose among three bonds. All have the same degree of default risk and mature in 10 years. The first is a zero-coupon bond that pays $1000 at maturity. The second has an 8 percent coupon rate and pays the $80 coupon once per year. The third has a 10 percent coupon rate and pays the $100 coupon once per year. If all three bonds are now priced to yield 8 percent to maturity, what are a) their price? If you expect their yield to maturity to be 8 percent at the beginning of b) next year, what will their price be then? What is your before-tax holding period return on each bonds? If your tax bracket is 30 percent on ordinary income and 20 percent on capital gain income, what will your after-tax rate of return be on each? Recalculate your answer to b) under the assumption that your expect the c) yields to maturity on each bond to be 7 percent at the beginning of next year.
Default Risk and Ratings
Rating companies
Moody’s Investor Service Standard & Poor’s DBRS
Rating Categories
Investment grade Speculative grade
Definitions of Each Bond Rating Class
Factors Used by Rating Companies
Coverage ratios Leverage ratio Liquidity ratios Profitability ratios Cash flow to debt
Financial Ratios and Default Risk by Rating Class, LongTerm Debt
Default Risk and Bond Pricing: Bond Indentures
Sinking funds: A way to call bonds early Subordination of future debt: Restrict additional borrowing Dividend restrictions: Force firm to retain assets rather than paying them out to shareholders Collateral: A particular asset bondholders receive if the firm defaults
Default Risk
Corporate bonds are subject to default risk
Promised or stated yield may be different from realized yield The state yield is the maximum possible yield to maturity of the bond
To compensate for the possibility of default, corporate bonds must offer a default premium
Default premium = Promised yield – Yield of an otherwise-identical government bond
Credit Default Swaps (CDS)
Acts like an insurance policy on the default risk of a corporate bond or loan Buyer pays annual premiums Issuer agrees to buy the bond in a default or pay the difference between par and market values to the CDS buyer Institutional bondholders, e.g. banks, used CDS to enhance creditworthiness of their loan portfolios, to manufacture AAA debt Can also be used to speculate that bond prices will fall
Yields on Long-Term Bonds
Default Risk and Bond Pricing
Credit Risk and Collateralized Debt Obligations (CDOs)
Major mechanism to reallocate credit risk in the fixedincome markets Loans are pooled together and split into tranches with different levels of default risk Mortgage-backed CDOs were an investment disaster in 2007-2009
The Term Structure of Interest Rates Chapter 13
The Yield Curve
The yield curve is a graph that displays the relationship between YTM and time to maturity Information on expected future short-term rates can be implied from the yield curve Yields Upward Sloping
Flat
Downward Sloping
Maturity
Short Rate 0 r 0,1
2
1
…………
t-1
r 1,2
Short rate: interest rate for one unit of time
Notation r t-1,t=interest rate from t-1 to t Revealed only at time t-1 e.g., r 0,1 = interest rate from time 0 to time 1 r 1,2 = interest rate from time 1 to time 2 … r t-1,t = interest rate from t-1 to t
t r t-1,t
Spot Rates
A spot rate (or zero rate) for maturity T is the rate of interest earned on an investment that provides a payoff only at T. Spot rate equals the yield to maturity of a zero coupon bond. Notation: y t = YTM for t-year zero coupon bond. E.g., y1 = YTM for a 1-year zero coupon bond y2 = YTM for a 2-year zero coupon bond … yt = YTM for a t-year zero coupon bond
P0 =
F T
(1 + yT ) T
where, FT is the principal payment at time T and y T is the spot rate for maturity T.
Short Rate and Spot Rate 0
1
r 0,1
2
r 1,2
……..
3
t-1
t
r t-1,t
r 2,3
reveal at time 2 reveal at time t-1
y1 Known at time 0
reveal at time 1
y2
y3
yt
…..
Derive Spot Rates using Short rates • If we know future short rates, we could easily derive spot rate. • The following equation must holds PV t =
1 (1 + r 0,1 )(1 + r 1, 2 )...(1 + r t −1,t )
=
1 (1 + yt )
t
1
⇒ yt = [(1 +r 0,1)(1 + r 1, 2 )...(1 + r t −1,t )]t − 1
where PVt = the present value of $1 paid at time t r 0,1 = one year short rate for year 1 r 1,2 = one year short rate for year 2 … r t-1,t = one year short rate for year t yt = spot rate with a maturity year t
However, in a world with uncertainty, we usually do not know what the short rate will be in the future.
Problem 13.1 • A 2-year zero coupon bond priced at $896.47, the year 1 short rate is 5% and the year 2 short rate is 6%. What is the 2-year spot rate?
Spot Rate Curve A spot curve is a graph of spot rates as a function of maturity.
The spot rate curve for the spot rates in Example 12.2
The Yield Curve and Future Interest Rates • Short Rates and Yield Curve Slope
When next year’s short rate, r 2 , is greater than this year’s short rate, r 1, the yield curve slopes up
May indicate rates are expected to rise
• When next year’s short rate, r 2 , is less than this year’s short rate, r 1, the yield curve slopes down – May indicate rates are expected to fall
Yield Curve: Bond Pricing
Yields on different maturity bonds are not all equal We need to consider each bond cash flow as a standalone zero-coupon bond Bond stripping and bond reconstitution offer opportunities for arbitrage The value of the bond should be the sum of the values of its parts
Problem 13.2 Valuing Coupon Bonds Yields and Prices to Maturity on Zero-Coupon Bonds ($1,000 Face Value):
Value a 3 year, 10% coupon bond using discount rates In the table.
Forward Rate • In reality, future short rates are uncertain and only spot rates are available to investor (y 1, y2,..yt are reported in the newspaper). • The forward rate is the future spot rate implied by today’s term structure of spot rates. 2 • Definition (1 + y 2 ) = (1 + y1 )(1 + f 1, 2 ) • More distant forward rates are calculated in a similar way: − (1 + y n ) n = (1 + y n −1 ) n 1 (1 + f n −1, n )
where f n-1, n is one year forward rate in year n
Problem 13.3: Forward Rate
4 yr spot rate= 9.993; 3yr spot rate= 9.660; f = ?
3,4
Forward Contract A forward contract: A contract today to borrow/lend at later date, at a rate pre-specified in the contract (forward rate). In the previous example, an investor today could have a forward contract that allows him to borrow/lend at beginning of year 4 for one year at rate 10.998%. A forward contract is used to hedge interest rate risk since future short rate is uncertain.
Synthetic Forward Loan Time 0
Action Buy 1 3-year zero Sell (1+f 4) 4-year zero
CF -1000/(1+9.66%) 3 = -758.32 (1+10.998%)x1000/(1+9.993%) 4 = 758.32
3
3-year zero matures
+1,000
4
4-year zero matures
-(1+10.998%)x1,000
Forward Rate Curve
A forward curve is a graph of forward rates all for the same maturity but with different forward period (different from spot rate curve!!).
Problem 13.4: Forward Rate Curve Zero-Coupon Rates 12% 11.75% 11.25% 10.00% 9.25%
Calculate 1 year forward rates
Bond Maturity 1 2 3 4 5
Problem 13.4: Forward Rate Curve
Note that both the spot and forward curves provide identical information. If you have either one, you can construct the other.
Theories of Term Structure
The Expectations Hypothesis:
The upward sloping yield curve (y 2>y1) means the expect future short rate is to be higher than current short rate. The downward sloping yield curve (y 2
Liquidity Preference: f n −1, n = E (r n −1, n ) + L n -1,n
f n −1, n = E [r n −1, n ]
Short-term investors dominate the market and they will demand a liquidity premium for the resale value risk associated with long-term bonds. Yield curve has an upward bias built into the long-term rates because of the liquidity (resale value ) risk premium
Segmented Market Hypothesis
Yields for a maturity segment is a function of demand within that maturity segment.
Interpreting the Term Structure The yield curve reflects expectations of future interest rates The forecasts of future rates are clouded by other factors, such as liquidity premiums An upward sloping curve could indicate:
Rates are expected to rise and/or Investors require large liquidity premiums to hold long term bonds
The yield curve is a good predictor of the business cycle
Long term rates tend to rise in anticipation of economic expansion Inverted yield curve may indicate that interest rates are expected to fall and signal a recession
Yields on Long-Term Versus Short-Term Government Securities: Term Spread, 1980-2012
Problem 13.5
a) b) c) d)
The current yield curve for default-free zero-coupon bonds is as follows: Maturity (year)
YTM(%)
1
10
2
11
3
12
What are the implied one-year forward rate? If market expectations are accurate, what will be the pure yield curve? If you buy a two-year zero-coupon bond now, what is the expected total rate of turn over the next year? What should be the current price of a three-year-maturity bond with a 12 percent coupon rate? If you purchase at that price, what would your total expected rate of return be over the next year?
Managing Bond Portfolios Chapter 14
Bond Pricing Relationships 1. 2.
Inverse relationship between price and yield Convexity: An increase in a bond’s yield to maturity results in a smaller price decline than the gain associated with a decrease in yield. Price
The first relationship says that, for a increase of yield ∆y, the price of bond will decrease by ∆P2. For a decrease of yield ∆y, the price of bond will increase by ∆P1.
P+∆P1
P
The second relationship says that ∆P2<∆P1 for a given ∆y
P-∆P2
Y-∆Y
y
Y+∆Y
YTM
Bond Pricing Relationships C
M
Initial YTM
A
12%
5
10.00%
1,075.82
5.00%
15.00%
899.44
-16.40%
B
12%
30
10.00%
1,188.54
5.00%
15.00%
803.02
-32.44%
C
3%
30
10.00%
340.12
5.00%
15.00%
212.08
-37.64%
D
3%
30
6.00%
587.06
5.00%
11.00%
304.50
-48.13%
Bond
3. 4. 5. 6.
Price
YTM (%)
New YTM
New Price
Price (%)
Long-term bonds tend to be more price sensitive than short-tem bonds (A vs. B) As maturity increases, price sensitivity to yield to maturity increases at a decreasing rate. (A vs. B) Price sensitivity is inversely related to a bond’s coupon rate for a given maturity (B vs. C) Price sensitivity is inversely related to the yield to maturity at which the bond is selling (C vs. D)
Bond Pricing Relationships 200%
150%
100% A B
50%
C D
0% -5% -50%
-100%
-4%
-3%
-2% -1%
0%
1%
2%
3%
4%
5%
Bond Maturity
Consider cash flows of bonds AA and BB Both bond stop paying cash after 5th year. The stated maturity are both 5 years, but the cash flow patterns are quite different. What relevant is the effective maturity of cash flow.
Macaulay Duration
Duration is a measure of the effective maturity of a bond. It is the weighted average of the times until each payment is received. T D = 1× w1 + 2 × w2 + 3 × w3 + ... + T × wT = ∑ t × wt t =1
The weight in the duration measure is the proportion of the PV of cash flows relative to the entire present value of the t bond (bond price). CF (1 + y) wt =
t
P r ice
Duration is shorter than maturity for all bonds except zero coupon bonds Duration is equal to maturity for zero coupon bonds
Duration Calculation: An Example 10 year coupon bond with 4% annual coupon rate (assume a 8% market Yield) (1) Year
1 2 3 4 5 6 7 8 9 10 Sum
(2) Cash Flow
$ 40 40 40 40 40 40 40 40 40 1,040
(3) PV ($1)
$ 0.9259 0.8573 0.7938 0.7350 0.6806 0.6302 0.5835 0.5403 0.5002 0.4632
(4) (2)x(3)
$ 37.04 34.29 31.75 29.40 27.22 25.21 23.34 21.61 20.01 481.73 $ 731.58
(5) (4)÷731.58
0.0506 0.0469 0.0434 0.0402 0.0372 0.0345 0.0319 0.0295 0.0274 0.6585 1.0000
(6) (1)×(5)
0.0506 0.0938 0.1302 0.1608 0.1860 0.2070 0.2233 0.2360 0.2466 6.5850 8.1193
Duration Calculation (Continued)
Duration=8.12 Modified Duration=8.12/(1+0.08)=7.52 A 1 percentage point increase in interest rate, Bond price falls by 7.52 ×1%=7.52%
Duration/Price Relationship ΔP
P
= − Duration ×
Δ(1 + y)
(1 + y)
: A measure of percentage change in bond price to change in the interest rate. Modified Duration = Macaulay Duration / (1+y) ΔP
P
= −Modified Duration ⋅ ∆(1 + y )
= − Modified Duration ⋅ ∆ y
where ∆P
p
= % chang in bond price
∆y = change in interest rate
Why is duration a key concept?
It’s a simple summary statistic of the effective average maturity of the portfolio; It is an essential tool in immunizing portfolios from interest rate risk; It is a measure of interest rate risk of a portfolio
Duration and Interest Rate Risk
Two bonds have duration of 1.8852 years
One is a 2-year, 8% semi-annual coupon bond with YTM=10% The other bond is a zero coupon bond with maturity of 1.8852 years
Duration of both bonds is 1.8852 x 2 = 3.7704 semiannual periods Modified D = 3.7704/1 + 0.05 = 3.591 periods Suppose the semiannual interest rate increases by 0.01%. Bond prices fall by ∆P = − D * ∆ y = -3.591 x 0.01% = -0.03591% P Bonds with equal D have the same interest rate sensitivity
The coupon bond, which initially sells at $964.540, falls to $964.1942, when its yield increases to 5.01%, its price declines 0.0359% The zero-coupon bond initially sells for $1,000/1.053.7704 = $831.9704. At the higher yield, it sells for $1,000/1.053.7704 = $831.6717, therefore its price also falls by 0.0359%
Problem 14.1 You will be paying $10,000 a year in education expenses at the end of the next two years. Bonds currently yield 8 percent. a) What is the present value and duration of your obligation? b) What maturity zero-coupon bond would immunize your obligation? c) Suppose you buy a zero-coupon bond with value and duration equal to your obligation. Now suppose that rates immediately increase to 9 percent. What happens to your net position? What if rate fall to 7 percent?
Convexity
Duration assumes that the price change is a linear function of changes in YTM.
Predicting bond price changes based on duration is only the first-order approximation.
The actual true relationship is convex function.
The error is small only for small changes in YTM.
Duration and Convexity Price Pricing error from convexity
P+∆P1 P+∆P1’
P P+∆P2 P+∆P2’
y-∆y
y
Yield
y+∆y
Yield change
Linear Relationship
Convexity
Price error
-∆y
P+∆P1’
P+∆P1
∆P1-∆P1’
+∆y
P+∆P2’
P+∆P2
∆P2-∆P2’
Duration: Corrected for Convexity • A more exact approximation of price changes is a second-order approximation, which involves measures of convexity.
CFt 2 Convexity = ( t + t ) 2 ∑ t P × (1 + y) t =1 (1 + y) 1
n
Correction for Convexity: ∆P
P
2 = − D ∗ ∆ y + 1 ⋅ Convexity ⋅ (∆ y )
2
